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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Mixture Model Fitting: an Expectation Maximization approach"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div align=\"center\">Author: **Antonino Ingargiola** ([github page](https://github.com/tritemio))</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Abstract\n",
      "*In this notebook I describe the fitting of a 2-component mixture sample (i.e. two Gaussians) \n",
      "using the [Expectation Maximization](http://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm)\n",
      "approach.*\n",
      "\n",
      "*I start showing that the direct minimization of the log-likelihood function is not\n",
      "very roboust and poses convergence problems. Then I introduce the Expectation Maximization method\n",
      "and apply it to the fitting of two gaussians. Finally I consider the heteroscetastic case\n",
      "(samples with different variance). Drawing an analogy with the \n",
      "[Weights Least Squares]() method (for a 1-compontent population), \n",
      "I propose a \"Weighted Expectation Maximization\" method\n",
      "that improves the fitting accuracy of a 2-component mixture population.*"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Introduction"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "When a sample is drawn from a number $k$ of different distributions we talk of a\n",
      "**[Mixture Model](http://en.wikipedia.org/wiki/Mixture_model)**.\n",
      "\n",
      "In the common case in which the $k$ distributions are Gaussian we talk of \n",
      "[Gaussian Mixture Model (GMM)](http://en.wikipedia.org/wiki/Mixture_model#Gaussian_mixture_model).\n",
      "\n",
      "Fitting a Mixure model, if the number $k$ of ditributions is known, can be done by:\n",
      "\n",
      "- [K-mean](http://en.wikipedia.org/wiki/K-means_clustering) or [K-medians](http://en.wikipedia.org/wiki/K-medians_clustering) algorithm. These are clustering algorithms and return only \n",
      "the centroids and the boundaries of the different componets (although variance can be of course computed empirically after clustering).\n",
      "- [Expectation Maximization (EM)](http://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm): \n",
      "finds the [Maximum Likelihood (ML)](http://en.wikipedia.org/wiki/Maximum_likelihood) estimated of the model parameters.\n",
      "\n",
      "In case the number of components $k$ is unknown can be esitmated with the [BIC Criterion](http://scikit-learn.org/stable/modules/mixture.html#selecting-the-number-of-components-in-a-classical-gmm). \n",
      "\n",
      "Another alternative is using the [Dirichlet Process GMM](http://scikit-learn.org/stable/modules/mixture.html#dpgmm).\n",
      "\n",
      "In this notebook we describe the EM method for a GMM distribution with only 2 components.\n",
      "\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "References"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* [Estimating Gaussian Mixture Densities with EM \u2013 A Tutorial](https://www.cs.duke.edu/courses/spring04/cps196.1/handouts/EM/tomasiEM.pdf) *(PDF)*\n",
      "* [EM for Gaussian Mixtures](http://www.slideshare.net/petitegeek/expectation-maximization-and-gaussian-mixture-models) *(slides)*\n",
      "* [What is the expectation maximization algorithm?](http://www.nature.com/nbt/journal/v26/n8/full/nbt1406.html?pagewanted=all)\n",
      "\n",
      "*  [Data Mining Algorithms In R/Expectation Maximization (EM)](http://en.wikibooks.org/wiki/Data_Mining_Algorithms_In_R/Clustering/Expectation_Maximization_%28EM%29)"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Existing Python implementations"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Python packages implementing EM for GMM:\n",
      "\n",
      "* [scikit-learn](http://scikit-learn.org/stable/modules/mixture.html)\n",
      "* [PyMix](http://www.pymix.org/pymix/index.php?n=PyMix.Tutorial)\n",
      "* [PyPR](http://pypr.sourceforge.net/mog.html)\n",
      "* [PyMC](http://pymc-devs.github.io/pymc/README.html)\n",
      "\n",
      "Short examples/scripts implementing EM:\n",
      "\n",
      "* [577735-expectation-maximization](http://code.activestate.com/recipes/577735-expectation-maximization/)"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      " Simple Mixture Model"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Problem introduction"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "from numpy import random\n",
      "from scipy.optimize import minimize, show_options"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As a didactical example we want to fit a mixture of two univariate Gaussian distributions.\n",
      "Let call $s = \\{s_i\\}$ a sample of $N$ elements extracted form the mixture distribution,\n",
      "and $s_{1} = \\{s_{1i}\\}$ and $s_{2} = \\{s_{2i}\\}$ the samples extracted from\n",
      "each single Gaussian distribution. In the following we will assume that $s_1$ and $s_2$\n",
      "are not known."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N = 1000\n",
      "a = 0.3\n",
      "s1 =  normal(0, 0.08, size=N*a)\n",
      "s2 = normal(0.6,0.12, size=N*(1-a))\n",
      "s = concatenate([s1,s2])\n",
      "hist(s, bins=20);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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PRl1dne1QRESkB7/j1YScY2cDTgewRMp8fjXnxV+abtXRoyd1RjIRAL/jVQ8JGe1gkSfq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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x2c48910>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The model for this sample is the linear combination of two Gaussian PDF:\n",
      "\n",
