From a7160ad3467e5f49981c9f83d24cce38ce9a091b Mon Sep 17 00:00:00 2001
From: andre-fuchs-uni-oldenburg
 <64791323+andre-fuchs-uni-oldenburg@users.noreply.github.com>
Date: Fri, 16 Sep 2022 11:46:34 +0200
Subject: [PATCH] Add files via upload

---
 Functions_v5-MATLAB/FIT_KM.m                  | 1932 ++++++++--------
 Functions_v5-MATLAB/KM_Calculation.m          |    2 +
 Functions_v5-MATLAB/KM_STP_optimization.m     |    4 +
 Functions_v5-MATLAB/KM_plot.m                 |    4 +
 Functions_v5-MATLAB/KM_plot_raw.m             |    4 +
 Functions_v5-MATLAB/OPTI_IFT_dij.m            |  774 +++----
 Functions_v5-MATLAB/PART_I.m                  |    6 +-
 Functions_v5-MATLAB/PART_II.m                 |    2 +-
 Functions_v5-MATLAB/PART_III.m                |    5 +-
 Functions_v5-MATLAB/Struc_flip_test.m         |    2 +
 Functions_v5-MATLAB/ZeroCrossings.m           |    3 +-
 Functions_v5-MATLAB/checkFT.m                 |  718 +++---
 Functions_v5-MATLAB/conditional_PDF_markov.m  |   37 +-
 Functions_v5-MATLAB/conditional_moment.m      |    1 +
 Functions_v5-MATLAB/frequency_filter.m        |  619 ++---
 Functions_v5-MATLAB/length_scales.m           | 2008 ++++++++---------
 Functions_v5-MATLAB/main.m                    |    3 +
 Functions_v5-MATLAB/plot_conditional_moment.m |  266 +--
 Functions_v5-MATLAB/plot_entropy.m            |    3 +
 Functions_v5-MATLAB/plot_increment_pdf.m      |    8 +-
 Functions_v5-MATLAB/plot_pointwise_KM.m       |   12 +-
 Functions_v5-MATLAB/spectrum.m                |    2 +
 22 files changed, 3246 insertions(+), 3169 deletions(-)

diff --git a/Functions_v5-MATLAB/FIT_KM.m b/Functions_v5-MATLAB/FIT_KM.m
index 8522c27..bdd520c 100644
--- a/Functions_v5-MATLAB/FIT_KM.m
+++ b/Functions_v5-MATLAB/FIT_KM.m
@@ -1,965 +1,969 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% This function performs the surface fit with a linear function for $D^{(1)}\left(u_r,r\right)$ 
-% and a parabolic function for $D^{(2)}\left(u_r,r\right)$ to the optimized and non-optimized KMC's.
-% Coefficients $d_{ij}(r)$ in the fits are functions of scale~$r$ of the form 
-% $\alpha (r/\lambda)^{\beta}+\gamma$. After fitting, this function plots the optimized $D^{(1,2)}$ 
-% and its surface fits. This function also plots the parameters $d_{11}$, $d_{20}$, $d_{21}$ and 
-% $d_{22}$ as a function of $\frac{r}{\lambda}$ for optimized and non-optimized 
-% $D^{(1,2)}\left(u_r,r\right)$.
-%
-% Arguments IN
-% evaluated_point = struct array calculated in the function 'conditional_moment'
-% increment_bin = number of bins
-% taylor_L = Taylor length scale in meters
-% Fs = Acquisition/Sampling Frequency in Hz
-% m_data = mean of the data
-% markov = markov length in number of samples
-% multi_point = weather to do multi-point analysis or not 1=Yes, 0=No
-% condition = condition for multi-point analysis
-%
-% Arguments OUT
-% co_KM_opti = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients using the surface fits
-% co_KM_opti_old = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients using the
-% surface fits without an offset
-% co_no_opti = Coefficients $d_{ij}(r)$ of the non-optimized Kramers-Moyal coefficients using the surface fits
-% fitresult_D1_conf = Confidence intervals for fit coefficients of D1
-% fitresult_D2_conf = Confidence intervals for fit coefficients of D2
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-function[co_KM_opti,co_KM_opti_old,co_no_opti,fitresult_D1_conf,fitresult_D2_conf]=FIT_KM(evaluated_point,increment_bin,taylor_L,Fs,m_data,markov,multi_point,condition,norm_ur,norm_r,save_path,save_name)
-
-% taylor_L=1;
-
-if multi_point==1
-    for k=1:length(condition)
-            my_field = sprintf('point_eval_%d', k);              
-            co_KM_opti.(my_field)               = struct('a',NaN(1,2),'ea',nan,'b',NaN(1,6),'eb',NaN(1,3));
-            co_KM_opti_old.(my_field)           = struct('a',NaN(1,2),'ea',nan,'b',NaN(1,6),'eb',NaN(1,3));
-            co_no_opti.(my_field)               = struct('a',NaN(1,2),'ea',nan,'b',NaN(1,6),'eb',NaN(1,3));
-    end
-    
-    for k=1:length(condition)
-            my_field    = sprintf('point_eval_%d', k);    
-            clear evaluated
-            evaluated   = evaluated_point.(my_field);
-            
-            tmp=0;
-            for scal    = 1:length(evaluated)
-                tmp     = tmp+sum(evaluated(scal).x_bin_not_nan>0);
-            end
-            
-            if tmp>0
-                xx      = nan(length(evaluated),increment_bin);
-                y       = nan(length(evaluated),increment_bin);
-                z1      = nan(length(evaluated),increment_bin);
-                z1e     = nan(length(evaluated),increment_bin);
-                z2      = nan(length(evaluated),increment_bin);
-                z2e     = nan(length(evaluated),increment_bin);
-                z1_opti = nan(length(evaluated),increment_bin);
-                z2_opti = nan(length(evaluated),increment_bin);
-                z4      = nan(length(evaluated),increment_bin);
-
-                % for i=find((evaluated.r)/taylor_L>4):length(evaluated) 
-                for i=1:length(evaluated)
-                    xx(i,1:sum(evaluated(i).x_bin_not_nan))=evaluated(i).y_mean_bin(evaluated(i).x_bin_not_nan);
-                    if norm_r==1
-                    y(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).r./taylor_L);
-                    else
-                    y(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).r);   
-                    end
-                    z1(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1(evaluated(i).x_bin_not_nan));
-                    z1e(i,1:sum(evaluated(i).x_bin_not_nan))=abs(1./(evaluated(i).eD1(evaluated(i).x_bin_not_nan)));
-
-                    z2(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2(evaluated(i).x_bin_not_nan));
-                    z2e(i,1:sum(evaluated(i).x_bin_not_nan))=abs(1./(evaluated(i).eD2(evaluated(i).x_bin_not_nan)));
-
-                    z1_opti(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1_opti);
-                    z2_opti(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2_opti);
-
-                    z4(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D4(evaluated(i).x_bin_not_nan));
-                end    
-
-                %TEST Beginn
-                % y=y-(((markov./Fs).*m_data)/taylor_L);
-                %TEst Ende
-
-
-                %% D1-Fit
-                [fitresult_D1, xData, yData, zData, weights]    = FIT_D1(xx, y, z1_opti, z1e);
-                fitresult_D1_conf                               = confint(fitresult_D1);
-
-                h(1) = figure;
-                % surf(xx, y, z1_opti,'EdgeColor','none')
-                scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-                % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-                hold on
-                surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D1,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
-%                 surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),...
-%                     (-0.36.*repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100).^(-1).*repmat(linspace(min(xData),max(xData),100),100,1)),'FaceColor',[0.501960813999176 0.501960813999176 0.501960813999176],'EdgeColor',[0.501960813999176 0.501960813999176 0.501960813999176])
-
-                % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
-                set(gca,'yScale','log') 
-                set(gcf, 'Color', 'w')
-                set(gca, 'FontSize',18)
-                if norm_ur==1
-                    xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-                else
-                    xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-                end
-                if norm_r==1
-                    ylabel('$r / \lambda$', 'interpreter','latex')
-                else
-                    ylabel('$r\ (m)$', 'interpreter','latex')
-                end
-                zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
-                % zlabel('$D1 (x,r) / [\frac{m}{s^2}]$', 'interpreter','latex')
-                colormap (parula(200))
-                % colorbar
-                grid off
-                xlim([min(xData) max(xData)])
-                ylim([min(y(:,1)) max(y(:,1))])
-                % zlim([-0.005 0.005])
-                axis square
-                set(gca,'YDir','reverse');
-                title('optimized KM, optimized fit','interpreter','latex');
-                % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
-                % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
-                rotate3d on
-                colorbar('north','TickLabelInterpreter','latex','AxisLocation','out');
-                fig_setup
-                legend off
-                txt = {'(a)'};
-                a=text(4,0.5,txt);
-                a.Units='normalized';
-
-                %%%%%  2019 %%%%%%%%
-                % optimized coefficients 
-                % D1(x,y) = (a1*y^ra1+a11)*x
-                co_KM_opti.(my_field).a  = [fitresult_D1.a1 fitresult_D1.a11];
-                co_KM_opti.(my_field).ea = [fitresult_D1.ra1];
-
-                % non-optimized coefficients 
-                [fitresult_D1_o, xData_o1, yData_o1, zData_o1, weights] = FIT_D1(xx, y, z1, z1e);
-                %%%%%  2019 %%%%%%%%
-                % D1(x,y) = (a1*y^ra1+a11)*x
-                co_no_opti.(my_field).a  = [fitresult_D1_o.a1 fitresult_D1_o.a11];
-                co_no_opti.(my_field).ea = [fitresult_D1_o.ra1];
-
-                % optimized coefficients no offset
-                % D1(x,y) = (a1*y^ra1)*x
-                ft = fittype( '(a1*y^ra1)*x', 'independent', {'x', 'y'}, 'dependent', 'z' );
-                opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
-                % opts.DiffMinChange = 1e-12;
-                % opts.Display = 'Off';
-                % opts.MaxFunEvals = 1000;
-                % opts.MaxIter = 1000;
-                % opts.TolFun = 1e-12;
-                % opts.TolX = 1e-12;
-                opts.Weights = weights;
-
-                % opts.StartPoint = [-0.1  -1];
-                % opts.Lower = [-2 -2];
-                % opts.Upper = [0 0];
-
-                opts.StartPoint = [0  0];
-                opts.Lower = [-1 -Inf];
-                opts.Upper = [1 0];
-
-                [fitresult_D1_old, gof] = fit( [xData, yData], zData, ft, opts );
-                co_KM_opti_old.(my_field).a  = [fitresult_D1_old.a1 0];
-                co_KM_opti_old.(my_field).ea = [fitresult_D1_old.ra1];
-
-                % figure
-                % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra0)
-                % hold on
-                % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra1)
-                % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^0)
-                % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a3.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra3)
-                % set(gca,'xScale','log') 
-                % axis square
-                % set(gcf, 'Color', 'w')
-                % % grid on
-                % xlabel('$r / \lambda$', 'interpreter','latex')
-                % legend('d_{10}(r)','d_{11}(r)','d_{12}(r)','d_{13}(r)')
-                % xlim([min(y(:,1)) max(y(:,1))])
-
-
-
-                %% D2-Fit
-                [fitresult_D2, xData, yData, zData, weights] = FIT_D2(xx, y, z2_opti, z2e);
-                fitresult_D2_conf                            = confint(fitresult_D2);
-                h(2) = figure;
-                % surf(xx, y, z2_opti,'EdgeColor','none')
-                scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-                % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-                hold on
-                surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D2,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
-%                 surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),...
-%                     (0.0144.*repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100).^(-1).*repmat(linspace(min(xData),max(xData),100),100,1).^(2)),'FaceColor',[0.501960813999176 0.501960813999176 0.501960813999176],'EdgeColor',[0.501960813999176 0.501960813999176 0.501960813999176])
-                % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
-                set(gca,'yScale','log') 
-                % set(gca,'zScale','log')  
-                set(gcf, 'Color', 'w')
-                set(gca, 'FontSize',18)
-                if norm_ur==1
-                    xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-                else
-                    xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-                end
-                if norm_r==1
-                    ylabel('$r / \lambda$', 'interpreter','latex')
-                else
-                    ylabel('$r\ (m)$', 'interpreter','latex')
-                end
-                % zlabel('$D2 (x,r) / [\frac{m^2}{s^3}]$', 'interpreter','latex')
-                zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
-                colormap (parula(200))
-                % colorbar
-                grid off
-                xlim([min(xData) max(xData)])
-                ylim([min(y(:,1)) max(y(:,1))])
-                % zlim([0 2*max(zData)])
-                % caxis([0 max(zData)])
-                axis square
-                % close
-                set(gca,'YDir','reverse');
-                set(gca,'TickLabelInterpreter','latex');
-                title('optimized KM, optimized fit','interpreter','latex');
-                % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
-                % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
-                rotate3d on
-                colorbar('north','TickLabelInterpreter','latex','AxisLocation','out');
-                fig_setup
-                legend off
-                txt = {'(b)'};
-                b=text(4,0.5,txt);
-                b.Units='normalized';
-
-                % optimized coefficients 
-                %%%%%  2019 %%%%%%%%
-                % D2(x,y) = (b0*y^rb0+b00) + (b1*y^rb1+b11)*x + (b2*y^rb2+b22)*x^2
-                co_KM_opti.(my_field).b  = [fitresult_D2.b0 fitresult_D2.b00 fitresult_D2.b1 fitresult_D2.b11 fitresult_D2.b2 fitresult_D2.b22];
-                co_KM_opti.(my_field).eb = [fitresult_D2.rb0 fitresult_D2.rb1 fitresult_D2.rb2 ];     % for D2
-
-                % non-optimized coefficients 
-                [fitresult_D2_o, xData_o2, yData_o2, zData_o2, weights] = FIT_D2(xx, y, z2, z2e);
-                %%%%%  2019 %%%%%%%%
-                % D2(x,y) = (b0*y^rb0+b00) + (b1*y^rb1+b11)*x + (b2*y^rb2+b22)*x^2
-                co_no_opti.(my_field).b  = [fitresult_D2_o.b0 fitresult_D2_o.b00 fitresult_D2_o.b1 fitresult_D2_o.b11 fitresult_D2_o.b2 fitresult_D2_o.b22];
-                co_no_opti.(my_field).eb = [ fitresult_D2_o.rb0 fitresult_D2_o.rb1 fitresult_D2_o.rb2 ];     % for D2
-
-
-                % optimized coefficients no offset
-                ft = fittype( '(b0*y^rb0) + (b1*y^rb1)*x + (b2*y^rb2)*x^2', 'independent', {'x', 'y'}, 'dependent', 'z' );
-                opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
-                % opts.DiffMinChange = 1e-12;
-                % opts.Display = 'Off';
-                % opts.MaxFunEvals = 1000;
-                % opts.MaxIter = 1000;
-                % opts.TolFun = 1e-12;
-                % opts.TolX = 1e-12;
-                opts.Weights = weights;
-
-                % opts.StartPoint = [0.2 -0.02  0.007 0.001 -1 -0.3];
-                % opts.Lower = [0 -1  0 0 -2 -2];
-                % opts.Upper = [1  1  1 2  2  0];
-
-                opts.StartPoint =   [0 0 0     0 0 0];
-                opts.Lower =        [-1 -1  -1   0 -Inf -Inf];
-                opts.Upper =        [1  1  1   Inf Inf 0];
-
-                [fitresult_D2_old, gof] = fit( [xData, yData], zData, ft, opts );
-                co_KM_opti_old.(my_field).b  = [fitresult_D2_old.b0 0 fitresult_D2_old.b1 0 fitresult_D2_old.b2 0];
-                co_KM_opti_old.(my_field).eb = [fitresult_D2_old.rb0 fitresult_D2_old.rb1 fitresult_D2_old.rb2 ];     % for D2
-
-
-                %% d_ij(r) plots
-                h(3) = figure;
-                subplot(2,2,1);
-                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-                    fitresult_D1.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra1+fitresult_D1.a11,'LineWidth',2)
-                hold on
-                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-                    fitresult_D1_o.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1_o.ra1+fitresult_D1_o.a11,'LineWidth',2)
-%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%                     fitresult_D1_old.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1_old.ra1,'LineWidth',2,'Color','k')
-%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%                     (-0.36).*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^(-1),'LineWidth',2,'Color',[0.501960813999176 0.501960813999176 0.501960813999176])
-                axis square
-                ylabel('$d_{11}$', 'interpreter','latex')          
-                if norm_r==1
-                xlabel('$r / \lambda$', 'interpreter','latex')
-                else
-                xlabel('$r/m $', 'interpreter','latex')
-                end 
-                xlim([min(y(:,1)) max(y(:,1))])
-                set(gca, 'FontSize',12)
-                %
-                subplot(2,2,2);
-                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-                    fitresult_D2.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb0+fitresult_D2.b00,'LineWidth',2)
-                hold on
-                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-                    fitresult_D2_o.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb0+fitresult_D2_o.b00,'LineWidth',2)
-
-%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%                     fitresult_D2_old.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb0,'LineWidth',2,'Color','k')
-                axis square
-                ylabel('$d_{20}$', 'interpreter','latex')
-                if norm_r==1
-                xlabel('$r / \lambda$', 'interpreter','latex')
-                else
-                xlabel('$r/m $', 'interpreter','latex')
-                end 
-                xlim([min(y(:,1)) max(y(:,1))])
-                set(gca, 'FontSize',12)
-                %
-                subplot(2,2,3);
-                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-                    fitresult_D2.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb1+fitresult_D2.b11,'LineWidth',2)
-                hold on
-                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-                    fitresult_D2_o.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb1+fitresult_D2_o.b11,'LineWidth',2)
-%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%                     fitresult_D2_old.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb1,'LineWidth',2,'Color','k')
-                axis square
-                ylabel('$d_{21}$', 'interpreter','latex')
-                if norm_r==1
-                xlabel('$r / \lambda$', 'interpreter','latex')
-                else
-                xlabel('$r/m $', 'interpreter','latex')
-                end 
-                xlim([min(y(:,1)) max(y(:,1))])
-                set(gca, 'FontSize',12)
-                %
-                subplot(2,2,4);
-                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-                    fitresult_D2.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb2+fitresult_D2.b22,'LineWidth',2)
-                hold on
-                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-                    fitresult_D2_o.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb2+fitresult_D2_o.b22,'LineWidth',2)
-%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%                     fitresult_D2_old.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb2,'LineWidth',2,'Color','k')
-%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%                     0.0144.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^(-1),'LineWidth',2,'Color',[0.501960813999176 0.501960813999176 0.501960813999176])
-                axis square
-                ylabel('$d_{22}$', 'interpreter','latex')
-                if norm_r==1
-                xlabel('$r / \lambda$', 'interpreter','latex')
-                else
-                xlabel('$r/m $', 'interpreter','latex')
-                end 
-                xlim([min(y(:,1)) max(y(:,1))])
-                set(gca, 'FontSize',14)
-                set(gca,'TickLabelInterpreter','latex');
-                set(gcf, 'Color', 'w')
-                %legend('optimized','non-optimized','Location','NorthWest')
-%                 legend({'Optimized','Non-optimized','Optimized no offset','K62'},'Position',[0.45 0.5 0.1 0.05])
-                legend({'Optimized','Non-optimized','K62'},'Position',[0.45 0.5 0.1 0.05])
-
-                close all
-                disp(['point:' num2str(k) '/'  num2str(length(condition)) ])
-            end
-    end
-else
-    evaluated   = evaluated_point;
-    xx          = nan(length(evaluated),increment_bin);
-    y           = nan(length(evaluated),increment_bin);
-    z1          = nan(length(evaluated),increment_bin);
-    z1e         = nan(length(evaluated),increment_bin);
-    z2          = nan(length(evaluated),increment_bin);
-    z2e         = nan(length(evaluated),increment_bin);
-    z1_opti     = nan(length(evaluated),increment_bin);
-    z2_opti     = nan(length(evaluated),increment_bin);
-    z4          = nan(length(evaluated),increment_bin);
-    
-    % for i=find((evaluated.r)/taylor_L>4):length(evaluated) 
-    for i=1:length(evaluated)
-        xx(i,1:sum(evaluated(i).x_bin_not_nan))=evaluated(i).y_mean_bin(evaluated(i).x_bin_not_nan);
-        if norm_r==1
-        y(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).r./taylor_L);
-        else
-        y(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).r);   
-        end
-        z1(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1(evaluated(i).x_bin_not_nan));
-        z1e(i,1:sum(evaluated(i).x_bin_not_nan))=abs(1./(evaluated(i).eD1(evaluated(i).x_bin_not_nan)));
-
-        z2(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2(evaluated(i).x_bin_not_nan));
-        z2e(i,1:sum(evaluated(i).x_bin_not_nan))=abs(1./(evaluated(i).eD2(evaluated(i).x_bin_not_nan)));
-
-        z1_opti(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1_opti);
-        z2_opti(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2_opti);
-
-        z4(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D4(evaluated(i).x_bin_not_nan));
-    end
-    
-%     z1_opti=z1_opti.*0.5;
-%     z2_opti=z2_opti.*0.5;
-
-    %TEST Beginn
-    % y=y-(((markov./Fs).*m_data)/taylor_L);
-    %TEst Ende
-
-
-    %% D1-Fit
-    [fitresult_D1, xData, yData, zData, weights]    = FIT_D1(xx, y, z1_opti, z1e);
-    fitresult_D1_conf                               = confint(fitresult_D1);
-
-    h(1) = figure;
-    % surf(xx, y, z1_opti,'EdgeColor','none')
-    scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    hold on
-    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D1,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
-%     surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),...
-%         (-0.36.*repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100).^(-1).*repmat(linspace(min(xData),max(xData),100),100,1)),'FaceColor',[0.501960813999176 0.501960813999176 0.501960813999176],'EdgeColor',[0.501960813999176 0.501960813999176 0.501960813999176])
-
-    % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
-    set(gca,'yScale','log') 
-    set(gcf, 'Color', 'w')
-    set(gca, 'FontSize',18)
-    if norm_ur==1
-        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-    else
-        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-    end
-    if norm_r==1
-        ylabel('$r / \lambda$', 'interpreter','latex')
-    else
-        ylabel('$r\ (m)$', 'interpreter','latex')
-    end
-    zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
-    % zlabel('$D1 (x,r) / [\frac{m}{s^2}]$', 'interpreter','latex')
-    colormap (parula(200))
-    % colorbar
-    grid off
-    xlim([min(xData) max(xData)])
-    ylim([min(y(:,1)) max(y(:,1))])
-    % zlim([-0.005 0.005])
-    axis square
-    set(gca,'YDir','reverse');
-%     title('optimized KM, optimized fit','interpreter','latex');
-    % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
-    % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
-    rotate3d on
-    colorbar('north','TickLabelInterpreter','latex','AxisLocation','out');
-    fig_setup
-    legend off
-    txt = {'(a)'};
-    a=text(4,0.5,txt);
-    a.Units='normalized';
-
-
-    %%%%%  2019 %%%%%%%%
-    % optimized coefficients 
-    % D1(x,y) = (a1*y^ra1+a11)*x
-    co_KM_opti.a  = [fitresult_D1.a1 fitresult_D1.a11];
-    co_KM_opti.ea = [fitresult_D1.ra1];
-
-    % non-optimized coefficients 
-    [fitresult_D1_o, xData_o1, yData_o1, zData_o1, weights] = FIT_D1(xx, y, z1, z1e);
-    %%%%%  2019 %%%%%%%%
-    % D1(x,y) = (a1*y^ra1+a11)*x
-    co_no_opti.a  = [fitresult_D1_o.a1 fitresult_D1_o.a11];
-    co_no_opti.ea = [fitresult_D1_o.ra1];
-
-    % optimized coefficients no offset
-    % D1(x,y) = (a1*y^ra1)*x
-    ft = fittype( '(a1*y^ra1)*x', 'independent', {'x', 'y'}, 'dependent', 'z' );
-    opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
-    % opts.DiffMinChange = 1e-12;
-    % opts.Display = 'Off';
-    % opts.MaxFunEvals = 1000;
-    % opts.MaxIter = 1000;
-    % opts.TolFun = 1e-12;
-    % opts.TolX = 1e-12;
-    opts.Weights = weights;
-
-    % opts.StartPoint = [-0.1  -1];
-    % opts.Lower = [-2 -2];
-    % opts.Upper = [0 0];
-
-    opts.StartPoint = [0  0];
-    opts.Lower = [-1 -Inf];
-    opts.Upper = [1 0];
-
-    [fitresult_D1_old, gof] = fit( [xData, yData], zData, ft, opts );
-    co_KM_opti_old.a  = [fitresult_D1_old.a1 0];
-    co_KM_opti_old.ea = [fitresult_D1_old.ra1];
-
-    % figure
-    % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra0)
-    % hold on
-    % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra1)
-    % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^0)
-    % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a3.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra3)
-    % set(gca,'xScale','log') 
-    % axis square
-    % set(gcf, 'Color', 'w')
-    % % grid on
-    % xlabel('$r / \lambda$', 'interpreter','latex')
-    % legend('d_{10}(r)','d_{11}(r)','d_{12}(r)','d_{13}(r)')
-    % xlim([min(y(:,1)) max(y(:,1))])
-
-
-
-    %% D2-Fit
-    [fitresult_D2, xData, yData, zData, weights] = FIT_D2(xx, y, z2_opti, z2e);
-    fitresult_D2_conf                            = confint(fitresult_D2);
-    h(2) = figure;
-    % surf(xx, y, z2_opti,'EdgeColor','none')
-    scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    hold on
-    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D2,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
-%     surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),...
-%         (0.0144.*repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100).^(-1).*repmat(linspace(min(xData),max(xData),100),100,1).^(2)),'FaceColor',[0.501960813999176 0.501960813999176 0.501960813999176],'EdgeColor',[0.501960813999176 0.501960813999176 0.501960813999176])
-    % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
-    set(gca,'yScale','log') 
-    % set(gca,'zScale','log')  
-    set(gcf, 'Color', 'w')
-    set(gca, 'FontSize',18)
-    if norm_ur==1
-        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-    else
-        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-    end
-    if norm_r==1
-        ylabel('$r / \lambda$', 'interpreter','latex')
-    else
-        ylabel('$r\ (m)$', 'interpreter','latex')
-    end
-    % zlabel('$D2 (x,r) / [\frac{m^2}{s^3}]$', 'interpreter','latex')
-    zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
-    colormap (parula(200))
-    % colorbar
-    grid off
-    xlim([min(xData) max(xData)])
-    ylim([min(y(:,1)) max(y(:,1))])
-    % zlim([0 2*max(zData)])
-    % caxis([0 max(zData)])
-    axis square
-    % close
-    set(gca,'YDir','reverse');
-    set(gca,'TickLabelInterpreter','latex');
-%     title('optimized KM, optimized fit','interpreter','latex');
-    % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
-    % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
-    rotate3d on
-    colorbar('north','TickLabelInterpreter','latex','AxisLocation','out');
-    fig_setup
-    legend off
-    txt = {'(b)'};
-    b=text(4,0.5,txt);
-    b.Units='normalized';
-
-    font=22;
-    a.FontSize = font;
-    b.FontSize = font;
-    pos_txt=[-0.22   0.8];
-    a.Position=pos_txt;
-    b.Position=[-0.25  0.8];
-
-    % optimized coefficients 
-    %%%%%  2019 %%%%%%%%
-    % D2(x,y) = (b0*y^rb0+b00) + (b1*y^rb1+b11)*x + (b2*y^rb2+b22)*x^2
-    co_KM_opti.b  = [fitresult_D2.b0 fitresult_D2.b00 fitresult_D2.b1 fitresult_D2.b11 fitresult_D2.b2 fitresult_D2.b22];
-    co_KM_opti.eb = [fitresult_D2.rb0 fitresult_D2.rb1 fitresult_D2.rb2 ];     % for D2
-
-    % non-optimized coefficients 
-    [fitresult_D2_o, xData_o2, yData_o2, zData_o2, weights] = FIT_D2(xx, y, z2, z2e);
-    %%%%%  2019 %%%%%%%%
-    % D2(x,y) = (b0*y^rb0+b00) + (b1*y^rb1+b11)*x + (b2*y^rb2+b22)*x^2
-    co_no_opti.b  = [fitresult_D2_o.b0 fitresult_D2_o.b00 fitresult_D2_o.b1 fitresult_D2_o.b11 fitresult_D2_o.b2 fitresult_D2_o.b22];
-    co_no_opti.eb = [ fitresult_D2_o.rb0 fitresult_D2_o.rb1 fitresult_D2_o.rb2 ];     % for D2
-
-
-    % optimized coefficients no offset
-    ft = fittype( '(b0*y^rb0) + (b1*y^rb1)*x + (b2*y^rb2)*x^2', 'independent', {'x', 'y'}, 'dependent', 'z' );
-    opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
-    % opts.DiffMinChange = 1e-12;
-    % opts.Display = 'Off';
-    % opts.MaxFunEvals = 1000;
-    % opts.MaxIter = 1000;
-    % opts.TolFun = 1e-12;
-    % opts.TolX = 1e-12;
-    opts.Weights = weights;
-
-    % opts.StartPoint = [0.2 -0.02  0.007 0.001 -1 -0.3];
-    % opts.Lower = [0 -1  0 0 -2 -2];
-    % opts.Upper = [1  1  1 2  2  0];
-
-    opts.StartPoint =   [0 0 0     0 0 0];
-    opts.Lower =        [-1 -1  -1   0 -Inf -Inf];
-    opts.Upper =        [1  1  1   Inf Inf 0];
-
-    [fitresult_D2_old, gof] = fit( [xData, yData], zData, ft, opts );
-    co_KM_opti_old.b  = [fitresult_D2_old.b0 0 fitresult_D2_old.b1 0 fitresult_D2_old.b2 0];
-    co_KM_opti_old.eb = [fitresult_D2_old.rb0 fitresult_D2_old.rb1 fitresult_D2_old.rb2 ];     % for D2
-
-
-    %% d_ij(r) plots
-    h(3) = figure;
-    subplot(2,2,1);
-    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-        fitresult_D1.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra1+fitresult_D1.a11,'LineWidth',2)
-    hold on
-    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-        fitresult_D1_o.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1_o.ra1+fitresult_D1_o.a11,'LineWidth',2)
-%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%         fitresult_D1_old.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1_old.ra1,'LineWidth',2,'Color','k')
-%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%         (-0.36).*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^(-1),'LineWidth',2,'Color',[0.501960813999176 0.501960813999176 0.501960813999176])
-    axis square
-    ylabel('$d_{11}$', 'interpreter','latex')
-    if norm_r==1
-    xlabel('$r / \lambda$', 'interpreter','latex')
-    else
-    xlabel('$r/m $', 'interpreter','latex')
-    end 
-    xlim([min(y(:,1)) max(y(:,1))])
-    set(gca, 'FontSize',14)
-    fig_setup
-    legend off
-    txt = {'(a)'};
-    a=text(4,0.5,txt);
-    a.Units='normalized';
-    %
-    subplot(2,2,2);
-    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-        fitresult_D2.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb0+fitresult_D2.b00,'LineWidth',2)
-    hold on
-    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-        fitresult_D2_o.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb0+fitresult_D2_o.b00,'LineWidth',2)
-
-%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%         fitresult_D2_old.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb0,'LineWidth',2,'Color','k')
-    axis square
-    ylabel('$d_{20}$', 'interpreter','latex')
-    if norm_r==1
-        xlabel('$r / \lambda$', 'interpreter','latex')
-    else
-        xlabel('$r\ (m)$', 'interpreter','latex')
-    end
-    xlim([min(y(:,1)) max(y(:,1))])
-    set(gca, 'FontSize',14)
-    fig_setup
-    legend off
-    txt = {'(b)'};
-    b=text(4,0.5,txt);
-    b.Units='normalized';
-
-    %
-    subplot(2,2,3);
-    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-        fitresult_D2.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb1+fitresult_D2.b11,'LineWidth',2)
-    hold on
-    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-        fitresult_D2_o.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb1+fitresult_D2_o.b11,'LineWidth',2)
-%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%         fitresult_D2_old.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb1,'LineWidth',2,'Color','k')
-    axis square
-    ylabel('$d_{21}$', 'interpreter','latex')
-    if norm_r==1
-        xlabel('$r / \lambda$', 'interpreter','latex')
-    else
-        xlabel('$r\ (m)$', 'interpreter','latex')
-    end
-    xlim([min(y(:,1)) max(y(:,1))])
-    set(gca, 'FontSize',14)
-    fig_setup
-    legend off
-    txt = {'(c)'};
-    c=text(4,0.5,txt);
-    c.Units='normalized';
-
-
-    %
-    subplot(2,2,4);
-    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-        fitresult_D2.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb2+fitresult_D2.b22,'LineWidth',2)
-    hold on
-    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-        fitresult_D2_o.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb2+fitresult_D2_o.b22,'LineWidth',2)
-%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%         fitresult_D2_old.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb2,'LineWidth',2,'Color','k')
-%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
-%         0.0144.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^(-1),'LineWidth',2,'Color',[0.501960813999176 0.501960813999176 0.501960813999176])
-    axis square
-    ylabel('$d_{22}$', 'interpreter','latex')
-    if norm_r==1
-        xlabel('$r / \lambda$', 'interpreter','latex')
-    else
-        xlabel('$r\ (m)$', 'interpreter','latex')
-    end
-    xlim([min(y(:,1)) max(y(:,1))])
-    set(gca, 'FontSize',14)
-    set(gca,'TickLabelInterpreter','latex');
-    set(gcf, 'Color', 'w')
-    %legend('optimized','non-optimized','Location','NorthWest')
-%     legend({'Optimized','Non-optimized','Optimized no offset','K62'},'Position',[0.45 0.5 0.1 0.05])
-    legend({'Optimized','Non-optimized','K62'},'Position',[0.45 0.5 0.1 0.05])
-    fig_setup
-% legend off
-    txt = {'(d)'};
-    d=text(4,0.5,txt);
-    d.Units='normalized';
-
-    font=22;
-    a.FontSize = font;
-    b.FontSize = font;
-    c.FontSize = font;
-    d.FontSize = font;
-
-    pos_txt=[-0.68   0.9];
-    a.Position=pos_txt;
-    b.Position=pos_txt;
-    c.Position=pos_txt;
-    d.Position=pos_txt;
-
-    %% 3D plots
-    figure
-    % surf(xx, y, z1_opti,'EdgeColor','none')
-    scatter3(xData_o1, yData_o1, zData_o1,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    hold on
-    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D1,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
-    % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
-    set(gca,'yScale','log') 
-    set(gcf, 'Color', 'w')
-    set(gca, 'FontSize',18)
-    if norm_ur==1
-        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-    else
-        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-    end
-    if norm_r==1
-        ylabel('$r / \lambda$', 'interpreter','latex')
-    else
-        ylabel('$r\ (m)$', 'interpreter','latex')
-    end
-    zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
-    % zlabel('$D1 (x,r) / [\frac{m}{s^2}]$', 'interpreter','latex')
-    colormap (parula(200))
-    % colorbar
-    grid off
-    xlim([min(xData) max(xData)])
-    ylim([min(y(:,1)) max(y(:,1))])
-    % zlim([-0.005 0.005])
-    axis square
-    set(gca,'YDir','reverse');
-    set(gca,'TickLabelInterpreter','latex');
-    title('non-optimized KM, optimized fit','interpreter','latex');
-    close
-
-
-    figure
-    % surf(xx, y, z2_opti,'EdgeColor','none')
-    scatter3(xData_o2,yData_o2,zData_o2,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    hold on
-    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D2,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
-    % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
-    set(gca,'yScale','log') 
-    % set(gca,'zScale','log')  
-    set(gcf, 'Color', 'w')
-    set(gca, 'FontSize',18)
-    if norm_ur==1
-        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-    else
-        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-    end
-    if norm_r==1
-        ylabel('$r / \lambda$', 'interpreter','latex')
-    else
-        ylabel('$r\ (m)$', 'interpreter','latex')
-    end
-    % zlabel('$D2 (x,r) / [\frac{m^2}{s^3}]$', 'interpreter','latex')
-    zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
-    colormap (parula(200))
-    % colorbar
-    grid off
-    xlim([min(xData) max(xData)])
-    ylim([min(y(:,1)) max(y(:,1))])
-    % zlim([0 2*max(zData)])
-    % caxis([0 max(zData)])
-    axis square
-    % close
-    set(gca,'YDir','reverse');
-    set(gca,'TickLabelInterpreter','latex');
-    % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
-    % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
-    title('non-optimized KM, optimized fit','interpreter','latex');
-    close
-
-    figure
-    scatter3(xData_o1, yData_o1, zData_o1,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    hold on
-    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D1_o,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
-    set(gca,'yScale','log') 
-    set(gcf, 'Color', 'w')
-    set(gca, 'FontSize',18)
-    if norm_ur==1
-        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-    else
-        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-    end
-    if norm_r==1
-        ylabel('$r / \lambda$', 'interpreter','latex')
-    else
-        ylabel('$r\ (m)$', 'interpreter','latex')
-    end
-    zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
-    colormap (parula(200))
-    grid off
-    xlim([min(xData) max(xData)])
-    ylim([min(y(:,1)) max(y(:,1))])
-    axis square
-    set(gca,'YDir','reverse');
-    set(gca,'TickLabelInterpreter','latex');
-    title('non-optimized KM, non-optimized fit','interpreter','latex');
-    close
-
-
-    figure
-    scatter3(xData_o2,yData_o2,zData_o2,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    hold on
-    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D2_o,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
-    set(gca,'yScale','log') 
-    set(gcf, 'Color', 'w')
-    set(gca, 'FontSize',18)
-    if norm_ur==1
-        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-    else
-        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-    end
-    if norm_r==1
-        ylabel('$r / \lambda$', 'interpreter','latex')
-    else
-        ylabel('$r\ (m)$', 'interpreter','latex')
-    end
-    zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
-    colormap (parula(200))
-    grid off
-    xlim([min(xData) max(xData)])
-    ylim([min(y(:,1)) max(y(:,1))])
-    axis square
-    set(gca,'YDir','reverse');
-    set(gca,'TickLabelInterpreter','latex');
-    title('non-optimized KM, non-optimized fit','interpreter','latex');
-    set(gca,'TickLabelInterpreter','latex');
-    close
-    
-    
-    
-
-    z1=nan(length(evaluated),increment_bin);
-    z2=nan(length(evaluated),increment_bin);
-    xx_no_opti  = nan(length(evaluated),increment_bin);
-    
-    for i=1:length(evaluated)
-%         xx_no_opti(i,:)=evaluated(i).y_mean_bin;
-%         z1(i,:)=(evaluated(i).D1);
-%         z2(i,:)=(evaluated(i).D2);
-        xx_no_opti=xx;
-        z1(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1(evaluated(i).x_bin_not_nan));
-        z2(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2(evaluated(i).x_bin_not_nan));
-    end
-    
-    [xData, yData, zData] = prepareSurfaceData( xx, y, z1_opti);
-    tmp_u=repmat(linspace(min(xData),max(xData),100),100,1);
-    tmp_r=repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100);
-
-    h(4) = figure;
-%     scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    scatter3(xData,yData,zData,'MarkerFaceColor',[0    0.4470    0.7410],'MarkerEdgeColor',[1 1 1])
-    hold on
-    [xData, yData, zData] = prepareSurfaceData( xx_no_opti, y, z1);
-    scatter3(xData,yData,zData,'MarkerFaceColor',[0.8500, 0.3250, 0.0980],'MarkerEdgeColor',[1 1 1])
-    set(gca,'yScale','log') 
-    set(gcf, 'Color', 'w')
-    set(gca, 'FontSize',18)
-    if norm_ur==1
-        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-    else
-        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-    end
-    if norm_r==1
-        ylabel('$r / \lambda$', 'interpreter','latex')
-    else
-        ylabel('$r\ (m)$', 'interpreter','latex')
-    end
-    zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
-    % zlabel('$D1 (x,r) / [\frac{m}{s^2}]$', 'interpreter','latex')
-    colormap (parula(200))
-    % colorbar
-    grid off
-    xlim([min(xData) max(xData)])
-    ylim([min(y(:,1)) max(y(:,1))])
-    % zlim([-0.005 0.005])
-    axis square
-    set(gca,'YDir','reverse');
-    rotate3d on
-    legend({'Optimized','Non-optimized'},'Position',[0.5 0.7 0.3 0.1])
-
-
-
-    [xData, yData, zData] = prepareSurfaceData( xx, y, z2_opti);
-    tmp_u=repmat(linspace(min(xData),max(xData),100),100,1);
-    tmp_r=repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100);
-    h(5) = figure;
-%     scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
-    scatter3(xData,yData,zData,'MarkerFaceColor',[0    0.4470    0.7410],'MarkerEdgeColor',[1 1 1])
-    hold on
-    [xData, yData, zData] = prepareSurfaceData( xx_no_opti, y, z2);
-    tmp_u=repmat(linspace(min(xData),max(xData),100),100,1);
-    tmp_r=repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100);
-    scatter3(xData,yData,zData,'MarkerFaceColor',[0.8500, 0.3250, 0.0980],'MarkerEdgeColor',[1 1 1])
-    % surf(tmp_u,tmp_r,...
-    %     (d20_R+d21_R.*tmp_u+d22_R.*tmp_u.^2)...
-    %     ,'FaceColor',[0.831372559070587 0.815686285495758 0.7843137383461])
-    set(gca,'yScale','log') 
-    % set(gca,'zScale','log')  
-    set(gcf, 'Color', 'w')
-    set(gca, 'FontSize',18)
-    if norm_ur==1
-        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-    else
-        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
-    end
-    if norm_r==1
-        ylabel('$r / \lambda$', 'interpreter','latex')
-    else
-        ylabel('$r\ (m)$', 'interpreter','latex')
-    end
-    % zlabel('$D2 (x,r) / [\frac{m^2}{s^3}]$', 'interpreter','latex')
-    zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
-    colormap (parula(200))
-    % colorbar
-    grid off
-    xlim([min(xData) max(xData)])
-    ylim([min(y(:,1)) max(y(:,1))])
-    % zlim([0 2*max(zData)])
-    caxis([0 max(zData)])
-    axis square
-    % close
-    set(gca,'YDir','reverse');
-    rotate3d on
-    legend({'Optimized','Non-optimized'},'Position',[0.5 0.7 0.3 0.1])
-    
-    if ischar(save_path)
-        savefig(h,fullfile(save_path,append(save_name,'_','KM_Fit.fig')),'compact')
-        for a = 1:length(h)
-            exportgraphics(h(a),fullfile(save_path,append(save_name,'_',sprintf('KM_Fit_%d.png', a))))
-        end
-    end 
-    
-end
-for i=1:5
-    tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-    uiwait(gcf);
-    delete(tmp_ui);
-    close
-end
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% This function performs the surface fit with a linear function for $D^{(1)}\left(u_r,r\right)$ 
+% and a parabolic function for $D^{(2)}\left(u_r,r\right)$ to the optimized and non-optimized KMC's.
