Estimating the decay parameter in Exponentially Weighted Moving Average (EWMA) model

[0xF0F0F0F]

Given the data y_t, t=1, ... ,N; I would like to estimate the decay parameter "lambda" in Exponentially Weighted Moving Average (EWMA) model, such that

y_{t+1} = \sum_{k=0}^K lambda^k y_{t - k} + errors_t,
where K<N.

I am wondering, can Yalmip solve this nonlinear least squares problem? Is it possible to make this problem more "friendly" by changing of the variables?

[johanlofberg]

I'm a bit confused by the notation/formulation. No driving input to the filter?

Regardless I don't see any simple fix. Surely there are methods in the associated literature if this is considered a standard setup

Having said that, for small problems a global solver has no problems solving this

lambda = 0.5;
y = [ .1 .1 .1];
for i = 1:100
    y = [sum([lambda lambda.^2 lambda^3].*y(1:3)) + randn*0.01 y];
end
plot(y)

theta = sdpvar(3,1);
objective = 0;
for i = 1:length(y)-3
    objective = objective + (y(i)-y(i+1:i+3)*theta)^2;
end
optimize([0<=theta<=1, theta(3) == theta(2)*theta(1), theta(2) == theta(1)^2],objective,sdpsettings('solver','bmibnb'))
* Starting YALMIP global branch & bound.
* Upper solver     : fmincon
* Lower solver     : MOSEK
* LP solver        : MOSEK
* -Extracting bounds from model
* -Perfoming root-node bound propagation
* -Trivial solution constructed (objective 0.082348)
* -Calling upper solver (found a solution, objective 0.0087714)
* -Branch-variables : 3
* -More root-node bound-propagation
* -Performing LP-based bound-propagation 
* -And some more root-node bound-propagation
* Starting the b&b process
 Node       Upper       Gap(%)       Lower     Open   Time
    1 :   8.77136E-03     0.00    8.77136E-03    0     0s  Poor lower bound  
* Finished.  Cost: 0.0087714 (lower bound: 0.0087714, relative gap 0%)
* Termination with all nodes pruned 
* Timing: 36% spent in upper solver (1 problems solved)
*         2% spent in lower solver (1 problems solved)
*         24% spent in LP-based domain reduction (16 problems solved)
*         2% spent in upper heuristics (10 candidates tried)

sdpvar lambda
theta = [lambda;lambda^2;lambda^3]
objective = 0;
for i = 1:length(y)-3
    objective = objective + (y(i)-y(i+1:i+3)*theta)^2;
end
>> optimize([0<=lambda<=1],objective,sdpsettings('solver','bmibnb'))
* Starting YALMIP global branch & bound.
* Upper solver     : fmincon
* Lower solver     : MOSEK
* LP solver        : MOSEK
* -Extracting bounds from model
* -Perfoming root-node bound propagation
* -Trivial solution constructed (objective 0.0087767)
* -Calling upper solver (found a solution, objective 0.0087714)
* -Branch-variables : 1
* -More root-node bound-propagation
* -Performing LP-based bound-propagation 
* -And some more root-node bound-propagation
* Starting the b&b process
 Node       Upper       Gap(%)       Lower     Open   Time
    1 :   8.77136E-03     0.06    8.20489E-03    2     0s    
* Finished.  Cost: 0.0087714 (lower bound: 0.0082049, relative gap 0.05617%)
* Termination with relative gap satisfied 
* Timing: 32% spent in upper solver (2 problems solved)
*         4% spent in lower solver (1 problems solved)
*         20% spent in LP-based domain reduction (8 problems solved)
*         1% spent in upper heuristics (7 candidates tried)

[0xF0F0F0F]

Many thanks! Johan