      "$$f(x|p) = \\frac{\\pi_1}{\\sigma_1\\sqrt{2\\pi}} {\\rm exp} \\left\\{ -\\frac{(x-\\mu_1)^2}{2\\sigma_1^2} \\right\\} +\n",
      "\\frac{\\pi_2}{\\sigma_2\\sqrt{2\\pi}} {\\rm exp} \\left\\{ -\\frac{(x-\\mu_2)^2}{2\\sigma_2^2} \\right\\}$$\n",
      "\n",
      "where $p = [\\mu_1, \\sigma_1, \\mu_2, \\sigma_2, \\pi_1]$. \n",
      "Note that $\\pi_2$ is not included in $p$ since $\\pi_2 = 1-\\pi_1$.\n",
      "\n",
      "\n",
      "\n",
      "In python we can define $f(x|p)$ using `normpdf()` implemented by *Numpy*: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def pdf_model(x, p):\n",
      "    mu1, sig1, mu2, sig2, pi_1 = p\n",
      "    return pi_1*normpdf(x, mu1, sig1) + (1-pi_1)*normpdf(x, mu2, sig2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This function will be used in the following sections."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Maximum Likelihood: direct maximization"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      ">**NOTE**\n",
      ">\n",
      ">*This section illustrate the direct Maximum Likelihood method (i.e. direct minimization\n",
      ">of the -log-likelihood function). \n",
      ">The purpose is to show that a direct minimization is much more complex and \n",
      ">less robust than the EM algorithm. You can skip this section if you are already conviced\n",
      ">and you want to read about the EM algorithm.*\n",
      "\n",
      "If we try to apply the the **Maximum Likelihood (ML)** method directly, we find that \n",
      "the log-likelihood function is quite difficult to minimize numerically.\n",
      "\n",
      "Given a sample $s = \\{s_i\\}$ of size $N$ extracted from the mixture distribution, the likelihood function is\n",
      "\n",
      "$$\\mathcal{L(p,s)} = \\prod_i f(s_i|p)$$\n",
      "\n",
      "and the log-likelihood function is:\n",
      "\n",
      "$$\\ln \\mathcal{L(p,s)} = \\sum_i \\ln f(s_i|p)$$\n",
      "\n",
      "Now, since $f(\\cdot)$ is the sum of two terms, the term $\\log f(s_i|p)$ can't be simplified (it's the log of a sum). \n",
      "So for each $s_i$ we must compute the log of the sum of two exponetial. \n",
      "It's clear that not only the computation will be slow but also the numerical errors will be amplified.\n",
      "Moreover, often the likelihood function has local maxima other than the global one\n",
      "(in other terms the function is not convex).\n",
      "\n",
      "In python the log-likelihood function can be defined as:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def log_likelihood_two_1d_gauss(p, sample):\n",
      "    return -log(pdf_model(sample, p)).sum()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we try to minimize it starting with a close-to-real intial guess $p_0 = [-0.2,0.2,0.8,0.2,0.5]$ we fail with the most common methods.\n",
      "\n",
      ">**Python NOTES:**\n",
      ">\n",
      ">   - Here we use the function `minimize()` from `scipy.optimization`. This function allows\n",
      ">   to choose several minimization methods. The list of (currently) available\n",
      ">   minimization methods is\n",
      "> `'Nelder-Mead'` *(simplex)*, `'Powell'`, `'CG'`, `'BFGS'`, `'Newton-CG'`,>\n",
      "> `'Anneal'`, `'L-BFGS-B'` *(like BFGS but bounded)*, `'TNC'`, `'COBYLA'`, `'SLSQPG'`.\n",
      ">   \n",
      ">   The documetation can be found \n",
      ">   [here](http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html#scipy.optimize.minimize)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Initial guess\n",
      "p0 = array([-0.2,0.2,0.8,0.2,0.5])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Minimization 1\n",
      "res = minimize(log_likelihood_two_1d_gauss, x0=p0, args=(s,), method='BFGS')\n",
      "#res # NOT CONVERGED"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Minimization 2\n",
      "res = minimize(log_likelihood_two_1d_gauss, x0=p0, args=(s,), method='powell',\n",
      "        options=dict(maxiter=10e3, maxfev=2e4))\n",
      "#res # NOT CONVERGED"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Minimization 3\n",
      "res = minimize(log_likelihood_two_1d_gauss, x0=p0, args=(s,), method='Nelder-Mead',\n",
      "        options=dict(maxiter=10e3, maxfev=2e4))\n",
      "res"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "  status: 0\n",
        "    nfev: 428\n",
        " success: True\n",
        "     fun: -191.90433284079629\n",
        "       x: array([ 0.00650312,  0.08413543,  0.59445438,  0.12201147,  0.30045492])\n",
        " message: 'Optimization terminated successfully.'\n",
        "     nit: 262"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "res.x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "array([ 0.00650312,  0.08413543,  0.59445438,  0.12201147,  0.30045492])"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Minimization 4\n",
      "res = minimize(log_likelihood_two_1d_gauss, x0=p0, args=(s,), method='L-BFGS-B',\n",
      "    bounds=[(-0.5,2),(0.01,0.5),(-0.5,2),(0.01,0.5),(0.01,0.99)])\n",
      "res"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "  status: 0\n",
        " success: True\n",
        "    nfev: 54\n",
        "     fun: -191.90437529910372\n",
        "       x: array([ 0.59445438,  0.12199155,  0.00648396,  0.08411654,  0.69951773])\n",
        " message: 'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'\n",
        "     jac: array([ 0.00275406,  0.00739533, -0.00014779, -0.00086118,  0.00196962])\n",
        "     nit: 31"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Finds additional options for the different solvers:\n",
      "#show_options('minimize', 'powell')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Executing the minimizations we see that we have convergence only using the `'Nelder-Mead'` (simplex) and the `'L-BFGS-B'`. \n",