+% Coefficients $d_{ij}(r)$ in the fits are functions of scale~$r$ of the form 
+% $\alpha (r/\lambda)^{\beta}+\gamma$. After fitting, this function plots the optimized $D^{(1,2)}$ 
+% and its surface fits. This function also plots the parameters $d_{11}$, $d_{20}$, $d_{21}$ and 
+% $d_{22}$ as a function of $\frac{r}{\lambda}$ for optimized and non-optimized 
+% $D^{(1,2)}\left(u_r,r\right)$.
+%
+% Arguments IN
+% evaluated_point = struct array calculated in the function 'conditional_moment'
+% increment_bin = number of bins
+% taylor_L = Taylor length scale in meters
+% Fs = Acquisition/Sampling Frequency in Hz
+% m_data = mean of the data
+% markov = markov length in number of samples
+% multi_point = weather to do multi-point analysis or not 1=Yes, 0=No
+% condition = condition for multi-point analysis
+% norm_ur = normalization of the data using $\sigma_\infty$? data 1=Yes, 0=No
+% norm_r = normalization of the scale using $\lambda$? data 1=Yes, 0=No
+% save_path = path for saving figures and files
+% save_name = name for saving files
+%
+% Arguments OUT
+% co_KM_opti = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients using the surface fits
+% co_KM_opti_old = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients using the
+% surface fits without an offset
+% co_no_opti = Coefficients $d_{ij}(r)$ of the non-optimized Kramers-Moyal coefficients using the surface fits
+% fitresult_D1_conf = Confidence intervals for fit coefficients of D1
+% fitresult_D2_conf = Confidence intervals for fit coefficients of D2
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+function[co_KM_opti,co_KM_opti_old,co_no_opti,fitresult_D1_conf,fitresult_D2_conf]=FIT_KM(evaluated_point,increment_bin,taylor_L,Fs,m_data,markov,multi_point,condition,norm_ur,norm_r,save_path,save_name)
+
+% taylor_L=1;
+
+if multi_point==1
+    for k=1:length(condition)
+            my_field = sprintf('point_eval_%d', k);              
+            co_KM_opti.(my_field)               = struct('a',NaN(1,2),'ea',nan,'b',NaN(1,6),'eb',NaN(1,3));
+            co_KM_opti_old.(my_field)           = struct('a',NaN(1,2),'ea',nan,'b',NaN(1,6),'eb',NaN(1,3));
+            co_no_opti.(my_field)               = struct('a',NaN(1,2),'ea',nan,'b',NaN(1,6),'eb',NaN(1,3));
+    end
+    
+    for k=1:length(condition)
+            my_field    = sprintf('point_eval_%d', k);    
+            clear evaluated
+            evaluated   = evaluated_point.(my_field);
+            
+            tmp=0;
+            for scal    = 1:length(evaluated)
+                tmp     = tmp+sum(evaluated(scal).x_bin_not_nan>0);
+            end
+            
+            if tmp>0
+                xx      = nan(length(evaluated),increment_bin);
+                y       = nan(length(evaluated),increment_bin);
+                z1      = nan(length(evaluated),increment_bin);
+                z1e     = nan(length(evaluated),increment_bin);
+                z2      = nan(length(evaluated),increment_bin);
+                z2e     = nan(length(evaluated),increment_bin);
+                z1_opti = nan(length(evaluated),increment_bin);
+                z2_opti = nan(length(evaluated),increment_bin);
+                z4      = nan(length(evaluated),increment_bin);
+
+                % for i=find((evaluated.r)/taylor_L>4):length(evaluated) 
+                for i=1:length(evaluated)
+                    xx(i,1:sum(evaluated(i).x_bin_not_nan))=evaluated(i).y_mean_bin(evaluated(i).x_bin_not_nan);
+                    if norm_r==1
+                    y(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).r./taylor_L);
+                    else
+                    y(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).r);   
+                    end
+                    z1(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1(evaluated(i).x_bin_not_nan));
+                    z1e(i,1:sum(evaluated(i).x_bin_not_nan))=abs(1./(evaluated(i).eD1(evaluated(i).x_bin_not_nan)));
+
+                    z2(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2(evaluated(i).x_bin_not_nan));
+                    z2e(i,1:sum(evaluated(i).x_bin_not_nan))=abs(1./(evaluated(i).eD2(evaluated(i).x_bin_not_nan)));
+
+                    z1_opti(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1_opti);
+                    z2_opti(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2_opti);
+
+                    z4(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D4(evaluated(i).x_bin_not_nan));
+                end    
+
+                %TEST Beginn
+                % y=y-(((markov./Fs).*m_data)/taylor_L);
+                %TEst Ende
+
+
+                %% D1-Fit
+                [fitresult_D1, xData, yData, zData, weights]    = FIT_D1(xx, y, z1_opti, z1e);
+                fitresult_D1_conf                               = confint(fitresult_D1);
+
+                h(1) = figure;
+                % surf(xx, y, z1_opti,'EdgeColor','none')
+                scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+                % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+                hold on
+                surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D1,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
+%                 surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),...
+%                     (-0.36.*repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100).^(-1).*repmat(linspace(min(xData),max(xData),100),100,1)),'FaceColor',[0.501960813999176 0.501960813999176 0.501960813999176],'EdgeColor',[0.501960813999176 0.501960813999176 0.501960813999176])
+
+                % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
+                set(gca,'yScale','log') 
+                set(gcf, 'Color', 'w')
+                set(gca, 'FontSize',18)
+                if norm_ur==1
+                    xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+                else
+                    xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+                end
+                if norm_r==1
+                    ylabel('$r / \lambda$', 'interpreter','latex')
+                else
+                    ylabel('$r\ (m)$', 'interpreter','latex')
+                end
+                zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
+                % zlabel('$D1 (x,r) / [\frac{m}{s^2}]$', 'interpreter','latex')
+                colormap (parula(200))
+                % colorbar
+                grid off
+                xlim([min(xData) max(xData)])
+                ylim([min(y(:,1)) max(y(:,1))])
+                % zlim([-0.005 0.005])
+                axis square
+                set(gca,'YDir','reverse');
+                title('optimized KM, optimized fit','interpreter','latex');
+                % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
+                % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
+                rotate3d on
+                colorbar('north','TickLabelInterpreter','latex','AxisLocation','out');
+                fig_setup
+                legend off
+                txt = {'(a)'};
+                a=text(4,0.5,txt);
+                a.Units='normalized';
+
+                %%%%%  2019 %%%%%%%%
+                % optimized coefficients 
+                % D1(x,y) = (a1*y^ra1+a11)*x
+                co_KM_opti.(my_field).a  = [fitresult_D1.a1 fitresult_D1.a11];
+                co_KM_opti.(my_field).ea = [fitresult_D1.ra1];
+
+                % non-optimized coefficients 
+                [fitresult_D1_o, xData_o1, yData_o1, zData_o1, weights] = FIT_D1(xx, y, z1, z1e);
+                %%%%%  2019 %%%%%%%%
+                % D1(x,y) = (a1*y^ra1+a11)*x
+                co_no_opti.(my_field).a  = [fitresult_D1_o.a1 fitresult_D1_o.a11];
+                co_no_opti.(my_field).ea = [fitresult_D1_o.ra1];
+
+                % optimized coefficients no offset
+                % D1(x,y) = (a1*y^ra1)*x
+                ft = fittype( '(a1*y^ra1)*x', 'independent', {'x', 'y'}, 'dependent', 'z' );
+                opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
+                % opts.DiffMinChange = 1e-12;
+                % opts.Display = 'Off';
+                % opts.MaxFunEvals = 1000;
+                % opts.MaxIter = 1000;
+                % opts.TolFun = 1e-12;
+                % opts.TolX = 1e-12;
+                opts.Weights = weights;
+
+                % opts.StartPoint = [-0.1  -1];
+                % opts.Lower = [-2 -2];
+                % opts.Upper = [0 0];
+
+                opts.StartPoint = [0  0];
+                opts.Lower = [-1 -Inf];
+                opts.Upper = [1 0];
+
+                [fitresult_D1_old, gof] = fit( [xData, yData], zData, ft, opts );
+                co_KM_opti_old.(my_field).a  = [fitresult_D1_old.a1 0];
+                co_KM_opti_old.(my_field).ea = [fitresult_D1_old.ra1];
+
+                % figure
+                % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra0)
+                % hold on
+                % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra1)
+                % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^0)
+                % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a3.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra3)
+                % set(gca,'xScale','log') 
+                % axis square
+                % set(gcf, 'Color', 'w')
+                % % grid on
+                % xlabel('$r / \lambda$', 'interpreter','latex')
+                % legend('d_{10}(r)','d_{11}(r)','d_{12}(r)','d_{13}(r)')
+                % xlim([min(y(:,1)) max(y(:,1))])
+
+
+
+                %% D2-Fit
+                [fitresult_D2, xData, yData, zData, weights] = FIT_D2(xx, y, z2_opti, z2e);
+                fitresult_D2_conf                            = confint(fitresult_D2);
+                h(2) = figure;
+                % surf(xx, y, z2_opti,'EdgeColor','none')
+                scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+                % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+                hold on
+                surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D2,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
+%                 surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),...
+%                     (0.0144.*repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100).^(-1).*repmat(linspace(min(xData),max(xData),100),100,1).^(2)),'FaceColor',[0.501960813999176 0.501960813999176 0.501960813999176],'EdgeColor',[0.501960813999176 0.501960813999176 0.501960813999176])
+                % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
+                set(gca,'yScale','log') 
+                % set(gca,'zScale','log')  
+                set(gcf, 'Color', 'w')
+                set(gca, 'FontSize',18)
+                if norm_ur==1
+                    xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+                else
+                    xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+                end
+                if norm_r==1
+                    ylabel('$r / \lambda$', 'interpreter','latex')
+                else
+                    ylabel('$r\ (m)$', 'interpreter','latex')
+                end
+                % zlabel('$D2 (x,r) / [\frac{m^2}{s^3}]$', 'interpreter','latex')
+                zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
+                colormap (parula(200))
+                % colorbar
+                grid off
+                xlim([min(xData) max(xData)])
+                ylim([min(y(:,1)) max(y(:,1))])
+                % zlim([0 2*max(zData)])
+                % caxis([0 max(zData)])
+                axis square
+                % close
+                set(gca,'YDir','reverse');
+                set(gca,'TickLabelInterpreter','latex');
+                title('optimized KM, optimized fit','interpreter','latex');
+                % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
+                % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
+                rotate3d on
+                colorbar('north','TickLabelInterpreter','latex','AxisLocation','out');
+                fig_setup
+                legend off
+                txt = {'(b)'};
+                b=text(4,0.5,txt);
+                b.Units='normalized';
+
+                % optimized coefficients 
+                %%%%%  2019 %%%%%%%%
+                % D2(x,y) = (b0*y^rb0+b00) + (b1*y^rb1+b11)*x + (b2*y^rb2+b22)*x^2
+                co_KM_opti.(my_field).b  = [fitresult_D2.b0 fitresult_D2.b00 fitresult_D2.b1 fitresult_D2.b11 fitresult_D2.b2 fitresult_D2.b22];
+                co_KM_opti.(my_field).eb = [fitresult_D2.rb0 fitresult_D2.rb1 fitresult_D2.rb2 ];     % for D2
+
+                % non-optimized coefficients 
+                [fitresult_D2_o, xData_o2, yData_o2, zData_o2, weights] = FIT_D2(xx, y, z2, z2e);
+                %%%%%  2019 %%%%%%%%
+                % D2(x,y) = (b0*y^rb0+b00) + (b1*y^rb1+b11)*x + (b2*y^rb2+b22)*x^2
+                co_no_opti.(my_field).b  = [fitresult_D2_o.b0 fitresult_D2_o.b00 fitresult_D2_o.b1 fitresult_D2_o.b11 fitresult_D2_o.b2 fitresult_D2_o.b22];
+                co_no_opti.(my_field).eb = [ fitresult_D2_o.rb0 fitresult_D2_o.rb1 fitresult_D2_o.rb2 ];     % for D2
+
+
+                % optimized coefficients no offset
+                ft = fittype( '(b0*y^rb0) + (b1*y^rb1)*x + (b2*y^rb2)*x^2', 'independent', {'x', 'y'}, 'dependent', 'z' );
+                opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
+                % opts.DiffMinChange = 1e-12;
+                % opts.Display = 'Off';
+                % opts.MaxFunEvals = 1000;
+                % opts.MaxIter = 1000;
+                % opts.TolFun = 1e-12;
+                % opts.TolX = 1e-12;
+                opts.Weights = weights;
+
+                % opts.StartPoint = [0.2 -0.02  0.007 0.001 -1 -0.3];
+                % opts.Lower = [0 -1  0 0 -2 -2];
+                % opts.Upper = [1  1  1 2  2  0];
+
+                opts.StartPoint =   [0 0 0     0 0 0];
+                opts.Lower =        [-1 -1  -1   0 -Inf -Inf];
+                opts.Upper =        [1  1  1   Inf Inf 0];
+
+                [fitresult_D2_old, gof] = fit( [xData, yData], zData, ft, opts );
+                co_KM_opti_old.(my_field).b  = [fitresult_D2_old.b0 0 fitresult_D2_old.b1 0 fitresult_D2_old.b2 0];
+                co_KM_opti_old.(my_field).eb = [fitresult_D2_old.rb0 fitresult_D2_old.rb1 fitresult_D2_old.rb2 ];     % for D2
+
+
+                %% d_ij(r) plots
+                h(3) = figure;
+                subplot(2,2,1);
+                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+                    fitresult_D1.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra1+fitresult_D1.a11,'LineWidth',2)
+                hold on
+                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+                    fitresult_D1_o.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1_o.ra1+fitresult_D1_o.a11,'LineWidth',2)
+%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%                     fitresult_D1_old.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1_old.ra1,'LineWidth',2,'Color','k')
+%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%                     (-0.36).*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^(-1),'LineWidth',2,'Color',[0.501960813999176 0.501960813999176 0.501960813999176])
+                axis square
+                ylabel('$d_{11}$', 'interpreter','latex')          
+                if norm_r==1
+                xlabel('$r / \lambda$', 'interpreter','latex')
+                else
+                xlabel('$r/m $', 'interpreter','latex')
+                end 
+                xlim([min(y(:,1)) max(y(:,1))])
+                set(gca, 'FontSize',12)
+                %
+                subplot(2,2,2);
+                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+                    fitresult_D2.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb0+fitresult_D2.b00,'LineWidth',2)
+                hold on
+                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+                    fitresult_D2_o.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb0+fitresult_D2_o.b00,'LineWidth',2)
+
+%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%                     fitresult_D2_old.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb0,'LineWidth',2,'Color','k')
+                axis square
+                ylabel('$d_{20}$', 'interpreter','latex')
+                if norm_r==1
+                xlabel('$r / \lambda$', 'interpreter','latex')
+                else
+                xlabel('$r/m $', 'interpreter','latex')
+                end 
+                xlim([min(y(:,1)) max(y(:,1))])
+                set(gca, 'FontSize',12)
+                %
+                subplot(2,2,3);
+                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+                    fitresult_D2.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb1+fitresult_D2.b11,'LineWidth',2)
+                hold on
+                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+                    fitresult_D2_o.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb1+fitresult_D2_o.b11,'LineWidth',2)
+%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%                     fitresult_D2_old.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb1,'LineWidth',2,'Color','k')
+                axis square
+                ylabel('$d_{21}$', 'interpreter','latex')
+                if norm_r==1
+                xlabel('$r / \lambda$', 'interpreter','latex')
+                else
+                xlabel('$r/m $', 'interpreter','latex')
+                end 
+                xlim([min(y(:,1)) max(y(:,1))])
+                set(gca, 'FontSize',12)
+                %
+                subplot(2,2,4);
+                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+                    fitresult_D2.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb2+fitresult_D2.b22,'LineWidth',2)
+                hold on
+                plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+                    fitresult_D2_o.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb2+fitresult_D2_o.b22,'LineWidth',2)
+%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%                     fitresult_D2_old.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb2,'LineWidth',2,'Color','k')
+%                 plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%                     0.0144.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^(-1),'LineWidth',2,'Color',[0.501960813999176 0.501960813999176 0.501960813999176])
+                axis square
+                ylabel('$d_{22}$', 'interpreter','latex')
+                if norm_r==1
+                xlabel('$r / \lambda$', 'interpreter','latex')
+                else
+                xlabel('$r/m $', 'interpreter','latex')
+                end 
+                xlim([min(y(:,1)) max(y(:,1))])
+                set(gca, 'FontSize',14)
+                set(gca,'TickLabelInterpreter','latex');
+                set(gcf, 'Color', 'w')
+                %legend('optimized','non-optimized','Location','NorthWest')
+%                 legend({'Optimized','Non-optimized','Optimized no offset','K62'},'Position',[0.45 0.5 0.1 0.05])
+                legend({'Optimized','Non-optimized','K62'},'Position',[0.45 0.5 0.1 0.05])
+
+                close all
+                disp(['point:' num2str(k) '/'  num2str(length(condition)) ])
+            end
+    end
+else
+    evaluated   = evaluated_point;
+    xx          = nan(length(evaluated),increment_bin);
+    y           = nan(length(evaluated),increment_bin);
+    z1          = nan(length(evaluated),increment_bin);
+    z1e         = nan(length(evaluated),increment_bin);
+    z2          = nan(length(evaluated),increment_bin);
+    z2e         = nan(length(evaluated),increment_bin);
+    z1_opti     = nan(length(evaluated),increment_bin);
+    z2_opti     = nan(length(evaluated),increment_bin);
+    z4          = nan(length(evaluated),increment_bin);
+    
+    % for i=find((evaluated.r)/taylor_L>4):length(evaluated) 
+    for i=1:length(evaluated)
+        xx(i,1:sum(evaluated(i).x_bin_not_nan))=evaluated(i).y_mean_bin(evaluated(i).x_bin_not_nan);
+        if norm_r==1
+        y(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).r./taylor_L);
+        else
+        y(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).r);   
+        end
+        z1(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1(evaluated(i).x_bin_not_nan));
+        z1e(i,1:sum(evaluated(i).x_bin_not_nan))=abs(1./(evaluated(i).eD1(evaluated(i).x_bin_not_nan)));
+
+        z2(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2(evaluated(i).x_bin_not_nan));
+        z2e(i,1:sum(evaluated(i).x_bin_not_nan))=abs(1./(evaluated(i).eD2(evaluated(i).x_bin_not_nan)));
+
+        z1_opti(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1_opti);
+        z2_opti(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2_opti);
+
+        z4(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D4(evaluated(i).x_bin_not_nan));
+    end
+    
+%     z1_opti=z1_opti.*0.5;
+%     z2_opti=z2_opti.*0.5;
+
+    %TEST Beginn
+    % y=y-(((markov./Fs).*m_data)/taylor_L);
+    %TEst Ende
+
+
+    %% D1-Fit
+    [fitresult_D1, xData, yData, zData, weights]    = FIT_D1(xx, y, z1_opti, z1e);
+    fitresult_D1_conf                               = confint(fitresult_D1);
+
+    h(1) = figure;
+    % surf(xx, y, z1_opti,'EdgeColor','none')
+    scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    hold on
+    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D1,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
+%     surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),...
+%         (-0.36.*repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100).^(-1).*repmat(linspace(min(xData),max(xData),100),100,1)),'FaceColor',[0.501960813999176 0.501960813999176 0.501960813999176],'EdgeColor',[0.501960813999176 0.501960813999176 0.501960813999176])
+
+    % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
+    set(gca,'yScale','log') 
+    set(gcf, 'Color', 'w')
+    set(gca, 'FontSize',18)
+    if norm_ur==1
+        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+    else
+        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+    end
+    if norm_r==1
+        ylabel('$r / \lambda$', 'interpreter','latex')
+    else
+        ylabel('$r\ (m)$', 'interpreter','latex')
+    end
+    zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
+    % zlabel('$D1 (x,r) / [\frac{m}{s^2}]$', 'interpreter','latex')
+    colormap (parula(200))
+    % colorbar
+    grid off
+    xlim([min(xData) max(xData)])
+    ylim([min(y(:,1)) max(y(:,1))])
+    % zlim([-0.005 0.005])
+    axis square
+    set(gca,'YDir','reverse');
+%     title('optimized KM, optimized fit','interpreter','latex');
+    % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
+    % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
+    rotate3d on
+    colorbar('north','TickLabelInterpreter','latex','AxisLocation','out');
+    fig_setup
+    legend off
+    txt = {'(a)'};
+    a=text(4,0.5,txt);
+    a.Units='normalized';
+
+
+    %%%%%  2019 %%%%%%%%
+    % optimized coefficients 
+    % D1(x,y) = (a1*y^ra1+a11)*x
+    co_KM_opti.a  = [fitresult_D1.a1 fitresult_D1.a11];
+    co_KM_opti.ea = [fitresult_D1.ra1];
+
+    % non-optimized coefficients 
+    [fitresult_D1_o, xData_o1, yData_o1, zData_o1, weights] = FIT_D1(xx, y, z1, z1e);
+    %%%%%  2019 %%%%%%%%
+    % D1(x,y) = (a1*y^ra1+a11)*x
+    co_no_opti.a  = [fitresult_D1_o.a1 fitresult_D1_o.a11];
+    co_no_opti.ea = [fitresult_D1_o.ra1];
+
+    % optimized coefficients no offset
+    % D1(x,y) = (a1*y^ra1)*x
+    ft = fittype( '(a1*y^ra1)*x', 'independent', {'x', 'y'}, 'dependent', 'z' );
+    opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
+    % opts.DiffMinChange = 1e-12;
+    % opts.Display = 'Off';
+    % opts.MaxFunEvals = 1000;
+    % opts.MaxIter = 1000;
+    % opts.TolFun = 1e-12;
+    % opts.TolX = 1e-12;
+    opts.Weights = weights;
+
+    % opts.StartPoint = [-0.1  -1];
+    % opts.Lower = [-2 -2];
+    % opts.Upper = [0 0];
+
+    opts.StartPoint = [0  0];
+    opts.Lower = [-1 -Inf];
+    opts.Upper = [1 0];
+
+    [fitresult_D1_old, gof] = fit( [xData, yData], zData, ft, opts );
+    co_KM_opti_old.a  = [fitresult_D1_old.a1 0];
+    co_KM_opti_old.ea = [fitresult_D1_old.ra1];
+
+    % figure
+    % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra0)
+    % hold on
+    % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra1)
+    % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^0)
+    % plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),fitresult_D1.a3.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra3)
+    % set(gca,'xScale','log') 
+    % axis square
+    % set(gcf, 'Color', 'w')
+    % % grid on
+    % xlabel('$r / \lambda$', 'interpreter','latex')
+    % legend('d_{10}(r)','d_{11}(r)','d_{12}(r)','d_{13}(r)')
+    % xlim([min(y(:,1)) max(y(:,1))])
+
+
+
+    %% D2-Fit
+    [fitresult_D2, xData, yData, zData, weights] = FIT_D2(xx, y, z2_opti, z2e);
+    fitresult_D2_conf                            = confint(fitresult_D2);
+    h(2) = figure;
+    % surf(xx, y, z2_opti,'EdgeColor','none')
+    scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    hold on
+    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D2,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
+%     surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),...
+%         (0.0144.*repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100).^(-1).*repmat(linspace(min(xData),max(xData),100),100,1).^(2)),'FaceColor',[0.501960813999176 0.501960813999176 0.501960813999176],'EdgeColor',[0.501960813999176 0.501960813999176 0.501960813999176])
+    % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
+    set(gca,'yScale','log') 
+    % set(gca,'zScale','log')  
+    set(gcf, 'Color', 'w')
+    set(gca, 'FontSize',18)
+    if norm_ur==1
+        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+    else
+        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+    end
+    if norm_r==1
+        ylabel('$r / \lambda$', 'interpreter','latex')
+    else
+        ylabel('$r\ (m)$', 'interpreter','latex')
+    end
+    % zlabel('$D2 (x,r) / [\frac{m^2}{s^3}]$', 'interpreter','latex')
+    zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
+    colormap (parula(200))
+    % colorbar
+    grid off
+    xlim([min(xData) max(xData)])
+    ylim([min(y(:,1)) max(y(:,1))])
+    % zlim([0 2*max(zData)])
+    % caxis([0 max(zData)])
+    axis square
+    % close
+    set(gca,'YDir','reverse');
+    set(gca,'TickLabelInterpreter','latex');
+%     title('optimized KM, optimized fit','interpreter','latex');
+    % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
+    % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
+    rotate3d on
+    colorbar('north','TickLabelInterpreter','latex','AxisLocation','out');
+    fig_setup
+    legend off
+    txt = {'(b)'};
+    b=text(4,0.5,txt);
+    b.Units='normalized';
+
+    font=22;
+    a.FontSize = font;
+    b.FontSize = font;
+    pos_txt=[-0.22   0.8];
+    a.Position=pos_txt;
+    b.Position=[-0.25  0.8];
+
+    % optimized coefficients 
+    %%%%%  2019 %%%%%%%%
+    % D2(x,y) = (b0*y^rb0+b00) + (b1*y^rb1+b11)*x + (b2*y^rb2+b22)*x^2
+    co_KM_opti.b  = [fitresult_D2.b0 fitresult_D2.b00 fitresult_D2.b1 fitresult_D2.b11 fitresult_D2.b2 fitresult_D2.b22];
+    co_KM_opti.eb = [fitresult_D2.rb0 fitresult_D2.rb1 fitresult_D2.rb2 ];     % for D2
+
+    % non-optimized coefficients 
+    [fitresult_D2_o, xData_o2, yData_o2, zData_o2, weights] = FIT_D2(xx, y, z2, z2e);
+    %%%%%  2019 %%%%%%%%
+    % D2(x,y) = (b0*y^rb0+b00) + (b1*y^rb1+b11)*x + (b2*y^rb2+b22)*x^2
+    co_no_opti.b  = [fitresult_D2_o.b0 fitresult_D2_o.b00 fitresult_D2_o.b1 fitresult_D2_o.b11 fitresult_D2_o.b2 fitresult_D2_o.b22];
+    co_no_opti.eb = [ fitresult_D2_o.rb0 fitresult_D2_o.rb1 fitresult_D2_o.rb2 ];     % for D2
+
+
+    % optimized coefficients no offset
+    ft = fittype( '(b0*y^rb0) + (b1*y^rb1)*x + (b2*y^rb2)*x^2', 'independent', {'x', 'y'}, 'dependent', 'z' );
+    opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
+    % opts.DiffMinChange = 1e-12;
+    % opts.Display = 'Off';
+    % opts.MaxFunEvals = 1000;
+    % opts.MaxIter = 1000;
+    % opts.TolFun = 1e-12;
+    % opts.TolX = 1e-12;
+    opts.Weights = weights;
+
+    % opts.StartPoint = [0.2 -0.02  0.007 0.001 -1 -0.3];
+    % opts.Lower = [0 -1  0 0 -2 -2];
+    % opts.Upper = [1  1  1 2  2  0];
+
+    opts.StartPoint =   [0 0 0     0 0 0];
+    opts.Lower =        [-1 -1  -1   0 -Inf -Inf];
+    opts.Upper =        [1  1  1   Inf Inf 0];
+
+    [fitresult_D2_old, gof] = fit( [xData, yData], zData, ft, opts );
+    co_KM_opti_old.b  = [fitresult_D2_old.b0 0 fitresult_D2_old.b1 0 fitresult_D2_old.b2 0];
+    co_KM_opti_old.eb = [fitresult_D2_old.rb0 fitresult_D2_old.rb1 fitresult_D2_old.rb2 ];     % for D2
+
+
+    %% d_ij(r) plots
+    h(3) = figure;
+    subplot(2,2,1);
+    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+        fitresult_D1.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1.ra1+fitresult_D1.a11,'LineWidth',2)
+    hold on
+    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+        fitresult_D1_o.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1_o.ra1+fitresult_D1_o.a11,'LineWidth',2)
+%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%         fitresult_D1_old.a1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D1_old.ra1,'LineWidth',2,'Color','k')
+%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%         (-0.36).*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^(-1),'LineWidth',2,'Color',[0.501960813999176 0.501960813999176 0.501960813999176])
+    axis square
+    ylabel('$d_{11}$', 'interpreter','latex')
+    if norm_r==1
+    xlabel('$r / \lambda$', 'interpreter','latex')
+    else
+    xlabel('$r/m $', 'interpreter','latex')
+    end 
+    xlim([min(y(:,1)) max(y(:,1))])
+    set(gca, 'FontSize',14)
+    fig_setup
+    legend off
+    txt = {'(a)'};
+    a=text(4,0.5,txt);
+    a.Units='normalized';
+    %
+    subplot(2,2,2);
+    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+        fitresult_D2.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb0+fitresult_D2.b00,'LineWidth',2)
+    hold on
+    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+        fitresult_D2_o.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb0+fitresult_D2_o.b00,'LineWidth',2)
+
+%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%         fitresult_D2_old.b0.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb0,'LineWidth',2,'Color','k')
+    axis square
+    ylabel('$d_{20}$', 'interpreter','latex')
+    if norm_r==1
+        xlabel('$r / \lambda$', 'interpreter','latex')
+    else
+        xlabel('$r\ (m)$', 'interpreter','latex')
+    end
+    xlim([min(y(:,1)) max(y(:,1))])
+    set(gca, 'FontSize',14)
+    fig_setup
+    legend off
+    txt = {'(b)'};
+    b=text(4,0.5,txt);
+    b.Units='normalized';
+
+    %
+    subplot(2,2,3);
+    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+        fitresult_D2.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb1+fitresult_D2.b11,'LineWidth',2)
+    hold on
+    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+        fitresult_D2_o.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb1+fitresult_D2_o.b11,'LineWidth',2)
+%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%         fitresult_D2_old.b1.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb1,'LineWidth',2,'Color','k')
+    axis square
+    ylabel('$d_{21}$', 'interpreter','latex')
+    if norm_r==1
+        xlabel('$r / \lambda$', 'interpreter','latex')
+    else
+        xlabel('$r\ (m)$', 'interpreter','latex')
+    end
+    xlim([min(y(:,1)) max(y(:,1))])
+    set(gca, 'FontSize',14)
+    fig_setup
+    legend off
+    txt = {'(c)'};
+    c=text(4,0.5,txt);
+    c.Units='normalized';
+
+
+    %
+    subplot(2,2,4);
+    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+        fitresult_D2.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2.rb2+fitresult_D2.b22,'LineWidth',2)
+    hold on
+    plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+        fitresult_D2_o.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_o.rb2+fitresult_D2_o.b22,'LineWidth',2)
+%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%         fitresult_D2_old.b2.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^fitresult_D2_old.rb2,'LineWidth',2,'Color','k')
+%     plot(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100),...
+%         0.0144.*logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).^(-1),'LineWidth',2,'Color',[0.501960813999176 0.501960813999176 0.501960813999176])
+    axis square
+    ylabel('$d_{22}$', 'interpreter','latex')
+    if norm_r==1
+        xlabel('$r / \lambda$', 'interpreter','latex')
+    else
+        xlabel('$r\ (m)$', 'interpreter','latex')
+    end
+    xlim([min(y(:,1)) max(y(:,1))])
+    set(gca, 'FontSize',14)
+    set(gca,'TickLabelInterpreter','latex');
+    set(gcf, 'Color', 'w')
+    %legend('optimized','non-optimized','Location','NorthWest')
+%     legend({'Optimized','Non-optimized','Optimized no offset','K62'},'Position',[0.45 0.5 0.1 0.05])
+    legend({'Optimized','Non-optimized','K62'},'Position',[0.45 0.5 0.1 0.05])
+    fig_setup
+% legend off
+    txt = {'(d)'};
+    d=text(4,0.5,txt);
+    d.Units='normalized';
+
+    font=22;
+    a.FontSize = font;
+    b.FontSize = font;
+    c.FontSize = font;
+    d.FontSize = font;
+
+    pos_txt=[-0.68   0.9];
+    a.Position=pos_txt;
+    b.Position=pos_txt;
+    c.Position=pos_txt;
+    d.Position=pos_txt;
+
+    %% 3D plots
+    figure
+    % surf(xx, y, z1_opti,'EdgeColor','none')
+    scatter3(xData_o1, yData_o1, zData_o1,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    hold on
+    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D1,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
+    % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
+    set(gca,'yScale','log') 
+    set(gcf, 'Color', 'w')
+    set(gca, 'FontSize',18)
+    if norm_ur==1
+        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+    else
+        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+    end
+    if norm_r==1
+        ylabel('$r / \lambda$', 'interpreter','latex')
+    else
+        ylabel('$r\ (m)$', 'interpreter','latex')
+    end
+    zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
+    % zlabel('$D1 (x,r) / [\frac{m}{s^2}]$', 'interpreter','latex')
+    colormap (parula(200))
+    % colorbar
+    grid off
+    xlim([min(xData) max(xData)])
+    ylim([min(y(:,1)) max(y(:,1))])
+    % zlim([-0.005 0.005])
+    axis square
+    set(gca,'YDir','reverse');
+    set(gca,'TickLabelInterpreter','latex');
+    title('non-optimized KM, optimized fit','interpreter','latex');
+    close
+
+
+    figure
+    % surf(xx, y, z2_opti,'EdgeColor','none')
+    scatter3(xData_o2,yData_o2,zData_o2,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    % scatter3(xData,yData,yData.*zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    hold on
+    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D2,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
+    % surf(repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100),feval(fitresult_D1,repmat(linspace(-4,4,100),100,1),repmat(linspace(min(y(:,1)),max(y(:,1)),100).',1,100)))
+    set(gca,'yScale','log') 
+    % set(gca,'zScale','log')  
+    set(gcf, 'Color', 'w')
+    set(gca, 'FontSize',18)
+    if norm_ur==1
+        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+    else
+        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+    end
+    if norm_r==1
+        ylabel('$r / \lambda$', 'interpreter','latex')
+    else
+        ylabel('$r\ (m)$', 'interpreter','latex')
+    end
+    % zlabel('$D2 (x,r) / [\frac{m^2}{s^3}]$', 'interpreter','latex')
+    zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
+    colormap (parula(200))
+    % colorbar
+    grid off
+    xlim([min(xData) max(xData)])
+    ylim([min(y(:,1)) max(y(:,1))])
+    % zlim([0 2*max(zData)])
+    % caxis([0 max(zData)])
+    axis square
+    % close
+    set(gca,'YDir','reverse');
+    set(gca,'TickLabelInterpreter','latex');
+    % title(['$(', num2str(fitresult_D1.a1,'%1.1E'), ' * r / \lambda^{' num2str(fitresult_D1.ra1,'%1.1E') '} ) * u_r $'],'interpreter','latex')   
+    % title(['$\left(',num2str(fitresult_D1.a1),'\cdot r/\lambda^{',num2str(fitresult_D1.ra1),'} +',num2str(fitresult_D1.a11),' \right)\cdot u_r $'],'interpreter','latex')   
+    title('non-optimized KM, optimized fit','interpreter','latex');
+    close
+
+    figure
+    scatter3(xData_o1, yData_o1, zData_o1,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    hold on
+    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D1_o,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
+    set(gca,'yScale','log') 
+    set(gcf, 'Color', 'w')
+    set(gca, 'FontSize',18)
+    if norm_ur==1
+        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+    else
+        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+    end
+    if norm_r==1
+        ylabel('$r / \lambda$', 'interpreter','latex')
+    else
+        ylabel('$r\ (m)$', 'interpreter','latex')
+    end
+    zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
+    colormap (parula(200))
+    grid off
+    xlim([min(xData) max(xData)])
+    ylim([min(y(:,1)) max(y(:,1))])
+    axis square
+    set(gca,'YDir','reverse');
+    set(gca,'TickLabelInterpreter','latex');
+    title('non-optimized KM, non-optimized fit','interpreter','latex');
+    close
+
+
+    figure
+    scatter3(xData_o2,yData_o2,zData_o2,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    hold on
+    surf(repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100),feval(fitresult_D2_o,repmat(linspace(min(xData),max(xData),100),100,1),repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100)),'EdgeColor','none')
+    set(gca,'yScale','log') 
+    set(gcf, 'Color', 'w')
+    set(gca, 'FontSize',18)
+    if norm_ur==1
+        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+    else
+        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+    end
+    if norm_r==1
+        ylabel('$r / \lambda$', 'interpreter','latex')
+    else
+        ylabel('$r\ (m)$', 'interpreter','latex')
+    end
+    zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
+    colormap (parula(200))
+    grid off
+    xlim([min(xData) max(xData)])
+    ylim([min(y(:,1)) max(y(:,1))])
+    axis square
+    set(gca,'YDir','reverse');
+    set(gca,'TickLabelInterpreter','latex');
+    title('non-optimized KM, non-optimized fit','interpreter','latex');
+    set(gca,'TickLabelInterpreter','latex');
+    close
+    
+    
+    
+
+    z1=nan(length(evaluated),increment_bin);
+    z2=nan(length(evaluated),increment_bin);
+    xx_no_opti  = nan(length(evaluated),increment_bin);
+    
+    for i=1:length(evaluated)
+%         xx_no_opti(i,:)=evaluated(i).y_mean_bin;
+%         z1(i,:)=(evaluated(i).D1);
+%         z2(i,:)=(evaluated(i).D2);
+        xx_no_opti=xx;
+        z1(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D1(evaluated(i).x_bin_not_nan));
+        z2(i,1:sum(evaluated(i).x_bin_not_nan))=(evaluated(i).D2(evaluated(i).x_bin_not_nan));
+    end
+    
+    [xData, yData, zData] = prepareSurfaceData( xx, y, z1_opti);
+    tmp_u=repmat(linspace(min(xData),max(xData),100),100,1);
+    tmp_r=repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100);
+
+    h(4) = figure;
+%     scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    scatter3(xData,yData,zData,'MarkerFaceColor',[0    0.4470    0.7410],'MarkerEdgeColor',[1 1 1])
+    hold on
+    [xData, yData, zData] = prepareSurfaceData( xx_no_opti, y, z1);
+    scatter3(xData,yData,zData,'MarkerFaceColor',[0.8500, 0.3250, 0.0980],'MarkerEdgeColor',[1 1 1])
+    set(gca,'yScale','log') 
+    set(gcf, 'Color', 'w')
+    set(gca, 'FontSize',18)
+    if norm_ur==1
+        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+    else
+        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+    end
+    if norm_r==1
+        ylabel('$r / \lambda$', 'interpreter','latex')
+    else
+        ylabel('$r\ (m)$', 'interpreter','latex')
+    end
+    zlabel('$D^{(1)} (u_r,r)$', 'interpreter','latex')
+    % zlabel('$D1 (x,r) / [\frac{m}{s^2}]$', 'interpreter','latex')
+    colormap (parula(200))
+    % colorbar
+    grid off
+    xlim([min(xData) max(xData)])
+    ylim([min(y(:,1)) max(y(:,1))])
+    % zlim([-0.005 0.005])
+    axis square
+    set(gca,'YDir','reverse');
+    rotate3d on
+    legend({'Optimized','Non-optimized'},'Position',[0.5 0.7 0.3 0.1])
+
+
+
+    [xData, yData, zData] = prepareSurfaceData( xx, y, z2_opti);
+    tmp_u=repmat(linspace(min(xData),max(xData),100),100,1);
+    tmp_r=repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100);
+    h(5) = figure;
+%     scatter3(xData,yData,zData,'MarkerFaceColor',[0 0 0],'MarkerEdgeColor',[1 1 1])
+    scatter3(xData,yData,zData,'MarkerFaceColor',[0    0.4470    0.7410],'MarkerEdgeColor',[1 1 1])
+    hold on
+    [xData, yData, zData] = prepareSurfaceData( xx_no_opti, y, z2);
+    tmp_u=repmat(linspace(min(xData),max(xData),100),100,1);
+    tmp_r=repmat(logspace(log10(min(y(:,1))),log10(max(y(:,1))),100).',1,100);
+    scatter3(xData,yData,zData,'MarkerFaceColor',[0.8500, 0.3250, 0.0980],'MarkerEdgeColor',[1 1 1])
+    % surf(tmp_u,tmp_r,...
+    %     (d20_R+d21_R.*tmp_u+d22_R.*tmp_u.^2)...
