X03. Introduction to Statistics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# X03: Introduction to Statistics\n",
    "\n",
    "Before we do any analysis or use any advanced techniques like machine learning, it's important to have a grasp of statistics and some of the functions that are available to us within Python. \n",
    "\n",
    "Here we are going to look at:\n",
    "\n",
    "- Item frequency tables\n",
    "- Histograms\n",
    "- Mean, median, standard deviation etc\n",
    "- Skew\n",
    "- p-values\n",
    "- Chi squared\n",
    "\n",
    "Which are just a few of the statistical functions that are available to us in the stats library within scipy.\n",
    "\n",
    "So first we'll load scipy.stats, numpy (so we can use some ot its mathematical functions) and matplotlib (so we can plot our results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import scipy.stats as sps\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll create a random integer uniform distribution, a random normal distribution and a random poisson distribution to look at first of all."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "uniform = np.random.random_integers(10,size=10000)\n",
    "normal = np.random.randn(10000)\n",
    "poisson = np.random.poisson(1,10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can then create a item frequency table of these distributions using the stats.itemfreq function in scipy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[   1, 1010],\n",
       "       [   2,  973],\n",
       "       [   3, 1006],\n",
       "       [   4,  992],\n",
       "       [   5, 1005],\n",
       "       [   6,  966],\n",
       "       [   7, 1028],\n",
       "       [   8,  965],\n",
       "       [   9, 1023],\n",
       "       [  10, 1032]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sps.itemfreq(uniform)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This has given us a table of all the numbers in the distribution and the number of times they appear in the array \"uniform\".\n",
    "\n",
    "Another way we can look at these distributions is using a histogram, which we can do within scipy, but we're going to do it straight into matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAEACAYAAABI5zaHAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nphPvz34e5AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5e807b5550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, bins, patches = plt.hist(normal, bins=50)    #n is the number of items in each bin, bins gives the bin edges"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which we can also normalise to 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXoAAAEACAYAAAC9Gb03AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nmeglqeVM9JLUcv8f4Jx+8QL04c0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5e807bcc18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, bins, patches = plt.hist(normal, bins=50, normed=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And plot the cumulative distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXQAAAEACAYAAACj0I2EAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nACgEgQ4AhSDQAaAQ/w9/OVcYPMdPQwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5e805737f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, bins, patches = plt.hist(normal, bins=50, normed=True, cumulative=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also look at some stats of this distribution, such as the mean, the median and the standard deviation using numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00023238530555500745"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(normal)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0098224699474755931"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.median(normal)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0029457351448781"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(normal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is one standard deviation, so we'd expect 68% of all of our points to lie within the range \n",
    "\n",
    "mean - std and mean + std\n",
    "\n",
    "If we wanted to know where 95% of all our points lie we would multiply this by two, so the range is \n",
    "\n",
    "mean - 2 x std to mean + 2 x std"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using scipy we can find the skew (how assymetric the distribution is)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sps.skew(normal)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All these stats quantify what our distribution looks like, but the best way to quantify this is to try to find a generalised function that fits the distribution well. To find out how well our distribution fits we can use variuos measures of goodness of fit, but here we're going to focus on chi^2, which is just calculated using the following equation:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\chi^2 = \\Sigma ((data - theory)/(variance))$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Where the variance is std^2. This is the expected error on any measurement, so for a single data point where the data and the theory were within the measurement errors we'd expect chi^2<=1. This means for our 50 bins (data points) above we'd expect chi^2<=50."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So let's try to fit a normal distribution to our \"normal\" array.\n",
    "\n",
    "First we need to define the type of function we want to fit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def normalfunction(x, A, mu, sigma):\n",
    "    return A*np.exp(-(x-mu)**2/(2.*sigma**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then we need to fit this Gaussian using scipy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f5e803bdc18>]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAEACAYAAABI5zaHAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nkruISAIpuYuIJJCSu4hIAim5i4gk0P8HNRze2jkNw2oAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f5e8045df98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.optimize import curve_fit\n",
    "\n",
    "#Build our histogram again\n",
    "n, bins, patches = plt.hist(normal, bins=50)   \n",
    "#Find our bin centres to use as our x values in the fit\n",
    "bincentres=bins[0:(len(bins)-1)]+np.diff(bins)/2\n",
    "\n",
    "#Fit the normal distribution function we created\n",
    "coefficients, covariance = curve_fit(normalfunction, bincentres, n)\n",
    "#Find the y values of our fitted function\n",
    "fittedFunction = normalfunction(bincentres,coefficients[0],coefficients[1],coefficients[2])  \n",
    "#Plot our fitted function over the top of our histogram to see how it looks\n",
    "plt.plot(bincentres, fittedFunction, 'r--', linewidth=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can check how good that fit is using chi^2 and the p-value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Power_divergenceResult(statistic=49.642671511145934, pvalue=0.44750267718076958)"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sps.chisquare(n, fittedFunction) #The statistic quoted is the chi^2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An excellent fit!"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}