      "methods.\n",
      "The latter is much ~10x faster as requires only 28 iteration and 40 function evaluation \n",
      "(instead of 256 iterations and 413 function evaluations).\n",
      "Refer to the **Scipy** documentation for a \n",
      "[description of the methods](http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html#scipy.optimize.minimize)."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Expectation Maximization"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Introduction"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The EM method allows to fit a statistical model in the case where the experimental data\n",
      "has unknown (latent) variables. In the context of a mixture-model fitting\n",
      "we can assume that the latent variables \"tell\" which component has generated each sample.\n",
      "\n",
      "With the EM method, we first \"assign\" each sample to each components of the distribution.\n",
      "After that, we can compute MLE estimators of parameters\n",
      "of each component of the mixture. For Gaussian components the MLE will be basically\n",
      "the empirical mean and variance.\n",
      "\n",
      "It is possible to do the assigment choosing, for each sample, the component that has \n",
      "the highest probability to generate the sample (**hard assignment**). \n",
      "\n",
      "Alternatively it's possible to compute, for each sample $s_i$, the \"fraction\" \n",
      "of $s_i$ generated by each component (**soft assignment**). \n",
      "\n",
      "In our example, for each sample $s_i$ we have two coefficients\n",
      "$\\gamma(i,1)$ and $\\gamma(i,2)$ that represent the fraction of $s_i$ that\n",
      "belongs to (respectively) component 1 and 2. The sum $\\gamma(i,1)+\\gamma(i,2) = 1$. \n",
      "$\\gamma()$ is called in litterature **responsibility fuction**.\n",
      "This soft assignment method is the one commonly used in the context of \n",
      "EM algorithms for mixture model fitting.\n",
      "\n",
      "Only the **soft assignment** scheme will be used is the following sections."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Algorithm"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Starting from the PDF $f_1()$ and $f_2()$ of the single components:\n",
      "\n",
      "$$f_1(x|p) = \\frac{1}{\\sigma_1\\sqrt{2\\pi}} {\\rm exp} \\left\\{ -\\frac{(x-\\mu_1)^2}{2\\sigma_1^2} \\right\\}\n",
      "\\qquad\n",
      "f_2(x|p) = \\frac{1}{\\sigma_2\\sqrt{2\\pi}} {\\rm exp} \\left\\{ -\\frac{(x-\\mu_2)^2}{2\\sigma_2^2} \\right\\}\n",
      "$$\n",
      "\n",
      "the mixture PDF is:\n",
      "\n",
      "$$ f(x|p) = \\pi_1 f_1(x|p) + \\pi_2 f_2(x|p) $$\n",
      "\n",
      "If we know (or guess initially) the parameters $p$, we can compute for each sample \n",
      "and each component the **responsibility function** defined as:\n",
      "    \n",
      "$$\\gamma(i, k) =  \\frac{\\pi_kf_k(s_i|p)}{f(s_i|p)}$$\n",
      "\n",
      "and starting from the \"effective\" number of samples for each category ($N_k$) we can compute the \n",
      "\"new\" estimation of parameters:\n",
      "    \n",
      "$$N_k = \\sum_{i=1}^N \\gamma(i, k)\n",
      "\\qquad\\qquad\n",
      "k=1,2\n",
      "\\quad\n",
      "({\\rm note\\;that} \\quad N_1 + N_2 = N)\n",
      "$$\n",
      "\n",
      "$$\\mu_k^{new} = \\frac{1}{N_k}\\sum_{i=1}^N \\gamma(i, k) \\cdot s_i$$\n",
      "\n",
      "$$\\sigma_k^{2\\,new} = \\frac{1}{N_k}\\sum_{i=1}^N \\gamma(i, k) \\cdot (s_i - \\mu_k^{new})^2$$\n",
      "\n",
      "$$\\pi_k^{new} = \\frac{N_k}{N}$$\n",
      "\n",
      "Now we just loop \n",
      "\n",
      " 1. recompute $\\gamma(i, k)$\n",
      " 2. estimate updated parameters\n",
      "\n",
      "until convergence.\n",
      "\n",
      "**REMARKS**\n",
      "\n",
      "- There is no minimization! It's just an iterative algorithm that theory \n",
      "  guarantee to converge to a (local) minimum.\n",
      "    \n",
      "- We don't even write the likelihood function but we obtain a ML estimation.\n",
      "    \n",
      "- Even if the components of the mixture are not Gaussian, the method works as far as it's possible\n",
      "  to determine the distribution parameters from empirical moments. For example:\n",
      "        \n",
      "     - Poisson distribution, MLE: $$\\hat{\\mu} = \\frac{1}{N}\\sum s_i$$\n",
      "     - Binomial distribution, MLE: $$\\hat{p} = \\frac{N_{success}}{N} $$"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Implementation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let implement this EM algorithm in python. For simplicity the iteration is stopped only after a fixed number of iteration:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "max_iter = 100\n",
      "\n",
      "# Initial guess of parameters and initializations\n",
      "p0 = array([-0.2,0.2,0.8,0.2,0.5])\n",
      "mu1, sig1, mu2, sig2, pi_1 = p0\n",
      "mu = array([mu1, mu2])\n",
      "sig = array([sig1, sig2])\n",
      "pi_ = array([pi_1, 1-pi_1])\n",
      "\n",
      "gamma = zeros((2, s.size))\n",
      "N_ = zeros(2)\n",
      "p_new = p0\n",
      "\n",
      "# EM loop\n",
      "counter = 0\n",
      "converged = False\n",
      "while not converged:\n",
      "    # Compute the responsibility func. and new parameters\n",
      "    for k in [0,1]:\n",
      "        gamma[k,:] = pi_[k]*normpdf(s, mu[k], sig[k])/pdf_model(s, p_new)\n",
      "        N_[k] = 1.*gamma[k].sum()\n",
      "        mu[k] = sum(gamma[k]*s)/N_[k]\n",
      "        sig[k] = sqrt( sum(gamma[k]*(s-mu[k])**2)/N_[k] )\n",