+    %     ,'FaceColor',[0.831372559070587 0.815686285495758 0.7843137383461])
+    set(gca,'yScale','log') 
+    % set(gca,'zScale','log')  
+    set(gcf, 'Color', 'w')
+    set(gca, 'FontSize',18)
+    if norm_ur==1
+        xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+    else
+        xlabel('$u_r\ (m/s)$', 'interpreter','latex')   
+    end
+    if norm_r==1
+        ylabel('$r / \lambda$', 'interpreter','latex')
+    else
+        ylabel('$r\ (m)$', 'interpreter','latex')
+    end
+    % zlabel('$D2 (x,r) / [\frac{m^2}{s^3}]$', 'interpreter','latex')
+    zlabel('$D^{(2)} (u_r,r)$', 'interpreter','latex')
+    colormap (parula(200))
+    % colorbar
+    grid off
+    xlim([min(xData) max(xData)])
+    ylim([min(y(:,1)) max(y(:,1))])
+    % zlim([0 2*max(zData)])
+    caxis([0 max(zData)])
+    axis square
+    % close
+    set(gca,'YDir','reverse');
+    rotate3d on
+    legend({'Optimized','Non-optimized'},'Position',[0.5 0.7 0.3 0.1])
+    
+    if ischar(save_path)
+        savefig(h,fullfile(save_path,append(save_name,'_','KM_Fit.fig')),'compact')
+        for a = 1:length(h)
+            exportgraphics(h(a),fullfile(save_path,append(save_name,'_',sprintf('KM_Fit_%d.png', a))))
+        end
+    end 
+    
+end
+for i=1:5
+    tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+    uiwait(gcf);
+    delete(tmp_ui);
+    close
+end
 end
\ No newline at end of file
diff --git a/Functions_v5-MATLAB/KM_Calculation.m b/Functions_v5-MATLAB/KM_Calculation.m
index 26a872a..fbf5584 100644
--- a/Functions_v5-MATLAB/KM_Calculation.m
+++ b/Functions_v5-MATLAB/KM_Calculation.m
@@ -24,9 +24,11 @@
 % m_data = mean of the data
 % multi_point = Multipoint condition 1=YES or 2=NO
 % condition = This input is for multipoint statistics
+% norm_r = normalization of the scale using $\lambda$? data 1=Yes, 0=No
 %
 % Arguments OUT
 % evaluated = a modified/updated struct 'evaluated' array
+% Enclosed in a "evaluated":
 % r is the scale in meters at which moments will be calculated and hence this r will
 % be the same at which D1 & D2 will be calculated===>r2
 % r_samp is nothing but the r in number of samples==>r2
diff --git a/Functions_v5-MATLAB/KM_STP_optimization.m b/Functions_v5-MATLAB/KM_STP_optimization.m
index a938bfc..fedd25a 100644
--- a/Functions_v5-MATLAB/KM_STP_optimization.m
+++ b/Functions_v5-MATLAB/KM_STP_optimization.m
@@ -25,6 +25,10 @@
 % tol_opti = Optimization: Tolerance of the range of Kramers-Moyal coefficients in % 
 %            It is the percentage(For Ex: 0.1 for 10 percent or 0.2 for 20 percent)of D1 or D2 
 %            within these limit which you want to optimize these coeffcients D1 & D2
+% norm_ur = normalization of the data using $\sigma_\infty$? data 1=Yes, 0=No
+% norm_r = normalization of the scale using $\lambda$? data 1=Yes, 0=No
+% save_path = path for saving figures and files
+% save_name = name for saving files
 %
 % Arguments OUT
 % evaluated = a modified/updated struct 'evaluated' array with optimized value of D1 & D2 
diff --git a/Functions_v5-MATLAB/KM_plot.m b/Functions_v5-MATLAB/KM_plot.m
index 00e616d..712d62c 100644
--- a/Functions_v5-MATLAB/KM_plot.m
+++ b/Functions_v5-MATLAB/KM_plot.m
@@ -11,6 +11,10 @@
 % taylor_L = Taylor length scale in meters
 % multi_point = Multipoint condition 1=YES or 2=NO
 % condition = This input is for multipoint statistics
+% norm_ur = normalization of the data using $\sigma_\infty$? data 1=Yes, 0=No
+% norm_r = normalization of the scale using $\lambda$? data 1=Yes, 0=No
+% save_path = path for saving figures and files
+% save_name = name for saving files
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 function KM_plot(co,evaluated,Fs,taylor_L,int_L,multi_point,condition,norm_ur,norm_r,save_path,save_name,varargin)
diff --git a/Functions_v5-MATLAB/KM_plot_raw.m b/Functions_v5-MATLAB/KM_plot_raw.m
index 4313e09..4750d8c 100644
--- a/Functions_v5-MATLAB/KM_plot_raw.m
+++ b/Functions_v5-MATLAB/KM_plot_raw.m
@@ -9,6 +9,10 @@
 % taylor_L = Taylor length scale in meters
 % point = Multipoint condition 1=YES or 2=NO
 % condition = This input is for multipoint statistics
+% norm_ur = normalization of the data using $\sigma_\infty$? data 1=Yes, 0=No
+% norm_r = normalization of the scale using $\lambda$? data 1=Yes, 0=No
+% save_path = path for saving figures and files
+% save_name = name for saving files
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 function KM_plot_raw(evaluated,Fs,taylor_L,point,condition,norm_ur,norm_r,save_path,save_name)
diff --git a/Functions_v5-MATLAB/OPTI_IFT_dij.m b/Functions_v5-MATLAB/OPTI_IFT_dij.m
index 9415a93..29d152c 100644
--- a/Functions_v5-MATLAB/OPTI_IFT_dij.m
+++ b/Functions_v5-MATLAB/OPTI_IFT_dij.m
@@ -1,386 +1,390 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% This function performs the pointwise optimization of Kramers-Moyal coefficients towards the 
-% integral fluctuation theorem will be done. Thereby the separation of scales/step increment 
-% (in samples) referred to the sequence from large to small scales in the cascade trajectory is set 
-% to a minimum of 1 sample and for the optimization, independent cascade trajectories (z=3) are used. 
-% Note, we use here a separation that is less than or equal to the Einstein-Markov length of $\Delta_{EM}$.
-% 
-% This function performs the optimization of $D^{(1,2)}\left(u_r,r\right)$ at each scale and at each 
-% value of velocity increment in order to satisfy the integral fluctuation theorem with minimum 
-% possible error and plots the optimized $d_{ij}$ as a function of $r$. The optimization procedure 
-% systematically changes $D^{(1,2)}\left(u_r,r\right)$ until the error function 
-% \begin{eqnarray}
-% \xi =|1-\langle e^{\mathrm{-}\Delta S_{tot}} \rangle_{max(N)}|
-% \end{eqnarray}
-% is minimized. 
-% 
-% Arguments IN
-% tol_D1          = Tolerance of the range of Kramers-Moyal coefficients in %% for the Optimization
-% tol_D2          = tol_D1;
-% evaluated = a modified/updated struct 'evaluated' array with in function 'KM_STP_optimization'
-% co_KM_opti = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients towards the best 
-%              Fokker-Planck equation to reproduce the conditional PDF's using the surface fits
-% fitresult_D1_conf = Confidence intervals for fit coefficients of D1 (co_KM_opti)
-% fitresult_D2_conf = Confidence intervals for fit coefficients of D2 (co_KM_opti)
-% int_L = Integral length scale in meters
-% taylor_L = Taylor length scale in meters
-% m_data = mean of the data
-% Fs = Acquisition/Sampling Frequency in Hz
-% data = filtered data 
-% z = This is the parameter which decides how to compute entropy using different methods using either 
-% overlapping or independent trajectories
-% z = 1 ==> Overlapping trajectories 
-% z = 3 ==> Independent trajectories
-% iter = The maximum number of iteration used for the optimization algorithms
-% markov = markov length in number of samples
-% trajec = 1 ==> The start/end of the cascade trajectory will be adjusted.
-% dr_ind = Separation of scales/step increment (in samples) referred to the sequence from large to small scales in the cascade trajectory
-% 
-% Arguments OUT
-% co_IFT_opti = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients towards the 
-%               integral fluctuation theorem using the surface fits
-% history = details of the optimization for every single iteration 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-function [co_IFT_opti,history]=OPTI_IFT_dij(tol_D1,tol_D2,evaluated,co_KM_opti,fitresult_D1_conf,fitresult_D2_conf,int_L,taylor_L,m_data,Fs,data,z,iter,markov,trajec,dr_ind,norm_ur,norm_r,save_path,save_name)
-set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
-set(groot, 'defaulttextinterpreter','latex');   
-set(groot, 'defaultLegendInterpreter','latex');
-set(groot, 'defaultAxesTitle','latex');
-
-% if z==1
-%     [Sm,Ds,DS,~,~,~,~]=checkFT(evaluated,int_L,taylor_L,Fs,data,m_data,z,co_KM_opti,1,markov,trajec,dr_ind);
-%     %Plot of Entropy
-%     plot_entropy(Sm,Ds,DS)
-%     set(gca, 'FontSize',10)
-%     data_length=str2double(inputdlg({'Data length for optimization in samples:'}));
-%     clear Sm Ds DS
-% end
-
-
-%% Initial D1 and D2 before optimization  
-clear r_norm
-r_norm           = nan(1,length(evaluated));
-
-if norm_r==1
-    for i=1:length(evaluated)
-        r_norm(1,i)  = (evaluated(i).r./taylor_L);
-    end
-else
-    for i=1:length(evaluated)
-        r_norm(1,i)  = (evaluated(i).r);
-    end
-end
-
-
-% fitresult_D1
-d11 = co_KM_opti.a(1).*r_norm.^co_KM_opti.ea(1)+co_KM_opti.a(2);
-% fitresult_D2
-d20 = co_KM_opti.b(1).*r_norm.^co_KM_opti.eb(1)+co_KM_opti.b(2);
-d21 = co_KM_opti.b(3).*r_norm.^co_KM_opti.eb(2)+co_KM_opti.b(4);
-% d21=0.*d21;
-d22 = co_KM_opti.b(5).*r_norm.^co_KM_opti.eb(3)+co_KM_opti.b(6);
-x0  = [d11 d20 d21 d22];
-
-
-%% Tolerance of the range of Kramers-Moyal coefficients
-clear x1 lb ub
-% x1(1:length(x0))    = nan;
-lb(1:length(x0))    = nan;
-ub(1:length(x0))    = nan;
-
-% percent of the range 
-% D1 Opti
-t                       = length(r_norm);
-lb(1:t)                 = x0(1:t)-abs(x0(1:t).*tol_D1); % Lower bound
-ub(1:t)                 = x0(1:t)+abs(x0(1:t).*tol_D1); % Upper bound
-
-% D2 Opti
-for i=1:3
-    lb(t*i+1:(t*i+1)+t-1)   = x0(t*i+1:(t*i+1)+t-1)-abs(x0(t*i+1:(t*i+1)+t-1).*tol_D2);
-    ub(t*i+1:(t*i+1)+t-1)   = x0(t*i+1:(t*i+1)+t-1)+abs(x0(t*i+1:(t*i+1)+t-1).*tol_D2);
-end
-
-
-% constant value 10% of max/min value
-% D1 Opti
-% t                       = length(r_norm);
-% lb(1:t)                 = x0(1:t)-(max(abs(x0(1:t))).*tol_D1);
-% ub(1:t)                 = x0(1:t)+(max(abs(x0(1:t))).*tol_D1);
- 
-% D2 Opti
-% for i=1:3
-%     lb(t*i+1:(t*i+1)+t-1)   = x0(t*i+1:(t*i+1)+t-1)-(max(abs(x0(t*i+1:(t*i+1)+t-1))).*tol_D2);
-%     ub(t*i+1:(t*i+1)+t-1)   = x0(t*i+1:(t*i+1)+t-1)+(max(abs(x0(t*i+1:(t*i+1)+t-1))).*tol_D2);
-% end
-
-
-% fix to min and max value
-% d11
-% lb(1:t)                 = min(lb(1:t));
-% ub(1:t)                 = max(ub(1:t));
-
-% d20
-% lb(t*1+1:(t*1+1)+t-1)   = min(lb(t*1+1:(t*1+1)+t-1));
-% ub(t*1+1:(t*1+1)+t-1)   = max(ub(t*1+1:(t*1+1)+t-1));
- 
-% d21
-% lb(t*2+1:(t*2+1)+t-1)   = min(lb(t*2+1:(t*2+1)+t-1));
-% ub(t*2+1:(t*2+1)+t-1)   = max(ub(t*2+1:(t*2+1)+t-1));
-
-% d22
-% lb(t*3+1:(t*3+1)+t-1)   = min(lb(t*3+1:(t*3+1)+t-1));
-% ub(t*3+1:(t*3+1)+t-1)   = max(ub(t*3+1:(t*3+1)+t-1));
-
-
-% using confidence intervals for fit coefficients of D1 and D2 (co_KM_opti)
-% D1 Opti
-% lb(1:2)    = [fitresult_D1_conf(1,1:2)];
-% lb(9)      = [fitresult_D1_conf(1,3)];
-% ub(1:2)    = [fitresult_D1_conf(2,1:2)];
-% ub(9)      = [fitresult_D1_conf(2,3)];
- 
-% D2 Opti
-% lb(3:8)       = [fitresult_D2_conf(1,1:6)];
-% lb(10:end)    = [fitresult_D2_conf(1,7:9)];
-% ub(3:8)       = [fitresult_D2_conf(2,1:6)];
-% ub(10:end)    = [fitresult_D2_conf(2,7:9)];
-
-% lb(isnan(lb))=0;
-% ub(isnan(ub))=0;
-
-
-%% The constraints were set in a physically and mathematically meaningful way: 
-% (see for more details \textbf{\textit{FIT\_KM}})
-% d11 <= 0
-if ub(1:t)>0
-    ub(ub(1:t)>0) = 0;
-end
-% d20 >= 0 
-if lb(t*1+1:(t*1+1)+t-1)<0
-    lb(lb(t*1+1:(t*1+1)+t-1)<0) = 0;
-end
-% d22 >= 0 
-if lb(t*3+1:(t*3+1)+t-1)<0
-    lb(lb(t*3+1:(t*3+1)+t-1)) = 0;
-end
-
-
-
-%% Selcet dij for the optimization
-if askYesno('Do you want to perform the pointwise optimization of d_11?', 'Yes')
-else
-    lb(1:t)                 = x0(1:t); % Lower bound
-    ub(1:t)                 = x0(1:t); % Upper bound
-end
-
-if askYesno('Do you want to perform the pointwise optimization of d_20?', 'Yes')
-else
-    lb(t*1+1:(t*1+1)+t-1)   = x0(t*1+1:(t*1+1)+t-1);
-    ub(t*1+1:(t*1+1)+t-1)   = x0(t*1+1:(t*1+1)+t-1);
-end
-
-if askYesno('Do you want to perform the pointwise optimization of d_21?', 'Yes')
-else
-    lb(t*2+1:(t*2+1)+t-1)   = x0(t*2+1:(t*2+1)+t-1);
-    ub(t*2+1:(t*2+1)+t-1)   = x0(t*2+1:(t*2+1)+t-1);
-end
-
-if askYesno('Do you want to perform the pointwise optimization of d_22?', 'Yes')
-else
-    lb(t*3+1:(t*3+1)+t-1)   = x0(t*3+1:(t*3+1)+t-1);
-    ub(t*3+1:(t*3+1)+t-1)   = x0(t*3+1:(t*3+1)+t-1);
-end
-
-
-%% Plot dij(r) and optimizaiton range
-figure
-subplot(2,2,1);
-plot(r_norm,...
-    lb(1:t),'LineWidth',2,'LineStyle','--')
-hold on
-plot(r_norm,...
-    d11,'k','LineWidth',2)
-plot(r_norm,...
-    ub(1:t),'LineWidth',2,'LineStyle','--')
-axis square
-ylabel('$d_{11}$', 'interpreter','latex')
-if norm_r==1
-    xlabel('$r / \lambda$', 'interpreter','latex')
-    else
-    xlabel('$r/m $', 'interpreter','latex')
-end
-xlim([min(r_norm) max(r_norm)])
-set(gca, 'FontSize',24)
-%
-subplot(2,2,2);
-plot(r_norm,...
-    lb(t*1+1:(t*1+1)+t-1),'LineWidth',2,'LineStyle','--')
-hold on
-plot(r_norm,...
-    d20,'k','LineWidth',2)
-plot(r_norm,...
-    ub(t*1+1:(t*1+1)+t-1),'LineWidth',2,'LineStyle','--')
-axis square
-ylabel('$d_{d20}$', 'interpreter','latex')
-if norm_r==1
-    xlabel('$r / \lambda$', 'interpreter','latex')
-    else
-    xlabel('$r/m $', 'interpreter','latex')
-end
-xlim([min(r_norm) max(r_norm)])
-set(gca, 'FontSize',24)
-%
-subplot(2,2,3);
-plot(r_norm,...
-    lb(t*2+1:(t*2+1)+t-1),'LineWidth',2,'LineStyle','--')
-hold on
-plot(r_norm,...
-    d21,'k','LineWidth',2)
-plot(r_norm,...
-    ub(t*2+1:(t*2+1)+t-1),'LineWidth',2,'LineStyle','--')
-axis square
-ylabel('$d_{d21}$', 'interpreter','latex')
-if norm_r==1
-    xlabel('$r / \lambda$', 'interpreter','latex')
-    else
-    xlabel('$r/m $', 'interpreter','latex')
-end
-xlim([min(r_norm) max(r_norm)])
-set(gca, 'FontSize',24)
-%
-subplot(2,2,4);
-plot(r_norm,...
-    lb(t*3+1:(t*3+1)+t-1),'LineWidth',2,'LineStyle','--')
-hold on
-plot(r_norm,...
-    d22,'k','LineWidth',2)
-plot(r_norm,...
-    ub(t*3+1:(t*3+1)+t-1),'LineWidth',2,'LineStyle','--')
-axis square
-ylabel('$d_{d22}$', 'interpreter','latex')
-if norm_r==1
-    xlabel('$r / \lambda$', 'interpreter','latex')
-    else
-    xlabel('$r/m $', 'interpreter','latex')
-end
-xlim([min(r_norm) max(r_norm)])
-set(gca, 'FontSize',24)
-set(gcf, 'Color', 'w')
-
-
-%% Fix length of trajectories
-if trajec==1
-        L       = 0;
-        % Generally each Overlapping/Non-Overlapping trajectory starts at L and
-        % ends at Lambda for the calculation of the entropy; but its a choice
-        % So we ask you at which biggest scale you want to start the trajectory
-        tmp     = str2double(inputdlg({'Adjustment: Start of trajectory 1=Yes, 0=No'}));
-        if tmp == 1
-            % Here, if you want to start your trajectories at scale smaller than L then
-            % you should enter a value between 0 & 1 
-            % For ex: if your L=1 meters and you enter 0.9 your trajectory will start at 0.9 meters
-            L   = str2double(inputdlg({'Start of trajectory in multiples of Integral length scale:'}))*int_L/taylor_L;
-        else
-            L   = int_L/taylor_L; % dimless integral scale
-        end
-        % So we ask you at which smallest scale you want to end the trajectory
-        tmp     = str2double(inputdlg({'Adjustment: End of trajectory 1=Yes, 0=No'}));
-        if tmp == 1
-            % Here, if you want to end your trajectories at scale bigger than lambda then
-            % you should enter a value between 1 and (L/lambda) 
-            % For ex: if your lambda=1 cm and if you enter 5 your
-            % trajectory will end at 5 cms
-            l   = str2double(inputdlg({'End of trajectory in multiples of Taylor length scale:'}))*taylor_L/taylor_L;
-        else
-            l   = evaluated(1).r/taylor_L;% dimless smallest scale by D1 and D2
-        end
-else % start=L, end=lambda
-    L   = int_L/taylor_L;
-    l   = 1;
-end
-
-x       = m_data.*(1:ceil(1.1*L*Fs*taylor_L/m_data)).'./Fs;
-x       = x/taylor_L; % dimless position
-Lind    = find(abs(x(:,1)-L  )==min(abs(x(:,1)-L  )), 1, 'last' ); % index for x showing the LAST  entry where x is within range lambda..L
-lind    = find(abs(x(:,1)-l  )==min(abs(x(:,1)-l  )), 1, 'first'); % index for x showing the FIRST entry where x is within range lambda..L
-
-% Index for calculating the increment using data
-Lind    = Lind+1;   
-lind    = lind+1;
-
-if lind<2
-    lind = 2;
-end
-
-rindful = 1:Lind;                       % index-vector for all scales
-rind    = fliplr(rindful(lind:Lind));   % index-vector only for scales from start to end of the cascade trajectory
-% rind = round(linspace(Lind,lind,floor(L/l)+1)); % index-vector for scales in inertial range in steps of lambda
-stp     = Lind;                         % index used to cut v(x) in non-overlapping pieces of length equal to integral scale
-% R(k).r   = x(rind); % scales from index-vector 
-if norm_r==1
-    r       = x(rind); % scales from index-vector 
-else
-    r       = x(rind).*taylor_L; % scales from index-vector 
-end
-clear x
-clear Sm Ds DS tmp_Sm tmp_Ds tmp_DS not_nan_index u x
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%% calc Entropy - overlapping trajectories
-% if z==1
-%     %% PDF of start/end of the cascade trajectory ==> Entropy of the system or the Shanon entropy
-%     rind_tmp    = rind(1:dr_ind:end);
-%     end_L       = size(data,1);
-%     tmp_size    = size(data,1)-stp+1;
-%     u_tmp       = nan( tmp_size, 2 );
-% 
-%     for i=0:end_L-stp
-%         %% u
-%         % u_tmp(i+1,:) = (data([rind_tmp(1)+i,rind_tmp(end)+i])- data(1+i)).';
-%         %% u_s
-%         u_tmp(i+1,:) = (data([rind_tmp(2)+i,rind_tmp(end-1)+i])- data(1+i)).';
-%     end   
-% 
-%     nbins=301;
-%     [fuL, uL]       = hist(u_tmp(:,1),nbins); 
-%     puL             = fuL / ( nansum(fuL) * mean(diff(uL)) ); % relative frequency to approximate p(u,r=L)
-% 
-%     [ful, ul]       = hist(u_tmp(:,2),uL); 
-%     pul             = ful / ( nansum(ful) * mean(diff(ul)) ); % relative frequency to approximate p(u,r=lambda)
-%     clear u_tmp
-% 
-%     [history] = runfmincon_overlap(x0,lb,ub,data(1:data_length),stp,r, uL, puL, ul, pul,rind,iter,r_norm,t,co_KM_opti);
-%     [history, x1] = runfmincon_overlap(x0,lb,ub,data(1:10^4),stp,r, uL, puL, ul, pul,rind,iter);
-% end
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%% calc Entropy - non-overlapping trajectories
-if z==3
-    tmp_size    = ceil(numel(data)/stp);
-    [u]         = create_us(data,r,rind,stp);
-%     [u]       = create_us(data(1:0.5*10^7),r,rind,stp);
-    [history]   = runfmincon(x0,lb,ub,u,r,tmp_size,iter,r_norm,t,co_KM_opti,markov,taylor_L,Fs,m_data,trajec,dr_ind,z);
-end
-
-
-% Find minimized  error function 
-index                       = find(history.fval==min(history.fval),1,'first');
-[fit_d11]                   = FIT_d1j(r_norm,history.x1_iter(index,1:t));
-[fit_d20, fit_d21, fit_d22] = FIT_d2j(r_norm,history.x1_iter(index,:),t);
-
-
-%% IFT optimzed dij coefficients
-co_IFT_opti.a  = [fit_d11.a1 fit_d11.a11];
-co_IFT_opti.ea = [fit_d11.ra1];
-co_IFT_opti.b  = [fit_d20.b0 fit_d20.b00 fit_d21.b1 fit_d21.b11 fit_d22.b2 fit_d22.b22];
-co_IFT_opti.eb = [fit_d20.rb0 fit_d21.rb1 fit_d22.rb2]; 
-
-
-%% Plot: comparision co_KM_opti and co_IFT_opti dij coefficients
-OPTI_IFT_plot(data,stp,r,rind,co_KM_opti,co_IFT_opti,evaluated,Fs,taylor_L,lb,ub,t,history,index,d11,d20,d21,d22,m_data,markov,trajec,dr_ind,z,norm_ur,norm_r,save_path,save_name)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% This function performs the pointwise optimization of Kramers-Moyal coefficients towards the 
+% integral fluctuation theorem will be done. Thereby the separation of scales/step increment 
+% (in samples) referred to the sequence from large to small scales in the cascade trajectory is set 
+% to a minimum of 1 sample and for the optimization, independent cascade trajectories (z=3) are used. 
+% Note, we use here a separation that is less than or equal to the Einstein-Markov length of $\Delta_{EM}$.
+% 
+% This function performs the optimization of $D^{(1,2)}\left(u_r,r\right)$ at each scale and at each 
+% value of velocity increment in order to satisfy the integral fluctuation theorem with minimum 
+% possible error and plots the optimized $d_{ij}$ as a function of $r$. The optimization procedure 
+% systematically changes $D^{(1,2)}\left(u_r,r\right)$ until the error function 
+% \begin{eqnarray}
+% \xi =|1-\langle e^{\mathrm{-}\Delta S_{tot}} \rangle_{max(N)}|
+% \end{eqnarray}
+% is minimized. 
+% 
+% Arguments IN
+% tol_D1          = Tolerance of the range of Kramers-Moyal coefficients in %% for the Optimization
+% tol_D2          = tol_D1;
+% evaluated = a modified/updated struct 'evaluated' array with in function 'KM_STP_optimization'
+% co_KM_opti = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients towards the best 
+%              Fokker-Planck equation to reproduce the conditional PDF's using the surface fits
+% fitresult_D1_conf = Confidence intervals for fit coefficients of D1 (co_KM_opti)
+% fitresult_D2_conf = Confidence intervals for fit coefficients of D2 (co_KM_opti)
+% int_L = Integral length scale in meters
+% taylor_L = Taylor length scale in meters
+% m_data = mean of the data
+% Fs = Acquisition/Sampling Frequency in Hz
+% data = filtered data 
+% z = This is the parameter which decides how to compute entropy using different methods using either 
+% overlapping or independent trajectories
+% z = 1 ==> Overlapping trajectories 
+% z = 3 ==> Independent trajectories
+% iter = The maximum number of iteration used for the optimization algorithms
+% markov = markov length in number of samples
+% trajec = 1 ==> The start/end of the cascade trajectory will be adjusted.
+% dr_ind = Separation of scales/step increment (in samples) referred to the sequence from large to small scales in the cascade trajectory
+% norm_ur = normalization of the data using $\sigma_\infty$? data 1=Yes, 0=No
+% norm_r = normalization of the scale using $\lambda$? data 1=Yes, 0=No
+% save_path = path for saving figures and files
+% save_name = name for saving files
+% 
+% Arguments OUT
+% co_IFT_opti = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients towards the 
+%               integral fluctuation theorem using the surface fits
+% history = details of the optimization for every single iteration 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+function [co_IFT_opti,history]=OPTI_IFT_dij(tol_D1,tol_D2,evaluated,co_KM_opti,fitresult_D1_conf,fitresult_D2_conf,int_L,taylor_L,m_data,Fs,data,z,iter,markov,trajec,dr_ind,norm_ur,norm_r,save_path,save_name)
+set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
+set(groot, 'defaulttextinterpreter','latex');   
+set(groot, 'defaultLegendInterpreter','latex');
+set(groot, 'defaultAxesTitle','latex');
+
+% if z==1
+%     [Sm,Ds,DS,~,~,~,~]=checkFT(evaluated,int_L,taylor_L,Fs,data,m_data,z,co_KM_opti,1,markov,trajec,dr_ind);
+%     %Plot of Entropy
+%     plot_entropy(Sm,Ds,DS)
+%     set(gca, 'FontSize',10)
+%     data_length=str2double(inputdlg({'Data length for optimization in samples:'}));
+%     clear Sm Ds DS
+% end
+
+
+%% Initial D1 and D2 before optimization  
+clear r_norm
+r_norm           = nan(1,length(evaluated));
+
+if norm_r==1
+    for i=1:length(evaluated)
+        r_norm(1,i)  = (evaluated(i).r./taylor_L);
+    end
+else
+    for i=1:length(evaluated)
+        r_norm(1,i)  = (evaluated(i).r);
+    end
+end
+
+
+% fitresult_D1
+d11 = co_KM_opti.a(1).*r_norm.^co_KM_opti.ea(1)+co_KM_opti.a(2);
+% fitresult_D2
+d20 = co_KM_opti.b(1).*r_norm.^co_KM_opti.eb(1)+co_KM_opti.b(2);
+d21 = co_KM_opti.b(3).*r_norm.^co_KM_opti.eb(2)+co_KM_opti.b(4);
+% d21=0.*d21;
+d22 = co_KM_opti.b(5).*r_norm.^co_KM_opti.eb(3)+co_KM_opti.b(6);
+x0  = [d11 d20 d21 d22];
+
+
+%% Tolerance of the range of Kramers-Moyal coefficients
+clear x1 lb ub
+% x1(1:length(x0))    = nan;
+lb(1:length(x0))    = nan;
+ub(1:length(x0))    = nan;
+
+% percent of the range 
+% D1 Opti
+t                       = length(r_norm);
+lb(1:t)                 = x0(1:t)-abs(x0(1:t).*tol_D1); % Lower bound
+ub(1:t)                 = x0(1:t)+abs(x0(1:t).*tol_D1); % Upper bound
+
+% D2 Opti
+for i=1:3
+    lb(t*i+1:(t*i+1)+t-1)   = x0(t*i+1:(t*i+1)+t-1)-abs(x0(t*i+1:(t*i+1)+t-1).*tol_D2);
+    ub(t*i+1:(t*i+1)+t-1)   = x0(t*i+1:(t*i+1)+t-1)+abs(x0(t*i+1:(t*i+1)+t-1).*tol_D2);
+end
+
+
+% constant value 10% of max/min value
+% D1 Opti
+% t                       = length(r_norm);
+% lb(1:t)                 = x0(1:t)-(max(abs(x0(1:t))).*tol_D1);
+% ub(1:t)                 = x0(1:t)+(max(abs(x0(1:t))).*tol_D1);
+ 
+% D2 Opti
+% for i=1:3
+%     lb(t*i+1:(t*i+1)+t-1)   = x0(t*i+1:(t*i+1)+t-1)-(max(abs(x0(t*i+1:(t*i+1)+t-1))).*tol_D2);
+%     ub(t*i+1:(t*i+1)+t-1)   = x0(t*i+1:(t*i+1)+t-1)+(max(abs(x0(t*i+1:(t*i+1)+t-1))).*tol_D2);
+% end
+
+
+% fix to min and max value
+% d11
+% lb(1:t)                 = min(lb(1:t));
+% ub(1:t)                 = max(ub(1:t));
+
+% d20
+% lb(t*1+1:(t*1+1)+t-1)   = min(lb(t*1+1:(t*1+1)+t-1));
+% ub(t*1+1:(t*1+1)+t-1)   = max(ub(t*1+1:(t*1+1)+t-1));
+ 
+% d21
+% lb(t*2+1:(t*2+1)+t-1)   = min(lb(t*2+1:(t*2+1)+t-1));
+% ub(t*2+1:(t*2+1)+t-1)   = max(ub(t*2+1:(t*2+1)+t-1));
+
+% d22
+% lb(t*3+1:(t*3+1)+t-1)   = min(lb(t*3+1:(t*3+1)+t-1));
+% ub(t*3+1:(t*3+1)+t-1)   = max(ub(t*3+1:(t*3+1)+t-1));
+
+
+% using confidence intervals for fit coefficients of D1 and D2 (co_KM_opti)
+% D1 Opti
+% lb(1:2)    = [fitresult_D1_conf(1,1:2)];
+% lb(9)      = [fitresult_D1_conf(1,3)];
+% ub(1:2)    = [fitresult_D1_conf(2,1:2)];
+% ub(9)      = [fitresult_D1_conf(2,3)];
+ 
+% D2 Opti
+% lb(3:8)       = [fitresult_D2_conf(1,1:6)];
+% lb(10:end)    = [fitresult_D2_conf(1,7:9)];
+% ub(3:8)       = [fitresult_D2_conf(2,1:6)];
+% ub(10:end)    = [fitresult_D2_conf(2,7:9)];
+
+% lb(isnan(lb))=0;
+% ub(isnan(ub))=0;
+
+
+%% The constraints were set in a physically and mathematically meaningful way: 
+% (see for more details \textbf{\textit{FIT\_KM}})
+% d11 <= 0
+if ub(1:t)>0
+    ub(ub(1:t)>0) = 0;
+end
+% d20 >= 0 
+if lb(t*1+1:(t*1+1)+t-1)<0
+    lb(lb(t*1+1:(t*1+1)+t-1)<0) = 0;
+end
+% d22 >= 0 
+if lb(t*3+1:(t*3+1)+t-1)<0
+    lb(lb(t*3+1:(t*3+1)+t-1)) = 0;
+end
+
+
+
+%% Selcet dij for the optimization
+if askYesno('Do you want to perform the pointwise optimization of d_11?', 'Yes')
+else
+    lb(1:t)                 = x0(1:t); % Lower bound
+    ub(1:t)                 = x0(1:t); % Upper bound
+end
+
+if askYesno('Do you want to perform the pointwise optimization of d_20?', 'Yes')
+else
+    lb(t*1+1:(t*1+1)+t-1)   = x0(t*1+1:(t*1+1)+t-1);
+    ub(t*1+1:(t*1+1)+t-1)   = x0(t*1+1:(t*1+1)+t-1);
+end
+
+if askYesno('Do you want to perform the pointwise optimization of d_21?', 'Yes')
+else
+    lb(t*2+1:(t*2+1)+t-1)   = x0(t*2+1:(t*2+1)+t-1);
+    ub(t*2+1:(t*2+1)+t-1)   = x0(t*2+1:(t*2+1)+t-1);
+end
+
+if askYesno('Do you want to perform the pointwise optimization of d_22?', 'Yes')
+else
+    lb(t*3+1:(t*3+1)+t-1)   = x0(t*3+1:(t*3+1)+t-1);
+    ub(t*3+1:(t*3+1)+t-1)   = x0(t*3+1:(t*3+1)+t-1);
+end
+
+
+%% Plot dij(r) and optimizaiton range
+figure
+subplot(2,2,1);
+plot(r_norm,...
+    lb(1:t),'LineWidth',2,'LineStyle','--')
+hold on
+plot(r_norm,...
+    d11,'k','LineWidth',2)
+plot(r_norm,...
+    ub(1:t),'LineWidth',2,'LineStyle','--')
+axis square
+ylabel('$d_{11}$', 'interpreter','latex')
+if norm_r==1
+    xlabel('$r / \lambda$', 'interpreter','latex')
+    else
+    xlabel('$r/m $', 'interpreter','latex')
+end
+xlim([min(r_norm) max(r_norm)])
+set(gca, 'FontSize',24)
+%
+subplot(2,2,2);
+plot(r_norm,...
+    lb(t*1+1:(t*1+1)+t-1),'LineWidth',2,'LineStyle','--')
+hold on
+plot(r_norm,...
+    d20,'k','LineWidth',2)
+plot(r_norm,...
+    ub(t*1+1:(t*1+1)+t-1),'LineWidth',2,'LineStyle','--')
+axis square
+ylabel('$d_{d20}$', 'interpreter','latex')
+if norm_r==1
+    xlabel('$r / \lambda$', 'interpreter','latex')
+    else
+    xlabel('$r/m $', 'interpreter','latex')
+end
+xlim([min(r_norm) max(r_norm)])
+set(gca, 'FontSize',24)
+%
+subplot(2,2,3);
+plot(r_norm,...
+    lb(t*2+1:(t*2+1)+t-1),'LineWidth',2,'LineStyle','--')
+hold on
+plot(r_norm,...
+    d21,'k','LineWidth',2)
+plot(r_norm,...
+    ub(t*2+1:(t*2+1)+t-1),'LineWidth',2,'LineStyle','--')
+axis square
+ylabel('$d_{d21}$', 'interpreter','latex')
+if norm_r==1
+    xlabel('$r / \lambda$', 'interpreter','latex')
+    else
+    xlabel('$r/m $', 'interpreter','latex')
+end
+xlim([min(r_norm) max(r_norm)])
+set(gca, 'FontSize',24)
+%
+subplot(2,2,4);
+plot(r_norm,...
+    lb(t*3+1:(t*3+1)+t-1),'LineWidth',2,'LineStyle','--')
+hold on
+plot(r_norm,...
+    d22,'k','LineWidth',2)
+plot(r_norm,...
+    ub(t*3+1:(t*3+1)+t-1),'LineWidth',2,'LineStyle','--')
+axis square
+ylabel('$d_{d22}$', 'interpreter','latex')
+if norm_r==1
+    xlabel('$r / \lambda$', 'interpreter','latex')
+    else
+    xlabel('$r/m $', 'interpreter','latex')
+end
+xlim([min(r_norm) max(r_norm)])
+set(gca, 'FontSize',24)
+set(gcf, 'Color', 'w')
+
+
+%% Fix length of trajectories
+if trajec==1
+        L       = 0;
+        % Generally each Overlapping/Non-Overlapping trajectory starts at L and
+        % ends at Lambda for the calculation of the entropy; but its a choice
+        % So we ask you at which biggest scale you want to start the trajectory
+        tmp     = str2double(inputdlg({'Adjustment: Start of trajectory 1=Yes, 0=No'}));
+        if tmp == 1
+            % Here, if you want to start your trajectories at scale smaller than L then
+            % you should enter a value between 0 & 1 
+            % For ex: if your L=1 meters and you enter 0.9 your trajectory will start at 0.9 meters
+            L   = str2double(inputdlg({'Start of trajectory in multiples of Integral length scale:'}))*int_L/taylor_L;
+        else
+            L   = int_L/taylor_L; % dimless integral scale
+        end
+        % So we ask you at which smallest scale you want to end the trajectory
+        tmp     = str2double(inputdlg({'Adjustment: End of trajectory 1=Yes, 0=No'}));
+        if tmp == 1
+            % Here, if you want to end your trajectories at scale bigger than lambda then
+            % you should enter a value between 1 and (L/lambda) 
+            % For ex: if your lambda=1 cm and if you enter 5 your
+            % trajectory will end at 5 cms
+            l   = str2double(inputdlg({'End of trajectory in multiples of Taylor length scale:'}))*taylor_L/taylor_L;
+        else
+            l   = evaluated(1).r/taylor_L;% dimless smallest scale by D1 and D2
+        end
+else % start=L, end=lambda
+    L   = int_L/taylor_L;
+    l   = 1;
+end
+
+x       = m_data.*(1:ceil(1.1*L*Fs*taylor_L/m_data)).'./Fs;
+x       = x/taylor_L; % dimless position
+Lind    = find(abs(x(:,1)-L  )==min(abs(x(:,1)-L  )), 1, 'last' ); % index for x showing the LAST  entry where x is within range lambda..L
+lind    = find(abs(x(:,1)-l  )==min(abs(x(:,1)-l  )), 1, 'first'); % index for x showing the FIRST entry where x is within range lambda..L
+
+% Index for calculating the increment using data
+Lind    = Lind+1;   
+lind    = lind+1;
+
+if lind<2
+    lind = 2;
+end
+
+rindful = 1:Lind;                       % index-vector for all scales
+rind    = fliplr(rindful(lind:Lind));   % index-vector only for scales from start to end of the cascade trajectory
+% rind = round(linspace(Lind,lind,floor(L/l)+1)); % index-vector for scales in inertial range in steps of lambda
+stp     = Lind;                         % index used to cut v(x) in non-overlapping pieces of length equal to integral scale
+% R(k).r   = x(rind); % scales from index-vector 
+if norm_r==1
+    r       = x(rind); % scales from index-vector 
+else
+    r       = x(rind).*taylor_L; % scales from index-vector 
+end
+clear x
+clear Sm Ds DS tmp_Sm tmp_Ds tmp_DS not_nan_index u x
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% calc Entropy - overlapping trajectories
+% if z==1
+%     %% PDF of start/end of the cascade trajectory ==> Entropy of the system or the Shanon entropy
+%     rind_tmp    = rind(1:dr_ind:end);
+%     end_L       = size(data,1);
+%     tmp_size    = size(data,1)-stp+1;
+%     u_tmp       = nan( tmp_size, 2 );
+% 
+%     for i=0:end_L-stp
+%         %% u
+%         % u_tmp(i+1,:) = (data([rind_tmp(1)+i,rind_tmp(end)+i])- data(1+i)).';
+%         %% u_s
+%         u_tmp(i+1,:) = (data([rind_tmp(2)+i,rind_tmp(end-1)+i])- data(1+i)).';
+%     end   
+% 
+%     nbins=301;
+%     [fuL, uL]       = hist(u_tmp(:,1),nbins); 
+%     puL             = fuL / ( nansum(fuL) * mean(diff(uL)) ); % relative frequency to approximate p(u,r=L)
+% 
+%     [ful, ul]       = hist(u_tmp(:,2),uL); 
+%     pul             = ful / ( nansum(ful) * mean(diff(ul)) ); % relative frequency to approximate p(u,r=lambda)
+%     clear u_tmp
+% 
+%     [history] = runfmincon_overlap(x0,lb,ub,data(1:data_length),stp,r, uL, puL, ul, pul,rind,iter,r_norm,t,co_KM_opti);
+%     [history, x1] = runfmincon_overlap(x0,lb,ub,data(1:10^4),stp,r, uL, puL, ul, pul,rind,iter);
+% end
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% calc Entropy - non-overlapping trajectories
+if z==3
+    tmp_size    = ceil(numel(data)/stp);
+    [u]         = create_us(data,r,rind,stp);
+%     [u]       = create_us(data(1:0.5*10^7),r,rind,stp);
+    [history]   = runfmincon(x0,lb,ub,u,r,tmp_size,iter,r_norm,t,co_KM_opti,markov,taylor_L,Fs,m_data,trajec,dr_ind,z);
+end
+
+
+% Find minimized  error function 
+index                       = find(history.fval==min(history.fval),1,'first');
+[fit_d11]                   = FIT_d1j(r_norm,history.x1_iter(index,1:t));
+[fit_d20, fit_d21, fit_d22] = FIT_d2j(r_norm,history.x1_iter(index,:),t);
+
+
+%% IFT optimzed dij coefficients
+co_IFT_opti.a  = [fit_d11.a1 fit_d11.a11];
+co_IFT_opti.ea = [fit_d11.ra1];
+co_IFT_opti.b  = [fit_d20.b0 fit_d20.b00 fit_d21.b1 fit_d21.b11 fit_d22.b2 fit_d22.b22];
+co_IFT_opti.eb = [fit_d20.rb0 fit_d21.rb1 fit_d22.rb2]; 
+
+
+%% Plot: comparision co_KM_opti and co_IFT_opti dij coefficients
+OPTI_IFT_plot(data,stp,r,rind,co_KM_opti,co_IFT_opti,evaluated,Fs,taylor_L,lb,ub,t,history,index,d11,d20,d21,d22,m_data,markov,trajec,dr_ind,z,norm_ur,norm_r,save_path,save_name)
 end
\ No newline at end of file
diff --git a/Functions_v5-MATLAB/PART_I.m b/Functions_v5-MATLAB/PART_I.m
index ac1e995..a0ddda7 100644
--- a/Functions_v5-MATLAB/PART_I.m
+++ b/Functions_v5-MATLAB/PART_I.m
@@ -95,7 +95,7 @@
 %% Estimation of fundamental length scales
 lenght_scales     = askYesno('Do you want to estimate the fundamental length scales?', 'Yes');
 if lenght_scales==1
-    C2 = askInput({sprintf('Enter the value of Kolmogorov constant, C_2 associated with second order structure function(in between 2.0 to 2.2)')}, {'2.1'});
+    C2 = askInput({sprintf('Enter the value of constant, C_2 associated with second order structure function:')}, {'2.0'});
     [int_L,taylor_L,int_L_calc, taylor_L_calc,epsi,epsi_calc,diss_scale,Ce, Ce_calc, Re, Re_lambda ] = length_scales(data_filter,Fs,low_freq,kin_vis,C2,increment_bin,m_data,save_path,save_name);
 else
     int_L       = str2double(inputdlg({'Enter the integral length scale in meter:'}));
@@ -117,7 +117,7 @@
 
 %% Wether to flip the Hot-wire data or not?
 flip_data       = askYesno('Do you want to check if you have to flip the data?', 'Yes');
-norm_ur         = askYesno('Do you want to perform the normalization of the data using $\sigma_\infty$?', 'Yes');
+norm_ur        = askYesno('Do you want to perform the normalization of the data using $\sigma_\infty$?', 'Yes');
 norm_r          = askYesno('Do you want to perform the normalization of the scale using $\lambda$?', 'Yes');
 if flip_data==1
     Struc_flip_test(Fs,data_filter,int_L,taylor_L,save_path,save_name)
@@ -145,4 +145,4 @@
     close(bar)
     save(fullfile(save_path,append(save_name,'_','evaluated.mat')),'-v7.3')
 end
-end
+end
\ No newline at end of file
diff --git a/Functions_v5-MATLAB/PART_II.m b/Functions_v5-MATLAB/PART_II.m
index 135c3cd..9ced8fb 100644
--- a/Functions_v5-MATLAB/PART_II.m
+++ b/Functions_v5-MATLAB/PART_II.m
@@ -66,7 +66,7 @@
 %% Plot conditional moments
 plot_moment     = askYesno('Do you want to plot conditional moments?', 'Yes');
 if plot_moment==1
-    plot_conditional_moment(askInput({sprintf('Plot: number of Scale:')}, {num2str(scale_steps)}),askInput({sprintf('Plot: number of a bin (value of the velocity increment u_r):')}, {num2str(ceil(increment_bin/2))}),evaluated,step_con_moment,markov,multi_point,increment_bin,condition,tol,data_filter,save_path,save_name)
+    plot_conditional_moment(askInput({sprintf('Plot: number of Scale:')}, {num2str(scale_steps)}),askInput({sprintf('Plot: number of a bin (value of the velocity increment u_r):')}, {num2str(ceil(increment_bin/2))}),evaluated,step_con_moment,markov,multi_point,increment_bin,condition,tol,data_filter,save_path,save_name);
 end
 waitbar(13/ikx,bar,'Calculation of Kramers-Moyal coefficients');
 
diff --git a/Functions_v5-MATLAB/PART_III.m b/Functions_v5-MATLAB/PART_III.m
index 16b0aa0..d453c9c 100644
--- a/Functions_v5-MATLAB/PART_III.m
+++ b/Functions_v5-MATLAB/PART_III.m
@@ -51,11 +51,12 @@
 
 %% Optimization of Kramers-Moyal coefficients Fit: IFT opti
 close all
-opti_IFT    = askYesno('Do you want to perform the pointwise optimization of Kramers-Moyal coefficients towards the integral fluctuation theorem?', 'Yes');
+opti_IFT    = askYesno('Do you want to perform the pointwise optimization of Kramers-Moyal coefficients relatively to the integral fluctuation theorem?', 'Yes');
 if opti_IFT==1
     bar = waitbar(20/ikx,'Optimization of Kramers-Moyal coefficients: IFT','Position', tmp_bar);
 %     z               = 3;
-    dr_ind          = 1;
+%     dr_ind          = 1;
+	dr_ind          = askInput({sprintf('Enter the separation of scales/step increment (in samples) referred to the sequence from large to small scales in the cascade trajectory:')},{num2str(1)});
     iter            = askInput({sprintf('Enter the maximum number of iteration used for the optimization algorithms:')}, {'5'});
     tol_D1          = askInput({sprintf('Optimization: Tolerance of the range of Kramers-Moyal coefficients in %%')}, {'10'})/100;
     tol_D2          = tol_D1;
diff --git a/Functions_v5-MATLAB/Struc_flip_test.m b/Functions_v5-MATLAB/Struc_flip_test.m
index 8933727..11757d2 100644
--- a/Functions_v5-MATLAB/Struc_flip_test.m
+++ b/Functions_v5-MATLAB/Struc_flip_test.m
@@ -14,6 +14,8 @@
 % Fs = Acquisition/Sampling Frequency in Hz
 % int_L = Integral length scale in meters
 % taylor_L = Taylors length scale in meters
+% save_path = path for saving figures and files
+% save_name = name for saving files
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 function Struc_flip_test(Fs,data,int_L,taylor_L,save_path,save_name)
diff --git a/Functions_v5-MATLAB/ZeroCrossings.m b/Functions_v5-MATLAB/ZeroCrossings.m
index ac1d013..6493afe 100644
--- a/Functions_v5-MATLAB/ZeroCrossings.m
+++ b/Functions_v5-MATLAB/ZeroCrossings.m
@@ -228,7 +228,8 @@
 %             EpsNew  = trapz(x_achse_k_spektrum_2pi(1:ind_2),y_tmp(1:ind_2));
 %         else   
 %             [xData, yData] = prepareCurveData( x_achse_k_spektrum_2pi(ind_1:ind_2), y_tmp(ind_1:ind_2));
-            EpsNew = trapz(x_achse_k_spektrum_2pi(1:ind_1),y_tmp(1:ind_1))+trapz(xData,feval(fitresult,xData));
+%             EpsNew = trapz(x_achse_k_spektrum_2pi(1:ind_1),y_tmp(1:ind_1))+trapz(xData,feval(fitresult,xData));
+			EpsNew = trapz(x_achse_k_spektrum_2pi(1:ind_1),y_tmp(1:ind_1))+trapz(x_achse_k_spektrum_2pi(ind_1:end),feval(fitresult,x_achse_k_spektrum_2pi(ind_1:end)));
             % not more than 5% should be modeld  
 %             p(1).*ind_1.^(p(3)+1)/(p(3)+1);
 %         end
diff --git a/Functions_v5-MATLAB/checkFT.m b/Functions_v5-MATLAB/checkFT.m
index 47b0463..38a97b3 100644
--- a/Functions_v5-MATLAB/checkFT.m
+++ b/Functions_v5-MATLAB/checkFT.m
@@ -1,359 +1,361 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% The calculation leading towards the integral fluctuation theorem will be done. In the spirit of 
-% non-equilibrium stochastic thermodynamics \cite{seifert2012stochastic} it is possible to associate 
-% with every individual cascade trajectory $\left[u(\cdot) \right]$ a total entropy variation 
-% $\Delta S_{tot}$. In this investigation it is assumed that a single cascade trajectory represents 
-% one realization of the turbulent cascade process and a large number of these trajectories reflect 
-% the statistics caused by the process.
