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Maximum Likelihood Estimation in Python with StatsModels
{"nbformat_minor": 0, "cells": [{"execution_count": 1, "cell_type": "code", "source": "%matplotlib inline", "outputs": [], "metadata": {"collapsed": true, "trusted": true}}, {"source": "---\ntitle: Maximum Likelihood Estimation of Custom Models in Python with StatsModels\ntags: Statistics, Python\n---", "cell_type": "markdown", "metadata": {}}, {"source": "Maximum likelihood estimation is a common method for fitting statistical models. In Python, it is quite possible to fit maximum likelihood models using just [`scipy.optimize`](http://docs.scipy.org/doc/scipy/reference/optimize.html). Over time, however, I have come to prefer the convenience provided by [`statsmodels`'](http://statsmodels.sourceforge.net/) [`GenericLikelihoodModel`](http://statsmodels.sourceforge.net/devel/dev/generated/statsmodels.base.model.GenericLikelihoodModel.html). In this post, I will show how easy it is to subclass `GenericLikelihoodModel` and take advantage of much of `statsmodels`' well-developed machinery for maximum likelihood estimation of custom models.", "cell_type": "markdown", "metadata": {}}, {"source": "#### Zero-inflated Poisson models", "cell_type": "markdown", "metadata": {}}, {"source": "The model we use for this demonstration is a [zero-inflated Poisson model](http://en.wikipedia.org/wiki/Zero-inflated_model#Zero-inflated_Poisson). This is a model for count data that generalizes the Poisson model by allowing for an overabundance of zero observations.\n\nThe model has two parameters, $\\pi$, the proportion of excess zero observations, and $\\lambda$, the mean of the Poisson distribution. We assume that observations from this model are generated as follows. First, a weighted coin with probability $\\pi$ of landing on heads is flipped. If the result is heads, the observation is zero. If the result is tails, the observation is generated from a Poisson distribution with mean $\\lambda$. Note that there are two ways for an observation to be zero under this model:\n\n1. the coin is heads, and\n2. the coin is tails, and the sample from the Poisson distribution is zero.\n\nIf $X$ has a zero-inflated Poisson distribution with parameters $\\pi$ and $\\lambda$, its probability mass function is given by\n\n$$\\begin{align*}\nP(X = 0)\n & = \\pi + (1 - \\pi)\\ e^{-\\lambda} \\\\\nP(X = x)\n & = (1 - \\pi)\\ e^{-\\lambda}\\ \\frac{\\lambda^x}{x!} \\textrm{ for } x > 0.\n\\end{align*}$$\n\nIn this post, we will use the parameter values $\\pi = 0.3$ and $\\lambda = 2$. The probability mass function of the zero-inflated Poisson distribution is shown below, next to a normal Poisson distribution, for comparison.", "cell_type": "markdown", "metadata": {}}, {"execution_count": 2, "cell_type": "code", "source": "from __future__ import division\n\nfrom matplotlib import pyplot as plt\nimport numpy as np\nfrom scipy import stats\nimport seaborn as sns\nfrom statsmodels.base.model import GenericLikelihoodModel", "outputs": [], "metadata": {"collapsed": true, "trusted": true}}, {"execution_count": 3, "cell_type": "code", "source": "np.random.seed(123456789)", "outputs": [], "metadata": {"collapsed": true, "trusted": true}}, {"execution_count": 4, "cell_type": "code", "source": "pi = 0.3\nlambda_ = 2.", "outputs": [], "metadata": {"collapsed": true, "trusted": true}}, {"execution_count": 5, "cell_type": "code", "source": "def zip_pmf(x, pi=pi, lambda_=lambda_):\n if pi < 0 or pi > 1 or lambda_ <= 0:\n return np.zeros_like(x)\n else:\n return (x == 0) * pi + (1 - pi) * stats.poisson.pmf(x, lambda_)", "outputs": [], "metadata": {"collapsed": false, "trusted": true}}, {"execution_count": 6, "cell_type": "code", "source": "fig, ax = plt.subplots(figsize=(8, 6))\n\nxs = np.arange(0, 10);\n\npalette = sns.color_palette()\n\nax.bar(2.5 * xs, stats.poisson.pmf(xs, lambda_), width=0.9, color=palette[0], label='Poisson');\nax.bar(2.5 * xs + 1, zip_pmf(xs), width=0.9, color=palette[1], label='Zero-inflated Poisson');\n\nax.set_xticks(2.5 * xs + 1);\nax.set_xticklabels(xs);\nax.set_xlabel('$x$');\n\nax.set_ylabel('$P(X = x)$');\n\nax.legend();", "outputs": [{"output_type": "display_data", "data": {"image/png": 