      "        pi_[k] = N_[k]/s.size\n",
      "    p_new = [mu[0], sig[0], mu[1], sig[1], pi_[0]]\n",
      "    assert abs(N_.sum() - N)/float(N) < 1e-6 \n",
      "    assert abs(pi_.sum() - 1) < 1e-6\n",
      "    \n",
      "    # Convergence check\n",
      "    counter += 1\n",
      "    converged = counter >= max_iter"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \"Means:   %6.3f  %6.3f\" % (p_new[0], p_new[2])\n",
      "print \"Std dev: %6.3f  %6.3f\" % (p_new[1], p_new[3])\n",
      "print \"Mix (1): %6.3f \" % p_new[4]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Means:    0.006   0.594\n",
        "Std dev:  0.084   0.122\n",
        "Mix (1):  0.300 \n"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print pi_.sum(), N_.sum()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1.0 1000.0\n"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "res.x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "array([ 0.59445438,  0.12199155,  0.00648396,  0.08411654,  0.69951773])"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Heteroscetastic Mixture Model"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Introduction"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the first examples the samples are extracted from a mixture of two gaussians. \n",
      "The samples of each gaussian component are \n",
      "[i.i.d.](http://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables) or, \n",
      "in other words each component is [homoscedastic](http://en.wikipedia.org/wiki/Homoscedasticity).\n",
      "\n",
      "In some cases, the components of the mixture are not **homoscedastic** but [**heteroscedastic**](http://en.wikipedia.org/wiki/Heteroscedasticity). This means that within a single component/population\n",
      "of the mixture each sample has a different variance. The variance is assumed to be known."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Non-mixture case: Weighted Least Squares"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let consider first the case of a single population $s = \\{s_i\\}$ (no mixture), \n",
      "in which all the samples have same mean $\\mu$ but different variances $\\sigma^2_i$. \n",
      "Assuming that the mean square error is **normally distributed** with variance $\\sigma_i$, than the \n",
      "[**Weighted Least Square**](http://en.wikipedia.org/wiki/Weighted_least_squares#Weighted_least_squares)\n",
      "method allows to obtain a **MLE** of the mean:\n",
      "\n",
      "$$ \\hat{\\mu}_{ML} = \\frac{\\sum_i w_i \\cdot s_i}{\\sum_i w_i} \n",
      "\\quad\\quad{\\rm and}\\quad\\quad\n",
      "w_i = \\frac{1}{\\sigma^2_i}\n",
      "$$\n",
      "\n",
      "We can still compute the **mean square error** of the samples:\n",
      "\n",
      "$$ \\sigma_m^2 = \\frac{1}{N}\\sum_i (s_i - \\hat{\\mu}_{ML})^2 $$\n",
      "\n",
      "but it will now have the meaning of \"distribution parameter\" (i.e. the variance) like \n",
      "in the case of i.i.d. gaussian samples. Nonetheless $\\sigma_m^2$\n",
      "provides [descriptive](https://en.wikipedia.org/wiki/Descriptive_statistics) \n",
      "information about the sample variability (or dispersion). \n",
      "\n",
      "We may also introduce other measures of variability by weighting the\n",
      "errors in different ways:\n",
      "\n",
      "$$ \\sigma_w^2 = \\frac{\\sum_i w_i (s_i - \\hat{\\mu}_{ML})^2 }{\\sum_i w_i} $$\n",
      "\n",
      "For example let say we want to compare the variability of the two population.\n",
      "An higher variability will mean necessary higher error in the estimator $\\hat{\\mu}$.\n",
      "\n",
      "But if we compute $\\hat{\\mu}_{ML}$ weighting each sample, then samples with high variance\n",
      "will affect less the average than samples with small variance. Therefore\n",
      "if we want to estimate the \"accuracy\" (or simply the variance) \n",
      "of $\\hat{\\mu}_{ML}$ then we need to weight **less** the **mean square errors** of\n",
      "samples with high variance. In this context seems natural to use\n",
      "$w_i = 1/\\sigma_i^2$ also to compute $\\sigma_w^2$.\n",
      "\n",
      "> **NOTE**\n",
      "> A practical consideration is due here. If the sample population is large enough then\n",
      "> a small variation in the sample dispersion will not significantly affect $\\hat{\\mu}$. \n",
      "> In fact, \n",
      "> the fitting error it's already small (due to the high # samples). \n",
      "> If the theorethical fitting error \n",
      "> is below a (application-specific) \"significativity threshold\" then \n",
      "> the effect of a \"small\" variance difference beween two population becomes\n",
      "> completely negligible. In this case other effects (for example model mismatch)\n",
      "> will dominate the fitting error.\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Mixture case: Weighted Expectation Maximization"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To generalize the EM method in case of samples with different variance we can apply different schemes."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "***SCHEME 1***\n",
      "\n",
      "$$\\gamma(i, k) = \\frac{\\pi_kf_k(s_i|p)}{f(s_i|p)} \\color{red}{w_i}$$\n",
      "\n",
      "\n",
      "$N_k$, $\\mu_k^{new}$, $\\sigma_k^{2\\,new}$, $\\pi_k^{new}$ computed as the no-weight case \n",
      "(but using the new definition of $\\gamma$).\n",
      "\n",
      "*Requirement:* $\\sum_i w_i = N$\n",
      "\n",
      "> **NOTE**\n",
      "> In this scheme we **weight more** samples with lower variance. They contribute more \n",
      "> to $N_k = \\sum_i \\gamma(k,i)$ (thus $\\pi_k = N_k/N$). \n",