-% The set of measured cascade trajectories results in a set of total entropy variation values 
-% $\Delta S_{tot}$. This function calculates the system entropy, medium entropy and the total 
-% entropy variation for all the independent cascade trajectories.
-% 
-% Arguments IN
-% evaluated = a modified/updated struct 'evaluated' array with in function 'KM_STP_optimization'
-% int_L = Integral length scale in meters
-% taylor_L = Taylor length scale in meters
-% Fs = Acquisition/Sampling Frequency in Hz
-% data = filtered data 
-% m_data = mean of the data
-% z = This is the parameter which decides how to compute entropy using different methods using either 
-% overlapping or independent trajectories
-% z = 1 ==> Overlapping trajectories 
-% z = 3 ==> Independent trajectories
-% co = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients, where first entry is 
-%      zeroth power and power laws for r-dependency with exponents co.ae and co.be, for each power 
-%      in D1 and D2 respectively
-% data_length = A value between 0 & 1 ==> how much of data you want to consider for this calculation
-% data_length = 1 for all the data i.e. data(1:end)
-% data_length = 0.5 for the data i.e data(1:end/2)   <== For 50% of the data 
-% markov = markov length in number of samples
-% trajec = 1 ==> The start/end of the cascade trajectory will be adjusted.
-% dr_ind = Separation of scales/step increment (in samples) referred to the sequence from large to small scales in the cascade trajectory
-% 
-% Arguments OUT
-% Sm = Entropy of the medium
-% Ds = Entropy of the system or the Shanon entropy
-% DS = Total entropy production=Sm+Ds
-% r =  Normalized scale vector from the start to end of the cascade trajectory
-% ind_trajec = index-vector of the trajectories (important vector for the recalculation of the trajectories from data)
-% rind = index-vector from the start to end of the cascade trajectory
-% dr = separation of scales/step increment (in normalized scale) referred to the sequence from 
-%      large to small scales in the cascade trajectory
-% r_s = mid-point scale vector (Stratonovich convention)
-% u = used trajectoires (only z=3 ==> Independent trajectories) 
-% A = action functional, pathintegral of Lagrangian
-% Lag,p,H,tmpvar,H1,H2,ur_s_tmp (are not yet included/used in the current version)
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-function[Sm,Ds,DS,r,ind_trajec,rind,dr,r_s,u,Lag,A,p,H,tmpvar,H1,H2,ur_s_tmp]=checkFT(evaluated,int_L,taylor_L,Fs,data,m_data,z,co,data_length,markov,trajec,dr_ind,norm_ur,norm_r)
-set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
-set(groot, 'defaulttextinterpreter','latex');   
-set(groot, 'defaultLegendInterpreter','latex');
-set(groot, 'defaultAxesTitle','latex');
-
-%% Fix length of trajectories
-if trajec==1
-        L       = 0;
-        % Generally each Overlapping/Non-Overlapping trajectory starts at L and
-        % ends at Lambda for the calculation of the entropy; but its a choice
-        % So we ask you at which biggest scale you want to start the trajectory
-        tmp     = str2double(inputdlg({'Adjustment: Start of trajectory 1=Yes, 0=No'}));
-        if tmp == 1
-            % Here, if you want to start your trajectories at scale smaller than L then
-            % you should enter a value between 0 & 1 
-            % For ex: if your L=1 meters and you enter 0.9 your trajectory will start at 0.9 meters
-            L   = str2double(inputdlg({'Start of trajectory in multiples of Integral length scale:'}))*int_L/taylor_L;
-        else
-            L   = int_L/taylor_L; % dimless integral scale
-        end
-        % So we ask you at which smallest scale you want to end the trajectory
-        tmp     = str2double(inputdlg({'Adjustment: End of trajectory 1=Yes, 0=No'}));
-        if tmp == 1
-            % Here, if you want to end your trajectories at scale bigger than lambda then
-            % you should enter a value between 1 and (L/lambda) 
-            % For ex: if your lambda=1 cm and if you enter 5 your
-            % trajectory will end at 5 cms
-            l   = str2double(inputdlg({'End of trajectory in multiples of Taylor length scale:'}))*taylor_L/taylor_L;
-        else
-            l   = evaluated(1).r/taylor_L;% dimless smallest scale by D1 and D2
-        end
-else % start=L, end=lambda
-    L   = int_L/taylor_L;
-    l   = 1;
-end
-
-x       = m_data.*(1:ceil(1.1*L*Fs*taylor_L/m_data)).'./Fs;
-x       = x/taylor_L; % dimless position
-Lind    = find(abs(x(:,1)-L  )==min(abs(x(:,1)-L  )), 1, 'last' ); % index for x showing the LAST  entry where x is within range lambda..L
-lind    = find(abs(x(:,1)-l  )==min(abs(x(:,1)-l  )), 1, 'first'); % index for x showing the FIRST entry where x is within range lambda..L
-
-% Index for calculating the increment using data
-Lind    = Lind+1;   
-lind    = lind+1;
-
-if lind<2
-    lind = 2;
-end
-
-rindful = 1:Lind;                       % index-vector for all scales
-rind    = fliplr(rindful(lind:Lind));   % index-vector only for scales from start to end of the cascade trajectory
-% rind = round(linspace(Lind,lind,floor(L/l)+1)); % index-vector for scales in inertial range in steps of lambda
-stp     = Lind;                         % index used to cut v(x) in non-overlapping pieces of length equal to integral scale
-% R(k).r   = x(rind); % scales from index-vector
-
-if norm_r==1
-    r       = x(rind); % scales from index-vector 
-else
-    r       = x(rind)*taylor_L; % scales from index-vector 
-end
-
-clear x
-clear Sm Ds DS tmp_Sm tmp_Ds tmp_DS not_nan_index u x
-
-
-
-%% calc Entropy - overlapping trajectories
-if z==1
-    %% PDF of start/end of the cascade trajectory ==> Entropy of the system or the Shanon entropy
-    rind_tmp    = rind(1:dr_ind:end);
-    end_L       = size(data,1);
-    tmp_size    = size(data,1)-stp+1;
-    u_tmp       = nan( tmp_size, 2 );
-
-    for i=0:end_L-stp
-        %% u
-        % u_tmp(i+1,:) = (data([rind_tmp(1)+i,rind_tmp(end)+i])- data(1+i)).';
-        %% u_s
-        u_tmp(i+1,:) = (data([rind_tmp(2)+i,rind_tmp(end-1)+i])- data(1+i)).';
-    end   
-
-    nbins=301;
-    [fuL, uL]       = hist(u_tmp(:,1),nbins); 
-    puL             = fuL / ( nansum(fuL) * mean(diff(uL)) ); % relative frequency to approximate p(u,r=L)
-
-    [ful, ul]       = hist(u_tmp(:,2),uL); 
-    pul             = ful / ( nansum(ful) * mean(diff(ul)) ); % relative frequency to approximate p(u,r=lambda)
-
-    figure
-    plot(uL,puL,'LineWidth',2)
-    hold on
-    plot(ul,pul,'LineWidth',2)
-    norm1 = pdf(fitdist(u_tmp(:,1),'Normal'),uL);
-    plot(uL,norm1,'LineWidth',2,'LineStyle','--','Color',[0.501960813999176 0.501960813999176 0.501960813999176])
-    norm1 = pdf(fitdist(u_tmp(:,2),'Normal'),uL);
-    plot(ul,norm1,'LineWidth',2,'LineStyle','--','Color',[0.501960813999176 0.501960813999176 0.501960813999176])
-    set(gca,'yScale','log') 
-    set(gcf, 'Color', 'w')
-    set(gca, 'FontSize',18)
-    xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
-    ylabel('$p(u_r / \sigma_\infty)$','interpreter','latex')
-    axis square
-    legend('u_L','u_\lambda');
-    xlim([min(uL) max(uL)])
-    ylim([0.0000001*max(puL) 2*max(pul)])
-    tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-    uiwait(gcf);
-    delete(tmp_ui);
-    close
-    clear u_tmp
-
-    
-    %% Preallocating (considering the available RAM)
-    data        = data(1:ceil(length(data)*data_length));
-    giga        = askInput({sprintf('Enter available RAM in GB:')},{num2str(8)}) * 1.07*10^9; %1 GB in bytes   
-    k_split     = ceil((12*4*length(data)*length(r))/giga)
-    data        = reshape(data((1:floor(numel(data)/k_split)*k_split)),[],k_split);
-
-    A           = nan( size(data,1)-stp+1,k_split );
-    Sm          = nan( size(data,1)-stp+1,k_split );
-    Ds          = nan( size(data,1)-stp+1,k_split );
-    DS          = nan( size(data,1)-stp+1,k_split );
-    ind_trajec  = nan( size(data,1)-stp+1,k_split );
-    %  Q_hk     = nan( size(data,1)-stp+1,k_split );
-    %  Q_med    = nan( size(data,1)-stp+1,k_split );
-    %  Q_ex     = nan( size(data,1)-stp+1,k_split );
-    %  Q_med_xi = nan; 
-
-    clear tmp
-    tmp(1:k_split) = struct('Sm_tmp',nan(size(data,1)-stp+1,1),'Ds_tmp',nan(size(data,1)-stp+1,1),'DS_tmp',nan(size(data,1)-stp+1,1), 'ind_trajec_tmp',nan(size(data,1)-stp+1,1), 'A_tmp',nan(size(data,1)-stp+1,1));
-    % tmp(1:k_split) = struct('Sm_tmp',nan(size(data,1)-stp+1,1),'Ds_tmp',nan(size(data,1)-stp+1,1),'DS_tmp',nan(size(data,1)-stp+1,1), 'ind_trajec_tmp',nan(size(data,1)-stp+1,1),...
-    %'Q_hk_tmp',nan(size(data,1)-stp+1,1),'Q_med_tmp',nan(size(data,1)-stp+1,1),'Q_ex',nan(size(data,1)-stp+1,1)); 
-    f = uifigure;
-    d = uiprogressdlg(f,'Title','Please Wait','ShowPercentage','on');
-    
-    
-    %% calc Entropy
-    for k=1:k_split
-        [u,ind_trajec_tmp]                                      = create_us_overlap(data(:,k),r,rind,stp);     
-        [Sm_tmp,Ds_tmp,DS_tmp,dr,r_s,~,~,~,A_tmp,~,~,~,~,~,~]   = calcDS(z,u,r.', co,markov,taylor_L,Fs,m_data,dr_ind, uL, puL, ul, pul);
-        tmp(k).A_tmp(1:length(A_tmp))                           = A_tmp;
-        tmp(k).Sm_tmp(1:length(Sm_tmp))                         = Sm_tmp;
-        tmp(k).Ds_tmp(1:length(Ds_tmp))                         = Ds_tmp;
-        tmp(k).DS_tmp(1:length(DS_tmp))                         = DS_tmp;
-        % [Q_hk_tmp, Q_med_tmp, Q_ex_tmp,~]                     = heat(u,r.', co);
-        % tmp(k).Q_hk_tmp(1:length(Q_hk_tmp))                   = Q_hk_tmp;
-        % tmp(k).Q_med_tmp(1:length(Q_med_tmp))                 = Q_med_tmp;
-        % tmp(k).Q_ex_tmp(1:length(Q_ex_tmp))                   = Q_ex_tmp;
-
-        if k==1
-            tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))     = ind_trajec_tmp;
-        else
-            % tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))   = (size(data,1)-stp)*(k-1)+ind_trajec_tmp+1;
-            % tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))   = tmp(k-1).ind_trajec_tmp(length(ind_trajec_tmp))+ind_trajec_tmp+1;
-            % tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))   = tmp(k-1).ind_trajec_tmp(length(ind_trajec_tmp))+stp+ind_trajec_tmp+1;
-            tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))     = tmp(k-1).ind_trajec_tmp(length(ind_trajec_tmp))+stp+ind_trajec_tmp;
-        end   
-        d.Value=k/k_split;
-		k/k_split
-    end
-    
-    close(d);
-    close(f);
-
-    for k=1:k_split
-        A(1:length(tmp(k).A_tmp),k)                             = tmp(k).A_tmp;
-        Sm(1:length(tmp(k).Sm_tmp),k)                           = tmp(k).Sm_tmp;
-        Ds(1:length(tmp(k).Ds_tmp),k)                           = tmp(k).Ds_tmp;
-        DS(1:length(tmp(k).DS_tmp),k)                           = tmp(k).DS_tmp;
-        ind_trajec(1:length(tmp(k).ind_trajec_tmp),k)           = tmp(k).ind_trajec_tmp;
-    %     Q_hk(1:length(tmp(k).Q_hk_tmp),k)                     = tmp(k).Q_hk_tmp;
-    %     Q_med(1:length(tmp(k).Q_med_tmp),k)                   = tmp(k).Q_med_tmp;
-    %     Q_ex(1:length(tmp(k).Q_ex_tmp),k)                     = tmp(k).Q_ex_tmp; 
-    end
-    clear tmp
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%% calc Entropy - non-overlapping trajectories
-elseif z==3
-    %% Preallocating (considering the available RAM)
-    tmp_size    = ceil(numel(data)/stp);
-    data        = data(1:ceil(length(data)*data_length));
-%     giga        = askInput({sprintf('Enter available RAM in GB:')},{num2str(8)}) * 1.07*10^9; %1 GB in bytes 
-    giga        = 8* 1.07*10^9;
-    k_split     = ceil((20*4*tmp_size*length(r(1:dr_ind:end)))/giga);
-    data        = reshape(data((1:floor(numel(data)/k_split)*k_split)),[],k_split);
-   
-    Sm          = nan( tmp_size, k_split );
-    Ds          = nan( tmp_size, k_split );
-    DS          = nan( tmp_size, k_split );
-    ind_trajec  = nan( tmp_size, k_split );
-    A           = nan( tmp_size, k_split );    
-%     Lag         = nan( tmp_size, k_split );
-%     p           = nan( tmp_size, k_split );
-%     H           = nan( tmp_size, k_split );
-%     tmpvar      = nan( tmp_size, k_split );
-%     H1          = nan( tmp_size, k_split );
-%     H2          = nan( tmp_size, k_split );
-%     u           = nan( tmp_size, k_split );
-%     Q_hk      = nan( tmp_size,1 );
-%     Q_med     = nan( tmp_size,1 );
-%     Q_ex      = nan( tmp_size,1 );
-%     not_nan_index = nan( tmp_size, k_split );
- 
-    
-    %% calc Entropy
-    f = uifigure;
-    d = uiprogressdlg(f,'Title','Please Wait','ShowPercentage','on');
-    for k=1:k_split
-        [u_tmp,ind_trajec_tmp]                  = create_us(data(:,k),r,rind,stp);
-        [Sm_tmp,Ds_tmp,DS_tmp,dr,r_s,~,~,L_tmp,A_tmp,p_tmp,H_tmp,tmpvar_tmp,H1_tmp,H2_tmp,ur_s_tmp]           = calcDS(z,u_tmp,r.',co,markov,taylor_L,Fs,m_data,dr_ind);
-    %   [Q_hk_tmp, Q_med_tmp, Q_ex_tmp,Q_med_xi] = heat(u_tmp,r.', co, r(end)); 
-
-        A(1:sum(~isnan(DS_tmp),1),k)            = A_tmp(~isnan(DS_tmp));
-        ind_trajec(1:sum(~isnan(DS_tmp),1),k)   = ind_trajec_tmp(~isnan(DS_tmp));
-        Sm(1:sum(~isnan(DS_tmp),1),k)           = Sm_tmp(~isnan(DS_tmp));
-        Ds(1:sum(~isnan(DS_tmp),1),k)           = Ds_tmp(~isnan(DS_tmp));
-        DS(1:sum(~isnan(DS_tmp),1),k)           = DS_tmp(~isnan(DS_tmp));
-        
-        if k_split==1
-%             Lag(1:sum(~isnan(DS_tmp),1),:)         = L_tmp(~isnan(DS_tmp),:);
-%             p(1:sum(~isnan(DS_tmp),1),:)           = p_tmp(~isnan(DS_tmp),:); 
-%             H(1:sum(~isnan(DS_tmp),1),:)           = H_tmp(~isnan(DS_tmp),:); 
-%             tmpvar(1:sum(~isnan(DS_tmp),1),:)      = tmpvar_tmp(~isnan(DS_tmp),:);
-            ur_s_tmp(1:sum(~isnan(DS_tmp),1),:)    = ur_s_tmp(~isnan(DS_tmp),:);
-%             H1(1:sum(~isnan(DS_tmp),1),:)          = H1_tmp(~isnan(DS_tmp),:);   
-%             H2(1:sum(~isnan(DS_tmp),1),:)          = H2_tmp(~isnan(DS_tmp),:);
-            u(1:sum(~isnan(DS_tmp),1),:)           = u_tmp(~isnan(DS_tmp),:);
-        end
-%             Q_hk(1:sum(~isnan(DS_tmp),1),k)           = Q_hk_tmp(~isnan(DS_tmp));
-%             Q_med(1:sum(~isnan(DS_tmp),1),k)          = Q_med_tmp(~isnan(DS_tmp));
-%             Q_ex(1:sum(~isnan(DS_tmp),1),k)           = Q_ex_tmp(~isnan(DS_tmp));
-%             not_nan                                   = (0:tmp_size).';
-%             not_nan_index(1:sum(~isnan(DS_tmp),1),k)  = not_nan(~isnan(DS_tmp));
-        clear ind_trajec_tmp Sm_tmp Ds_tmp DS_tmp u_tmp
-
-     d.Value=k/k_split;
-    end
-    close(d);
-    close(f);
-end
-
-A       = reshape(A,[],1);
-Sm      = reshape(Sm,[],1);
-Ds      = reshape(Ds,[],1);
-DS      = reshape(DS,[],1);
-% Q_hk  = reshape(Q_hk,[],1);
-% Q_med = reshape(Q_med,[],1);
-% Q_ex  = reshape(Q_ex,[],1);
-
-% cut-out NaN's due to vanishing or negative D2
-DS(isinf(DS))   = nan;
-A               = A(~isnan(DS));
-
-% if z==1
-    Lag             = nan;
-    p               = nan;
-    H               = nan;
-    tmpvar          = nan;
-    H1              = nan;
-    H2              = nan;
-    if k_split>1
-        ur_s_tmp        = nan;
-        u               = nan;
-    end
-% end
-
-% if z==3
-    % L      = reshape(L,[],1);
-    % p      = reshape(p,[],1);
-    % H      = reshape(H,[],1);
-    % tmpvar = reshape(tmpvar,[],1);
-    % H1     = reshape(H1,[],1);
-    % H2     = reshape(H2,[],1);
-% end
-% if z==3
-%     Lag             = Lag(~isnan(DS),:);
-%     p               = p(~isnan(DS),:);
-%     H               = H(~isnan(DS),:);
-%     tmpvar          = tmpvar(~isnan(DS),:);
-%     ur_s_tmp        = ur_s_tmp(~isnan(DS),:);
-%     H1              = H1(~isnan(DS),:);
-%     H2              = H2(~isnan(DS),:);
- if k_split==1
-     ur_s_tmp        = ur_s_tmp(~isnan(DS),:);
-     u               = u(~isnan(DS),:);
- end
-% end
-
-Sm              = Sm(~isnan(DS));
-Ds              = Ds(~isnan(DS));
-% Q_hk  = Q_hk(~isnan(DS));
-% Q_med = Q_med(~isnan(DS));
-% Q_ex  = Q_ex(~isnan(DS));
-
-ind_trajec      = reshape(ind_trajec,[],1);
-ind_trajec      = ind_trajec(~isnan(DS));
-% not_nan_index = ~isnan(DS);
-
-DS              = DS(~isnan(DS));
-
-if z==1
-    Sm          = Sm(1:end-1);
-    Ds          = Ds(1:end-1);
-    DS          = DS(1:end-1); 
-    ind_trajec  = ind_trajec(1:end-1);
-end
-r_s = r_s(1,:);
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% The calculation leading towards the integral fluctuation theorem will be done. In the spirit of 
+% non-equilibrium stochastic thermodynamics \cite{seifert2012stochastic} it is possible to associate 
+% with every individual cascade trajectory $\left[u(\cdot) \right]$ a total entropy variation 
+% $\Delta S_{tot}$. In this investigation it is assumed that a single cascade trajectory represents 
+% one realization of the turbulent cascade process and a large number of these trajectories reflect 
+% the statistics caused by the process.
+% The set of measured cascade trajectories results in a set of total entropy variation values 
+% $\Delta S_{tot}$. This function calculates the system entropy, medium entropy and the total 
+% entropy variation for all the independent cascade trajectories.
+% 
+% Arguments IN
+% evaluated = a modified/updated struct 'evaluated' array with in function 'KM_STP_optimization'
+% int_L = Integral length scale in meters
+% taylor_L = Taylor length scale in meters
+% Fs = Acquisition/Sampling Frequency in Hz
+% data = filtered data 
+% m_data = mean of the data
+% z = This is the parameter which decides how to compute entropy using different methods using either 
+% overlapping or independent trajectories
+% z = 1 ==> Overlapping trajectories 
+% z = 3 ==> Independent trajectories
+% co = Coefficients $d_{ij}(r)$ of the optimized Kramers-Moyal coefficients, where first entry is 
+%      zeroth power and power laws for r-dependency with exponents co.ae and co.be, for each power 
+%      in D1 and D2 respectively
+% data_length = A value between 0 & 1 ==> how much of data you want to consider for this calculation
+% data_length = 1 for all the data i.e. data(1:end)
+% data_length = 0.5 for the data i.e data(1:end/2)   <== For 50% of the data 
+% markov = markov length in number of samples
+% trajec = 1 ==> The start/end of the cascade trajectory will be adjusted.
+% dr_ind = Separation of scales/step increment (in samples) referred to the sequence from large to small scales in the cascade trajectory
+% norm_ur = normalization of the data using $\sigma_\infty$? data 1=Yes, 0=No
+% norm_r = normalization of the scale using $\lambda$? data 1=Yes, 0=No
+% 
+% Arguments OUT
+% Sm = Entropy of the medium
+% Ds = Entropy of the system or the Shanon entropy
+% DS = Total entropy production=Sm+Ds
+% r =  Normalized scale vector from the start to end of the cascade trajectory
+% ind_trajec = index-vector of the trajectories (important vector for the recalculation of the trajectories from data)
+% rind = index-vector from the start to end of the cascade trajectory
+% dr = separation of scales/step increment (in normalized scale) referred to the sequence from 
+%      large to small scales in the cascade trajectory
+% r_s = mid-point scale vector (Stratonovich convention)
+% u = used trajectoires (only z=3 ==> Independent trajectories) 
+% A = action functional, pathintegral of Lagrangian
+% Lag,p,H,tmpvar,H1,H2,ur_s_tmp (are not yet included/used in the current version)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+function[Sm,Ds,DS,r,ind_trajec,rind,dr,r_s,u,Lag,A,p,H,tmpvar,H1,H2,ur_s_tmp]=checkFT(evaluated,int_L,taylor_L,Fs,data,m_data,z,co,data_length,markov,trajec,dr_ind,norm_ur,norm_r)
+set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
+set(groot, 'defaulttextinterpreter','latex');   
+set(groot, 'defaultLegendInterpreter','latex');
+set(groot, 'defaultAxesTitle','latex');
+
+%% Fix length of trajectories
+if trajec==1
+        L       = 0;
+        % Generally each Overlapping/Non-Overlapping trajectory starts at L and
+        % ends at Lambda for the calculation of the entropy; but its a choice
+        % So we ask you at which biggest scale you want to start the trajectory
+        tmp     = str2double(inputdlg({'Adjustment: Start of trajectory 1=Yes, 0=No'}));
+        if tmp == 1
+            % Here, if you want to start your trajectories at scale smaller than L then
+            % you should enter a value between 0 & 1 
+            % For ex: if your L=1 meters and you enter 0.9 your trajectory will start at 0.9 meters
+            L   = str2double(inputdlg({'Start of trajectory in multiples of Integral length scale:'}))*int_L/taylor_L;
+        else
+            L   = int_L/taylor_L; % dimless integral scale
+        end
+        % So we ask you at which smallest scale you want to end the trajectory
+        tmp     = str2double(inputdlg({'Adjustment: End of trajectory 1=Yes, 0=No'}));
+        if tmp == 1
+            % Here, if you want to end your trajectories at scale bigger than lambda then
+            % you should enter a value between 1 and (L/lambda) 
+            % For ex: if your lambda=1 cm and if you enter 5 your
+            % trajectory will end at 5 cms
+            l   = str2double(inputdlg({'End of trajectory in multiples of Taylor length scale:'}))*taylor_L/taylor_L;
+        else
+            l   = evaluated(1).r/taylor_L;% dimless smallest scale by D1 and D2
+        end
+else % start=L, end=lambda
+    L   = int_L/taylor_L;
+    l   = 1;
+end
+
+x       = m_data.*(1:ceil(1.1*L*Fs*taylor_L/m_data)).'./Fs;
+x       = x/taylor_L; % dimless position
+Lind    = find(abs(x(:,1)-L  )==min(abs(x(:,1)-L  )), 1, 'last' ); % index for x showing the LAST  entry where x is within range lambda..L
+lind    = find(abs(x(:,1)-l  )==min(abs(x(:,1)-l  )), 1, 'first'); % index for x showing the FIRST entry where x is within range lambda..L
+
+% Index for calculating the increment using data
+Lind    = Lind+1;   
+lind    = lind+1;
+
+if lind<2
+    lind = 2;
+end
+
+rindful = 1:Lind;                       % index-vector for all scales
+rind    = fliplr(rindful(lind:Lind));   % index-vector only for scales from start to end of the cascade trajectory
+% rind = round(linspace(Lind,lind,floor(L/l)+1)); % index-vector for scales in inertial range in steps of lambda
+stp     = Lind;                         % index used to cut v(x) in non-overlapping pieces of length equal to integral scale
+% R(k).r   = x(rind); % scales from index-vector
+
+if norm_r==1
+    r       = x(rind); % scales from index-vector 
+else
+    r       = x(rind)*taylor_L; % scales from index-vector 
+end
+
+clear x
+clear Sm Ds DS tmp_Sm tmp_Ds tmp_DS not_nan_index u x
+
+
+
+%% calc Entropy - overlapping trajectories
+if z==1
+    %% PDF of start/end of the cascade trajectory ==> Entropy of the system or the Shanon entropy
+    rind_tmp    = rind(1:dr_ind:end);
+    end_L       = size(data,1);
+    tmp_size    = size(data,1)-stp+1;
+    u_tmp       = nan( tmp_size, 2 );
+
+    for i=0:end_L-stp
+        %% u
+        % u_tmp(i+1,:) = (data([rind_tmp(1)+i,rind_tmp(end)+i])- data(1+i)).';
+        %% u_s
+        u_tmp(i+1,:) = (data([rind_tmp(2)+i,rind_tmp(end-1)+i])- data(1+i)).';
+    end   
+
+    nbins=301;
+    [fuL, uL]       = hist(u_tmp(:,1),nbins); 
+    puL             = fuL / ( nansum(fuL) * mean(diff(uL)) ); % relative frequency to approximate p(u,r=L)
+
+    [ful, ul]       = hist(u_tmp(:,2),uL); 
+    pul             = ful / ( nansum(ful) * mean(diff(ul)) ); % relative frequency to approximate p(u,r=lambda)
+
+    figure
+    plot(uL,puL,'LineWidth',2)
+    hold on
+    plot(ul,pul,'LineWidth',2)
+    norm1 = pdf(fitdist(u_tmp(:,1),'Normal'),uL);
+    plot(uL,norm1,'LineWidth',2,'LineStyle','--','Color',[0.501960813999176 0.501960813999176 0.501960813999176])
+    norm1 = pdf(fitdist(u_tmp(:,2),'Normal'),uL);
+    plot(ul,norm1,'LineWidth',2,'LineStyle','--','Color',[0.501960813999176 0.501960813999176 0.501960813999176])
+    set(gca,'yScale','log') 
+    set(gcf, 'Color', 'w')
+    set(gca, 'FontSize',18)
+    xlabel('$u_r / \sigma_\infty$', 'interpreter','latex')
+    ylabel('$p(u_r / \sigma_\infty)$','interpreter','latex')
+    axis square
+    legend('u_L','u_\lambda');
+    xlim([min(uL) max(uL)])
+    ylim([0.0000001*max(puL) 2*max(pul)])
+    tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+    uiwait(gcf);
+    delete(tmp_ui);
+    close
+    clear u_tmp
+
+    
+    %% Preallocating (considering the available RAM)
+    data        = data(1:ceil(length(data)*data_length));
+    giga        = askInput({sprintf('Enter available RAM in GB:')},{num2str(8)}) * 1.07*10^9; %1 GB in bytes   
+    k_split     = ceil((12*4*length(data)*length(r))/giga)
+    data        = reshape(data((1:floor(numel(data)/k_split)*k_split)),[],k_split);
+
+    A           = nan( size(data,1)-stp+1,k_split );
+    Sm          = nan( size(data,1)-stp+1,k_split );
+    Ds          = nan( size(data,1)-stp+1,k_split );
+    DS          = nan( size(data,1)-stp+1,k_split );
+    ind_trajec  = nan( size(data,1)-stp+1,k_split );
+    %  Q_hk     = nan( size(data,1)-stp+1,k_split );
+    %  Q_med    = nan( size(data,1)-stp+1,k_split );
+    %  Q_ex     = nan( size(data,1)-stp+1,k_split );
+    %  Q_med_xi = nan; 
+
+    clear tmp
+    tmp(1:k_split) = struct('Sm_tmp',nan(size(data,1)-stp+1,1),'Ds_tmp',nan(size(data,1)-stp+1,1),'DS_tmp',nan(size(data,1)-stp+1,1), 'ind_trajec_tmp',nan(size(data,1)-stp+1,1), 'A_tmp',nan(size(data,1)-stp+1,1));
+    % tmp(1:k_split) = struct('Sm_tmp',nan(size(data,1)-stp+1,1),'Ds_tmp',nan(size(data,1)-stp+1,1),'DS_tmp',nan(size(data,1)-stp+1,1), 'ind_trajec_tmp',nan(size(data,1)-stp+1,1),...