      "> Also $\\mu_k^{new}$ and to $\\sigma_k^{2\\,new}$ are \"weighted means\": we simply redistribute the weights\n",
      "> including also the information on the variance. \n",
      "> The effect on $\\sigma_k^{2\\,new}$ is that the contribution of high variance samples is less, \n",
      "> so the extimated variance is smaller. This helps to resolve better the sub-populations.\n",
      "> The reduction in the estimated variance is \"consistent\" (not too big) because it is a\n",
      "> weighted mean: a sample with high variance \"weights\" less in the sum, but also contribute\n",
      "> less to the sum of weights (i.e. the normalization of the weighted mean).\n",
      "\n",
      "* **Question**: $N_1 + N_2 = N$ ?\n",
      "    * YES: \n",
      "        1. We set as requirement:$\\quad\\sum_i w_i = N$. \n",
      "        2. From the $N_k$ definition: $\\quad N_1+N_2 = \\sum_{i,k} \\gamma(i,k)$.\n",
      "        3. For each $i$: $\\quad\\gamma(i,1) + \\gamma(i,2) = w_i\\quad$ \n",
      "            *(this sum is 1 in the no-weight case)*\n",
      "        4. $\\Rightarrow \\quad N_1+N_2 =\\sum_{i,k} \\gamma(i,k) = \\sum_i w_i = N$\n",
      "* **Question**: $\\pi_1 + \\pi_2 = 1$? \n",
      "    * YES: Since $\\pi_k = N_k/N$ and $N_1+N_2=N$, \n",
      "        $\\quad\\Rightarrow\\quad \\pi_1 + \\pi_2 = N_1/N+N_2/N = 1$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "***SCHEME 2***\n",
      "\n",
      "Here we compute the mean $\\mu_k^{new}$ exactly as in SCHEME1, \"reweighting\" the weights with $w_i$ (here \n",
      "the equation is explicit, in SCHEME1 we included $w_i$ in $\\gamma$):\n",
      "\n",
      "$$\\mu_k^{new} = \\frac{\\sum_{i=1}^N w_i\\gamma(i, k) \\cdot s_i}{N_k \\sum_i w_i}$$\n",
      "\n",
      "No other modification is done compared to the no-weight case\n",
      "\n",
      "> **NOTES**\n",
      ">\n",
      "> This approach is less \"invasive\" and has the advantage to give real empirical variance\n",
      "> (or mean square error) of each sub-population. In contrast with the no-weight case\n",
      "> this apprach gives better estimate of the mean. However (like the no-weight) is less\n",
      "> [efficient](http://en.wikipedia.org/wiki/Efficiency_(statistics). \n",
      "> In ohter words it uses \"less\" information than SCHEME1 \n",
      "> (and requires more iterations to converge).\n",
      "\n",
      "> **WARNING**\n",
      "> This method has serious converge problems if initial guess is not very close\n",
      "> to the real value."
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "The code"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First we define a functions to generate the sample populations:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.stats import expon\n",
      "\n",
      "def sim_single_population(mu, N=1000, max_sigma=0.5, mean_sigma=0.08):\n",
      "    \"\"\"Extract samples from a normal distribution \n",
      "    with variance distributed as an exponetial distribution\n",
      "    \"\"\"\n",
      "    exp_min_size = 1./max_sigma**2\n",
      "    exp_mean_size = 1./mean_sigma**2\n",
      "    sigma = 1/sqrt(expon.rvs(loc=exp_min_size, scale=exp_mean_size, size=N))\n",
      "    return normal(mu, scale=sigma, size=N), sigma"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N = 1000\n",
      "a = 0.3\n",
      "s1, sig1 = sim_single_population(0, N=N*a)\n",
      "s2, sig2 = sim_single_population(0.5, N=N*(1-a))\n",
      "s = concatenate([s1, s2])\n",
      "sigma_tot = concatenate([sig1, sig2])\n",
      "hist(s, bins=r_[-1:2:0.025], alpha=0.3, color='g', histtype='stepfilled');\n",
      "ax = twinx(); ax.grid(False)\n",
      "ax.plot(s, 0.1/sigma_tot, 'o', mew=0, ms=2, alpha=0.6, color='b')\n",
      "xlim(-0.5, 1.5); title('Simulated sample (to be fitted)')\n",
      "print \"Means:   %6.3f  %6.3f\" % (s1.mean(), s2.mean())\n",
      "print \"Std dev: %6.3f  %6.3f\" % (sqrt((sig1**2).mean()), sqrt((sig2**2).mean()))\n",
      "print \"Mix (1): %6.3f \" % a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Means:   -0.007   0.496\n",
        "Std dev:  0.144   0.146\n",
        "Mix (1):  0.300 \n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Dbh/Xg2b4Ep2Ov//NHznr+WfyUqw/4YqUl5cjOTkZMTExGDp0KD755BOLMvn5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\nAIMIfwAwiPAHAIMIfwAw6AQZTjMNgb6U7AAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x36daa10>"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Then we use again the EM algoritm (with optional addition of weights) and print/plot the results:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "max_iter = 300\n",
      "weights = 1./sigma_tot**2\n",
      "\n",
      "# Renormalizing the weights so they sum to N\n",
      "weights *= 1.*weights.size/weights.sum()\n",
      "\n",
      "# No weights case\n",
      "#weights = ones(s.size)\n",
      "\n",
      "# Initial guess of parameters and initializations\n",
      "p0 = array([-0.05,0.1,0.6,0.1,0.5])\n",
      "mu1, sig1, mu2, sig2, pi_1 = p0\n",
      "mu = array([mu1, mu2])\n",
      "sig = array([sig1, sig2])\n",
      "pi_ = array([pi_1, 1-pi_1])\n",
      "\n",
      "gamma = zeros((2, s.size))\n",
      "N_ = zeros(2)\n",
      "p_new = p0\n",
      "\n",
      "# EM loop\n",
      "counter = 0\n",
      "converged = False\n",
      "while not converged:\n",
      "    # Compute the responsibility func. and new parameters\n",
      "    for k in [0,1]:\n",
      "        gamma[k,:] = weights*pi_[k]*normpdf(s, mu[k], sig[k])/pdf_model(s, p_new) # SCHEME1\n",
      "        #gamma[k,:] = pi_[k]*normpdf(s, mu[k], sig[k])/pdf_model(s, p_new)         # SCHEME2\n",
      "        N_[k] = gamma[k,:].sum()\n",
      "        mu[k] = sum(gamma[k]*s)/N_[k] # SCHEME1\n",
      "        #mu[k] = sum(weights*gamma[k]*s)/sum(weights*gamma[k]) # SCHEME2\n",
      "        sig[k] = sqrt( sum(gamma[k]*(s-mu[k])**2)/N_[k] )\n",
      "        pi_[k] = 1.*N_[k]/N\n",
      "    p_new = [mu[0], sig[0], mu[1], sig[1], pi_[0]]\n",
      "    assert abs(N_.sum() - N)/float(N) < 1e-6 \n",
      "    assert abs(pi_.sum() - 1) < 1e-6\n",
      "    \n",
      "    # Convergence check\n",
      "    counter += 1\n",
      "    converged = counter >= max_iter"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 61
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \">> NO WEIGHTS\"\n",
      "print \"Means:   %6.3f  %6.3f\" % (p_new[0], p_new[2])\n",
      "print \"Std dev: %6.3f  %6.3f\" % (p_new[1], p_new[3])\n",
      "print \"Mix (1): %6.3f \" % p_new[4]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        ">> NO WEIGHTS\n",
        "Means:   -0.010   0.509\n",
        "Std dev:  0.126   0.113\n",
        "Mix (1):  0.315 \n"
       ]
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \">> WEIGHTED SCHEME1\"\n",
      "print \"Means:   %7.4f  %7.4f\" % (p_new[0], p_new[2])\n",
      "print \"Std dev: %7.4f  %7.4f\" % (p_new[1], p_new[3])\n",
      "print \"Mix (1): %7.4f \" % p_new[4]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        ">> WEIGHTED SCHEME1\n",
        "Means:   -0.0105   0.5035\n",
        "Std dev:  0.0748   0.0724\n",
        "Mix (1):  0.2850 \n"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \">> WEIGHTED SCHEME2\"\n",
      "print \"Means:   %6.3f  %6.3f\" % (p_new[0], p_new[2])\n",
      "print \"Std dev: %6.3f  %6.3f\" % (p_new[1], p_new[3])\n",
      "print \"Mix (1): %6.3f \" % p_new[4]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        ">> WEIGHTED SCHEME2\n",
        "Means:   -0.009   0.503\n",
        "Std dev:  0.125   0.113\n",
        "Mix (1):  0.314 \n"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "title('NO WEIGHTS')\n",
      "#hist(s, bins=r_[-1:2:0.05], normed=True);\n",
      "x = r_[-1:2:0.01]\n",
      "plot(x, pdf_model(x, p_new), color='k', lw=2); grid(True)\n",
      "plot(s, 0.1/sigma_tot, 'o', mew=0, ms=2, alpha=0.5);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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LJ4i2AnnuEiG3uJ/Qa5XTOqNszRxWR1P3Ti4LZsvt2RQid/3shTx3wiqSkxsRF+f6ho4g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\nThAEIUP+P9lKwyBnogTWAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x3314710>"
       ]
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "title('WEIGHTED SCHEME1')\n",
      "#hist(s, bins=r_[-1:2:0.05], normed=True);\n",
      "x = r_[-1:2:0.01]\n",
      "plot(x, pdf_model(x, p_new), color='k', lw=2); grid(True)\n",
      "plot(s, 0.1/sigma_tot, 'o', mew=0, ms=2, alpha=0.5);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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P7DjD8DMPVCrzrItVq1YhIiICgPEfS2xsLPtJRTpIrttnzpxxKXlsbZeWHoGR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\nA0vHcfssJ8yoAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x36d7190>"
       ]
      }
     ],
     "prompt_number": 58
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "title('WEIGHTED SCHEME2')\n",
      "#hist(s, bins=r_[-1:2:0.05], normed=True);\n",
      "x = r_[-1:2:0.01]\n",
      "plot(x, pdf_model(x, p_new), color='k', lw=2); grid(True)\n",
      "plot(s, 0.1/sigma_tot, 'o', mew=0, ms=2, alpha=0.5);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2uuc5Ir/84gVgDoKCgjBxYrOMui+wqChGkMOv1cewpx8UFIRLly7hl19+gYL9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\nQRAS5P8BGnypsyexzGgAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x3428ad0>"
       ]
      }
     ],
     "prompt_number": 60
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Compare EM to other methods"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "from scipy.stats import poisson, expon, binom\n",
      "from scipy.optimize import minimize, leastsq\n",
      "from scipy.special import erf"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "WARNING: pylab import has clobbered these variables: ['poisson']\n",
        "`%pylab --no-import-all` prevents importing * from pylab and numpy\n"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It seems natural to empirically bechmark the performance of the EM algorithm\n",
      "compared to other common fitting methods. The most simple method is the histogram\n",
      "fitting. Another method is curve-fitting the empirical CDF\n",
      "([ECDF](http://en.wikipedia.org/wiki/Empirical_distribution_function)) which \n",
      "does not require binning.\n",
      "\n",
      "Let define the fitting function we want to compare:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def fit_two_peaks_EM(sample, sigma, weights=False, p0=array([0.1,0.2,0.6,0.2,0.5]), \n",
      "        max_iter=300, tollerance=1e-3):\n",
      "    \n",
      "    if not weights: w = ones(sample.size)\n",
      "    else: w = 1./(sigma**2)\n",
      "    w *= 1.*w.size/w.sum() # renormalization so they sum to N\n",
      "    \n",
      "    # Initial guess of parameters and initializations\n",
      "    mu = array([p0[0], p0[2]])\n",
      "    sig = array([p0[1], p0[3]])\n",
      "    pi_ = array([p0[4], 1-p0[4]])\n",
      "    \n",
      "    gamma, N_ = zeros((2, sample.size)), zeros(2)\n",
      "    p_new = array(p0)\n",
      "    N = sample.size\n",
      "    \n",
      "    # EM loop\n",
      "    counter = 0\n",
      "    converged, stop_iteration = False, False\n",
      "    while not stop_iteration:\n",
      "        p_old = p_new\n",
      "        # Compute the responsibility func. and new parameters\n",