+    %'Q_hk_tmp',nan(size(data,1)-stp+1,1),'Q_med_tmp',nan(size(data,1)-stp+1,1),'Q_ex',nan(size(data,1)-stp+1,1)); 
+    f = uifigure;
+    d = uiprogressdlg(f,'Title','Please Wait','ShowPercentage','on');
+    
+    
+    %% calc Entropy
+    for k=1:k_split
+        [u,ind_trajec_tmp]                                      = create_us_overlap(data(:,k),r,rind,stp);     
+        [Sm_tmp,Ds_tmp,DS_tmp,dr,r_s,~,~,~,A_tmp,~,~,~,~,~,~]   = calcDS(z,u,r.', co,markov,taylor_L,Fs,m_data,dr_ind, uL, puL, ul, pul);
+        tmp(k).A_tmp(1:length(A_tmp))                           = A_tmp;
+        tmp(k).Sm_tmp(1:length(Sm_tmp))                         = Sm_tmp;
+        tmp(k).Ds_tmp(1:length(Ds_tmp))                         = Ds_tmp;
+        tmp(k).DS_tmp(1:length(DS_tmp))                         = DS_tmp;
+        % [Q_hk_tmp, Q_med_tmp, Q_ex_tmp,~]                     = heat(u,r.', co);
+        % tmp(k).Q_hk_tmp(1:length(Q_hk_tmp))                   = Q_hk_tmp;
+        % tmp(k).Q_med_tmp(1:length(Q_med_tmp))                 = Q_med_tmp;
+        % tmp(k).Q_ex_tmp(1:length(Q_ex_tmp))                   = Q_ex_tmp;
+
+        if k==1
+            tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))     = ind_trajec_tmp;
+        else
+            % tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))   = (size(data,1)-stp)*(k-1)+ind_trajec_tmp+1;
+            % tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))   = tmp(k-1).ind_trajec_tmp(length(ind_trajec_tmp))+ind_trajec_tmp+1;
+            % tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))   = tmp(k-1).ind_trajec_tmp(length(ind_trajec_tmp))+stp+ind_trajec_tmp+1;
+            tmp(k).ind_trajec_tmp(1:length(ind_trajec_tmp))     = tmp(k-1).ind_trajec_tmp(length(ind_trajec_tmp))+stp+ind_trajec_tmp;
+        end   
+        d.Value=k/k_split;
+		k/k_split
+    end
+    
+    close(d);
+    close(f);
+
+    for k=1:k_split
+        A(1:length(tmp(k).A_tmp),k)                             = tmp(k).A_tmp;
+        Sm(1:length(tmp(k).Sm_tmp),k)                           = tmp(k).Sm_tmp;
+        Ds(1:length(tmp(k).Ds_tmp),k)                           = tmp(k).Ds_tmp;
+        DS(1:length(tmp(k).DS_tmp),k)                           = tmp(k).DS_tmp;
+        ind_trajec(1:length(tmp(k).ind_trajec_tmp),k)           = tmp(k).ind_trajec_tmp;
+    %     Q_hk(1:length(tmp(k).Q_hk_tmp),k)                     = tmp(k).Q_hk_tmp;
+    %     Q_med(1:length(tmp(k).Q_med_tmp),k)                   = tmp(k).Q_med_tmp;
+    %     Q_ex(1:length(tmp(k).Q_ex_tmp),k)                     = tmp(k).Q_ex_tmp; 
+    end
+    clear tmp
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% calc Entropy - non-overlapping trajectories
+elseif z==3
+    %% Preallocating (considering the available RAM)
+    tmp_size    = ceil(numel(data)/stp);
+    data        = data(1:ceil(length(data)*data_length));
+%     giga        = askInput({sprintf('Enter available RAM in GB:')},{num2str(8)}) * 1.07*10^9; %1 GB in bytes 
+    giga        = 8* 1.07*10^9;
+    k_split     = ceil((20*4*tmp_size*length(r(1:dr_ind:end)))/giga);
+    data        = reshape(data((1:floor(numel(data)/k_split)*k_split)),[],k_split);
+   
+    Sm          = nan( tmp_size, k_split );
+    Ds          = nan( tmp_size, k_split );
+    DS          = nan( tmp_size, k_split );
+    ind_trajec  = nan( tmp_size, k_split );
+    A           = nan( tmp_size, k_split );    
+%     Lag         = nan( tmp_size, k_split );
+%     p           = nan( tmp_size, k_split );
+%     H           = nan( tmp_size, k_split );
+%     tmpvar      = nan( tmp_size, k_split );
+%     H1          = nan( tmp_size, k_split );
+%     H2          = nan( tmp_size, k_split );
+%     u           = nan( tmp_size, k_split );
+%     Q_hk      = nan( tmp_size,1 );
+%     Q_med     = nan( tmp_size,1 );
+%     Q_ex      = nan( tmp_size,1 );
+%     not_nan_index = nan( tmp_size, k_split );
+ 
+    
+    %% calc Entropy
+    f = uifigure;
+    d = uiprogressdlg(f,'Title','Please Wait','ShowPercentage','on');
+    for k=1:k_split
+        [u_tmp,ind_trajec_tmp]                  = create_us(data(:,k),r,rind,stp);
+        [Sm_tmp,Ds_tmp,DS_tmp,dr,r_s,~,~,L_tmp,A_tmp,p_tmp,H_tmp,tmpvar_tmp,H1_tmp,H2_tmp,ur_s_tmp]           = calcDS(z,u_tmp,r.',co,markov,taylor_L,Fs,m_data,dr_ind);
+    %   [Q_hk_tmp, Q_med_tmp, Q_ex_tmp,Q_med_xi] = heat(u_tmp,r.', co, r(end)); 
+
+        A(1:sum(~isnan(DS_tmp),1),k)            = A_tmp(~isnan(DS_tmp));
+        ind_trajec(1:sum(~isnan(DS_tmp),1),k)   = ind_trajec_tmp(~isnan(DS_tmp));
+        Sm(1:sum(~isnan(DS_tmp),1),k)           = Sm_tmp(~isnan(DS_tmp));
+        Ds(1:sum(~isnan(DS_tmp),1),k)           = Ds_tmp(~isnan(DS_tmp));
+        DS(1:sum(~isnan(DS_tmp),1),k)           = DS_tmp(~isnan(DS_tmp));
+        
+        if k_split==1
+%             Lag(1:sum(~isnan(DS_tmp),1),:)         = L_tmp(~isnan(DS_tmp),:);
+%             p(1:sum(~isnan(DS_tmp),1),:)           = p_tmp(~isnan(DS_tmp),:); 
+%             H(1:sum(~isnan(DS_tmp),1),:)           = H_tmp(~isnan(DS_tmp),:); 
+%             tmpvar(1:sum(~isnan(DS_tmp),1),:)      = tmpvar_tmp(~isnan(DS_tmp),:);
+            ur_s_tmp(1:sum(~isnan(DS_tmp),1),:)    = ur_s_tmp(~isnan(DS_tmp),:);
+%             H1(1:sum(~isnan(DS_tmp),1),:)          = H1_tmp(~isnan(DS_tmp),:);   
+%             H2(1:sum(~isnan(DS_tmp),1),:)          = H2_tmp(~isnan(DS_tmp),:);
+            u(1:sum(~isnan(DS_tmp),1),:)           = u_tmp(~isnan(DS_tmp),:);
+        end
+%             Q_hk(1:sum(~isnan(DS_tmp),1),k)           = Q_hk_tmp(~isnan(DS_tmp));
+%             Q_med(1:sum(~isnan(DS_tmp),1),k)          = Q_med_tmp(~isnan(DS_tmp));
+%             Q_ex(1:sum(~isnan(DS_tmp),1),k)           = Q_ex_tmp(~isnan(DS_tmp));
+%             not_nan                                   = (0:tmp_size).';
+%             not_nan_index(1:sum(~isnan(DS_tmp),1),k)  = not_nan(~isnan(DS_tmp));
+        clear ind_trajec_tmp Sm_tmp Ds_tmp DS_tmp u_tmp
+
+     d.Value=k/k_split;
+    end
+    close(d);
+    close(f);
+end
+
+A       = reshape(A,[],1);
+Sm      = reshape(Sm,[],1);
+Ds      = reshape(Ds,[],1);
+DS      = reshape(DS,[],1);
+% Q_hk  = reshape(Q_hk,[],1);
+% Q_med = reshape(Q_med,[],1);
+% Q_ex  = reshape(Q_ex,[],1);
+
+% cut-out NaN's due to vanishing or negative D2
+DS(isinf(DS))   = nan;
+A               = A(~isnan(DS));
+
+% if z==1
+    Lag             = nan;
+    p               = nan;
+    H               = nan;
+    tmpvar          = nan;
+    H1              = nan;
+    H2              = nan;
+    if k_split>1
+        ur_s_tmp        = nan;
+        u               = nan;
+    end
+% end
+
+% if z==3
+    % L      = reshape(L,[],1);
+    % p      = reshape(p,[],1);
+    % H      = reshape(H,[],1);
+    % tmpvar = reshape(tmpvar,[],1);
+    % H1     = reshape(H1,[],1);
+    % H2     = reshape(H2,[],1);
+% end
+% if z==3
+%     Lag             = Lag(~isnan(DS),:);
+%     p               = p(~isnan(DS),:);
+%     H               = H(~isnan(DS),:);
+%     tmpvar          = tmpvar(~isnan(DS),:);
+%     ur_s_tmp        = ur_s_tmp(~isnan(DS),:);
+%     H1              = H1(~isnan(DS),:);
+%     H2              = H2(~isnan(DS),:);
+ if k_split==1
+     ur_s_tmp        = ur_s_tmp(~isnan(DS),:);
+     u               = u(~isnan(DS),:);
+ end
+% end
+
+Sm              = Sm(~isnan(DS));
+Ds              = Ds(~isnan(DS));
+% Q_hk  = Q_hk(~isnan(DS));
+% Q_med = Q_med(~isnan(DS));
+% Q_ex  = Q_ex(~isnan(DS));
+
+ind_trajec      = reshape(ind_trajec,[],1);
+ind_trajec      = ind_trajec(~isnan(DS));
+% not_nan_index = ~isnan(DS);
+
+DS              = DS(~isnan(DS));
+
+if z==1
+    Sm          = Sm(1:end-1);
+    Ds          = Ds(1:end-1);
+    DS          = DS(1:end-1); 
+    ind_trajec  = ind_trajec(1:end-1);
+end
+r_s = r_s(1,:);
 end
\ No newline at end of file
diff --git a/Functions_v5-MATLAB/conditional_PDF_markov.m b/Functions_v5-MATLAB/conditional_PDF_markov.m
index fcfd9d0..fbb0000 100644
--- a/Functions_v5-MATLAB/conditional_PDF_markov.m
+++ b/Functions_v5-MATLAB/conditional_PDF_markov.m
@@ -21,6 +21,9 @@
 % second_condi = increment of the second condition
 % num_bin = Number of bins 
 % min_events = minimum number of events to consider in a single bin
+% norm_ur = normalization of the scale using $\lambda$? data 1=Yes, 0=No
+% save_path = path for saving figures and files
+% save_name = name for saving files
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 function conditional_PDF_markov(data,Fs,m_data,markov,second_condi,num_bin,min_events,norm_ur,save_path,save_name)
@@ -84,12 +87,15 @@ function conditional_PDF_markov(data,Fs,m_data,markov,second_condi,num_bin,min_e
 %% single conditional probability
 [P_AIB,P_BIA,P_AnB,P_A,P_B,binP_AIB,x_mean_bin,y_mean_bin,events_1,counter_A,counter_B] = distribution(tau2,tau1,incr1,incr2,num_bin,dx,dy,x,y,dA);
 
+P_AIB_div	= P_AIB;
+P_AIB_div(counter_A<min_events,counter_B<min_events)=0;
+
 %cut out nan's and min_events
 y_input     = y_mean_bin(counter_B>=min_events);
 x_input     = x_mean_bin(counter_A>=min_events);
 
-P_AIB       = P_AIB(counter_A>=min_events,counter_B>=min_events);
-P_AnB       = P_AnB(counter_A>=min_events,counter_B>=min_events);
+P_AIB      = P_AIB(counter_A>=min_events,counter_B>=min_events);
+P_AnB     = P_AnB(counter_A>=min_events,counter_B>=min_events);
 P_A         = P_A(counter_A>=min_events);
 P_B         = P_B(counter_B>=min_events);
                          
@@ -121,6 +127,8 @@ function conditional_PDF_markov(data,Fs,m_data,markov,second_condi,num_bin,min_e
 % cut for the second conditon (incr3) 
 index = find(incr3>-tol+second_condi & incr3<tol+second_condi);
 [P_AIB_2,P_BIA_2,P_AnB_2,P_A_2,P_B_2,binP_AIB_2,x_mean_bin_2,y_mean_bin_2,events_2,counter_A_2,counter_B_2] = distribution_double(tau2,tau1,incr1(index),incr2(index),num_bin,dx,dy,x,y,dA);   
+P_AIB_div_2=P_AIB_2;
+P_AIB_div_2(counter_A_2<min_events,counter_B_2<min_events)=0;
 
 % cut out nan's and min_events
 y_input_2=y_mean_bin_2(counter_B_2>=min_events);
@@ -148,6 +156,31 @@ function conditional_PDF_markov(data,Fs,m_data,markov,second_condi,num_bin,min_e
 
 legend({'$p(u_{r_2}|u_{r_1})$','$p(u_{r_2}|u_{r_1},u_{r_0})$'}, 'interpreter','latex','Location','northwest')
 
+
+
+
+% Titel hier einfügen
+%% weighted mean square error function in logarithmic space; conditional probability densities
+d_M_1   = 0;
+d_M_2   = 0;
+for j =1:num_bin
+    for i =1:num_bin
+         if (P_AIB_div_2(i,j) > 0) && (P_AIB_div(i,j) > 0)     
+            d_M_1 = d_M_1 + ((P_AIB_div_2(i,j) + P_AIB_div(i,j)).*...
+                            (log(P_AIB_div_2(i,j))    - log(P_AIB_div(i,j))).^2);                        
+            d_M_2 = d_M_2 + ((P_AIB_div_2(i,j) + P_AIB_div(i,j)).*...
+                            (log(P_AIB_div_2(i,j)).^2 + log(P_AIB_div(i,j)).^2));
+        end
+%         if (P_verbund_n(i,j) < 0)
+%             d_M_1 = d_M_1 + 10.^(10);                        
+%         end
+    end
+end
+fval     = d_M_1/d_M_2;
+
+title(['$\xi_{\Delta_{EM}}$=' num2str(fval,2)])
+
+
 fig_setup
 txt = {'(a)'};
 a=text(4,0.5,txt);
diff --git a/Functions_v5-MATLAB/conditional_moment.m b/Functions_v5-MATLAB/conditional_moment.m
index 5742cc3..065f3c8 100644
--- a/Functions_v5-MATLAB/conditional_moment.m
+++ b/Functions_v5-MATLAB/conditional_moment.m
@@ -25,6 +25,7 @@
 % Arguments OUT
 % evaluated = struct array containing all the information about conditional moments for each scale and for each bin
 % step_con_moment = the steps at which the value of conditional moments are calculated
+% Enclosed in a "evaluated":
 % r is the scale in meters at which moments will be calculated and hence this r will
 % be the same at which D1 & D2 will be calculated===>r2
 % r_samp is the r in number of samples==>r2
diff --git a/Functions_v5-MATLAB/frequency_filter.m b/Functions_v5-MATLAB/frequency_filter.m
index fc32d86..a6ebac6 100644
--- a/Functions_v5-MATLAB/frequency_filter.m
+++ b/Functions_v5-MATLAB/frequency_filter.m
@@ -1,310 +1,311 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
-% This function returns the filtered data using the low pass filter at the previously set frequency
-% (using $\textbf{\textit{butter}}$ and $\textbf{\textit{filtfilt}}$ function of MATLAB). The 
-% filtered data named as $\textbf{data\_filter}$ for all the further data post-processing. If the 
-% filtering was negated in the previous step, $\textbf{data\_filter}$ and $\textbf{data}$ are equal.
-% In addition, different representation/normalization of the energy spectrum density with respect to
-% frequency $f$, scale $r$, wave number $k$ will also be plotted. 
-%
-% Arguments IN
-% data = 1D array of data to which you would like to apply the low pass filter
-% Fs = Acquisition/Sampling Frequency in Hz
-% low_freq = Frequency at which you would like to use the low-pass filter in Hz
-% kin_vis = Kinematic viscosity of fluid in m^2/sec
-%
-% Arguments OUT
-% tmps = this is the filtered data which will be returned by this function
-% x_achse_k_spektrum_2pi_filter = 2 pi k waven umbers(1/m) after filtering
-% E_k = Wavenumber Spectrum (m^3/sec^2) after filtering
-% x_achse_spektrum_meter_filter = k Spektrum; Scales,r (m) 
-% Dr_avg_filter = Dissipation Spectrum; scale domain; after filtering
-% Dk_avg_filter = Dissipation Spectrum; wave number domain; after filtering
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
-function [tmps,x_achse_f_spektrum_filter,spek_smooth_filter,x_achse_k_spektrum_2pi_filter,E_k,x_achse_spektrum_meter_filter,Dr_avg_filter,Dk_avg_filter] = frequency_filter(data, Fs, low_freq,I,kin_vis,filter,save_path,save_name)
-set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
-set(groot, 'defaulttextinterpreter','latex');   
-set(groot, 'defaultLegendInterpreter','latex');
-set(groot, 'defaultAxesTitle','latex');
-
-close
-if mod(length(data),2)==1
-    data=data(1:end-1); % This is to make length of data array even
-end  
-
-m_data=nanmean(data); % Mean of the data
-if filter==1
-    % butter is MATLAB function to design a filter of desired order and at desired cut-off frequency
-    order=20;
-    [b,a] = butter(order,low_freq/(Fs/2),'low');
-    % tmps will be the array returned by this function
-    tmps = filtfilt (b, a, data); % For filtfilt  see the MATLAB documentation
-
-    % For a fixed cutoff frequency used in butter filter there is a threashold of order of butter 
-    % filter at which we can apply the filter successfully.
-    % Next loop tries to design a highest order of butter filter for a fixed cut-off low frequency
-    % We repeat line 16,17 & 18 in the following while loop
-    while (sum(isnan(tmps))>0 || (length(tmps)-nnz(tmps))>0 || var(tmps)>var(data))
-        % sum(isnan(tmps))>0 ==> To avoid nan entries in filtered data
-        % (length(tmps)-nnz(tmps))>0 ==> To avoid zeros in filterd data 
-        % var(tmps)>var(data) ==>variance of the filtered data will always be
-        % lesser than the variance of the unfilterd data
-        clear tmps;
-        order=order-1;
-        clear b a;
-        [b,a] = butter(order,low_freq/(Fs/2),'low');
-        tmps = filtfilt (b, a, data); %tmps will be the array returned by this function
-    end 
-    % PSD of the fluctuations after filtering; mean value does not play a role in spectrum
-    data_filter     = tmps-m_data;
-else
-    tmps            = data;
-end
-
-% PSD of the fluctuations before filtering; mean value does not play a role in spectrum
-data          = data-m_data; 
-L               = length(data); % length of the data
-
-% spek              = abs(fft(data,L)).^2/L; % Power spectral density(PSD)
-spek    = abs(fft(data,L)).^2/(L*Fs);        % Energy spectral density(ESD) using a fft
-spek    = 2.*spek(2:L/2+1);                  % FFt will yield half number of unique points 
-
-f       = Fs/2*linspace(0,1,L/2+1).';        % Nyquist frequency of the signal==Fs/2
-f       = f(2:end);                          % Remove zero Hz component
-
-% moving average with equally spaced frequency interval in log-space 
-intervall = unique(round(logspace(0,log10(L/2),3001)),'stable');
-
-% moving average with equally spaced frequency interval in lin-space 
-% intervall=unique(round(linspace(1,L/2,plot_length)),'stable');
-
-plot_length = length(intervall);             % Number of points for plotting averaged spectrum
-
-% Initializing the arrays/preallocation
-spek_smooth         = zeros(plot_length-2,1);
-x_achse_f_spektrum  = zeros(plot_length-2,1);
-
-% Averaging of spectrum and hence smoothing
-for i=2:(plot_length-1)
-    x_achse_f_spektrum(i-1,1) = mean(f(intervall(i-1):intervall(i+1)));
-    spek_smooth(i-1,1)        = mean(spek(intervall(i-1):intervall(i+1)));
-end
-% ------------------------Windowing / averaging end ------------------------ 
-
-
-
-%%%%%%%%% Filter data
-%% PSD of the fluctuations after filtering
-if filter==0
-    data_filter     = data;    
-end  
-    
-spek_filter              = abs(fft(data_filter,L)).^2/(L*Fs);   % Energy spectral density(ESD)
-spek_filter              = 2.*spek_filter(1:L/2+1);             % FFt will yield half number of unique points
-                                                                
-f_filter = Fs/2*linspace(0,1,L/2+1).';                          % Nyquist frequency of the signal
-
-% %moving average with equally spaced frequency interval in log-space
-% intervall=unique(round(logspace(0,log10(L/2),plot_length)),'stable');
-% plot_length=length(intervall);
-
-spek_smooth_filter          = zeros(plot_length-2,1);
-x_achse_f_spektrum_filter   = zeros(plot_length-2,1);
-
-% Averaging of spectrum and hence smoothing
-for i=2:(plot_length-1)
-    spek_smooth_filter(i-1,1)           = mean(spek_filter(intervall(i-1):intervall(i+1)));
-    x_achse_f_spektrum_filter(i-1,1)    = mean(f_filter(intervall(i-1):intervall(i+1)));
-end
-% ------------------------Windowing / averaging end ------------------------  
-
-
-id          = round(logspace(0,log10(length(spek)),10^5));
-id(end)     = length(spek);
-f_plot      = f(id);
-spek_plot   = spek(id);
-
-%%%PLOTTING
-% Plot of Frequency Vs. Energy spectral density(ESD)
-h(1) = figure;
-% Plot of Frequency Vs. Energy spectral density(ESD) without smoothing
-loglog(f_plot,spek_plot)
-hold on
-%Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing
-loglog(x_achse_f_spektrum,spek_smooth,'LineWidth',2)
-% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing & Filtering
-loglog(x_achse_f_spektrum_filter,spek_smooth_filter,'LineWidth',2)
-set(groot, 'defaultLegendInterpreter','latex');
-% xlabel('f / Hz','interpreter','latex');
-% ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
-xlabel('$f\ (Hz)$','interpreter','latex');
-ylabel('$E\ (m^2/s)$', 'interpreter','latex')
-title(['Low-pass filter frequency =',num2str(low_freq,'%1.0f'), ' Hz'],'interpreter','latex')
-set(gca,'FontSize',18)
-set(gcf, 'Color', 'w')
-xlim([min(x_achse_f_spektrum_filter) max(x_achse_f_spektrum_filter)]*2)
-ylim([min(spek) max(spek)]*2)
-axis square 
-% vline(low_freq,'k')
-
-% vline(m_data/int_L,'k')
-% vline(m_data/taylor_L,'k')
-set(groot, 'defaultAxesTickLabelInterpreter','latex');
-% title(['$ti=$',num2str(std(data)/m_data,'%1.3f')],'interpreter','latex') %% Turbulence Intensity of the flow
-legend('raw','averaged','low-pass filtered','Location','southwest')
-set(gca,'XTick',[10^-4 10^-2 10^0  10^2  10^4]);
-
-
-% savefig(fullfile(save_path,append(save_name,'_','spectrum_filter.fig')),'compact')
-fig_setup
-
-f_start            = find(f_plot>I(1), 1,'first');
-f_end              = find(f_plot<=I(2), 1,'last');
-
-[xData, yData] = prepareCurveData( f_plot(f_start:f_end), spek_plot(f_start:f_end));
-ft = fittype( 'a*x^(-5/3)', 'independent', 'x', 'dependent', 'y' );
-opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
-[fitresult, gof] = fit( xData, yData, ft, opts );
-
-plot(f_plot,feval(fitresult,f_plot),'Color',[0 0 0],'LineWidth',2,'LineStyle','--')
-legend('raw','averaged','low-pass filtered','$f^{(-5/3)}$')
-ylim([min(spek_plot) max(spek_plot)]*4)
-
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf); 
-delete(tmp_ui);
-
-if ischar(save_path)
-   % savefig(fullfile(save_path,append(save_name,'_','spectrum_var_domain.fig')))
-%    savefig(h,fullfile(save_path,append(save_name,'_','spectrum_filter.fig')),'compact')
-%    for a = 1:length(h)
-       exportgraphics(h(1),fullfile(save_path,append(save_name,'_',sprintf('spectrum_filter_%d.png', 1))))
-%        export_fig(h(a), fullfile(save_path,append(save_name,'_',sprintf('spectrum_filter_%d.png', a))));
-%    end
-end
-
-
-% Wavenumber Spectrum (m^3/sec^2)
-E_k = ((spek_smooth_filter.*m_data)./(2*pi)); 
-% 2-pi - k Spektrum; Wavenumbers(1/m)
-x_achse_k_spektrum_2pi_filter = (2.*pi.*x_achse_f_spektrum_filter)./m_data;
-% k Spektrum; Scales,r (m)
-x_achse_spektrum_meter_filter = m_data./(2*pi*x_achse_f_spektrum_filter); 
-
-Dr_avg_filter=(spek_smooth_filter.*x_achse_f_spektrum_filter.^2)./m_data;
-Dk_avg_filter=15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi_filter).^2;
-
-
-h(2) = figure;
-set(gcf, 'Color', 'w')
-subplot(2,2,1)
-loglog(x_achse_f_spektrum_filter,spek_smooth_filter,'-','LineWidth',1)%% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing & Filtering
-% xlabel('f / Hz','interpreter','latex');
-% ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
-xlabel('$f\ (Hz)$','interpreter','latex');
-ylabel('$E\ (m^2/s)$', 'interpreter','latex')
-xlim([min(x_achse_f_spektrum_filter) 2*max(x_achse_f_spektrum_filter)])
-ylim([min(spek_smooth_filter) max(spek_smooth_filter)]*2)
-title('frequency domain','interpreter','latex')
-set(gca,'FontSize',12)
-set(groot, 'defaultAxesTickLabelInterpreter','latex');
-% plot([10^3 10^3],[10^-6 10^-2],'y = x ','-','LineWidth',1) 
-% h = vline(10^3,'k','The Answer')
-% text(245,spek_smooth(find(x_achse_f_spektrum>245,1,'first')),'\leftarrow sin(\pi) = 0','FontSize',16)
-set(gca,'XTick',sort([10^-2 10^0  10^2 10^4 low_freq]));
-hold on
-vline(low_freq,'k')
-
-txt = {'(a)'};
-a=text(4,0.5,txt);
-a.Units='normalized';
-
-subplot(2,2,2)
-loglog(x_achse_k_spektrum_2pi_filter,E_k,'-','LineWidth',1)%% Plot of Wavenumber(k) Vs. Energy(k)(m^3/sec^2) with smoothing & Filtering
-% loglog(x_achse_k_spektrum_2pi_filter,x_achse_k_spektrum_2pi_filter.^(-5/3).*(max(spek_smooth).*200),'--','color','k','LineWidth',2)
-% xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
-% ylabel('$E(k)=\frac{E(f) <u>}{2 \pi}$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
-xlabel('$k=\frac{2 \pi f}{<u>}\ (1/m)$','interpreter','latex');
-ylabel('$\frac{E <u>}{2 \pi}\ (m^3/s^2)$', 'interpreter','latex')
-xlim([min(x_achse_k_spektrum_2pi_filter) 2*max(x_achse_k_spektrum_2pi_filter)])
-ylim([min(E_k) max(E_k)]*2)
-title('wave number domain','interpreter','latex')
-set(gca,'FontSize',12)
-set(gca,'XTick',sort([10^-2 10^0  10^2 10^4 round(2*pi*low_freq/m_data)]));
-set(gca,'XTick',sort([10^-2 10^0  10^2 round(2*pi*low_freq/m_data)]));
-hold on
-vline(2*pi*low_freq/m_data,'k')
-txt = {'(b)'};
-b=text(4,0.5,txt);
-b.Units='normalized';
-
-subplot4=subplot(2,2,3);
-loglog(x_achse_spektrum_meter_filter,Dr_avg_filter,'-','LineWidth',1)%% Plot of Scales,r(m) Vs. Energy(r)(m/sec^2) with smoothing & Filtering
-% xlabel('$r=\frac{<u>}{2 \pi f} / m$', 'interpreter','latex');
-% ylabel('$E(r)=\frac{E(f) f^2}{<u>}$ / $\frac{m}{s^2}$', 'interpreter','latex')
-xlabel('$r=\frac{<u>}{2 \pi f}\ (m)$','interpreter','latex');
-ylabel('$\frac{E\ f^2}{<u>}\ (m/s^2)$', 'interpreter','latex')
-set(subplot4, 'Xdir', 'reverse')
-xlim([0.5* min(x_achse_spektrum_meter_filter) max(x_achse_spektrum_meter_filter)])
-ylim([min((spek_smooth_filter.*x_achse_f_spektrum_filter.^2)./m_data) max((spek_smooth_filter.*x_achse_f_spektrum_filter.^2)./m_data)]*2)
-set(gca,'FontSize',12)
-set(gca,'XTick',sort([10^-2 10^0  10^2 10^4 m_data/(2*pi*low_freq)]));
-title('dissipation spectrum','interpreter','latex')
-hold on
-vline(m_data/(2*pi*low_freq),'k')
-txt = {'(c)'};
-c=text(4,0.5,txt);
-c.Units='normalized';
-
-
-% Dissipation Spectrum
-% kin_vis = 15.328737178409*10^-6;
-subplot(2,2,4)
-loglog(x_achse_k_spektrum_2pi_filter,Dk_avg_filter,'-','LineWidth',1)%% Plot of Wavenumber(k) Vs. Dissipation(k)(m/sec^2) with smoothing & Filtering
-% xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
-% ylabel('E(k)$\cdot k^2$ / $\frac{m}{s^2}$', 'interpreter','latex')
-xlabel('$k=\frac{2 \pi f}{<u>}\ (1/m)$','interpreter','latex');
-ylabel('$\frac{E <u>}{2 \pi}\ k^2 \ (m/s^2)$', 'interpreter','latex')
-
-xlim([min(x_achse_k_spektrum_2pi_filter) 2*max(x_achse_k_spektrum_2pi_filter)])
-ylim([min(15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi_filter).^2) max(15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi_filter).^2)]*2)
-title('dissipation spectrum','interpreter','latex')
-set(gca,'FontSize',12)
-set(gca,'XTick',sort([10^-2 10^0  10^2 10^4 round(2*pi*low_freq/m_data)]));
-set(gca,'XTick',sort([10^-2 10^0  10^2 round(2*pi*low_freq/m_data)]));
-hold on
-vline(2*pi*low_freq/m_data,'k')
-txt = {'(d)'};
-d=text(4,0.5,txt);
-d.Units='normalized';
-
-% font=22;
-% a.FontSize = font;
-% b.FontSize = font;
-% c.FontSize = font;
-% d.FontSize = font;
-
-pos_txt=[-0.25   0.9];
-a.Position=pos_txt;
-b.Position=pos_txt;
-c.Position=pos_txt;
-d.Position=pos_txt;
-
-datacursormode on
-
-
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf); 
-delete(tmp_ui);
-
-if ischar(save_path)
-   % savefig(fullfile(save_path,append(save_name,'_','spectrum_var_domain.fig')))
-   savefig(h,fullfile(save_path,append(save_name,'_','spectrum_filter.fig')),'compact')
-%    for a = 1:length(h)
-       exportgraphics(h(2),fullfile(save_path,append(save_name,'_',sprintf('spectrum_filter_%d.png', 2))))
-%        export_fig(h(a), fullfile(save_path,append(save_name,'_',sprintf('spectrum_filter_%d.png', a))));
-%    end
-end
-close all
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+% This function returns the filtered data using the low pass filter at the previously set frequency
+% (using $\textbf{\textit{butter}}$ and $\textbf{\textit{filtfilt}}$ function of MATLAB). The 
+% filtered data named as $\textbf{data\_filter}$ for all the further data post-processing. If the 
+% filtering was negated in the previous step, $\textbf{data\_filter}$ and $\textbf{data}$ are equal.
+% In addition, different representation/normalization of the energy spectrum density with respect to
+% frequency $f$, scale $r$, wave number $k$ will also be plotted. 
+%
+% Arguments IN
+% data = 1D array of data to which you would like to apply the low pass filter
+% Fs = Acquisition/Sampling Frequency in Hz
+% low_freq = Frequency at which you would like to use the low-pass filter in Hz
+% kin_vis = Kinematic viscosity of fluid in m^2/sec
+%
+% Arguments OUT
+% tmps = this is the filtered data which will be returned by this function
+% x_achse_k_spektrum_2pi_filter = 2 pi k waven umbers(1/m) after filtering
+% spek_smooth_filter = Frequency Spectrum after filtering
+% E_k = Wavenumber Spectrum (m^3/sec^2) after filtering
+% x_achse_spektrum_meter_filter = r Spektrum; Scales,r (m)  
+% Dr_avg_filter = Dissipation Spectrum; scale domain; after filtering
+% Dk_avg_filter = Dissipation Spectrum; wave number domain; after filtering
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+function [tmps,x_achse_f_spektrum_filter,spek_smooth_filter,x_achse_k_spektrum_2pi_filter,E_k,x_achse_spektrum_meter_filter,Dr_avg_filter,Dk_avg_filter] = frequency_filter(data, Fs, low_freq,I,kin_vis,filter,save_path,save_name)
+set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
+set(groot, 'defaulttextinterpreter','latex');   
+set(groot, 'defaultLegendInterpreter','latex');
+set(groot, 'defaultAxesTitle','latex');
+
+close
+if mod(length(data),2)==1
+    data=data(1:end-1); % This is to make length of data array even
+end  
+
+m_data=nanmean(data); % Mean of the data
+if filter==1
+    % butter is MATLAB function to design a filter of desired order and at desired cut-off frequency
+    order=20;
+    [b,a] = butter(order,low_freq/(Fs/2),'low');
+    % tmps will be the array returned by this function
+    tmps = filtfilt (b, a, data); % For filtfilt  see the MATLAB documentation
+
+    % For a fixed cutoff frequency used in butter filter there is a threashold of order of butter 
+    % filter at which we can apply the filter successfully.
+    % Next loop tries to design a highest order of butter filter for a fixed cut-off low frequency
+    % We repeat line 16,17 & 18 in the following while loop
+    while (sum(isnan(tmps))>0 || (length(tmps)-nnz(tmps))>0 || var(tmps)>var(data))
+        % sum(isnan(tmps))>0 ==> To avoid nan entries in filtered data
+        % (length(tmps)-nnz(tmps))>0 ==> To avoid zeros in filterd data 
+        % var(tmps)>var(data) ==>variance of the filtered data will always be
+        % lesser than the variance of the unfilterd data
+        clear tmps;
+        order=order-1;
+        clear b a;
+        [b,a] = butter(order,low_freq/(Fs/2),'low');
+        tmps = filtfilt (b, a, data); %tmps will be the array returned by this function
+    end 
+    % PSD of the fluctuations after filtering; mean value does not play a role in spectrum
+    data_filter     = tmps-m_data;
+else
+    tmps            = data;
+end
+
+% PSD of the fluctuations before filtering; mean value does not play a role in spectrum
+data          = data-m_data; 
+L               = length(data); % length of the data
+
+% spek              = abs(fft(data,L)).^2/L; % Power spectral density(PSD)
+spek    = abs(fft(data,L)).^2/(L*Fs);        % Energy spectral density(ESD) using a fft
+spek    = 2.*spek(2:L/2+1);                  % FFt will yield half number of unique points 
+
+f       = Fs/2*linspace(0,1,L/2+1).';        % Nyquist frequency of the signal==Fs/2
+f       = f(2:end);                          % Remove zero Hz component
+
+% moving average with equally spaced frequency interval in log-space 
+intervall = unique(round(logspace(0,log10(L/2),3001)),'stable');
+
+% moving average with equally spaced frequency interval in lin-space 
+% intervall=unique(round(linspace(1,L/2,plot_length)),'stable');
+
+plot_length = length(intervall);             % Number of points for plotting averaged spectrum
+
+% Initializing the arrays/preallocation
+spek_smooth         = zeros(plot_length-2,1);
+x_achse_f_spektrum  = zeros(plot_length-2,1);
+
+% Averaging of spectrum and hence smoothing
+for i=2:(plot_length-1)
+    x_achse_f_spektrum(i-1,1) = mean(f(intervall(i-1):intervall(i+1)));
+    spek_smooth(i-1,1)        = mean(spek(intervall(i-1):intervall(i+1)));
+end
+% ------------------------Windowing / averaging end ------------------------ 
+
+
+
+%%%%%%%%% Filter data
+%% PSD of the fluctuations after filtering
+if filter==0
+    data_filter     = data;    
+end  
+    
+spek_filter              = abs(fft(data_filter,L)).^2/(L*Fs);   % Energy spectral density(ESD)
+spek_filter              = 2.*spek_filter(1:L/2+1);             % FFt will yield half number of unique points
+                                                                
+f_filter = Fs/2*linspace(0,1,L/2+1).';                          % Nyquist frequency of the signal
+
+% %moving average with equally spaced frequency interval in log-space
+% intervall=unique(round(logspace(0,log10(L/2),plot_length)),'stable');
+% plot_length=length(intervall);
+
+spek_smooth_filter          = zeros(plot_length-2,1);
+x_achse_f_spektrum_filter   = zeros(plot_length-2,1);
+
+% Averaging of spectrum and hence smoothing
+for i=2:(plot_length-1)
+    spek_smooth_filter(i-1,1)           = mean(spek_filter(intervall(i-1):intervall(i+1)));
+    x_achse_f_spektrum_filter(i-1,1)    = mean(f_filter(intervall(i-1):intervall(i+1)));
+end
+% ------------------------Windowing / averaging end ------------------------  
+
+
+id          = round(logspace(0,log10(length(spek)),10^5));
+id(end)     = length(spek);
+f_plot      = f(id);
+spek_plot   = spek(id);
+
+%%%PLOTTING
+% Plot of Frequency Vs. Energy spectral density(ESD)
+h(1) = figure;
+% Plot of Frequency Vs. Energy spectral density(ESD) without smoothing
+loglog(f_plot,spek_plot)
+hold on
+%Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing
+loglog(x_achse_f_spektrum,spek_smooth,'LineWidth',2)
+% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing & Filtering
+loglog(x_achse_f_spektrum_filter,spek_smooth_filter,'LineWidth',2)
+set(groot, 'defaultLegendInterpreter','latex');
+% xlabel('f / Hz','interpreter','latex');
+% ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
+xlabel('$f\ (Hz)$','interpreter','latex');
+ylabel('$E\ (m^2/s)$', 'interpreter','latex')
+title(['Low-pass filter frequency =',num2str(low_freq,'%1.0f'), ' Hz'],'interpreter','latex')
+set(gca,'FontSize',18)
+set(gcf, 'Color', 'w')
+xlim([min(x_achse_f_spektrum_filter) max(x_achse_f_spektrum_filter)]*2)
+ylim([min(spek) max(spek)]*2)
+axis square 
+% vline(low_freq,'k')
+
+% vline(m_data/int_L,'k')
+% vline(m_data/taylor_L,'k')
+set(groot, 'defaultAxesTickLabelInterpreter','latex');
+% title(['$ti=$',num2str(std(data)/m_data,'%1.3f')],'interpreter','latex') %% Turbulence Intensity of the flow
+legend('raw','averaged','averaged \& low-pass filtered','Location','southwest')
+set(gca,'XTick',[10^-4 10^-2 10^0  10^2  10^4]);
+
+
+% savefig(fullfile(save_path,append(save_name,'_','spectrum_filter.fig')),'compact')
+fig_setup
+
+f_start            = find(f_plot>I(1), 1,'first');
+f_end              = find(f_plot<=I(2), 1,'last');
+
+[xData, yData] = prepareCurveData( f_plot(f_start:f_end), spek_plot(f_start:f_end));
+ft = fittype( 'a*x^(-5/3)', 'independent', 'x', 'dependent', 'y' );
+opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
+[fitresult, gof] = fit( xData, yData, ft, opts );
+
+plot(f_plot,feval(fitresult,f_plot),'Color',[0 0 0],'LineWidth',2,'LineStyle','--')
+legend('raw','averaged','averaged \& low-pass filtered','$f^{(-5/3)}$','Location','southwest')
+ylim([min(spek_plot) max(spek_plot)]*4)
+
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf); 
+delete(tmp_ui);
+
+if ischar(save_path)
+   % savefig(fullfile(save_path,append(save_name,'_','spectrum_var_domain.fig')))
+%    savefig(h,fullfile(save_path,append(save_name,'_','spectrum_filter.fig')),'compact')
+%    for a = 1:length(h)
+       exportgraphics(h(1),fullfile(save_path,append(save_name,'_',sprintf('spectrum_filter_%d.png', 1))))
+%        export_fig(h(a), fullfile(save_path,append(save_name,'_',sprintf('spectrum_filter_%d.png', a))));
+%    end
+end
+
+
+% Wavenumber Spectrum (m^3/sec^2)
+E_k = ((spek_smooth_filter.*m_data)./(2*pi)); 
+% 2-pi - k Spektrum; Wavenumbers(1/m)
+x_achse_k_spektrum_2pi_filter = (2.*pi.*x_achse_f_spektrum_filter)./m_data;
+% k Spektrum; Scales,r (m)
+x_achse_spektrum_meter_filter = m_data./(2*pi*x_achse_f_spektrum_filter); 
+
+Dr_avg_filter=(spek_smooth_filter.*x_achse_f_spektrum_filter.^2)./m_data;
+Dk_avg_filter=15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi_filter).^2;
+
+
+h(2) = figure;
+set(gcf, 'Color', 'w')
+subplot(2,2,1)
+loglog(x_achse_f_spektrum_filter,spek_smooth_filter,'-','LineWidth',1)%% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing & Filtering
+% xlabel('f / Hz','interpreter','latex');
+% ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
+xlabel('$f\ (Hz)$','interpreter','latex');
+ylabel('$E\ (m^2/s)$', 'interpreter','latex')
+xlim([min(x_achse_f_spektrum_filter) 2*max(x_achse_f_spektrum_filter)])
+ylim([min(spek_smooth_filter) max(spek_smooth_filter)]*2)
+title('frequency domain','interpreter','latex')
+set(gca,'FontSize',12)
+set(groot, 'defaultAxesTickLabelInterpreter','latex');
+% plot([10^3 10^3],[10^-6 10^-2],'y = x ','-','LineWidth',1) 
+% h = vline(10^3,'k','The Answer')
+% text(245,spek_smooth(find(x_achse_f_spektrum>245,1,'first')),'\leftarrow sin(\pi) = 0','FontSize',16)
+set(gca,'XTick',sort([10^-2 10^0  10^2 10^4 low_freq]));
+hold on
+vline(low_freq,'k')
+
+txt = {'(a)'};
+a=text(4,0.5,txt);
+a.Units='normalized';
+
+subplot(2,2,2)
+loglog(x_achse_k_spektrum_2pi_filter,E_k,'-','LineWidth',1)%% Plot of Wavenumber(k) Vs. Energy(k)(m^3/sec^2) with smoothing & Filtering
+% loglog(x_achse_k_spektrum_2pi_filter,x_achse_k_spektrum_2pi_filter.^(-5/3).*(max(spek_smooth).*200),'--','color','k','LineWidth',2)
+% xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
+% ylabel('$E(k)=\frac{E(f) <u>}{2 \pi}$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
+xlabel('$k=\frac{2 \pi f}{<u>}\ (1/m)$','interpreter','latex');
+ylabel('$\frac{E <u>}{2 \pi}\ (m^3/s^2)$', 'interpreter','latex')
+xlim([min(x_achse_k_spektrum_2pi_filter) 2*max(x_achse_k_spektrum_2pi_filter)])
+ylim([min(E_k) max(E_k)]*2)
+title('wave number domain','interpreter','latex')
+set(gca,'FontSize',12)
+set(gca,'XTick',sort([10^-2 10^0  10^2 10^4 round(2*pi*low_freq/m_data)]));
+set(gca,'XTick',sort([10^-2 10^0  10^2 round(2*pi*low_freq/m_data)]));
+hold on
+vline(2*pi*low_freq/m_data,'k')
+txt = {'(b)'};
+b=text(4,0.5,txt);
+b.Units='normalized';
+
+subplot4=subplot(2,2,3);
+loglog(x_achse_spektrum_meter_filter,Dr_avg_filter,'-','LineWidth',1)%% Plot of Scales,r(m) Vs. Energy(r)(m/sec^2) with smoothing & Filtering
+% xlabel('$r=\frac{<u>}{2 \pi f} / m$', 'interpreter','latex');
+% ylabel('$E(r)=\frac{E(f) f^2}{<u>}$ / $\frac{m}{s^2}$', 'interpreter','latex')
+xlabel('$r=\frac{<u>}{2 \pi f}\ (m)$','interpreter','latex');
+ylabel('$\frac{E\ f^2}{<u>}\ (m/s^2)$', 'interpreter','latex')
+set(subplot4, 'Xdir', 'reverse')
+xlim([0.5* min(x_achse_spektrum_meter_filter) max(x_achse_spektrum_meter_filter)])
+ylim([min((spek_smooth_filter.*x_achse_f_spektrum_filter.^2)./m_data) max((spek_smooth_filter.*x_achse_f_spektrum_filter.^2)./m_data)]*2)
+set(gca,'FontSize',12)
+set(gca,'XTick',sort([10^-2 10^0  10^2 10^4 m_data/(2*pi*low_freq)]));
+title('dissipation spectrum','interpreter','latex')
+hold on
+vline(m_data/(2*pi*low_freq),'k')
+txt = {'(c)'};
+c=text(4,0.5,txt);
+c.Units='normalized';
+
+
+% Dissipation Spectrum
+% kin_vis = 15.328737178409*10^-6;
+subplot(2,2,4)
+loglog(x_achse_k_spektrum_2pi_filter,Dk_avg_filter,'-','LineWidth',1)%% Plot of Wavenumber(k) Vs. Dissipation(k)(m/sec^2) with smoothing & Filtering
+% xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
+% ylabel('E(k)$\cdot k^2$ / $\frac{m}{s^2}$', 'interpreter','latex')
+xlabel('$k=\frac{2 \pi f}{<u>}\ (1/m)$','interpreter','latex');
+ylabel('$\frac{E <u>}{2 \pi}\ k^2 \ (m/s^2)$', 'interpreter','latex')
+
+xlim([min(x_achse_k_spektrum_2pi_filter) 2*max(x_achse_k_spektrum_2pi_filter)])
+ylim([min(15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi_filter).^2) max(15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi_filter).^2)]*2)
+title('dissipation spectrum','interpreter','latex')
+set(gca,'FontSize',12)
+set(gca,'XTick',sort([10^-2 10^0  10^2 10^4 round(2*pi*low_freq/m_data)]));
+set(gca,'XTick',sort([10^-2 10^0  10^2 round(2*pi*low_freq/m_data)]));
+hold on
+vline(2*pi*low_freq/m_data,'k')
+txt = {'(d)'};
+d=text(4,0.5,txt);
+d.Units='normalized';
+
+% font=22;
+% a.FontSize = font;
+% b.FontSize = font;
+% c.FontSize = font;
+% d.FontSize = font;
+
+pos_txt=[-0.25   0.9];
+a.Position=pos_txt;
+b.Position=pos_txt;
+c.Position=pos_txt;
+d.Position=pos_txt;
+
+datacursormode on
+
+
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf); 
+delete(tmp_ui);
+
+if ischar(save_path)
+   % savefig(fullfile(save_path,append(save_name,'_','spectrum_var_domain.fig')))
+   savefig(h,fullfile(save_path,append(save_name,'_','spectrum_filter.fig')),'compact')
+%    for a = 1:length(h)
+       exportgraphics(h(2),fullfile(save_path,append(save_name,'_',sprintf('spectrum_filter_%d.png', 2))))
+%        export_fig(h(a), fullfile(save_path,append(save_name,'_',sprintf('spectrum_filter_%d.png', a))));
+%    end
+end
+close all
 end
\ No newline at end of file
diff --git a/Functions_v5-MATLAB/length_scales.m b/Functions_v5-MATLAB/length_scales.m
index 9e9cce9..427c457 100644
--- a/Functions_v5-MATLAB/length_scales.m
+++ b/Functions_v5-MATLAB/length_scales.m
@@ -1,1005 +1,1005 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
-% In this function the integral length scale $L$, Taylor length scale $\lambda$ and Kolmogorov 
-% length scale are estimated using different methods of calculation. Within this function, pop-up 
-% dialog boxes will be generated to enter the values of the integral, Taylor length scale in $m$ 
-% and energy dissipation rate in $m^2/s^3$ on which the further processing of data (solving the 
-% FPE and extracting cascade trajectories) should be referred. The entered length scales will be 
-% round towards the nearest integer sample. The proposed value in the pop-up dialog box is the 
-% median length scale for all methods.