      "        for k in [0,1]:\n",
      "            gamma[k,:] = w*pi_[k]*normpdf(sample, mu[k], sig[k])/pdf_model(sample, p_new) # SCHEME1\n",
      "            #gamma[k,:] = pi_[k]*normpdf(sample, mu[k], sig[k])/pdf_model(sample, p_new) # SCHEME2\n",
      "            N_[k] = gamma[k,:].sum()\n",
      "            mu[k] = sum(gamma[k]*sample)/N_[k] # SCHEME1\n",
      "            #mu[k] = sum(w*gamma[k]*sample)/sum(w*gamma[k]) # SCHEME2\n",
      "            sig[k] = sqrt( sum(gamma[k]*(sample-mu[k])**2)/N_[k] )\n",
      "            pi_[k] = 1.*N_[k]/N\n",
      "        p_new = array([mu[0], sig[0], mu[1], sig[1], pi_[0]])\n",
      "        assert abs(N_.sum() - N)/float(N) < 1e-6 \n",
      "        assert abs(pi_.sum() - 1) < 1e-6\n",
      "        \n",
      "        # Convergence check\n",
      "        counter += 1\n",
      "        max_variation = max((p_new-p_old)/p_old)\n",
      "        converged = True if max_variation < tollerance else False\n",
      "        stop_iteration = converged or (counter >= max_iter)\n",
      "    #print \"Iterations:\", counter\n",
      "    if not converged: print \"WARNING: Not converged\"\n",
      "    return p_new"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 64
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def fit_two_gauss_mix_hist(s, bins=r_[-0.5:1.5:0.001], weights=None, \n",
      "            p0=[0,0.1,0.5,0.1,0.5]):\n",
      "    \"\"\"Fit the (optionally weighted) histogram with a 2-Gaussian mix PDF.\n",
      "    \"\"\"\n",
      "    H = histogram(s, bins=bins, weights=weights, normed=True)\n",
      "    x = .5*(H[1][1:]+H[1][:-1])\n",
      "    y = H[0]\n",
      "    assert x.size == y.size\n",
      "    \n",
      "    err_func = lambda p, x_, y_: (y_ - pdf_model(x_, p))\n",
      "    res = leastsq(err_func, x0=p0, args=(x,y), full_output=1)\n",
      "    #print res\n",
      "    return array(res[0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 65
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def fit_two_gauss_mix_cdf(s, p0=[0.2,1,0.8,1,0.3], weights=None):\n",
      "    \"\"\"Fit the sample s with two gaussians.\n",
      "    \"\"\"\n",
      "    ## Empirical CDF\n",
      "    ecdf = [sort(s), arange(0.5,s.size+0.5)*1./s.size]\n",
      "    \n",
      "    ## CDF for a Gaussian distribution, and for a 2-Gaussian mix\n",
      "    gauss_cdf = lambda x, mu, sigma: 0.5*(1+erf((x-mu)/(sqrt(2)*sigma)))\n",
      "    two_gauss_cdf = lambda x, m1, s1, m2, s2, a:\\\n",
      "            a*gauss_cdf(x,m1,s1)+(1-a)*gauss_cdf(x,m2,s2)\n",
      "\n",
      "    ## Fitting the empirical CDF\n",
      "    fit_func = lambda p, x: two_gauss_cdf(x, *p)\n",
      "    err_func = lambda p, x, y: fit_func(p, x) - y\n",
      "    p,v = leastsq(err_func, x0=p0, args=(ecdf[0],ecdf[1]))\n",
      "    return array(p)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 66
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A simple function to draw samples from a mixture of 2 Gaussians:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def sim_two_gauss_mix(p, N=1000): \n",
      "    s1 =  normal(p[0], p[1], size=N*p[4])\n",
      "    s2 = normal(p[2], p[3], size=N*(1-p[4]))\n",
      "    s = concatenate([s1,s2])\n",
      "    return s"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 67
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now define a function perform the fitting an **high number** populations:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def test_accuracy(N_test=200, **kwargs):\n",
      "    \n",
      "    fit_kwargs = dict()\n",
      "    if 'p0' in kwargs: \n",
      "        fit_kwargs.update(p0=kwargs.pop('p0'))\n",
      "    if 'max_iter' in kwargs: \n",
      "        fit_kwargs.update(max_iter=kwargs.pop('max_iter'))\n",
      "        \n",
      "    pop_kwargs = dict()\n",
      "    pop_kwargs.update(kwargs)\n",
      "    \n",
      "    P_em = zeros((N_test, 5))\n",
      "    P_h = zeros((N_test, 5))\n",
      "    P_cdf = zeros((N_test, 5))\n",
      "    for i_test in xrange(N_test):\n",
      "        s = sim_two_gauss_mix(**pop_kwargs)\n",
      "        \n",
      "        P_em[i_test,:] = fit_two_peaks_EM(s, sigma=None, weights=False, **fit_kwargs)\n",
      "        if 'max_iter' in fit_kwargs: fit_kwargs.pop('max_iter')\n",
      "        P_h[i_test,:] = fit_two_gauss_mix_hist(s, bins=r_[-0.5:1.5:0.01], weights=None, **fit_kwargs)\n",
      "        P_cdf[i_test,:] = fit_two_gauss_mix_cdf(s)\n",
      "        \n",
      "    return P_em, P_h, P_cdf"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 68
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot_accuracy(P_LIST, labels, name=\"Fit comparison\", ip=0, \n",
      "                  bins=(r_[0.2:0.5:0.004]+0.002)):\n",
      "    print \"METHOD\\t   MEAN  STD.DEV.\"\n",
      "    for P, label in zip(P_LIST, labels):\n",
      "        hist(P[:,ip], bins=bins, histtype='stepfilled', alpha=0.5, label=label);\n",
      "        print \"%s:\\t %6.2f %6.2f\" % (label, P[:,ip].mean()*100, P[:,ip].std()*100)\n",
      "    legend(); grid(True); title(name);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 72
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Study of different populations"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**CASE 1**: Half-mixed populations"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s = sim_two_gauss_mix(N=1000, p=[0,0.15,0.5,0.15,0.3])\n",
      "hist(s, bins=r_[-0.4:1:0.05]);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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SpEm9mrUnRvIPHjwYjzzyCPr27Yu+ffti/PjxOHTokC2GvJH8u3btwhtvvAEA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\nPBGRg3HIExE52P8Ds+aKddS0JxgAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x32fc410>"