-% COMMENT: All velocity increments in this script are taken as right hand increment i.e. 
-% dV=V(t2)-V(t1) where t2>t1; t1,t2 in seconds
-% 
-% Arguments IN
-% data = it is the filtered data 
-% Fs = Acquisition/Sampling Frequency in Hz
-% low_freq = Frequency at which you would like to use the low-pass filter in Hz
-% kin_vis = Kinematic viscosity of fluid in m^2/sec
-% C2 = Kolmogorov constant, C2 associated with second order structure function(in between 2.0 to 2.2)
-%
-% Arguments OUT
-% int_L = Integral length scale in meters
-% taylor_L = Taylors length scale in meters
-% diss_scale = Kolmogorv lenght scale in meters
-% int_L_calc = 1D array of integral length scale calculated by different methods/formula
-% taylor_L_calc = 1D array of Taylor's length scale calculated by different methods/formula
-% epsi = mean energy dissipation rate
-% epsi_calc = 1D array of mean energy dissipation rate calculated by different methods/formula
-% Ce = is the non-dimentional / normalized energy dissipation rate
-% Ce_calc = 1D array of Ce calculated by different methods/formula
-% Re = Reynolds Number  based on integral length scale
-% Re_lambda= Reynolds Number based on Taylor's length scale
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
-function [int_L,taylor_L,int_L_calc, taylor_L_calc,epsi, epsi_calc,diss_scale,Ce, Ce_calc, Re, Re_lambda]=length_scales(data,Fs,low_freq,kin_vis,C2,increment_bin,m_data,save_path,save_name)
-set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
-set(groot, 'defaulttextinterpreter','latex');   
-set(groot, 'defaultLegendInterpreter','latex');
-set(groot, 'defaultAxesTitle','latex');
-
-maxnumattempts      = 2;
-
-
-% Pressure calculation: 
-% http://www.peacesoftware.de/einigewerte/luft_e.html
-% https://www.einheiten-umrechnen.de/einheiten-rechner.php?typ=druck
-% The higher the pressure the lower the kinematic viscosity, 
-% the higher the temperature the higher the kinematic viscosity.
-% High Pressure: 1040hPa = 1,04 Bar, at 0 Celsius: 13.00*10^-6 at 20 Celsius: 14.73*10^-6
-%                1000hPa = 1 Bar,    at 0 Celsius: 13.52*10^-6 at 20 Celsius: 15.32*10^-6
-% Low pressure:  960hPa  = 0,96 Bar, at 0 Celsius: 17.24*10^-6 at 20 Celsius: 14.73*10^-6
-
-% SHREK: Helium 
-% http://hbcponline.com/faces/documents/06_45/06_45_0001.xhtml?search=true
-% 10^-6 *10*10^-4*3.259/0.1448402
-% kin_vis = 3.259*10^-6 / (0.1448402*10^3) % at 2 Kelvin
-% kin_vis = 1.468*10^-6 / (0.1456217*10^3) % at 2.5 Kelvin
-% kin_vis = 2.1*10^-8
-
-if mod(length(data),2)==1
-    data=data(1:end-1);% This is to make length of data array even
-end  
-
-% mean of the filtered data which is nothing but the mean of unfiltered data==>Filtering does not change the mean value
-% m_data          = mean(data); 
-% standard deviation of the filtered data which will always be lesser than the standard deviation of the unfiltered data
-uprime          = std(data); 
-% variance of the data
-uprimesquared   = uprime^2;
-
-% PSD of the fluctuations after filtering; mean value does not play a role in spectrum
-daten           = data-mean(data);
-L               = length(daten); %Length of the data
-
-% spek              = abs(fft(daten,L)).^2/L;           % Power spectral density(PSD)
-spek            = abs(fft(daten,L)).^2/(L*Fs);          % Energy spectral density(ESD) using a fft
-spek            = 2.*spek(2:L/2+1);                     % FFt will yield half number of unique points
-
-f = Fs/2*linspace(0,1,L/2+1).';                         % Nyquist frequency of the signal==Fs/2
-f = f(2:end);                                           % Remove zero Hz component
-
-
-% moving average with equally spaced frequency interval in log-space 
-plot_length         = increment_bin*10;
-if mod(plot_length,2) == 0
-    plot_length = plot_length +1;
-end
-
-intervall       = unique(round(logspace(0,log10(L/2),plot_length)),'stable');
-plot_length     = length(intervall);
-
-% Initializing the arrays/preallocation
-spek_smooth         = zeros(plot_length-2,1);
-x_achse_f_spektrum  = zeros(plot_length-2,1);
-
-% Averaging of spectrum and hence smoothing
-for i=2:(plot_length-1)
-    x_achse_f_spektrum(i-1,1) = mean(f(intervall(i-1):intervall(i+1)));
-    spek_smooth(i-1,1)        = mean(spek(intervall(i-1):intervall(i+1)));
-end
-% ------------------------Windowing / averaging end ------------------------  
-
-E_k                     = ((spek_smooth.*m_data)./(2*pi));
-% 2-pi - k Spektrum; Wavenumbers(1/m)
-x_achse_k_spektrum_2pi  = (2.*pi.*x_achse_f_spektrum)./m_data;
-
-
-
-% Wavenumber Spectrum (m^3/sec^2)
-k           = (2.*pi.*f(1:end))./m_data;
-Ek          = ((spek(1:end).*m_data)./(2.*pi));
-low_freq_k  = (2*pi*low_freq)/m_data;
-
-
-h(1) = figure;
-% Plot of Frequency Vs. Power Spectral Density(m^2/sec) without smoothing
-loglog(f,spek) 
-hold on
-% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing
-loglog(x_achse_f_spektrum,spek_smooth,'-','LineWidth',2)
-xlabel('f / Hz','interpreter','latex');
-ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
-xlabel('$f\ (Hz)$','interpreter','latex');
-ylabel('$E\ (m^2/s)$', 'interpreter','latex')
-
-% title('energy density spectrum','interpreter','latex')
-set(gca,'FontSize',10)
-set(gcf, 'Color', 'w')
-% xlim([min(x_achse_f_spektrum) max(x_achse_f_spektrum)]*2)
-xlim([min(x_achse_f_spektrum) 10^2])
-grid on
-axis square
-% datacursormode on
-fig_setup
-legend off
-grid off
-set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
-waitfor(msgbox('Use interactive data cursor to select the range of frequency which will be used to linearly extrapolate the value of ESD at a frequency of 0 Hz (see readme).'));
-% tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-% uiwait(gcf); 
-% delete(tmp_ui);
-
-
-
-%% First Method==> Calculating Ef0 with using Robust property
-% Calculate the integral length scale (macro scale) Roach Eqn 15
-% In this method we will ask you to choose two values of frequencies in Hz such as 
-% f_start & f_end ===> [f_start<f_end ]<== 
-% Must condition and so we will extrapolat a line designed bz f_start & f_end to find
-% energy content at the limit f-->0
-trying              = true;
-attemptcount        = 1;
-while trying
-    try
-        [I,yy]  = ginput(2);
-        if I(1)>I(2)
-            I   = flip(I);
-            yy  = flip(yy);
-        end     
-        tmp_ui = uicontrol('Position',[10 10 100 40],'String','Processing','Callback','uiresume(gcbf)');
-
-        f_start_plot     = askInput({sprintf('Lowest frequency of constant range in Hz:')}, {num2str(I(1),'%.8f')});
-        f_end_plot       = askInput({sprintf('Highest frequency of constant range in Hz:')}, {num2str(I(2),'%.8f')});
-        f_start     = find(f>f_start_plot, 1,'first');
-        f_end       = find(f<=f_end_plot, 1,'last');
-
-        % Ef0       = mean(spek(f_start:f_end));
-        % Ef0       = median(spek(f_start:f_end));
-
-        % IntegralLengthScale_1   = (Ef0*m_data)/(4*uprimesquared);           % Integral Scale in meters
-
-        %Second Method==> Calculating Ef0 from fit without using Robust property
-        [xData, yData] = prepareCurveData( f(f_start:f_end),(spek(f_start:f_end)));
-        
-        ft = fittype( 'poly1' );
-        opts = fitoptions( 'Method', 'LinearLeastSquares' );
-        % opts.Algorithm = 'Levenberg-Marquardt';
-        % opts.Display = 'Off';
-        opts.Lower = [0 0 ];
-        % opts.StartPoint = [0.5 0.1];
-        [fitresult, gof]        = fit( xData, yData, ft, opts );
-        IntegralLengthScale_2   = (fitresult.p2*m_data)/(4*uprimesquared);   % Integral Scale in meters
-        fitresult.p1=0;
-        plot(f,feval(fitresult,f),'k','LineWidth',2,'LineStyle','-')
-        vline(f_start_plot,'k')              % Construct a verticle line at f_start
-        vline(f_end_plot,'k')                % Construct a verticle line at f_end
-        legend('raw','averaged','Location','southwest')
-        
-        trying = false;
-    catch
-        waitfor(msgbox('Please try again: f_start should be smaller than f_end and the decimal separator should be "."(period).'));   
-        if attemptcount>maxnumattempts
-            waitfor(msgbox('Function failed.', 'Error','error'));   
-            error('Function failed after %d times',maxnumattempts)
-        end
-        attemptcount=attemptcount+1;
-    end  
-end       
-      
-        
-% %Third Method==> Calculating Ef0 from fit with using Robust property='LAR' specifies the least absolute residual method.
-% [xData, yData] = prepareCurveData( f(f_start:f_end),spek(f_start:f_end));
-% ft = fittype( 'a*x*0+b', 'independent', 'x', 'dependent', 'y' );
-% opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
-% opts.Algorithm = 'Levenberg-Marquardt';
-% opts.Display = 'Off';
-% opts.Robust = 'LAR';
-% opts.StartPoint         = [0.0 0.01];
-% [fitresult, gof]        = fit( xData, yData, ft, opts );
-% IntegralLengthScale_3   = (fitresult.b*m_data)/(4*uprimesquared);
-% 
-% %%Fourth Method==> Calculating Ef0 from fit with using Robust property='Bisquare' specifies the bisquare weights method.
-% [xData, yData] = prepareCurveData( f(f_start:f_end),spek(f_start:f_end));
-% ft = fittype( 'a*x*0+b', 'independent', 'x', 'dependent', 'y' );
-% opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
-% opts.Algorithm = 'Levenberg-Marquardt';
-% opts.Display = 'Off';
-% opts.Robust = 'Bisquare';
-% opts.StartPoint         = [0.0 0.01];
-% [fitresult, gof]        = fit( xData, yData, ft, opts );
-% IntegralLengthScale_4   = (fitresult.b*m_data)/(4*uprimesquared);
-
-
-
-%% Dissipation from different methods
-trying              = true;
-attemptcount        = 1;
-while trying
-    try
-        clear r tau step stepp 
-        r                   = unique(round(logspace(log10((1./Fs).*m_data*10^6),log10(20*IntegralLengthScale_2*10^6),increment_bin))./10^6,'stable');
-
-        tau(1:length(r))    = nan;
-        step(1:length(r))   = nan;
-
-        for z=1:length(r)
-            tau(z)          = r(z)/m_data;
-            step(z)         = round(tau(z)*Fs);  
-        end
-        stepp              = unique(step);
-        stepp              = stepp(stepp>0);
-
-        clear r R S_exp_3 S_exp_2
-        r(1:length(stepp)) = nan;
-        R(1:length(stepp)) = nan;
-        S_exp_3(1:length(stepp))=nan; % Third order structure function
-        S_exp_2(1:length(stepp))=nan; % Second order structure function
-
-        parfor z=1:length(stepp)
-        %     z/length(stepp)   
-            tau         = stepp(z)              % In steps, just a number
-            r(z)        = (tau./Fs).*m_data;    % in meters
-            incr        = (data(tau+1:L) - data(1:L-tau)).';   % velocity Increment in steps of tau
-            S_exp_2(z)  = mean(incr.^2);        % Second order structure function
-            S_exp_3(z)  = mean(incr.^3);        % Third order structure function   
-            % Autocorr André
-            R(1,z)      = mean(daten(tau+1:L).*daten(1:L-tau))/uprimesquared;
-        end
-
-        % HIT = Homogeneous Isotropic Turbulence
-        eps_2   = (1./(r)).*((S_exp_2./C2).^(3/2)); % Estimation of dissipation from S_exp_2 using HIT
-        eps_3   = -(5/4).*(S_exp_3./r);             % Estimation of dissipation from S_exp_3 using HIT 
-
-        I0      = find( eps_2==max(eps_2),1);       % Finding the maximum value of dissipation in inertial range
-        I1      = find(-eps_3==max(-eps_3),1);      % Finding the maximum value of dissipation in inertial range
-
-        % dissipation rate m^2.s^-3 from S2
-        epsi_S2 = mean(eps_2(1,I0-2:I0+2));         % finding the mean amongst 5 points closest to peak value;
-        % dissipation rate m^2.s^-3 from S3
-        epsi_S3 = mean(-eps_3(1,I1-2:I1+2));        % finding the mean amongst 5 points closest to peak value;
-        
-        trying = false;
-    catch
-%         waitfor(msgbox(' f_start should be smaller than f_end should be. ')); 
-        increment_bin = askInput({sprintf('Please try again by increasing the number of Bin''s used (must be larger than:):')}, {num2str(increment_bin,'%1.0f')});      
-        if attemptcount>maxnumattempts
-            waitfor(msgbox('Function failed.', 'Error','error'));   
-            error('Function failed after %d times',maxnumattempts)
-        end
-        attemptcount=attemptcount+1;
-    end  
-end       
-        
-delete(tmp_ui);
-
-
-% figure
-% loglog(r,S_exp_2)
-
-
-%% Integral length scale from Auto-corelation or structure function 
-%% Autocorr André
-low_freq_r  =  m_data/low_freq;  
-
-% Full integration 
-int_L_full  = trapz(r,R);
-% trapz(r,abs(R))
-
-% integrate up to the first zero crossing
-tmp_zero=(find(R<0,1,'first'))-1;
-if isempty(tmp_zero)
-   int_L_0     = nan;
-else
-    % r(tmp_zero)
-    int_L_0     = trapz(r(1:tmp_zero),R(1:tmp_zero));
-end
-
-% integrate up to the first 1/e crossing
-tmp_exp=(find(R<1/exp(1),1,'first'));
-if isempty(tmp_exp)
-    int_L_1_exp=nan;
-    theo_exp_Funktion = nan;
-else
-    int_L_1_exp = trapz(r(1:tmp_exp),R(1:tmp_exp));
-    % local minimum 
-    % index1 = find(diff(sign(diff(R))) ~= 0,2,'first')+1;
-    % r(index1)
-    % int_L_min=trapz(r(1:index1),abs(R(1:index1)));
-
-    %exp Fit to R; Fitregion r=1/exp(1):2*low_freq_r
-    tmp_up      = (find(r>4*low_freq_r,1,'first'));
-    tmp_low     = (find(R<1/exp(1),1,'first'));
-    
-    if tmp_up<tmp_low
-        [xData, yData] = prepareCurveData( r(tmp_up:tmp_low), R (tmp_up:tmp_low));
-        ft = fittype( 'exp1' );
-        % ft = fittype( 'exp(-x/b)', 'independent', 'x', 'dependent', 'y' )
-        % exp(-(x+a)/b)
-        opts                = fitoptions( 'Method', 'NonlinearLeastSquares' );
-        opts.Algorithm      = 'Levenberg-Marquardt';
-        opts.Display        = 'Off';
-        opts.Robust         = 'Bisquare';
-        [fitresult, gof]    = fit( xData, yData, ft, opts );
-        theo_exp_Funktion   = ((fitresult.a/-fitresult.b));
-        % theo_exp_Funktion=sum(R(1:tmp_low).*diff(r(1:tmp_low+1)));
-        % theo_exp_Funktion=(fitresult.a/-fitresult.b * exp(-fitresult.b*r(tmp_up)));
-        % theo_exp_Funktion=sum(R(1:tmp_up).*diff(r(1:tmp_up+1)))+(fitresult.a/-fitresult.b * exp(fitresult.b*r(tmp_up)));
-        % theo_exp_Funktion=sum(R(2:tmp_up).*diff(r(1:tmp_up)))+(fitresult.a/-fitresult.b * exp(fitresult.b*r(tmp_up)));
-        
-        % Plot the Auto-correlation coefficient Vs. Scale, r and exp fit
-        h(2) = figure;
-        if ~isempty(tmp_exp)
-            plot(r,fitresult.a*exp(fitresult.b*r),'-r','LineWidth',2)
-            % plot( fitresult, r, R )
-            hold on
-        end
-        plot(r,R,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-        ylabel('$R_{\widetilde{u}\widetilde{u}}$', 'interpreter','latex')
-        xlabel('r/m', 'interpreter','latex')
-        xlabel('$r\ (m)$','interpreter','latex');
-        % title(['integral length scale L = ' num2str(theo_exp_Funktion) 'm'])
-        % title(['first zero crossing = ' num2str(min(r(find(R<0)-1))) 'm'])
-        legend('exp fit', 'Location', 'NorthEast' );
-        axis square
-        set(gcf, 'Color', 'w')
-        set(gca, 'FontSize',18)
-        % xlim([0 150*low_freq_r])
-        if ~isempty(tmp_exp)
-            xlim([0 4*r(tmp_low)])
-            vline(r(tmp_low),'k')
-            vline(r(tmp_up),'k')
-        % vline(r(find(R<1/exp(1),1,'first')),'r')
-        end
-        fig_setup
-    else
-        theo_exp_Funktion=nan;  
-    end
-end
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf);
-delete(tmp_ui);
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Processing','Callback','uiresume(gcbf)');
-pause(1)
-
-
-
-%% Following calculation ensures that auto-correlation fluctuates around zero. For that you need to take the value of standard deviation of velocity calculated from the 2nd order structure function
-Ruu     = 1-(S_exp_2./(mean(S_exp_2((r>10*IntegralLengthScale_2)))));   % Auto-correlation coefficient Using std from S_exp_2 ==>This is a dimentionless quantity
-Ruu     = [1,Ruu];                                                      % Auto-correlation is 1 at scale=0
-r_Ruu   = [0,r];                                                        % Auto-correlation is 1 at scale=0
-LL      = (cumsum(Ruu(1,1:end-1).*diff(r_Ruu)))';                       % Cumulative integration of Ruu with respect to r
-pks     = findpeaks(LL,3);                                              % Finding the peaks in LL
-Npks    = size(pks,1);
-
-if Npks == 0 
-IntegralLengthScale_5 = trapz(r_Ruu,Ruu);       % If no peaks are found then simply take the area under the curve which will be in meters
-else  
-IntegralLengthScale_5 = pks(1);                 % in meters At this length we should reach an asymptotic value on the plot of r Vs. LL
-end
-
-lambda_S2   = (sqrt(15*kin_vis*uprimesquared./epsi_S2));    % Taylors length scale in meters from epsi_S2 and hence from S_exp_2
-lambda_S3   = (sqrt(15*kin_vis*uprimesquared./epsi_S3));    % Taylors length scale in meters from epsi_S3 and hence from S_exp_3
-
-
-
-%% Integral length scale from auto-corelation function 'xcorr' of MATLAB 
-% tau                     = 1/Fs;             % time in seconds/Time per sample in secs
-% [Ruu_xcorr,tau_xcorr]   = xcorr(data - mean(data),size(data,1)/2,'unbiased');
-% Ruu_xcorr               = Ruu_xcorr(tau_xcorr>=0);
-% tau_xcorr               = tau_xcorr(tau_xcorr>=0);
-% Ruu_xcorr               = Ruu_xcorr./max(Ruu_xcorr);
-% dist_xcorr              = tau_xcorr.*tau.*m_data;
-% Ruu_xcorr               = Ruu_xcorr';
-% LL_xcorr                = (cumsum(Ruu_xcorr(1,1:end-1).*diff(dist_xcorr(1,1:end))));
-% pks_xcorr               = findpeaks(LL_xcorr);
-% Npks_xcorr              = size(pks_xcorr,1);
-% 
-% if Npks_xcorr == 0 
-% IntegralLengthScale_6   = trapz(dist_xcorr,Ruu_xcorr);  % If no peaks are found then simply take the area under the curve
-% else  
-% IntegralLengthScale_6   = pks_xcorr(1);                 % in meters at this length we should reach an asymptotic value on plot of r Vs. LL
-% end
-% 
-% figure;
-% plot(r,R,'-k','LineWidth',2)
-% hold on
-% plot(r_Ruu,Ruu,'LineWidth',2)
-% plot(dist_xcorr,Ruu_xcorr,'LineWidth',2)
-% xlim([0 1500*low_freq_r])
-
-
-% figure;
-% plot(r_Ruu(1:end-1),LL,'LineWidth',2)
-% hold on
-% plot(dist_xcorr(1:end-1),LL_xcorr,'LineWidth',2)
-
-
-
-%% Zero-crossing method: Martin Obligado
-delete(tmp_ui);
-[int_L_zero_cros,lambda_zero_cros,lambda1_zero_cros,Eps_zero_cros,h]=ZeroCrossings(data,Fs,m_data,kin_vis,low_freq,x_achse_k_spektrum_2pi,E_k,h);
-
-%% 
-% int_L_calc  = [IntegralLengthScale_2,int_L_full,int_L_0,int_L_1_exp,theo_exp_Funktion,IntegralLengthScale_5,IntegralLengthScale_6,int_L_zero_cros];%% meter
-int_L_calc  = [IntegralLengthScale_2,int_L_full,int_L_0,int_L_1_exp,theo_exp_Funktion,IntegralLengthScale_5,int_L_zero_cros];%% meter
-
-% Choose the best estimate of integral and Taylor length scale on which you would like to rely on
-h(3) = figure;
-plot(int_L_calc,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-xlabel('method', 'interpreter','latex')
-ylabel('L/m', 'interpreter','latex')
-ylabel('$L\ (m)$','interpreter','latex');
-axis square
-set(gcf, 'Color', 'w')
-set(gca, 'FontSize',18)
-fig_setup
-legend off
-set(gca,'XTick',[1 2 3 4 5 6 7 8]);
-datacursormode on
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf);
-delete(tmp_ui);
-int_L    = askInput({sprintf('Enter the integral length scale in meter. Median integral length scale in meter:')}, {num2str(nanmedian(int_L_calc),'%.5f')});
-
-
-
-
-
-
-%% Taylor length by different methods
-%% using Autocorrelation function
-trying              = true;
-attemptcount        = 1;
-while trying
-    try
-        figure;
-        plot(r_Ruu,Ruu,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-        ylabel('$R_{\widetilde{u}\widetilde{u}}$', 'interpreter','latex')
-        xlabel('r/m', 'interpreter','latex')
-        xlabel('$r\ (m)$','interpreter','latex');
-        axis square
-        set(gcf, 'Color', 'w')
-        set(gca, 'FontSize',18)
-        ylim([0.7 1.05])
-        xlim([0 r_Ruu(find(Ruu<0.7,1,'first'))])
-        datacursormode on
-        waitfor(msgbox('Use interactive data cursor to select the range of scales which will be used to estimate Taylor lenght scalee (see readme).'));
-        [I,yy]  = ginput(1);
-        i_tmp = askInput({sprintf(['Fit Region Auto-correlation Coefficient e.g. = ' ,num2str(0.98,'%1.2f')])}, {num2str(yy(1),'%.2f')});
-        close
-
-     
-        [xData, yData] = prepareCurveData( r_Ruu(1:(find(R<i_tmp,1,'first'))), R (1:(find(R<i_tmp,1,'first'))));
-        ft = fittype( '1-(x^2/a^2)', 'independent', 'x', 'dependent', 'y' );
-        opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
-        opts.Lower = 0;
-        opts.StartPoint = 0.002;
-        [fitresult, gof] = fit( xData, yData, ft, opts );
-           
-        h(4) = figure;
-        plot(r_Ruu,feval(fitresult,r_Ruu),'r','LineWidth',2)
-        hold on
-        plot(r_Ruu,Ruu,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-        ylabel('$R_{\widetilde{u}\widetilde{u}}$', 'interpreter','latex')
-        xlabel('r/m', 'interpreter','latex')
-        xlabel('$r\ (m)$','interpreter','latex');
-        axis square
-        set(gcf, 'Color', 'w')
-        set(gca, 'FontSize',18)
-        ylim([0.7 1.05])
-        xlim([0 r_Ruu(find(Ruu<0.7,1,'first'))])
-        vline(r_Ruu((find(R<i_tmp,1,'first'))),'k')
-        lambda_autocorr=fitresult.a;
-        % legend('','$R_{\widetilde{u}\widetilde{u}}(r)=1-\frac{r^2}{\lambda^2}$', 'Location', 'NorthEast' );
-        legend('$R_{\widetilde{u}\widetilde{u}}(r)=1-\frac{r^2}{\lambda^2}$', 'Location', 'NorthEast' );
-        title(['$\lambda$ = ' num2str(lambda_autocorr,'%1.4f') ' m'],'interpreter','latex')
-        fig_setup
-               
-        trying = false;
-    catch
-        waitfor(msgbox('Please try again'));   
-%         askInput({sprintf('Please try again by increasing the number of Bin''s used:')}, {num2str(increment_bin,'%1.0f')});
-        if attemptcount>maxnumattempts
-            waitfor(msgbox('Function failed.', 'Error','error'));   
-            error('Function failed after %d times',maxnumattempts)
-        end
-        attemptcount=attemptcount+1;
-    end  
-end       
-
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf);
-delete(tmp_ui); 
-
-
-
-
-%% numerical differentiation
-%<(du/dx)^2>
-du_dx           = mean((diff(daten)./(m_data*(1/Fs))).^2);
-lambda_direct   = sqrt(uprimesquared/du_dx);
-
-% Aronson and Löfdahl
-tmp = sqrt((uprimesquared.*r.^2)./S_exp_2);
-h(5) = figure;
-plot(r,tmp,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-hold on
-xlabel('r/m', 'interpreter','latex')
-xlabel('$r\ (m)$','interpreter','latex');
-ylabel('$\sqrt{\frac{u''^2 r^2}{S^{2}(r)}}/m$', 'interpreter','latex')
-ylabel('$\sqrt{\frac{u''^2 r^2}{S^{2}}}\ (m)$', 'interpreter','latex')
-axis square
-set(gcf, 'Color', 'w')
-set(gca, 'FontSize',18)
-xlim([0 25*low_freq_r])
-datacursormode on
-waitfor(msgbox('Use interactive data cursor to select the range of scales which will be used to estimate Taylor lenght scalee (see readme).'));
-
-tmp_up      = (find(r>4*low_freq_r,1,'first'));
-vline(r(tmp_up),'k')
-
-trying              = true;
-attemptcount        = 1;
-while trying
-    try
-        [I,yy]  = ginput(1);
-        % i_tmp       = str2double(inputdlg({'Fit Region rmin ='}));
-        % tmp_up      = (find(r>i_tmp,1,'first'));
-        i_tmp       = askInput({sprintf(['Fit Region r_max e.g. = ' ,num2str(20*low_freq_r,'%0.2f')])}, {num2str(I(1),'%.2f')});
-        tmp_low     = (find(r<i_tmp,1,'last'));
-        [xData, yData] = prepareCurveData(r(tmp_up:tmp_low),tmp(tmp_up:tmp_low));
-        ft = fittype( 'poly1' );
-        [fitresult, gof] = fit( xData, yData, ft );
-        plot(linspace(0,1,1001),feval(fitresult,linspace(0,1,1001)),'r','LineWidth',2)
-        fitresult.p1=0;
-        plot(linspace(0,1,1001),feval(fitresult,linspace(0,1,1001)),'k','LineWidth',2,'LineStyle','-')
-        vline(r(tmp_low),'k')
-        A_L_lambda =   fitresult.p2;
-        fig_setup
-        legend off
-        
-        trying = false;
-    catch
-        waitfor(msgbox('Please try again'));   
-        if attemptcount>maxnumattempts
-            waitfor(msgbox('Function failed.', 'Error','error'));   
-            error('Function failed after %d times',maxnumattempts)
-        end
-        attemptcount=attemptcount+1;
-    end  
-end               
-        
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf);
-delete(tmp_ui);
-
-
-% Dissipationspectrum
-k_end                   = find(k<=low_freq_k, 1,'last');
-lambda_spec_low_freq    = sqrt(trapz(k(1:k_end),Ek(1:k_end))/trapz(k(1:k_end),Ek(1:k_end).*k(1:k_end).^2));
-lambda_spec_full        = sqrt(uprimesquared/trapz(k,Ek.*k.^2));
-% trapz(k,2.*Ek.*k.^2) anstatt trapz(k,Ek.*k.^2)
-% lambda_1=sqrt(uprimesquared/trapz(k,2.*Ek.*k.^2))
-% lambda_2=sqrt(trapz(k,2.*Ek)/trapz(k,2.*Ek.*k.^2))
-
-% take everthing of the specturm
-% lambda_1=sqrt(uprimesquared/trapz(k,Ek.*k.^2));
-% lambda_1=sqrt(uprimesquared/trapz(x_achse_k_spektrum_2pi(2:end),E_k.*x_achse_k_spektrum_2pi(2:end).^2))
-% lambda_2=sqrt(trapz(k,Ek)/trapz(x_achse_k_spektrum_2pi(2:end),E_k.*x_achse_k_spektrum_2pi(2:end).^2))
-
-% find Minimum at high frequencies
-% k_start           = find(k>0.2*10^4, 1,'first');
-% k_end             = find(k<=5*10^4, 1,'last');
-% lambda_1_kurz     = sqrt(uprimesquared/trapz(k(1:find(spek==min(spek(k_start:k_end)))),Ek(1:find(spek==min(spek(k_start:k_end)))).*k(1:find(spek==min(spek(k_start:k_end)))).^2));
-
-% lambda_4  = sqrt(uprimesquared/du_dx_4);
-% lambda_6    = sqrt((10*kin_vis/m_data) * (uprimesquared/(uprimesquared/(m_data*(1/Fs)))));
-
-% taylor_L_calc = [lambda_autocorr,lambda_direct,A_L_lambda,lambda_spec_low_freq,lambda_spec_full, lambda_S2, lambda_S3,lambda_zero_cros,lambda1_zero_cros];%% meter
-taylor_L_calc = [lambda_autocorr,lambda_direct,A_L_lambda,lambda_spec_low_freq,lambda_spec_full,lambda_zero_cros,lambda1_zero_cros];%% meter
-
-
-h(6) = figure;
-plot(taylor_L_calc,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-xlabel('method', 'interpreter','latex')
-ylabel('$\lambda/m$', 'interpreter','latex')
-ylabel('$\lambda\ (m)$', 'interpreter','latex')
-axis square
-set(gcf, 'Color', 'w')
-set(gca, 'FontSize',18)
-fig_setup
-legend off
-set(gca,'XTick',[1 2 3 4 5 6 7]);
-datacursormode on
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf);
-delete(tmp_ui);
-taylor_L = askInput({sprintf('Enter the Taylor length scale in meter. Median Taylor length scale in meter:')}, {num2str(nanmedian(taylor_L_calc),'%.5f')});
-
-
-
-%% Dissipation from different methods
-% epsi_1  = 15*kin_vis*trapz(k(1:k_end),Ek(1:k_end).*k(1:k_end).^2); %% From dissipation Spectrum untill cutoff frequency
-% epsi_2  = 15*kin_vis*trapz(k,Ek.*k.^2); %%From entire dissipation Spectrum
-
-% epsi_2  = trapz(k,2.*kin_vis.*Ek.*k.^2);
-% epsi_3  = 15*kin_vis*du_dx_3;
-% epsi_4  = 15*kin_vis*du_dx_4;
-% epsi_5  = 15*kin_vis*du_dx_5;
-
-% epsi_5 = (m_data/2) * (mean(diff(daten.^2)/(m_data*(1/Fs))));
-% epsi_6 = (3*m_data/2) * (var(diff(daten)./(m_data*(1/Fs))));
-% epsi_7 = 15*kin_vis*(uprimesquared/lambda_1^2); 
-% epsi_7a= 30*kin_vis*(uprimesquared/lambda_1^2) %longitudinale microscale
-% epsi_7b= 15*kin_vis*(uprimesquared/(lambda_1/sqrt(2))^2) %trasnversale microscale
-% epsi_8 = m_data^3/int_L
-
-% epsi_1/m_data
-% var(daten)/(m_data*(1/Fs))
-% std(daten)/(m_data*(1/Fs))
-% epsi_calc   = [epsi_1, epsi_2, epsi_3, epsi_5, epsi_S2,epsi_S3];
-
-
-h(7) = figure;
-semilogx(r,eps_2,'LineWidth',2)
-hold on
-semilogx(r,-eps_3,'LineWidth',2)
-xlim([min(r)*0.5 max(r)*2])
-xlabel('r/m', 'interpreter','latex')
-xlabel('$r\ (m)$','interpreter','latex');
-ylabel('$\epsilon(r)/\frac{m^2}{s^3}$', 'interpreter','latex')
-ylabel('$\epsilon\ (m^2/s^3)$','interpreter','latex');
-set(gca,'FontSize',16)
-set(gcf, 'Color', 'w')
-fig_setup
-axis square
-% vline(r(I0),'r')                          
-% vline(r(I1),'k')                          
-
-plot(linspace(min(r)*0.5,max(r)*2,1000),eps_2(I0).*ones(1000,1),'k','LineWidth',2)
-plot(linspace(min(r)*0.5,max(r)*2,1000),-eps_3(I1).*ones(1000,1),'k','LineWidth',2)
-legend('$\frac{1}{r}\left[\frac{S^{2}(r)}{C_{2}}\right]^{3/2}$','$-\frac{5}{4}\left[\frac{S^{3}(r)}{r}\right]$','Location','south')
-
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf);
-delete(tmp_ui);
-
-etaflow_S2  = ((kin_vis^3/epsi_S2)^(1/4));                  % Kolmogorov length scale in meters from epsi_S2 and hence from S_exp_2
-etaflow_S3  = ((kin_vis^3/epsi_S3)^(1/4));                  % Kolmogorov length scale in meters from epsi_S3 and hence from S_exp_3
-
-
-
-%% Spectrum-Fit
-epsi_calc   = [epsi_S2,epsi_S3,(15*kin_vis*(uprimesquared/taylor_L^2))];
-
-h(8) = figure;
-loglog(x_achse_k_spektrum_2pi,E_k,'-','LineWidth',2)
-hold on
-a=find(x_achse_k_spektrum_2pi<=(2*pi*((1/(int_L/m_data))/(2*pi))/m_data), 1,'last');
-b=find(x_achse_k_spektrum_2pi>=(2*pi*((1/(taylor_L/m_data))/(2*pi))/m_data), 1,'first');
-
-% b=find(x_achse_k_spektrum_2pi>=(1)/diss_scale, 1,'first'); 
-% a=1;
-% b=length(x_achse_k_spektrum_2pi);
-[xData, yData] = prepareCurveData( x_achse_k_spektrum_2pi(a:b), E_k(a:b) );
-% [xData, yData] = prepareCurveData( x_achse_f_spektrum,spek_smooth);
-% [xData, yData] = prepareCurveData(k, Ek );
-
-opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
-opts.DiffMinChange = 1e-12;
-opts.Display = 'Off';
-opts.MaxFunEvals = 1200;
-opts.MaxIter = 1200;
-
-ft = fittype( 'a*b^(2/3)*(x-d)^(-5/3)', 'independent', 'x', 'dependent', 'y' );
-% opts.Lower = [0 median(epsi_calc) -Inf];
-% opts.Upper = [Inf median(epsi_calc) Inf];
-opts.Lower = [0 0 -Inf];
-opts.Upper = [Inf Inf Inf];
-opts.StartPoint = [0.5 nanmedian(epsi_calc) 0];
-try 
-    [fitresult, gof] = fit( xData, yData, ft, opts );
-    ft2 = fittype( 'a*b^(2/3)*(x)^(-5/3)', 'independent', 'x', 'dependent', 'y' );
-    % opts.Lower = [0 nanmedian(epsi_calc)];
-    % opts.Upper = [Inf nanmedian(epsi_calc)];
-    opts.Lower = [0 0];
-    opts.Upper = [Inf Inf];
-    opts.StartPoint = [0.5 nanmedian(epsi_calc)];
-    [fitresult2, gof] = fit( xData, yData, ft2, opts );
-    plot(x_achse_k_spektrum_2pi, feval(fitresult,x_achse_k_spektrum_2pi),'-','LineWidth',2)
-    plot(x_achse_k_spektrum_2pi, feval(fitresult2,x_achse_k_spektrum_2pi),'-','LineWidth',2)
-catch 
-end
-set(gca,'XScale','log');
-set(gca,'YScale','log');
-xlabel('$k= / \frac{1}{m}$', 'interpreter','latex');
-ylabel('$E(k)$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
-xlabel('$k\ (1/m)$','interpreter','latex');
-ylabel('$\frac{E <u>}{2 \pi}\ (m^3/s^2)$', 'interpreter','latex')
-
-xlim([min(x_achse_k_spektrum_2pi) 2*max(x_achse_k_spektrum_2pi)])
-ylim([min(E_k) 2*max(E_k)])
-axis square
-% vline(x_achse_k_spektrum_2pi(a),'k')
-% vline(x_achse_k_spektrum_2pi(b),'k')
-% vline((1)/diss_scale,'k')
-set(gcf, 'Color', 'w')
-legend('$\frac{E <u>}{2 \pi}$','$C_k\langle \epsilon \rangle^{2/3}(k-k_0)^{-5/3}$','$C_k\langle \epsilon \rangle^{2/3}k^{-5/3}$','Location','south')
-fig_setup
-set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
-% fitresult;
-% fitresult2;
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf);
-delete(tmp_ui);
-
-% figure
-% plot( fitresult, xData, yData );
-% hold on
-% plot( fitresult2, xData, yData );
-% plot( fitresult, xData, yData );
-% loglog(x_achse_k_spektrum_2pi,E_k)
-% set(gca,'XScale','log');
-% set(gca,'YScale','log');
-
-
-epsi_calc   = [epsi_S2,epsi_S3, (15*kin_vis*(uprimesquared/taylor_L^2)), fitresult2.b,fitresult.b, Eps_zero_cros];
-
-h(9) = figure;
-plot(epsi_calc,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-xlabel('method', 'interpreter','latex')
-ylabel('$\langle \epsilon \rangle/\frac{m^2}{s^3}$', 'interpreter','latex')
-ylabel('$\langle \epsilon \rangle\ (m^2/s^3)$', 'interpreter','latex')
-axis square
-set(gcf, 'Color', 'w')
-set(gca, 'FontSize',18)
-fig_setup
-legend off
-set(gca,'XTick',[1 2 3 4 5 6 7 ]);
-datacursormode on
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf);
-delete(tmp_ui);
-epsi = askInput({sprintf('Enter the energy dissipation rate in $m^2/s^3$. Median energy dissipation rate in $m^2/s^3$:')}, {num2str(nanmedian(epsi_calc),'%.5f')});
-
-
-
-
-Re          = m_data*int_L/kin_vis;
-% Re_lambda   = uprime*lambda_1_kurz_low_freq/kin_vis;
-Re_lambda   = uprime*taylor_L/kin_vis; 
-% Re_lambda = uprime*lambda_1/kin_vis;
-
-% Calculate the dissipation length scale (micro scale) Roach Eqn 8
-% Z=((2*pi^2)/(m_data^2*uprimesquared))*trapz((f(2:end).^2),spek(2:L/2+1)); % Use trapezoid rule for integration
-% DissipationScale=sqrt(1/Z)
-
-DissipationScale_1      = ((kin_vis^3)/epsi)^0.25;
-% DissipationScale_2    = sqrt(1/(trapz(f,spek.*f.^2)*((2*pi^2)/(m_data^2*uprimesquared))))
-% DissipationScale_3      = int_L/(Re^0.75);
-
-% kolmogorov_calc         = [DissipationScale_1, etaflow_S2,etaflow_S3];%% meter
-diss_scale         = DissipationScale_1;
-
-%%Non dimentional dissipation constant
-Ce      = 15*(int_L/taylor_L)/Re_lambda;
-Ce_b    = epsi*int_L/uprime^3;
-
-% Ce      = 15*(int_L/lambda_1_kurz_low_freq)/Re_lambda;
-% Ce_1    = epsi_1*int_L/uprime^3;
-% Ce_2    = epsi_2*int_L/uprime^3;
-% Ce_3    = epsi_3*int_L/uprime^3;
-% Ce_5    = epsi_5*int_L/uprime^3;
-
-% Ce_calc =[Ce, Ce_1, Ce_2, Ce_3, Ce_5];
-Ce_calc = [Ce, Ce_b];
-Ce      = mean(Ce_calc);
-
-
-%% Plot Spectrum
-h(10) = figure;
-set(gcf, 'Color', 'w')
-yyaxis left
-loglog(x_achse_k_spektrum_2pi,E_k,'-','LineWidth',2)
-% hold on
-% loglog(x_achse_k_spektrum_2pi,(x_achse_k_spektrum_2pi.^(-5/3).*ones(length(E_k)).*10),'-','LineWidth',1,'color','k')
-% loglog(x_achse_k_spektrum_2pi,(x_achse_k_spektrum_2pi.^(-2).*ones(length(E_k)).*0.7.*10^2),'-','LineWidth',1,'color','k')
-% loglog(x_achse_k_spektrum_2pi,(x_achse_k_spektrum_2pi.^(-3).*ones(length(E_k)).*0.6.*10^5),'-','LineWidth',1,'color','k')
-% xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
-xlabel('$k= / \frac{1}{m}$', 'interpreter','latex');
-% ylabel('$E(k)=\frac{E(f) <u>}{2 \pi}$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
-ylabel('$E(k)$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
-%xlabel('k');
-%ylabel('E(k)')
-xlabel('$k\ (1/m)$','interpreter','latex');
-ylabel('$\frac{E <u>}{2 \pi}\ (m^3/s^2)$', 'interpreter','latex')
-xlim([min(x_achse_k_spektrum_2pi) 2*max(x_achse_k_spektrum_2pi)])
-% ylim([10^-10 1])
-axis square
-hold on
-yyaxis right
-% loglog(x_achse_k_spektrum_2pi,15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi).^2,'-','LineWidth',2)
-loglog(x_achse_k_spektrum_2pi,E_k.*(x_achse_k_spektrum_2pi).^2,'-','LineWidth',2)
-ylabel('E(k)$\cdot k^2$ / $\frac{m}{s^2}$', 'interpreter','latex')
-ylabel('$\frac{E <u>}{2 \pi} k^2\ (m/s^2)$', 'interpreter','latex')
-
-set(gca,'YScale','log');
-set(gca,'FontSize',24)
-% vline(low_freq_k,'r')
-% vline((1/(int_L/m_data))/m_data,'k')
-vline((2*pi*((1/(int_L/m_data))/(2*pi))/m_data),'k')
-vline((2*pi*((1/(taylor_L/m_data))/(2*pi))/m_data),'k')
-vline((2*pi*((1/(diss_scale/m_data))/(2*pi))/m_data),'k')
-fig_setup
-set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
-legend off
-
-
-
-h(11) = figure;
-set(gcf, 'Color', 'w')
-yyaxis left
-% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing
-loglog(x_achse_f_spektrum,spek_smooth,'-','LineWidth',2)
-xlabel('f / Hz','interpreter','latex');
-ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
-xlabel('$f\ (Hz)$','interpreter','latex');
-ylabel('$E\ (m^2/s)$', 'interpreter','latex')
-set(gca,'FontSize',10)
-set(gcf, 'Color', 'w')
-xlim([min(x_achse_f_spektrum) max(x_achse_f_spektrum)]*2)
-% xlim([min(x_achse_f_spektrum) 10^2])
-grid on
-axis square
-datacursormode on
-fig_setup
-grid off
-legend off
-set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
-vline((1/(int_L/m_data))/(2*pi),'k')
-vline((1/(taylor_L/m_data))/(2*pi),'k')
-vline((1/(diss_scale/m_data))/(2*pi),'k')
-
-yyaxis right
-% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing
-loglog(x_achse_f_spektrum,spek_smooth.*x_achse_f_spektrum.^(5/3),'-','LineWidth',2)
-ylabel('$E(f)f^{5/3}$ / $\frac{m^2}{s}$', 'interpreter','latex')
-ylabel('$E\ f^{5/3}\ (m^2/s)$', 'interpreter','latex')
-xlim([min(x_achse_f_spektrum) max(x_achse_f_spektrum)]*2)
-ylim([min(spek_smooth.*x_achse_f_spektrum.^(5/3)) max(spek_smooth.*x_achse_f_spektrum.^(5/3))]*2)
-
-
-
-
-% k Spektrum; Scales,r (m)
-x_achse_spektrum_meter  = m_data./(2*pi*x_achse_f_spektrum);
-Dr_avg_filter=(spek_smooth.*x_achse_f_spektrum.^2)./m_data;
-Dk_avg_filter=15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi).^2;
-
-
-
-figure;
-set(gcf, 'Color', 'w')
-subplot(2,2,1)
-loglog(x_achse_f_spektrum,spek_smooth,'-','LineWidth',1)%% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing & Filtering
-xlabel('f / Hz','interpreter','latex');
-ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
-xlabel('$f\ (Hz)$','interpreter','latex');
-ylabel('$E\ (m^2/s)$', 'interpreter','latex')
-xlim([min(x_achse_f_spektrum) 2*max(x_achse_f_spektrum)])
-ylim([min(spek_smooth) max(spek_smooth)]*2)
-title('frequency domain','interpreter','latex')
-set(gca,'FontSize',12)
-set(groot, 'defaultAxesTickLabelInterpreter','latex');
-% plot([10^3 10^3],[10^-6 10^-2],'y = x ','-','LineWidth',1) 
-% h = vline(10^3,'k','The Answer')
-% text(245,spek_smooth(find(x_achse_f_spektrum>245,1,'first')),'\leftarrow sin(\pi) = 0','FontSize',16)
-set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
-hold on
-vline((1/(int_L/m_data))/(2*pi),'k')
-vline((1/(taylor_L/m_data))/(2*pi),'k')
-vline((1/(diss_scale/m_data))/(2*pi),'k')
-txt = {'(a)'};
-a=text(4,0.5,txt);
-a.Units='normalized';
-
-subplot(2,2,2)
-loglog(x_achse_k_spektrum_2pi,E_k,'-','LineWidth',1)%% Plot of Wavenumber(k) Vs. Energy(k)(m^3/sec^2) with smoothing & Filtering
-% loglog(x_achse_k_spektrum_2pi,x_achse_k_spektrum_2pi.^(-5/3).*(max(spek_smooth).*200),'--','color','k','LineWidth',2)
-xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
-ylabel('$E(k)=\frac{E(f) <u>}{2 \pi}$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
-xlabel('$k=\frac{2 \pi f}{<u>}\ (1/m)$','interpreter','latex');
-ylabel('$\frac{E <u>}{2 \pi}\ (m^3/s^2)$', 'interpreter','latex')
-xlim([min(x_achse_k_spektrum_2pi) 2*max(x_achse_k_spektrum_2pi)])
-ylim([min(E_k) max(E_k)]*2)
-title('wave number domain','interpreter','latex')
-set(gca,'FontSize',12)
-set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
-hold on
-vline((2*pi*((1/(int_L/m_data))/(2*pi))/m_data),'k')
-vline((2*pi*((1/(taylor_L/m_data))/(2*pi))/m_data),'k')
-vline((2*pi*((1/(diss_scale/m_data))/(2*pi))/m_data),'k')
-txt = {'(b)'};
-b=text(4,0.5,txt);
-b.Units='normalized';
-
-subplot4=subplot(2,2,3);
-loglog(x_achse_spektrum_meter,Dr_avg_filter,'-','LineWidth',1)%% Plot of Scales,r(m) Vs. Energy(r)(m/sec^2) with smoothing & Filtering
-xlabel('$r=\frac{<u>}{2 \pi f} / m$', 'interpreter','latex');
-ylabel('$E(r)=\frac{E(f) f^2}{<u>}$ / $\frac{m}{s^2}$', 'interpreter','latex')
-xlabel('$r=\frac{<u>}{2 \pi f}\ (m)$','interpreter','latex');
-ylabel('$\frac{E\ f^2}{<u>}\ (m/s^2)$', 'interpreter','latex')
-set(subplot4, 'Xdir', 'reverse')
-xlim([0.5* min(x_achse_spektrum_meter) max(x_achse_spektrum_meter)])
-ylim([min((spek_smooth.*x_achse_f_spektrum.^2)./m_data) max((spek_smooth.*x_achse_f_spektrum.^2)./m_data)]*2)
-set(gca,'FontSize',12)
-set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
-title('dissipation spectrum','interpreter','latex')
-vline(int_L,'k')
-vline(taylor_L,'k')
-vline(diss_scale,'k')
-txt = {'(c)'};
-c=text(4,0.5,txt);
-c.Units='normalized';
-
-
-
-% Dissipation Spectrum
-% kin_vis = 15.328737178409*10^-6;
-subplot(2,2,4)
-loglog(x_achse_k_spektrum_2pi,Dk_avg_filter,'-','LineWidth',1)%% Plot of Wavenumber(k) Vs. Dissipation(k)(m/sec^2) with smoothing & Filtering
-xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
-ylabel('E(k)$\cdot k^2$ / $\frac{m}{s^2}$', 'interpreter','latex')
-xlabel('$k=\frac{2 \pi f}{<u>}\ (1/m)$','interpreter','latex');
-ylabel('$\frac{E <u>}{2 \pi}\ k^2 \ (m/s^2)$', 'interpreter','latex')
-xlim([min(x_achse_k_spektrum_2pi) 2*max(x_achse_k_spektrum_2pi)])
-ylim([min(15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi).^2) max(15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi).^2)]*2)
-title('dissipation spectrum','interpreter','latex')
-set(gca,'FontSize',12)
-set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
-datacursormode on
-hold on
-vline((2*pi*((1/(int_L/m_data))/(2*pi))/m_data),'k')
-vline((2*pi*((1/(taylor_L/m_data))/(2*pi))/m_data),'k')
-vline((2*pi*((1/(diss_scale/m_data))/(2*pi))/m_data),'k')
-txt = {'(d)'};
-d=text(4,0.5,txt);
-d.Units='normalized';
-
-% font=22;
-% a.FontSize = font;
-% b.FontSize = font;
-% c.FontSize = font;
-% d.FontSize = font;
-
-pos_txt=[-0.25   0.9];
-a.Position=pos_txt;
-b.Position=pos_txt;
-c.Position=pos_txt;
-d.Position=pos_txt;
-
-
-
-
-
-
-
-
-if ischar(save_path)
-    savefig(h,fullfile(save_path,append(save_name,'_','length_scales.fig')),'compact')
-    for a = 1:length(h)
-       exportgraphics(h(a),fullfile(save_path,append(save_name,'_',sprintf('length_scales_%d.png', a))))
-    end
-end
-close all
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+% In this function the integral length scale $L$, Taylor length scale $\lambda$ and Kolmogorov 
+% length scale are estimated using different methods of calculation. Within this function, pop-up 
+% dialog boxes will be generated to enter the values of the integral, Taylor length scale in $m$ 
+% and energy dissipation rate in $m^2/s^3$ on which the further processing of data (solving the 
+% FPE and extracting cascade trajectories) should be referred. The entered length scales will be 
+% round towards the nearest integer sample. The proposed value in the pop-up dialog box is the 
+% median length scale for all methods.