       ]
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P_em, P_h, P_cdf = test_accuracy(N_test=2000, N=200, p=[0,0.15,0.5,0.15,0.3],\n",
      "                                              p0=[0,0.1,0.6,0.1,0.5])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stderr",
       "text": [
        "-c:37: RuntimeWarning: divide by zero encountered in divide\n",
        "/home/anto/.local/lib/python2.7/site-packages/scipy/optimize/minpack.py:402: RuntimeWarning: Number of calls to function has reached maxfev = 1200.\n",
        "  warnings.warn(errors[info][0], RuntimeWarning)\n"
       ]
      }
     ],
     "prompt_number": 71
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_accuracy([P_em, P_h, P_cdf], labels=['EM', 'hist', 'CDF'], ip=0,\n",
      "              bins=(r_[-.2:.2:0.01]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "METHOD\t   MEAN  STD.DEV.\n",
        "EM:\t   0.64   4.17"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "hist:\t   1.79   6.78\n",
        "CDF:\t  10.04  14.28\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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/f9TdEezCy787LznlQEWfEEI8CBV9Cbz0+XjIyUNGgJ9x+tTTlxcvOeVgs+g/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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x331ac10>"
       ]
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_accuracy([P_em, P_h, P_cdf], labels=['EM', 'hist', 'CDF'], ip=2,\n",
      "              bins=(r_[.4:.6:0.005]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "METHOD\t   MEAN  STD.DEV.\n",
        "EM:\t  50.20   2.00"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "hist:\t  50.24   2.31\n",
        "CDF:\t  48.48  10.96\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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paWkAgJycHOzcuRM5OTnYu3cvHn30UTid8l9TSRnMmyqHOX1lsW+qT5tBf+LE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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x332c910>"
       ]
      }
     ],
     "prompt_number": 74
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class=warn>\n",
      "**EM** is better both for bias and accuracy (CDF is worst)\n",
      "</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**CASE 2**: Separated populations"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sample_parameters = dict(N=200, p=[0,0.15,0.8,0.15,0.7])\n",
      "s = sim_two_gauss_mix(**sample_parameters)\n",
      "hist(s, bins=r_[-0.4:1.4:0.05]);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x36e7310>"
       ]
      }
     ],
     "prompt_number": 76
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P_em, P_h, P_cdf = test_accuracy(N_test=2000, p0=[0,0.1,0.6,0.1,0.5],\n",
      "                                 **sample_parameters)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 77
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_accuracy([P_em, P_h, P_cdf], labels=['EM', 'hist', 'CDF'], ip=0,\n",
      "              bins=(r_[-.1:.1:0.005]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "METHOD\t   MEAN  STD.DEV.\n",
        "EM:\t  -0.06   1.31"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "hist:\t  -0.04   1.60\n",
        "CDF:\t  -0.08   1.53\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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mymFNX1nsm+rDpE9E5EKY9DWAdVPlcJ6+stg31YdJn4jIhTDpawDrpsphTV9Z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\nyIUw6RMRuRAmfSIiF/L/AV6xmQ+bCXtbAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x3bb7b50>"
       ]
      }
     ],
     "prompt_number": 78
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_accuracy([P_em, P_h, P_cdf], labels=['EM', 'hist', 'CDF'], ip=2,\n",
      "              bins=(r_[.7:.9:0.005]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "METHOD\t   MEAN  STD.DEV.\n",
        "EM:\t  80.04   2.04"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "hist:\t  80.03   2.58\n",
        "CDF:\t  79.02   6.03\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MrcRExvioV6I7anWKcffdd7Nt2zZD26pVq5gzZw55eXmkpaWxatUqAHJzc9m4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\n+kII0Y1I0hdCiG7k/wGBes4eiH02/QAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x2d585d0>"
       ]
      }
     ],
     "prompt_number": 79
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class=warn>\n",
      "**EM** is better both for bias and accuracy (CDF is worst)\n",
      "</div>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.core.display import HTML\n",
      "def css_styling():\n",
      "    styles = open(\"./styles/custom.css\", \"r\").read()\n",
      "    return HTML(styles)\n",
      "css_styling()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<style>\n",
        "    div.warn {    \n",
        "        background-color: #fcf2f2;\n",
        "        border-color: #dFb5b4;\n",
        "        border-left: 5px solid #dfb5b4;\n",
        "        padding: 0.5em;\n",
        "        }\n",
        "        \n",
        "    .rendered_html code {\n",
        "\tbackground-color: #f7f7f7;\n",
        "\tborder: 1px solid #dddddd;\n",
        "\tcolor: black;\n",
        "\tline-height: 1.1em;\n",
        "    \tpadding: 0.15em;\n",
        "\tfont-size: 85%;\n",
        "    }\n",
        "    .warning{\n",
        "        color: rgb( 240, 20, 20 )\n",
        "        } \n",
        "</style>\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 82,
       "text": [
        "<IPython.core.display.HTML at 0x36e8510>"
       ]
      }
     ],
     "prompt_number": 82
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}