+% COMMENT: All velocity increments in this script are taken as right hand increment i.e. 
+% dV=V(t2)-V(t1) where t2>t1; t1,t2 in seconds
+% 
+% Arguments IN
+% data = it is the filtered data 
+% Fs = Acquisition/Sampling Frequency in Hz
+% low_freq = Frequency at which you would like to use the low-pass filter in Hz
+% kin_vis = Kinematic viscosity of fluid in m^2/sec
+% C2 = constant, C2 associated with second order structure function(in between 2.0 to 2.2)
+%
+% Arguments OUT
+% int_L = Integral length scale in meters
+% taylor_L = Taylors length scale in meters
+% int_L_calc = 1D array of integral length scale calculated by different methods/formula
+% taylor_L_calc = 1D array of Taylor's length scale calculated by different methods/formula
+% epsi = mean energy dissipation rate
+% epsi_calc = 1D array of mean energy dissipation rate calculated by different methods/formula
+% diss_scale = Kolmogorv lenght scale in meters
+% Ce = is the non-dimentional / normalized energy dissipation rate
+% Ce_calc = 1D array of Ce calculated by different methods/formula
+% Re = Reynolds Number  based on integral length scale
+% Re_lambda= Reynolds Number based on Taylor's length scale
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+function [int_L,taylor_L,int_L_calc, taylor_L_calc,epsi, epsi_calc,diss_scale,Ce, Ce_calc, Re, Re_lambda]=length_scales(data,Fs,low_freq,kin_vis,C2,increment_bin,m_data,save_path,save_name)
+set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
+set(groot, 'defaulttextinterpreter','latex');   
+set(groot, 'defaultLegendInterpreter','latex');
+set(groot, 'defaultAxesTitle','latex');
+
+maxnumattempts      = 2;
+
+
+% Pressure calculation: 
+% http://www.peacesoftware.de/einigewerte/luft_e.html
+% https://www.einheiten-umrechnen.de/einheiten-rechner.php?typ=druck
+% The higher the pressure the lower the kinematic viscosity, 
+% the higher the temperature the higher the kinematic viscosity.
+% High Pressure: 1040hPa = 1,04 Bar, at 0 Celsius: 13.00*10^-6 at 20 Celsius: 14.73*10^-6
+%                1000hPa = 1 Bar,    at 0 Celsius: 13.52*10^-6 at 20 Celsius: 15.32*10^-6
+% Low pressure:  960hPa  = 0,96 Bar, at 0 Celsius: 17.24*10^-6 at 20 Celsius: 14.73*10^-6
+
+% SHREK: Helium 
+% http://hbcponline.com/faces/documents/06_45/06_45_0001.xhtml?search=true
+% 10^-6 *10*10^-4*3.259/0.1448402
+% kin_vis = 3.259*10^-6 / (0.1448402*10^3) % at 2 Kelvin
+% kin_vis = 1.468*10^-6 / (0.1456217*10^3) % at 2.5 Kelvin
+% kin_vis = 2.1*10^-8
+
+if mod(length(data),2)==1
+    data=data(1:end-1);% This is to make length of data array even
+end  
+
+% mean of the filtered data which is nothing but the mean of unfiltered data==>Filtering does not change the mean value
+% m_data          = mean(data); 
+% standard deviation of the filtered data which will always be lesser than the standard deviation of the unfiltered data
+uprime          = std(data); 
+% variance of the data
+uprimesquared   = uprime^2;
+
+% PSD of the fluctuations after filtering; mean value does not play a role in spectrum
+daten           = data-mean(data);
+L               = length(daten); %Length of the data
+
+% spek              = abs(fft(daten,L)).^2/L;           % Power spectral density(PSD)
+spek            = abs(fft(daten,L)).^2/(L*Fs);          % Energy spectral density(ESD) using a fft
+spek            = 2.*spek(2:L/2+1);                     % FFt will yield half number of unique points
+
+f = Fs/2*linspace(0,1,L/2+1).';                         % Nyquist frequency of the signal==Fs/2
+f = f(2:end);                                           % Remove zero Hz component
+
+
+% moving average with equally spaced frequency interval in log-space 
+plot_length         = increment_bin*10;
+if mod(plot_length,2) == 0
+    plot_length = plot_length +1;
+end
+
+intervall       = unique(round(logspace(0,log10(L/2),plot_length)),'stable');
+plot_length     = length(intervall);
+
+% Initializing the arrays/preallocation
+spek_smooth         = zeros(plot_length-2,1);
+x_achse_f_spektrum  = zeros(plot_length-2,1);
+
+% Averaging of spectrum and hence smoothing
+for i=2:(plot_length-1)
+    x_achse_f_spektrum(i-1,1) = mean(f(intervall(i-1):intervall(i+1)));
+    spek_smooth(i-1,1)        = mean(spek(intervall(i-1):intervall(i+1)));
+end
+% ------------------------Windowing / averaging end ------------------------  
+
+E_k                     = ((spek_smooth.*m_data)./(2*pi));
+% 2-pi - k Spektrum; Wavenumbers(1/m)
+x_achse_k_spektrum_2pi  = (2.*pi.*x_achse_f_spektrum)./m_data;
+
+
+
+% Wavenumber Spectrum (m^3/sec^2)
+k           = (2.*pi.*f(1:end))./m_data;
+Ek          = ((spek(1:end).*m_data)./(2.*pi));
+low_freq_k  = (2*pi*low_freq)/m_data;
+
+
+h(1) = figure;
+% Plot of Frequency Vs. Power Spectral Density(m^2/sec) without smoothing
+loglog(f,spek) 
+hold on
+% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing
+loglog(x_achse_f_spektrum,spek_smooth,'-','LineWidth',2)
+xlabel('f / Hz','interpreter','latex');
+ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
+xlabel('$f\ (Hz)$','interpreter','latex');
+ylabel('$E\ (m^2/s)$', 'interpreter','latex')
+
+% title('energy density spectrum','interpreter','latex')
+set(gca,'FontSize',10)
+set(gcf, 'Color', 'w')
+% xlim([min(x_achse_f_spektrum) max(x_achse_f_spektrum)]*2)
+xlim([min(x_achse_f_spektrum) 10^2])
+grid on
+axis square
+% datacursormode on
+fig_setup
+legend off
+grid off
+set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
+waitfor(msgbox('Use interactive data cursor to select the range of frequency which will be used to linearly extrapolate the value of ESD at a frequency of 0 Hz (see readme).'));
+% tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+% uiwait(gcf); 
+% delete(tmp_ui);
+
+
+
+%% First Method==> Calculating Ef0 with using Robust property
+% Calculate the integral length scale (macro scale) Roach Eqn 15
+% In this method we will ask you to choose two values of frequencies in Hz such as 
+% f_start & f_end ===> [f_start<f_end ]<== 
+% Must condition and so we will extrapolat a line designed bz f_start & f_end to find
+% energy content at the limit f-->0
+trying              = true;
+attemptcount        = 1;
+while trying
+    try
+        [I,yy]  = ginput(2);
+        if I(1)>I(2)
+            I   = flip(I);
+            yy  = flip(yy);
+        end     
+        tmp_ui = uicontrol('Position',[10 10 100 40],'String','Processing','Callback','uiresume(gcbf)');
+
+        f_start_plot     = askInput({sprintf('Lowest frequency of constant range in Hz:')}, {num2str(I(1),'%.8f')});
+        f_end_plot       = askInput({sprintf('Highest frequency of constant range in Hz:')}, {num2str(I(2),'%.8f')});
+        f_start     = find(f>f_start_plot, 1,'first');
+        f_end       = find(f<=f_end_plot, 1,'last');
+
+        % Ef0       = mean(spek(f_start:f_end));
+        % Ef0       = median(spek(f_start:f_end));
+
+        % IntegralLengthScale_1   = (Ef0*m_data)/(4*uprimesquared);           % Integral Scale in meters
+
+        %Second Method==> Calculating Ef0 from fit without using Robust property
+        [xData, yData] = prepareCurveData( f(f_start:f_end),(spek(f_start:f_end)));
+        
+        ft = fittype( 'poly1' );
+        opts = fitoptions( 'Method', 'LinearLeastSquares' );
+        % opts.Algorithm = 'Levenberg-Marquardt';
+        % opts.Display = 'Off';
+        opts.Lower = [0 0 ];
+        % opts.StartPoint = [0.5 0.1];
+        [fitresult, gof]        = fit( xData, yData, ft, opts );
+        IntegralLengthScale_2   = (fitresult.p2*m_data)/(4*uprimesquared);   % Integral Scale in meters
+        fitresult.p1=0;
+        plot(f,feval(fitresult,f),'k','LineWidth',2,'LineStyle','-')
+        vline(f_start_plot,'k')              % Construct a verticle line at f_start
+        vline(f_end_plot,'k')                % Construct a verticle line at f_end
+        legend('raw','averaged','Location','southwest')
+        
+        trying = false;
+    catch
+        waitfor(msgbox('Please try again: f_start should be smaller than f_end and the decimal separator should be "."(period).'));   
+        if attemptcount>maxnumattempts
+            waitfor(msgbox('Function failed.', 'Error','error'));   
+            error('Function failed after %d times',maxnumattempts)
+        end
+        attemptcount=attemptcount+1;
+    end  
+end       
+      
+        
+% %Third Method==> Calculating Ef0 from fit with using Robust property='LAR' specifies the least absolute residual method.
+% [xData, yData] = prepareCurveData( f(f_start:f_end),spek(f_start:f_end));
+% ft = fittype( 'a*x*0+b', 'independent', 'x', 'dependent', 'y' );
+% opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
+% opts.Algorithm = 'Levenberg-Marquardt';
+% opts.Display = 'Off';
+% opts.Robust = 'LAR';
+% opts.StartPoint         = [0.0 0.01];
+% [fitresult, gof]        = fit( xData, yData, ft, opts );
+% IntegralLengthScale_3   = (fitresult.b*m_data)/(4*uprimesquared);
+% 
+% %%Fourth Method==> Calculating Ef0 from fit with using Robust property='Bisquare' specifies the bisquare weights method.
+% [xData, yData] = prepareCurveData( f(f_start:f_end),spek(f_start:f_end));
+% ft = fittype( 'a*x*0+b', 'independent', 'x', 'dependent', 'y' );
+% opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
+% opts.Algorithm = 'Levenberg-Marquardt';
+% opts.Display = 'Off';
+% opts.Robust = 'Bisquare';
+% opts.StartPoint         = [0.0 0.01];
+% [fitresult, gof]        = fit( xData, yData, ft, opts );
+% IntegralLengthScale_4   = (fitresult.b*m_data)/(4*uprimesquared);
+
+
+
+%% Dissipation from different methods
+trying              = true;
+attemptcount        = 1;
+while trying
+    try
+        clear r tau step stepp 
+        r                   = unique(round(logspace(log10((1./Fs).*m_data*10^6),log10(20*IntegralLengthScale_2*10^6),increment_bin))./10^6,'stable');
+
+        tau(1:length(r))    = nan;
+        step(1:length(r))   = nan;
+
+        for z=1:length(r)
+            tau(z)          = r(z)/m_data;
+            step(z)         = round(tau(z)*Fs);  
+        end
+        stepp              = unique(step);
+        stepp              = stepp(stepp>0);
+
+        clear r R S_exp_3 S_exp_2
+        r(1:length(stepp)) = nan;
+        R(1:length(stepp)) = nan;
+        S_exp_3(1:length(stepp))=nan; % Third order structure function
+        S_exp_2(1:length(stepp))=nan; % Second order structure function
+
+        parfor z=1:length(stepp)
+        %     z/length(stepp)   
+            tau         = stepp(z)              % In steps, just a number
+            r(z)        = (tau./Fs).*m_data;    % in meters
+            incr        = (data(tau+1:L) - data(1:L-tau)).';   % velocity Increment in steps of tau
+            S_exp_2(z)  = mean(incr.^2);        % Second order structure function
+            S_exp_3(z)  = mean(incr.^3);        % Third order structure function   
+            % Autocorr André
+            R(1,z)      = mean(daten(tau+1:L).*daten(1:L-tau))/uprimesquared;
+        end
+
+        % HIT = Homogeneous Isotropic Turbulence
+        eps_2   = (1./(r)).*((S_exp_2./C2).^(3/2)); % Estimation of dissipation from S_exp_2 using HIT
+        eps_3   = -(5/4).*(S_exp_3./r);             % Estimation of dissipation from S_exp_3 using HIT 
+
+        I0      = find( eps_2==max(eps_2),1);       % Finding the maximum value of dissipation in inertial range
+        I1      = find(-eps_3==max(-eps_3),1);      % Finding the maximum value of dissipation in inertial range
+
+        % dissipation rate m^2.s^-3 from S2
+        epsi_S2 = mean(eps_2(1,I0-2:I0+2));         % finding the mean amongst 5 points closest to peak value;
+        % dissipation rate m^2.s^-3 from S3
+        epsi_S3 = mean(-eps_3(1,I1-2:I1+2));        % finding the mean amongst 5 points closest to peak value;
+        
+        trying = false;
+    catch
+%         waitfor(msgbox(' f_start should be smaller than f_end should be. ')); 
+        increment_bin = askInput({sprintf('Please try again by increasing the number of Bin''s used (must be larger than:):')}, {num2str(increment_bin,'%1.0f')});      
+        if attemptcount>maxnumattempts
+            waitfor(msgbox('Function failed.', 'Error','error'));   
+            error('Function failed after %d times',maxnumattempts)
+        end
+        attemptcount=attemptcount+1;
+    end  
+end       
+        
+delete(tmp_ui);
+
+
+% figure
+% loglog(r,S_exp_2)
+
+
+%% Integral length scale from Auto-corelation or structure function 
+%% Autocorr André
+low_freq_r  =  m_data/low_freq;  
+
+% Full integration 
+int_L_full  = trapz(r,R);
+% trapz(r,abs(R))
+
+% integrate up to the first zero crossing
+tmp_zero=(find(R<0,1,'first'))-1;
+if isempty(tmp_zero)
+   int_L_0     = nan;
+else
+    % r(tmp_zero)
+    int_L_0     = trapz(r(1:tmp_zero),R(1:tmp_zero));
+end
+
+% integrate up to the first 1/e crossing
+tmp_exp=(find(R<1/exp(1),1,'first'));
+if isempty(tmp_exp)
+    int_L_1_exp=nan;
+    theo_exp_Funktion = nan;
+else
+    int_L_1_exp = trapz(r(1:tmp_exp),R(1:tmp_exp));
+    % local minimum 
+    % index1 = find(diff(sign(diff(R))) ~= 0,2,'first')+1;
+    % r(index1)
+    % int_L_min=trapz(r(1:index1),abs(R(1:index1)));
+
+    %exp Fit to R; Fitregion r=1/exp(1):2*low_freq_r
+    tmp_up      = (find(r>4*low_freq_r,1,'first'));
+    tmp_low     = (find(R<1/exp(1),1,'first'));
+    
+    if tmp_up<tmp_low
+        [xData, yData] = prepareCurveData( r(tmp_up:tmp_low), R (tmp_up:tmp_low));
+        ft = fittype( 'exp1' );
+        % ft = fittype( 'exp(-x/b)', 'independent', 'x', 'dependent', 'y' )
+        % exp(-(x+a)/b)
+        opts                = fitoptions( 'Method', 'NonlinearLeastSquares' );
+        opts.Algorithm      = 'Levenberg-Marquardt';
+        opts.Display        = 'Off';
+        opts.Robust         = 'Bisquare';
+        [fitresult, gof]    = fit( xData, yData, ft, opts );
+        theo_exp_Funktion   = ((fitresult.a/-fitresult.b));
+        % theo_exp_Funktion=sum(R(1:tmp_low).*diff(r(1:tmp_low+1)));
+        % theo_exp_Funktion=(fitresult.a/-fitresult.b * exp(-fitresult.b*r(tmp_up)));
+        % theo_exp_Funktion=sum(R(1:tmp_up).*diff(r(1:tmp_up+1)))+(fitresult.a/-fitresult.b * exp(fitresult.b*r(tmp_up)));
+        % theo_exp_Funktion=sum(R(2:tmp_up).*diff(r(1:tmp_up)))+(fitresult.a/-fitresult.b * exp(fitresult.b*r(tmp_up)));
+        
+        % Plot the Auto-correlation coefficient Vs. Scale, r and exp fit
+        h(2) = figure;
+        if ~isempty(tmp_exp)
+            plot(r,fitresult.a*exp(fitresult.b*r),'-r','LineWidth',2)
+            % plot( fitresult, r, R )
+            hold on
+        end
+        plot(r,R,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
+        ylabel('$R_{\widetilde{u}\widetilde{u}}$', 'interpreter','latex')
+        xlabel('r/m', 'interpreter','latex')
+        xlabel('$r\ (m)$','interpreter','latex');
+        % title(['integral length scale L = ' num2str(theo_exp_Funktion) 'm'])
+        % title(['first zero crossing = ' num2str(min(r(find(R<0)-1))) 'm'])
+        legend('exp fit', 'Location', 'NorthEast' );
+        axis square
+        set(gcf, 'Color', 'w')
+        set(gca, 'FontSize',18)
+        % xlim([0 150*low_freq_r])
+        if ~isempty(tmp_exp)
+            xlim([0 4*r(tmp_low)])
+            vline(r(tmp_low),'k')
+            vline(r(tmp_up),'k')
+        % vline(r(find(R<1/exp(1),1,'first')),'r')
+        end
+        fig_setup
+    else
+        theo_exp_Funktion=nan;  
+    end
+end
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf);
+delete(tmp_ui);
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Processing','Callback','uiresume(gcbf)');
+pause(1)
+
+
+
+%% Following calculation ensures that auto-correlation fluctuates around zero. For that you need to take the value of standard deviation of velocity calculated from the 2nd order structure function
+Ruu     = 1-(S_exp_2./(mean(S_exp_2((r>10*IntegralLengthScale_2)))));   % Auto-correlation coefficient Using std from S_exp_2 ==>This is a dimentionless quantity
+Ruu     = [1,Ruu];                                                      % Auto-correlation is 1 at scale=0
+r_Ruu   = [0,r];                                                        % Auto-correlation is 1 at scale=0
+LL      = (cumsum(Ruu(1,1:end-1).*diff(r_Ruu)))';                       % Cumulative integration of Ruu with respect to r
+pks     = findpeaks(LL,3);                                              % Finding the peaks in LL
+Npks    = size(pks,1);
+
+if Npks == 0 
+IntegralLengthScale_5 = trapz(r_Ruu,Ruu);       % If no peaks are found then simply take the area under the curve which will be in meters
+else  
+IntegralLengthScale_5 = pks(1);                 % in meters At this length we should reach an asymptotic value on the plot of r Vs. LL
+end
+
+lambda_S2   = (sqrt(15*kin_vis*uprimesquared./epsi_S2));    % Taylors length scale in meters from epsi_S2 and hence from S_exp_2
+lambda_S3   = (sqrt(15*kin_vis*uprimesquared./epsi_S3));    % Taylors length scale in meters from epsi_S3 and hence from S_exp_3
+
+
+
+%% Integral length scale from auto-corelation function 'xcorr' of MATLAB 
+% tau                     = 1/Fs;             % time in seconds/Time per sample in secs
+% [Ruu_xcorr,tau_xcorr]   = xcorr(data - mean(data),size(data,1)/2,'unbiased');
+% Ruu_xcorr               = Ruu_xcorr(tau_xcorr>=0);
+% tau_xcorr               = tau_xcorr(tau_xcorr>=0);
+% Ruu_xcorr               = Ruu_xcorr./max(Ruu_xcorr);
+% dist_xcorr              = tau_xcorr.*tau.*m_data;
+% Ruu_xcorr               = Ruu_xcorr';
+% LL_xcorr                = (cumsum(Ruu_xcorr(1,1:end-1).*diff(dist_xcorr(1,1:end))));
+% pks_xcorr               = findpeaks(LL_xcorr);
+% Npks_xcorr              = size(pks_xcorr,1);
+% 
+% if Npks_xcorr == 0 
+% IntegralLengthScale_6   = trapz(dist_xcorr,Ruu_xcorr);  % If no peaks are found then simply take the area under the curve
+% else  
+% IntegralLengthScale_6   = pks_xcorr(1);                 % in meters at this length we should reach an asymptotic value on plot of r Vs. LL
+% end
+% 
+% figure;
+% plot(r,R,'-k','LineWidth',2)
+% hold on
+% plot(r_Ruu,Ruu,'LineWidth',2)
+% plot(dist_xcorr,Ruu_xcorr,'LineWidth',2)
+% xlim([0 1500*low_freq_r])
+
+
+% figure;
+% plot(r_Ruu(1:end-1),LL,'LineWidth',2)
+% hold on
+% plot(dist_xcorr(1:end-1),LL_xcorr,'LineWidth',2)
+
+
+
+%% Zero-crossing method: Martin Obligado
+delete(tmp_ui);
+[int_L_zero_cros,lambda_zero_cros,lambda1_zero_cros,Eps_zero_cros,h]=ZeroCrossings(data,Fs,m_data,kin_vis,low_freq,x_achse_k_spektrum_2pi,E_k,h);
+
+%% 
+% int_L_calc  = [IntegralLengthScale_2,int_L_full,int_L_0,int_L_1_exp,theo_exp_Funktion,IntegralLengthScale_5,IntegralLengthScale_6,int_L_zero_cros];%% meter
+int_L_calc  = [IntegralLengthScale_2,int_L_full,int_L_0,int_L_1_exp,theo_exp_Funktion,IntegralLengthScale_5,int_L_zero_cros];%% meter
+
+% Choose the best estimate of integral and Taylor length scale on which you would like to rely on
+h(3) = figure;
+plot(int_L_calc,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
+xlabel('method', 'interpreter','latex')
+ylabel('L/m', 'interpreter','latex')
+ylabel('$L\ (m)$','interpreter','latex');
+axis square
+set(gcf, 'Color', 'w')
+set(gca, 'FontSize',18)
+fig_setup
+legend off
+set(gca,'XTick',[1 2 3 4 5 6 7 8]);
+datacursormode on
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf);
+delete(tmp_ui);
+int_L    = askInput({sprintf('Enter the integral length scale in meter. Median integral length scale in meter:')}, {num2str(nanmedian(int_L_calc),'%.5f')});
+
+
+
+
+
+
+%% Taylor length by different methods
+%% using Autocorrelation function
+trying              = true;
+attemptcount        = 1;
+while trying
+    try
+        figure;
+        plot(r_Ruu,Ruu,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
+        ylabel('$R_{\widetilde{u}\widetilde{u}}$', 'interpreter','latex')
+        xlabel('r/m', 'interpreter','latex')
+        xlabel('$r\ (m)$','interpreter','latex');
+        axis square
+        set(gcf, 'Color', 'w')
+        set(gca, 'FontSize',18)
+        ylim([0.7 1.05])
+        xlim([0 r_Ruu(find(Ruu<0.7,1,'first'))])
+        datacursormode on
+        waitfor(msgbox('Use interactive data cursor to select the range of scales which will be used to estimate Taylor lenght scalee (see readme).'));
+        [I,yy]  = ginput(1);
+        i_tmp = askInput({sprintf(['Fit Region Auto-correlation Coefficient e.g. = ' ,num2str(0.98,'%1.2f')])}, {num2str(yy(1),'%.2f')});
+        close
+
+     
+        [xData, yData] = prepareCurveData( r_Ruu(1:(find(R<i_tmp,1,'first'))), R (1:(find(R<i_tmp,1,'first'))));
+        ft = fittype( '1-(x^2/a^2)', 'independent', 'x', 'dependent', 'y' );
+        opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
+        opts.Lower = 0;
+        opts.StartPoint = 0.002;
+        [fitresult, gof] = fit( xData, yData, ft, opts );
+           
+        h(4) = figure;
+        plot(r_Ruu,feval(fitresult,r_Ruu),'r','LineWidth',2)
+        hold on
+        plot(r_Ruu,Ruu,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
+        ylabel('$R_{\widetilde{u}\widetilde{u}}$', 'interpreter','latex')
+        xlabel('r/m', 'interpreter','latex')
+        xlabel('$r\ (m)$','interpreter','latex');
+        axis square
+        set(gcf, 'Color', 'w')
+        set(gca, 'FontSize',18)
+        ylim([0.7 1.05])
+        xlim([0 r_Ruu(find(Ruu<0.7,1,'first'))])
+        vline(r_Ruu((find(R<i_tmp,1,'first'))),'k')
+        lambda_autocorr=fitresult.a;
+        % legend('','$R_{\widetilde{u}\widetilde{u}}(r)=1-\frac{r^2}{\lambda^2}$', 'Location', 'NorthEast' );
+        legend('$R_{\widetilde{u}\widetilde{u}}(r)=1-\frac{r^2}{\lambda^2}$', 'Location', 'NorthEast' );
+        title(['$\lambda$ = ' num2str(lambda_autocorr,'%1.4f') ' m'],'interpreter','latex')
+        fig_setup
+               
+        trying = false;
+    catch
+        waitfor(msgbox('Please try again'));   
+%         askInput({sprintf('Please try again by increasing the number of Bin''s used:')}, {num2str(increment_bin,'%1.0f')});
+        if attemptcount>maxnumattempts
+            waitfor(msgbox('Function failed.', 'Error','error'));   
+            error('Function failed after %d times',maxnumattempts)
+        end
+        attemptcount=attemptcount+1;
+    end  
+end       
+
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf);
+delete(tmp_ui); 
+
+
+
+
+%% numerical differentiation
+%<(du/dx)^2>
+du_dx           = mean((diff(daten)./(m_data*(1/Fs))).^2);
+lambda_direct   = sqrt(uprimesquared/du_dx);
+
+% Aronson and Löfdahl
+tmp = sqrt((uprimesquared.*r.^2)./S_exp_2);
+h(5) = figure;
+plot(r,tmp,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
+hold on
+xlabel('r/m', 'interpreter','latex')
+xlabel('$r\ (m)$','interpreter','latex');
+ylabel('$\sqrt{\frac{u''^2 r^2}{S^{2}(r)}}/m$', 'interpreter','latex')
+ylabel('$\sqrt{\frac{u''^2 r^2}{S^{2}}}\ (m)$', 'interpreter','latex')
+axis square
+set(gcf, 'Color', 'w')
+set(gca, 'FontSize',18)
+xlim([0 25*low_freq_r])
+datacursormode on
+waitfor(msgbox('Use interactive data cursor to select the range of scales which will be used to estimate Taylor lenght scalee (see readme).'));
+
+tmp_up      = (find(r>4*low_freq_r,1,'first'));
+vline(r(tmp_up),'k')
+
+trying              = true;
+attemptcount        = 1;
+while trying
+    try
+        [I,yy]  = ginput(1);
+        % i_tmp       = str2double(inputdlg({'Fit Region rmin ='}));
+        % tmp_up      = (find(r>i_tmp,1,'first'));
+        i_tmp       = askInput({sprintf(['Fit Region r_max e.g. = ' ,num2str(20*low_freq_r,'%0.2f')])}, {num2str(I(1),'%.2f')});
+        tmp_low     = (find(r<i_tmp,1,'last'));
+        [xData, yData] = prepareCurveData(r(tmp_up:tmp_low),tmp(tmp_up:tmp_low));
+        ft = fittype( 'poly1' );
+        [fitresult, gof] = fit( xData, yData, ft );
+        plot(linspace(0,1,1001),feval(fitresult,linspace(0,1,1001)),'r','LineWidth',2)
+        fitresult.p1=0;
+        plot(linspace(0,1,1001),feval(fitresult,linspace(0,1,1001)),'k','LineWidth',2,'LineStyle','-')
+        vline(r(tmp_low),'k')
+        A_L_lambda =   fitresult.p2;
+        fig_setup
+        legend off
+        
+        trying = false;
+    catch
+        waitfor(msgbox('Please try again'));   
+        if attemptcount>maxnumattempts
+            waitfor(msgbox('Function failed.', 'Error','error'));   
+            error('Function failed after %d times',maxnumattempts)
+        end
+        attemptcount=attemptcount+1;
+    end  
+end               
+        
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf);
+delete(tmp_ui);
+
+
+% Dissipationspectrum
+k_end                   = find(k<=low_freq_k, 1,'last');
+lambda_spec_low_freq    = sqrt(trapz(k(1:k_end),Ek(1:k_end))/trapz(k(1:k_end),Ek(1:k_end).*k(1:k_end).^2));
+lambda_spec_full        = sqrt(uprimesquared/trapz(k,Ek.*k.^2));
+% trapz(k,2.*Ek.*k.^2) anstatt trapz(k,Ek.*k.^2)
+% lambda_1=sqrt(uprimesquared/trapz(k,2.*Ek.*k.^2))
+% lambda_2=sqrt(trapz(k,2.*Ek)/trapz(k,2.*Ek.*k.^2))
+
+% take everthing of the specturm
+% lambda_1=sqrt(uprimesquared/trapz(k,Ek.*k.^2));
+% lambda_1=sqrt(uprimesquared/trapz(x_achse_k_spektrum_2pi(2:end),E_k.*x_achse_k_spektrum_2pi(2:end).^2))
+% lambda_2=sqrt(trapz(k,Ek)/trapz(x_achse_k_spektrum_2pi(2:end),E_k.*x_achse_k_spektrum_2pi(2:end).^2))
+
+% find Minimum at high frequencies
+% k_start           = find(k>0.2*10^4, 1,'first');
+% k_end             = find(k<=5*10^4, 1,'last');
+% lambda_1_kurz     = sqrt(uprimesquared/trapz(k(1:find(spek==min(spek(k_start:k_end)))),Ek(1:find(spek==min(spek(k_start:k_end)))).*k(1:find(spek==min(spek(k_start:k_end)))).^2));
+
+% lambda_4  = sqrt(uprimesquared/du_dx_4);
+% lambda_6    = sqrt((10*kin_vis/m_data) * (uprimesquared/(uprimesquared/(m_data*(1/Fs)))));
+
+% taylor_L_calc = [lambda_autocorr,lambda_direct,A_L_lambda,lambda_spec_low_freq,lambda_spec_full, lambda_S2, lambda_S3,lambda_zero_cros,lambda1_zero_cros];%% meter
+taylor_L_calc = [lambda_autocorr,lambda_direct,A_L_lambda,lambda_spec_low_freq,lambda_spec_full,lambda_zero_cros,lambda1_zero_cros];%% meter
+
+
+h(6) = figure;
+plot(taylor_L_calc,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
+xlabel('method', 'interpreter','latex')
+ylabel('$\lambda/m$', 'interpreter','latex')
+ylabel('$\lambda\ (m)$', 'interpreter','latex')
+axis square
+set(gcf, 'Color', 'w')
+set(gca, 'FontSize',18)
+fig_setup
+legend off
+set(gca,'XTick',[1 2 3 4 5 6 7]);
+datacursormode on
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf);
+delete(tmp_ui);
+taylor_L = askInput({sprintf('Enter the Taylor length scale in meter. Median Taylor length scale in meter:')}, {num2str(nanmedian(taylor_L_calc),'%.5f')});
+
+
+
+%% Dissipation from different methods
+% epsi_1  = 15*kin_vis*trapz(k(1:k_end),Ek(1:k_end).*k(1:k_end).^2); %% From dissipation Spectrum untill cutoff frequency
+% epsi_2  = 15*kin_vis*trapz(k,Ek.*k.^2); %%From entire dissipation Spectrum
+
+% epsi_2  = trapz(k,2.*kin_vis.*Ek.*k.^2);
+% epsi_3  = 15*kin_vis*du_dx_3;
+% epsi_4  = 15*kin_vis*du_dx_4;
+% epsi_5  = 15*kin_vis*du_dx_5;
+
+% epsi_5 = (m_data/2) * (mean(diff(daten.^2)/(m_data*(1/Fs))));
+% epsi_6 = (3*m_data/2) * (var(diff(daten)./(m_data*(1/Fs))));
+% epsi_7 = 15*kin_vis*(uprimesquared/lambda_1^2); 
+% epsi_7a= 30*kin_vis*(uprimesquared/lambda_1^2) %longitudinale microscale
+% epsi_7b= 15*kin_vis*(uprimesquared/(lambda_1/sqrt(2))^2) %trasnversale microscale
+% epsi_8 = m_data^3/int_L
+
+% epsi_1/m_data
+% var(daten)/(m_data*(1/Fs))
+% std(daten)/(m_data*(1/Fs))
+% epsi_calc   = [epsi_1, epsi_2, epsi_3, epsi_5, epsi_S2,epsi_S3];
+
+
+h(7) = figure;
+semilogx(r,eps_2,'LineWidth',2)
+hold on
+semilogx(r,-eps_3,'LineWidth',2)
+xlim([min(r)*0.5 max(r)*2])
+xlabel('r/m', 'interpreter','latex')
+xlabel('$r\ (m)$','interpreter','latex');
+ylabel('$\epsilon(r)/\frac{m^2}{s^3}$', 'interpreter','latex')
+ylabel('$\epsilon\ (m^2/s^3)$','interpreter','latex');
+set(gca,'FontSize',16)
+set(gcf, 'Color', 'w')
+fig_setup
+axis square
+% vline(r(I0),'r')                          
+% vline(r(I1),'k')                          
+
+plot(linspace(min(r)*0.5,max(r)*2,1000),eps_2(I0).*ones(1000,1),'k','LineWidth',2)
+plot(linspace(min(r)*0.5,max(r)*2,1000),-eps_3(I1).*ones(1000,1),'k','LineWidth',2)
+legend('$\frac{1}{r}\left[\frac{S^{2}(r)}{C_{2}}\right]^{3/2}$','$-\frac{5}{4}\left[\frac{S^{3}(r)}{r}\right]$','Location','south')
+
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf);
+delete(tmp_ui);
+
+etaflow_S2  = ((kin_vis^3/epsi_S2)^(1/4));                  % Kolmogorov length scale in meters from epsi_S2 and hence from S_exp_2
+etaflow_S3  = ((kin_vis^3/epsi_S3)^(1/4));                  % Kolmogorov length scale in meters from epsi_S3 and hence from S_exp_3
+
+
+
+%% Spectrum-Fit
+epsi_calc   = [epsi_S2,epsi_S3,(15*kin_vis*(uprimesquared/taylor_L^2))];
+
+h(8) = figure;
+loglog(x_achse_k_spektrum_2pi,E_k,'-','LineWidth',2)
+hold on
+a=find(x_achse_k_spektrum_2pi<=(2*pi*((1/(int_L/m_data))/(2*pi))/m_data), 1,'last');
+b=find(x_achse_k_spektrum_2pi>=(2*pi*((1/(taylor_L/m_data))/(2*pi))/m_data), 1,'first');
+
+% b=find(x_achse_k_spektrum_2pi>=(1)/diss_scale, 1,'first'); 
+% a=1;
+% b=length(x_achse_k_spektrum_2pi);
+[xData, yData] = prepareCurveData( x_achse_k_spektrum_2pi(a:b), E_k(a:b) );
+% [xData, yData] = prepareCurveData( x_achse_f_spektrum,spek_smooth);
+% [xData, yData] = prepareCurveData(k, Ek );
+
+opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
+opts.DiffMinChange = 1e-12;
+opts.Display = 'Off';
+opts.MaxFunEvals = 1200;
+opts.MaxIter = 1200;
+
+ft = fittype( 'a*b^(2/3)*(x-d)^(-5/3)', 'independent', 'x', 'dependent', 'y' );
+% opts.Lower = [0 median(epsi_calc) -Inf];
+% opts.Upper = [Inf median(epsi_calc) Inf];
+opts.Lower = [0 0 -Inf];
+opts.Upper = [Inf Inf Inf];
+opts.StartPoint = [0.5 nanmedian(epsi_calc) 0];
+try 
+    [fitresult, gof] = fit( xData, yData, ft, opts );
+    ft2 = fittype( 'a*b^(2/3)*(x)^(-5/3)', 'independent', 'x', 'dependent', 'y' );
+    % opts.Lower = [0 nanmedian(epsi_calc)];
+    % opts.Upper = [Inf nanmedian(epsi_calc)];
+    opts.Lower = [0 0];
+    opts.Upper = [Inf Inf];
+    opts.StartPoint = [0.5 nanmedian(epsi_calc)];
+    [fitresult2, gof] = fit( xData, yData, ft2, opts );
+    plot(x_achse_k_spektrum_2pi, feval(fitresult,x_achse_k_spektrum_2pi),'-','LineWidth',2)
+    plot(x_achse_k_spektrum_2pi, feval(fitresult2,x_achse_k_spektrum_2pi),'-','LineWidth',2)
+catch 
+end
+set(gca,'XScale','log');
+set(gca,'YScale','log');
+xlabel('$k= / \frac{1}{m}$', 'interpreter','latex');
+ylabel('$E(k)$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
+xlabel('$k\ (1/m)$','interpreter','latex');
+ylabel('$\frac{E <u>}{2 \pi}\ (m^3/s^2)$', 'interpreter','latex')
+
+xlim([min(x_achse_k_spektrum_2pi) 2*max(x_achse_k_spektrum_2pi)])
+ylim([min(E_k) 2*max(E_k)])
+axis square
+% vline(x_achse_k_spektrum_2pi(a),'k')
+% vline(x_achse_k_spektrum_2pi(b),'k')
+% vline((1)/diss_scale,'k')
+set(gcf, 'Color', 'w')
+legend('$\frac{E <u>}{2 \pi}$','$C_k\langle \epsilon \rangle^{2/3}(k-k_0)^{-5/3}$','$C_k\langle \epsilon \rangle^{2/3}k^{-5/3}$','Location','south')
+fig_setup
+set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
+% fitresult;
+% fitresult2;
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf);
+delete(tmp_ui);
+
+% figure
+% plot( fitresult, xData, yData );
+% hold on
+% plot( fitresult2, xData, yData );
+% plot( fitresult, xData, yData );
+% loglog(x_achse_k_spektrum_2pi,E_k)
+% set(gca,'XScale','log');
+% set(gca,'YScale','log');
+
+
+epsi_calc   = [epsi_S2,epsi_S3, (15*kin_vis*(uprimesquared/taylor_L^2)), fitresult2.b,fitresult.b, Eps_zero_cros];
+
+h(9) = figure;
+plot(epsi_calc,'MarkerEdgeColor','k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
+xlabel('method', 'interpreter','latex')
+ylabel('$\langle \epsilon \rangle/\frac{m^2}{s^3}$', 'interpreter','latex')
+ylabel('$\langle \epsilon \rangle\ (m^2/s^3)$', 'interpreter','latex')
+axis square
+set(gcf, 'Color', 'w')
+set(gca, 'FontSize',18)
+fig_setup
+legend off
+set(gca,'XTick',[1 2 3 4 5 6 7 ]);
+datacursormode on
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf);
+delete(tmp_ui);
+epsi = askInput({sprintf('Enter the energy dissipation rate in $m^2/s^3$. Median energy dissipation rate in $m^2/s^3$:')}, {num2str(nanmedian(epsi_calc),'%.5f')});
+
+
+
+
+Re          = m_data*int_L/kin_vis;
+% Re_lambda   = uprime*lambda_1_kurz_low_freq/kin_vis;
+Re_lambda   = uprime*taylor_L/kin_vis; 
+% Re_lambda = uprime*lambda_1/kin_vis;
+
+% Calculate the dissipation length scale (micro scale) Roach Eqn 8
+% Z=((2*pi^2)/(m_data^2*uprimesquared))*trapz((f(2:end).^2),spek(2:L/2+1)); % Use trapezoid rule for integration
+% DissipationScale=sqrt(1/Z)
+
+DissipationScale_1      = ((kin_vis^3)/epsi)^0.25;
+% DissipationScale_2    = sqrt(1/(trapz(f,spek.*f.^2)*((2*pi^2)/(m_data^2*uprimesquared))))
+% DissipationScale_3      = int_L/(Re^0.75);
+
+% kolmogorov_calc         = [DissipationScale_1, etaflow_S2,etaflow_S3];%% meter
+diss_scale         = DissipationScale_1;
+
+%%Non dimentional dissipation constant
+Ce      = 15*(int_L/taylor_L)/Re_lambda;
+Ce_b    = epsi*int_L/uprime^3;
+
+% Ce      = 15*(int_L/lambda_1_kurz_low_freq)/Re_lambda;
+% Ce_1    = epsi_1*int_L/uprime^3;
+% Ce_2    = epsi_2*int_L/uprime^3;
+% Ce_3    = epsi_3*int_L/uprime^3;
+% Ce_5    = epsi_5*int_L/uprime^3;
+
+% Ce_calc =[Ce, Ce_1, Ce_2, Ce_3, Ce_5];
+Ce_calc = [Ce, Ce_b];
+Ce      = mean(Ce_calc);
+
+
+%% Plot Spectrum
+h(10) = figure;
+set(gcf, 'Color', 'w')
+yyaxis left
+loglog(x_achse_k_spektrum_2pi,E_k,'-','LineWidth',2)
+% hold on
+% loglog(x_achse_k_spektrum_2pi,(x_achse_k_spektrum_2pi.^(-5/3).*ones(length(E_k)).*10),'-','LineWidth',1,'color','k')
+% loglog(x_achse_k_spektrum_2pi,(x_achse_k_spektrum_2pi.^(-2).*ones(length(E_k)).*0.7.*10^2),'-','LineWidth',1,'color','k')
+% loglog(x_achse_k_spektrum_2pi,(x_achse_k_spektrum_2pi.^(-3).*ones(length(E_k)).*0.6.*10^5),'-','LineWidth',1,'color','k')
+% xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
+xlabel('$k= / \frac{1}{m}$', 'interpreter','latex');
+% ylabel('$E(k)=\frac{E(f) <u>}{2 \pi}$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
+ylabel('$E(k)$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
+%xlabel('k');
+%ylabel('E(k)')
+xlabel('$k\ (1/m)$','interpreter','latex');
+ylabel('$\frac{E <u>}{2 \pi}\ (m^3/s^2)$', 'interpreter','latex')
+xlim([min(x_achse_k_spektrum_2pi) 2*max(x_achse_k_spektrum_2pi)])
+% ylim([10^-10 1])
+axis square
+hold on
+yyaxis right
+% loglog(x_achse_k_spektrum_2pi,15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi).^2,'-','LineWidth',2)
+loglog(x_achse_k_spektrum_2pi,E_k.*(x_achse_k_spektrum_2pi).^2,'-','LineWidth',2)
+ylabel('E(k)$\cdot k^2$ / $\frac{m}{s^2}$', 'interpreter','latex')
+ylabel('$\frac{E <u>}{2 \pi} k^2\ (m/s^2)$', 'interpreter','latex')
+
+set(gca,'YScale','log');
+set(gca,'FontSize',24)
+% vline(low_freq_k,'r')
+% vline((1/(int_L/m_data))/m_data,'k')
+vline((2*pi*((1/(int_L/m_data))/(2*pi))/m_data),'k')
+vline((2*pi*((1/(taylor_L/m_data))/(2*pi))/m_data),'k')
+vline((2*pi*((1/(diss_scale/m_data))/(2*pi))/m_data),'k')
+fig_setup
+set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
+legend off
+
+
+
+h(11) = figure;
+set(gcf, 'Color', 'w')
+yyaxis left
+% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing
+loglog(x_achse_f_spektrum,spek_smooth,'-','LineWidth',2)
+xlabel('f / Hz','interpreter','latex');
+ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
+xlabel('$f\ (Hz)$','interpreter','latex');
+ylabel('$E\ (m^2/s)$', 'interpreter','latex')
+set(gca,'FontSize',10)
+set(gcf, 'Color', 'w')
+xlim([min(x_achse_f_spektrum) max(x_achse_f_spektrum)]*2)
+% xlim([min(x_achse_f_spektrum) 10^2])
+grid on
+axis square
+datacursormode on
+fig_setup
+grid off
+legend off
+set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
+vline((1/(int_L/m_data))/(2*pi),'k')
+vline((1/(taylor_L/m_data))/(2*pi),'k')
+vline((1/(diss_scale/m_data))/(2*pi),'k')
+
+yyaxis right
+% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing
+loglog(x_achse_f_spektrum,spek_smooth.*x_achse_f_spektrum.^(5/3),'-','LineWidth',2)
+ylabel('$E(f)f^{5/3}$ / $\frac{m^2}{s}$', 'interpreter','latex')
+ylabel('$E\ f^{5/3}\ (m^2/s)$', 'interpreter','latex')
+xlim([min(x_achse_f_spektrum) max(x_achse_f_spektrum)]*2)
+ylim([min(spek_smooth.*x_achse_f_spektrum.^(5/3)) max(spek_smooth.*x_achse_f_spektrum.^(5/3))]*2)
+
+
+
+
+% k Spektrum; Scales,r (m)
+x_achse_spektrum_meter  = m_data./(2*pi*x_achse_f_spektrum);
+Dr_avg_filter=(spek_smooth.*x_achse_f_spektrum.^2)./m_data;
+Dk_avg_filter=15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi).^2;
+
+
+
+figure;
+set(gcf, 'Color', 'w')
+subplot(2,2,1)
+loglog(x_achse_f_spektrum,spek_smooth,'-','LineWidth',1)%% Plot of Frequency Vs. Power Spectral Density(m^2/sec) with smoothing & Filtering
+xlabel('f / Hz','interpreter','latex');
+ylabel('E(f) / $\frac{m^2}{s}$', 'interpreter','latex')
+xlabel('$f\ (Hz)$','interpreter','latex');
+ylabel('$E\ (m^2/s)$', 'interpreter','latex')
+xlim([min(x_achse_f_spektrum) 2*max(x_achse_f_spektrum)])
+ylim([min(spek_smooth) max(spek_smooth)]*2)
+title('frequency domain','interpreter','latex')
+set(gca,'FontSize',12)
+set(groot, 'defaultAxesTickLabelInterpreter','latex');
+% plot([10^3 10^3],[10^-6 10^-2],'y = x ','-','LineWidth',1) 
+% h = vline(10^3,'k','The Answer')
+% text(245,spek_smooth(find(x_achse_f_spektrum>245,1,'first')),'\leftarrow sin(\pi) = 0','FontSize',16)
+set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
+hold on
+vline((1/(int_L/m_data))/(2*pi),'k')
+vline((1/(taylor_L/m_data))/(2*pi),'k')
+vline((1/(diss_scale/m_data))/(2*pi),'k')
+txt = {'(a)'};
+a=text(4,0.5,txt);
+a.Units='normalized';
+
+subplot(2,2,2)
+loglog(x_achse_k_spektrum_2pi,E_k,'-','LineWidth',1)%% Plot of Wavenumber(k) Vs. Energy(k)(m^3/sec^2) with smoothing & Filtering
+% loglog(x_achse_k_spektrum_2pi,x_achse_k_spektrum_2pi.^(-5/3).*(max(spek_smooth).*200),'--','color','k','LineWidth',2)
+xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
+ylabel('$E(k)=\frac{E(f) <u>}{2 \pi}$ / $\frac{m^3}{s^2}$', 'interpreter','latex')
+xlabel('$k=\frac{2 \pi f}{<u>}\ (1/m)$','interpreter','latex');
+ylabel('$\frac{E <u>}{2 \pi}\ (m^3/s^2)$', 'interpreter','latex')
+xlim([min(x_achse_k_spektrum_2pi) 2*max(x_achse_k_spektrum_2pi)])
+ylim([min(E_k) max(E_k)]*2)
+title('wave number domain','interpreter','latex')
+set(gca,'FontSize',12)
+set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
+hold on
+vline((2*pi*((1/(int_L/m_data))/(2*pi))/m_data),'k')
+vline((2*pi*((1/(taylor_L/m_data))/(2*pi))/m_data),'k')
+vline((2*pi*((1/(diss_scale/m_data))/(2*pi))/m_data),'k')
+txt = {'(b)'};
+b=text(4,0.5,txt);
+b.Units='normalized';
+
+subplot4=subplot(2,2,3);
+loglog(x_achse_spektrum_meter,Dr_avg_filter,'-','LineWidth',1)%% Plot of Scales,r(m) Vs. Energy(r)(m/sec^2) with smoothing & Filtering
+xlabel('$r=\frac{<u>}{2 \pi f} / m$', 'interpreter','latex');
+ylabel('$E(r)=\frac{E(f) f^2}{<u>}$ / $\frac{m}{s^2}$', 'interpreter','latex')
+xlabel('$r=\frac{<u>}{2 \pi f}\ (m)$','interpreter','latex');
+ylabel('$\frac{E\ f^2}{<u>}\ (m/s^2)$', 'interpreter','latex')
+set(subplot4, 'Xdir', 'reverse')
+xlim([0.5* min(x_achse_spektrum_meter) max(x_achse_spektrum_meter)])
+ylim([min((spek_smooth.*x_achse_f_spektrum.^2)./m_data) max((spek_smooth.*x_achse_f_spektrum.^2)./m_data)]*2)
+set(gca,'FontSize',12)
+set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
+title('dissipation spectrum','interpreter','latex')
+vline(int_L,'k')
+vline(taylor_L,'k')
+vline(diss_scale,'k')
+txt = {'(c)'};
+c=text(4,0.5,txt);
+c.Units='normalized';
+
+
+
+% Dissipation Spectrum
+% kin_vis = 15.328737178409*10^-6;
+subplot(2,2,4)
+loglog(x_achse_k_spektrum_2pi,Dk_avg_filter,'-','LineWidth',1)%% Plot of Wavenumber(k) Vs. Dissipation(k)(m/sec^2) with smoothing & Filtering
+xlabel('$k=\frac{2 \pi f}{<u>} / m^{-1}$', 'interpreter','latex');
+ylabel('E(k)$\cdot k^2$ / $\frac{m}{s^2}$', 'interpreter','latex')
+xlabel('$k=\frac{2 \pi f}{<u>}\ (1/m)$','interpreter','latex');
+ylabel('$\frac{E <u>}{2 \pi}\ k^2 \ (m/s^2)$', 'interpreter','latex')
+xlim([min(x_achse_k_spektrum_2pi) 2*max(x_achse_k_spektrum_2pi)])
+ylim([min(15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi).^2) max(15.*kin_vis.*E_k.*(x_achse_k_spektrum_2pi).^2)]*2)
+title('dissipation spectrum','interpreter','latex')
+set(gca,'FontSize',12)
+set(gca,'XTick',[10^-2 10^0  10^2  10^4]);
+datacursormode on
+hold on
+vline((2*pi*((1/(int_L/m_data))/(2*pi))/m_data),'k')
+vline((2*pi*((1/(taylor_L/m_data))/(2*pi))/m_data),'k')
+vline((2*pi*((1/(diss_scale/m_data))/(2*pi))/m_data),'k')
+txt = {'(d)'};
+d=text(4,0.5,txt);
+d.Units='normalized';
+
+% font=22;
+% a.FontSize = font;
+% b.FontSize = font;
+% c.FontSize = font;
+% d.FontSize = font;
+
+pos_txt=[-0.25   0.9];
+a.Position=pos_txt;
+b.Position=pos_txt;
+c.Position=pos_txt;
+d.Position=pos_txt;
+
+
+
+
+
+
+
+
+if ischar(save_path)
+    savefig(h,fullfile(save_path,append(save_name,'_','length_scales.fig')),'compact')
+    for a = 1:length(h)
+       exportgraphics(h(a),fullfile(save_path,append(save_name,'_',sprintf('length_scales_%d.png', a))))
+    end
+end
+close all
 end
\ No newline at end of file
diff --git a/Functions_v5-MATLAB/main.m b/Functions_v5-MATLAB/main.m
index a16a60d..aebb737 100644
--- a/Functions_v5-MATLAB/main.m
+++ b/Functions_v5-MATLAB/main.m
@@ -15,6 +15,9 @@
 
 pre_analysis       = askYesno('Would you like to load a pre-analysis?', 'No');
 
+% answer = 'PART I: Standard turbulence analysis';
+% pre_analysis       = 0;
+
 switch answer
     case 'PART I: Standard turbulence analysis'
         [data,ikx,bar,tmp_bar,save_path,save_name,Fs,m_data,kin_vis,increment_bin,f, E_f_no_filter, f_avg_no_filter, E_avg_no_filter, P_avg_no_filter,filter,...
diff --git a/Functions_v5-MATLAB/plot_conditional_moment.m b/Functions_v5-MATLAB/plot_conditional_moment.m
index a0c0d65..1a270bf 100644
--- a/Functions_v5-MATLAB/plot_conditional_moment.m
+++ b/Functions_v5-MATLAB/plot_conditional_moment.m
@@ -1,134 +1,134 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% This function plots the first and second conditional moments as a function of the scale 
-% separation. For this purpose, a scale and the number of a bin (value of the increment) condition 
-% must be specified. In addition, a linear extrapolation in $\Delta r$ (solid black line) of the 
-% first and second order conditional moments is plotted (see Chapter: Estimation of Kramers-Moyal 
-% coefficients).
-%
-% Arguments IN
-% scal = Plot conditional moment number of Scale
-% bin_num = Number of a bin (value of the velocity increment u_r)
-% evaluated = struct array containing all the information about conditional moments for each scale and for each bin
-% step_con_moment = the steps at which the value of conditional moments are calculated
-% markov = markov length in number of samples
-% multi_point = Multipoint condition 1=YES or 2=NO
-% increment_bin = number of bins
-% condition = This input is for multipoint statistics
-% tol = This input is for multipoint statistics
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-function plot_conditional_moment(scal,bin_num,evaluated,step_con_moment,markov,multi_point,increment_bin,condition,tol,data,save_path,save_name)
-set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
-set(groot, 'defaulttextinterpreter','latex');   
-set(groot, 'defaultLegendInterpreter','latex');
-set(groot, 'defaultAxesTitle','latex');
-
-if multi_point==1
-    k=askInput({sprintf('Plot: Multi-point number of point condition:')}, {num2str(ceil(length(condition)/2))});
-    my_field = sprintf('point_eval_%d', k);  
-    evaluated=evaluated.(my_field);
-end
-
-if (evaluated(scal).r_samp-1)<3*markov
-    step_con_moment_tmp     = unique(round(logspace(0,log10(evaluated(scal).r_samp-1),12)),'stable');
-else  
-    step_con_moment_tmp     = unique(round(logspace(0,log10(3*markov),12)),'stable');
-% step_con_moment_tmp     = round(linspace(0,2*markov,5));
-% step_con_moment_tmp      = round(linspace(markov,2*markov,5));
-end
-
-tau2                = evaluated(scal).r_samp;
-bin_events          = NaN(increment_bin,length(step_con_moment_tmp));
-M11                 = nan; 
-M21                 = nan;
-M31                 = nan;
-M41                 = nan;
-eM1                 = nan;
-eM2                 = nan;
-
-for jj=length(step_con_moment_tmp):-1:1
-    if multi_point==1
-        [incr1,incr2]  = Increment_point((tau2 - step_con_moment_tmp(jj)),tau2,data,condition(k),tol);
-    elseif multi_point==0
-        [incr1,incr2]  = Increment((tau2 - step_con_moment_tmp(jj)),tau2,data);
-    end
-
-    [dx,dy,x,y,dA] = limit(incr1,incr2,increment_bin);
-    if length(x)==increment_bin & length(y)==increment_bin
-       %     [M11_direct(:,jj),M21_direct(:,jj),M41_direct(:,jj),x_mean_bin_direct,y_mean_bin_direct,events] = conditional_moments_direct(tau1,tau2,incr1,incr2,aufl,min_events);
-       %     bin_events_direct(:,jj)=events.';
-        [P_AIB,P_BIA,P_AnB,P_A,P_B,binP_AIB,evaluated(scal).x_mean_bin,evaluated(scal).y_mean_bin,events,counter_A,counter_B] = distribution((tau2 - step_con_moment_tmp(jj)),tau2,incr1,incr2,increment_bin,dx,dy,x,y,dA);
-        bin_events(:,jj)=events.';
-        %----------------------------- conditional moments --------------------------------
-        for i=1:increment_bin
-            if (tau2 - step_con_moment_tmp(jj))>tau2
-                   M11(i,jj)    = nansum(binP_AIB(:,i).^1.*P_AIB(:,i).*dy);
-                   M21(i,jj)    = nansum(binP_AIB(:,i).^2.*P_AIB(:,i).*dy);
-                   M31(i,jj)    = nansum(binP_AIB(:,i).^3.*P_AIB(:,i).*dy);
-                   M41(i,jj)    = nansum(binP_AIB(:,i).^4.*P_AIB(:,i).*dy);
-            else
-                   M11(i,jj)    = nansum(binP_AIB(:,i).^1.*P_AIB(:,i).*dx);
-                   M21(i,jj)    = nansum(binP_AIB(:,i).^2.*P_AIB(:,i).*dx);
-                   M31(i,jj)    = nansum(binP_AIB(:,i).^3.*P_AIB(:,i).*dx);
-                   M41(i,jj)    = nansum(binP_AIB(:,i).^4.*P_AIB(:,i).*dx);
-            end
-        end
-        %Fehler von den Momenten
-        eM1= real(sqrt(abs((M21 - (M11.^2))./bin_events)));
-        eM2= real(sqrt(abs((M41 - (M21.^2))./bin_events)));
-    end
-end
-
-h(1) = figure;
-plot(step_con_moment,evaluated(scal).M11(bin_num,:),'Color',[0.00,0.45,0.74],'MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-hold
-plot(step_con_moment,evaluated(scal).M21(bin_num,:),'Color',[0.85,0.33,0.10],'MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
- 
-plot(step_con_moment_tmp,M11(bin_num,:),'k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-plot(step_con_moment_tmp,M21(bin_num,:),'k','MarkerSize',6,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-
-xlabel('$\Delta r/sample$','Interpreter','latex')
-% ylabel('Conditional Moment')
-set(gca,'FontSize',18)
-set(gcf, 'Color', 'w')
-axis square
-fig_setup
-xlim([min(step_con_moment_tmp)-2 max(step_con_moment_tmp)+2])      
-ylim([min([M11(bin_num,:) M21(bin_num,:)]) max([M11(bin_num,:) M21(bin_num,:)])])      
-
-
-[xData, yData] = prepareCurveData( step_con_moment,evaluated(scal).M11(bin_num,:));
-ft = fittype( 'poly1' );
-opts = fitoptions( 'Method', 'LinearLeastSquares' );
-[fitresult, gof]        = fit( xData, yData, ft, opts );
-plot(linspace(0,2*max(step_con_moment_tmp),100),feval(fitresult,linspace(0,2*max(step_con_moment_tmp),100)),'k','LineWidth',2,'LineStyle','-')
-
-[xData, yData] = prepareCurveData( step_con_moment,evaluated(scal).M21(bin_num,:));
-ft = fittype( 'poly1' );
-opts = fitoptions( 'Method', 'LinearLeastSquares' );
-[fitresult, gof]        = fit( xData, yData, ft, opts );
-plot(linspace(0,2*max(step_con_moment_tmp),100),feval(fitresult,linspace(0,2*max(step_con_moment_tmp),100)),'k','LineWidth',2,'LineStyle','-')
-
-vline(min(step_con_moment),'k')
-vline(max(step_con_moment),'k')
-
-plot(step_con_moment,evaluated(scal).M11(bin_num,:),'Color',[0.00,0.45,0.74],'MarkerSize',10,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-plot(step_con_moment,evaluated(scal).M21(bin_num,:),'Color',[0.85,0.33,0.10],'MarkerSize',10,'Marker','o', 'LineStyle','none','LineWidth',1.5)
-
-legend('$M^{(1)}\left(u_r,r\right)$ ','$M^{(2)}\left(u_r,r\right)$ ','Location','northwest')
-
-if ischar(save_path)
-    savefig(h,fullfile(save_path,append(save_name,'_','con_moment.fig')),'compact')
-    for a = 1:length(h)
-        exportgraphics(h(a),fullfile(save_path,append(save_name,'_',sprintf('con_moment_%d.png', a))))
-    end
-end
-
-set(gca,'xScale','lin') 
-set(gca,'yScale','lin') 
-tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
-uiwait(gcf);
-delete(tmp_ui);
-close
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% This function plots the first and second conditional moments as a function of the scale 
+% separation. For this purpose, a scale and the number of a bin (value of the increment) condition 
+% must be specified. In addition, a linear extrapolation in $\Delta r$ (solid black line) of the 
+% first and second order conditional moments is plotted (see Chapter: Estimation of Kramers-Moyal 
+% coefficients).
+%
+% Arguments IN
+% scal = Plot conditional moment number of Scale
+% bin_num = Number of a bin (value of the velocity increment u_r)
+% evaluated = struct array containing all the information about conditional moments for each scale and for each bin
+% step_con_moment = the steps at which the value of conditional moments are calculated
+% markov = markov length in number of samples
+% multi_point = Multipoint condition 1=YES or 2=NO
+% increment_bin = number of bins
+% condition = This input is for multipoint statistics
+% tol = This input is for multipoint statistics
+% data_filter = filtered data 
+% save_path = path for saving figures and files
+% save_name = name for saving files
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+function plot_conditional_moment(scal,bin_num,evaluated,step_con_moment,markov,multi_point,increment_bin,condition,tol,data,save_path,save_name)
+set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
+set(groot, 'defaulttextinterpreter','latex');   
+set(groot, 'defaultLegendInterpreter','latex');
+set(groot, 'defaultAxesTitle','latex');
+
+if multi_point==1
+    k=askInput({sprintf('Plot: Multi-point number of point condition:')}, {num2str(ceil(length(condition)/2))});
+    my_field = sprintf('point_eval_%d', k);  
+    evaluated=evaluated.(my_field);
+end
+
+if (evaluated(scal).r_samp-1)<3*markov
+    step_con_moment_tmp     = unique(round(logspace(0,log10(evaluated(scal).r_samp-1),12)),'stable');
+else  
+    step_con_moment_tmp     = unique(round(logspace(0,log10(3*markov),12)),'stable');
+% step_con_moment_tmp     = round(linspace(0,2*markov,5));
+% step_con_moment_tmp      = round(linspace(markov,2*markov,5));
+end
+
+tau2                = evaluated(scal).r_samp;
+bin_events          = NaN(increment_bin,length(step_con_moment_tmp));
+M11                 = nan; 
+M21                 = nan;
+M31                 = nan;
+M41                 = nan;
+eM1                 = nan;
+eM2                 = nan;
+
+for jj=length(step_con_moment_tmp):-1:1
+    if multi_point==1
+        [incr1,incr2]  = Increment_point((tau2 - step_con_moment_tmp(jj)),tau2,data,condition(k),tol);
+    elseif multi_point==0
+        [incr1,incr2]  = Increment((tau2 - step_con_moment_tmp(jj)),tau2,data);
+    end
+
+    [dx,dy,x,y,dA] = limit(incr1,incr2,increment_bin);
+    if length(x)==increment_bin & length(y)==increment_bin
+       %     [M11_direct(:,jj),M21_direct(:,jj),M41_direct(:,jj),x_mean_bin_direct,y_mean_bin_direct,events] = conditional_moments_direct(tau1,tau2,incr1,incr2,aufl,min_events);
+       %     bin_events_direct(:,jj)=events.';
+        [P_AIB,P_BIA,P_AnB,P_A,P_B,binP_AIB,evaluated(scal).x_mean_bin,evaluated(scal).y_mean_bin,events,counter_A,counter_B] = distribution((tau2 - step_con_moment_tmp(jj)),tau2,incr1,incr2,increment_bin,dx,dy,x,y,dA);
+        bin_events(:,jj)=events.';
+        %----------------------------- conditional moments --------------------------------
+        for i=1:increment_bin
+            if (tau2 - step_con_moment_tmp(jj))>tau2
+                   M11(i,jj)    = nansum(binP_AIB(:,i).^1.*P_AIB(:,i).*dy);
+                   M21(i,jj)    = nansum(binP_AIB(:,i).^2.*P_AIB(:,i).*dy);
+                   M31(i,jj)    = nansum(binP_AIB(:,i).^3.*P_AIB(:,i).*dy);
+                   M41(i,jj)    = nansum(binP_AIB(:,i).^4.*P_AIB(:,i).*dy);
+            else
+                   M11(i,jj)    = nansum(binP_AIB(:,i).^1.*P_AIB(:,i).*dx);
+                   M21(i,jj)    = nansum(binP_AIB(:,i).^2.*P_AIB(:,i).*dx);
+                   M31(i,jj)    = nansum(binP_AIB(:,i).^3.*P_AIB(:,i).*dx);
+                   M41(i,jj)    = nansum(binP_AIB(:,i).^4.*P_AIB(:,i).*dx);
+            end
+        end
+        %Fehler von den Momenten
+        eM1= real(sqrt(abs((M21 - (M11.^2))./bin_events)));
+        eM2= real(sqrt(abs((M41 - (M21.^2))./bin_events)));
+    end
+end
+
+h(1) = figure;
+plot(step_con_moment,evaluated(scal).M11(bin_num,:),'Color',[0.00,0.45,0.74],'MarkerSize',8,'Marker','o', 'LineStyle','none','LineWidth',2)
+hold
+plot(step_con_moment,evaluated(scal).M21(bin_num,:),'Color',[0.85,0.33,0.10],'MarkerSize',8,'Marker','+', 'LineStyle','none','LineWidth',2)
+
+plot(step_con_moment_tmp,M11(bin_num,:),'k','MarkerSize',8,'Marker','o', 'LineStyle','none','LineWidth',1.5)
+plot(step_con_moment_tmp,M21(bin_num,:),'k','MarkerSize',8,'Marker','+', 'LineStyle','none','LineWidth',1.5)
+
+xlabel('$\Delta r/sample$','Interpreter','latex')
+% ylabel('Conditional Moment')
+set(gca,'FontSize',18)
+set(gcf, 'Color', 'w')
+axis square
+fig_setup
+xlim([min(step_con_moment_tmp)-2 max(step_con_moment_tmp)+2])      
+ylim([min([M11(bin_num,:) M21(bin_num,:)]) max([M11(bin_num,:) M21(bin_num,:)])])      
+
+
+[xData, yData] = prepareCurveData( step_con_moment,evaluated(scal).M11(bin_num,:));
+ft = fittype( 'poly1' );
+opts = fitoptions( 'Method', 'LinearLeastSquares' );
+[fitresult, gof]        = fit( xData, yData, ft, opts );
+plot(linspace(0,2*max(step_con_moment_tmp),100),feval(fitresult,linspace(0,2*max(step_con_moment_tmp),100)),'k','LineWidth',1.5,'LineStyle','-')
+
+[xData, yData] = prepareCurveData( step_con_moment,evaluated(scal).M21(bin_num,:));
+ft = fittype( 'poly1' );
+opts = fitoptions( 'Method', 'LinearLeastSquares' );
+[fitresult, gof]        = fit( xData, yData, ft, opts );
+plot(linspace(0,2*max(step_con_moment_tmp),100),feval(fitresult,linspace(0,2*max(step_con_moment_tmp),100)),'k','LineWidth',1.5,'LineStyle','-')
+
+vline(min(step_con_moment),'k')
+vline(max(step_con_moment),'k')
+
+legend('$M^{(1)}\left(u_r,r\right)$ ','$M^{(2)}\left(u_r,r\right)$ ','Location','northwest')
+
+if ischar(save_path)
+    savefig(h,fullfile(save_path,append(save_name,'_','con_moment.fig')),'compact')
+    for a = 1:length(h)
+        exportgraphics(h(a),fullfile(save_path,append(save_name,'_',sprintf('con_moment_%d.png', a))))
+    end
+end
+
+set(gca,'xScale','lin') 
+set(gca,'yScale','lin') 
+tmp_ui=uicontrol('Position',[10 10 100 40],'String','Continue','Callback','uiresume(gcbf)');
+uiwait(gcf);
+delete(tmp_ui);
+close
 end
\ No newline at end of file
diff --git a/Functions_v5-MATLAB/plot_entropy.m b/Functions_v5-MATLAB/plot_entropy.m
index 80a1558..3275850 100644
--- a/Functions_v5-MATLAB/plot_entropy.m
+++ b/Functions_v5-MATLAB/plot_entropy.m
@@ -12,6 +12,9 @@
 % Sm = Entropy of the medium
 % Ds = Entropy of the system or the Shanon entropy
 % DS = Total entropy production=Sm+Ds
+% increment_bin = number of bins
+% save_path = path for saving figures and files
+% save_name = name for saving files
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 function plot_entropy(Sm,Ds,DS,increment_bin,save_path,save_name)
diff --git a/Functions_v5-MATLAB/plot_increment_pdf.m b/Functions_v5-MATLAB/plot_increment_pdf.m
index e43be28..8345244 100644
--- a/Functions_v5-MATLAB/plot_increment_pdf.m
+++ b/Functions_v5-MATLAB/plot_increment_pdf.m
@@ -2,7 +2,7 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % This function plots the probability density function (PDF) of the velocity increments 
 % at the scale $r=L$, $r=\lambda$ and  $r=\eta$. The colored dashed line correspond to Castaing fits 
-% (form factor $\lambda^2$ \cite{Castaing_1990}) and grey dashed line to Gaussian fits.
+% (form factor $\Lambda_r$ \cite{Castaing_1990}) and grey dashed line to Gaussian fits.
 %
 % Arguments IN
 % data = 1D array of data to which you would like to apply the low pass filter
@@ -122,7 +122,7 @@ function plot_increment_pdf(data,nbins,save_path,save_name,int_L,taylor_L,m_data
     norm1 = pdf(fitdist(incr3,'Normal'),uL);
     plot(uL,norm1,'LineWidth',2,'LineStyle','--','Color',[0.501960813999176 0.501960813999176 0.501960813999176])
     legend({'$u_L$','$u_\lambda$','$u_\eta$'},'Interpreter','latex','Location','northeast','FontSize',18)  
-    legend('$u_L$','$u_\lambda$','$u_\eta$',['Cast. $\lambda^2=$',num2str(Lam_sq_L,'%1.2f')],['Cast. $\lambda^2=$',num2str(Lam_sq_l,'%1.2f')],['Cast. $\lambda^2=$',num2str(Lam_sq_e,'%1.2f')],'Gaussian fit','FontSize',18);
+    legend('$u_L$','$u_\lambda$','$u_\eta$',['Cast. $\Lambda_r=$',num2str(Lam_sq_L,'%1.2f')],['Cast. $\Lambda_r=$',num2str(Lam_sq_l,'%1.2f')],['Cast. $\Lambda_r=$',num2str(Lam_sq_e,'%1.2f')],'Gaussian fit','FontSize',18);
     ylim([0.000001*max(puL) 2*max(pue)])
     title(['$S(u_L)$ = ',num2str(skewness(incr1),'%1.2f'), ';',...
           ' $S(u_\lambda)$ = ',num2str(skewness(incr2),'%1.2f'), ';',...
@@ -172,8 +172,8 @@ function plot_increment_pdf(data,nbins,save_path,save_name,int_L,taylor_L,m_data
     norm1 = pdf(fitdist(incr2,'Normal'),uL);
     plot(uL,norm1,'LineWidth',2,'LineStyle','--','Color',[0.501960813999176 0.501960813999176 0.501960813999176])
 %     legend({'$u_L$','$u_\lambda$'},'Interpreter','latex','Location','northeast','FontSize',18)
-    legend('$u_L$','$u_\lambda$',['Cast. $\lambda^2=$',num2str(Lam_sq_L,'%1.2f')],['Cast. $\lambda^2=$',num2str(Lam_sq_l,'%1.2f')]);
-    
+    legend('$u_L$','$u_\lambda$',['Cast. $\Lambda_r=$',num2str(Lam_sq_L,'%1.2f')],['Cast. $\Lambda_r=$',num2str(Lam_sq_l,'%1.2f')]);
+
     title(['$S(u_L)$ = ',num2str(skewness(incr1),'%1.2f'), ';',...
           ' $S(u_\lambda)$ = ',num2str(skewness(incr2),'%1.2f'), ';',...
     ],'interpreter','latex')
diff --git a/Functions_v5-MATLAB/plot_pointwise_KM.m b/Functions_v5-MATLAB/plot_pointwise_KM.m
index ca5e3bd..71c4895 100644
--- a/Functions_v5-MATLAB/plot_pointwise_KM.m
+++ b/Functions_v5-MATLAB/plot_pointwise_KM.m
@@ -6,6 +6,12 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 function plot_pointwise_KM(y,x,y_mean_bin,x_mean_bin,x_bin_not_nan,x0,x1,P_B,P_AnB,P_AIB,counter_A,counter_B,min_events,eD1,eD2,markov,fval_start,fval,tau1,tau2,m_data,Fs,taylor_L,tol,dy,increment_bin,ub,lb,norm_ur,norm_r,save_path,save_name)
+set(groot, 'defaultAxesTickLabelInterpreter','latex'); 
+set(groot, 'defaulttextinterpreter','latex');   
+set(groot, 'defaultLegendInterpreter','latex');
+set(groot, 'defaultAxesTitle','latex');
+
+
 y_input     = y_mean_bin(counter_B>=min_events);
 x_input     = x_mean_bin(counter_A>=min_events);
 
@@ -351,7 +357,7 @@ function plot_pointwise_KM(y,x,y_mean_bin,x_mean_bin,x_bin_not_nan,x0,x1,P_B,P_A
 ylabel(['$u_{r''=' num2str(tau1/markov,'%1.1f') '\Delta_{EM}}$'], 'interpreter','latex')
 end   
 fig_setup
-title(['$\xi$=' num2str(fval_start,2) '; ' '$\xi_{opti}$=' num2str(fval,2)])
+title(['$\xi_{stp}$=' num2str(fval_start,2) '; ' '$\xi_{stp,opti}$=' num2str(fval,2)])
 xPos = 0.0;
 yPos = 0.0;
 plot([xPos xPos], get(gca,'ylim'),'LineWidth',2,'LineStyle','--','Color','k'); % Adapts to y limits of current axes
@@ -388,7 +394,7 @@ function plot_pointwise_KM(y,x,y_mean_bin,x_mean_bin,x_bin_not_nan,x0,x1,P_B,P_A
 % axis([-5 5 -5 5])
 axis square
 set(gcf, 'Color', 'w')
-title(['$\xi$=' num2str(fval_start,2) '; ' '$\xi_{opti}$=' num2str(fval,2)])
+title(['$\xi_{stp}$=' num2str(fval_start,2) '; ' '$\xi_{stp,opti}$=' num2str(fval,2)])
 xPos = 0.0;
 yPos = 0.0;
 %vertical cut
@@ -436,7 +442,7 @@ function plot_pointwise_KM(y,x,y_mean_bin,x_mean_bin,x_bin_not_nan,x0,x1,P_B,P_A
 set(gcf, 'Color', 'w')
 % legend('experiment','optimized','Location','NorthWest')
 legend('$p_{exp}$','$p_{stp,opti}$','Location','NorthWest','Position',[0.281785714285714 0.657642855417161 0.165338666098458 0.113571428117298])
-title(['$\xi$=' num2str(fval_start,2) '; ' '$\xi_{opti}$=' num2str(fval,2)])
+title(['$\xi_{stp}$=' num2str(fval_start,2) '; ' '$\xi_{stp,opti}$=' num2str(fval,2)])
 % title(['directly=' num2str(fval_start,2) '; ' 'optimized=' num2str(fval,2)])
 grid off
 % xlabel(['$u_{r_2=' num2str(tau2/markov,'%1.1f') '\Delta_{EM}}/ \sigma_\infty$'], 'interpreter','latex')
diff --git a/Functions_v5-MATLAB/spectrum.m b/Functions_v5-MATLAB/spectrum.m
index 1e4b598..1a54934 100644
--- a/Functions_v5-MATLAB/spectrum.m
+++ b/Functions_v5-MATLAB/spectrum.m
@@ -7,6 +7,8 @@
 %
 % Arguments IN
 % data = 1D array for which you would like to plot power spectram density(PSD)
+% Fs= Sampling frequency in Hz
+% increment_bin = The number of bins in which you would like to divide your data
 % 
 % Arguments OUT
 % f = Frequency without smoothing