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QJKkZFXKefmbOBeb2mnZzj8fPsfVu/O2OlSRJ/fOKfJIklYSlL0lSSVj6kiSV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\nSSXx/wEKf0WcLgB7iwAAAABJRU5ErkJggg==\n", "text/plain": "<matplotlib.figure.Figure at 0x7faf75d57910>"}, "metadata": {}}], "metadata": {"collapsed": false, "trusted": true}}, {"source": "#### Maximum likelihood estimation", "cell_type": "markdown", "metadata": {}}, {"source": "First we generate 1,000 observations from the zero-inflated model.", "cell_type": "markdown", "metadata": {}}, {"execution_count": 7, "cell_type": "code", "source": "N = 1000", "outputs": [], "metadata": {"collapsed": true, "trusted": true}}, {"execution_count": 8, "cell_type": "code", "source": "inflated_zero = stats.bernoulli.rvs(pi, size=N)\nx = (1 - inflated_zero) * stats.poisson.rvs(lambda_, size=N)", "outputs": [], "metadata": {"collapsed": false, "trusted": true}}, {"execution_count": 9, "cell_type": "code", "source": "fig, ax = plt.subplots(figsize=(8, 6))\n\nax.hist(x, width=0.8, bins=np.arange(x.max() + 1), normed=True);\n\nax.set_xticks(np.arange(x.max() + 1) + 0.4);\nax.set_xticklabels(np.arange(x.max() + 1));\nax.set_xlabel('$x$');\n\nax.set_ylabel('Proportion of samples');", "outputs": [{"output_type": "display_data", "data": {"image/png": 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"text/plain": "<matplotlib.figure.Figure at 0x7faf80e98a50>"}, "metadata": {}}], "metadata": {"collapsed": false, "trusted": true}}, {"source": "We are now ready to estimate $\\pi$ and $\\lambda$ by maximum likelihood. To do so, we define a class that inherits from `statsmodels`' `GenericLikelihoodModel` as follows.", "cell_type": "markdown", "metadata": {}}, {"execution_count": 10, "cell_type": "code", "source": "class ZeroInflatedPoisson(GenericLikelihoodModel):\n def __init__(self, endog, exog=None, **kwds):\n if exog is None:\n exog = np.zeros_like(endog)\n \n super(ZeroInflatedPoisson, self).__init__(endog, exog, **kwds)\n \n def nloglikeobs(self, params):\n pi = params[0]\n lambda_ = params[1]\n\n return -np.log(zip_pmf(self.endog, pi=pi, lambda_=lambda_))\n \n def fit(self, start_params=None, maxiter=10000, maxfun=5000, **kwds):\n if start_params is None:\n lambda_start = self.endog.mean()\n excess_zeros = (self.endog == 0).mean() - stats.poisson.pmf(0, lambda_start)\n \n start_params = np.array([excess_zeros, lambda_start])\n \n return super(ZeroInflatedPoisson, self).fit(start_params=start_params,\n maxiter=maxiter, maxfun=maxfun, **kwds)", "outputs": [], "metadata": {"collapsed": false, "trusted": true}}, {"source": "The key component of this class is the method `nloglikeobs`, which returns the negative log likelihood of each observed value in `endog`. Secondarily, we must also supply reasonable initial guesses of the parameters in `fit`. Obtaining the maximum likelihood estimate is now simple.", "cell_type": "markdown", "metadata": {}}, {"execution_count": 11, "cell_type": "code", "source": "model = ZeroInflatedPoisson(x)\nresults = model.fit()", "outputs": [{"output_type": "stream", "name": "stdout", "text": "Optimization terminated successfully.\n Current function value: 1.586641\n Iterations: 37\n Function evaluations: 70\n"}], "metadata": {"collapsed": false, "trusted": true}}, {"source": "We see that we have estimated the parameters fairly well.", "cell_type": "markdown", "metadata": {}}, {"execution_count": 12, "cell_type": "code", "source": "pi_mle, lambda_mle = results.params\n\npi_mle, lambda_mle", "outputs": [{"execution_count": 12, "output_type": "execute_result", "data": {"text/plain": "(0.31542487710071976, 2.0451304204850853)"}, "metadata": {}}], "metadata": {"collapsed": false, "trusted": true}}, {"source": "There are many advantages to buying into the `statsmodels` ecosystem and subclassing `GenericLikelihoodModel`. The already-written `statsmodels` code handles storing the observations and the interaction with `scipy.optimize` for us. (It is possible to control the use of `scipy.optimize` through keyword arguments to `fit`.)\n\nWe also gain access to many of `statsmodels`' built in model analysis tools. For example, we can use bootstrap resampling to estimate the variation in our parameter estimates.", "cell_type": "markdown", "metadata": {}}, {"execution_count": 13, "cell_type": "code", "source": "boot_mean, boot_std, boot_samples = results.bootstrap(nrep=500, store=True)\nboot_pis = boot_samples[:, 0]\nboot_lambdas = boot_samples[:, 1]", "outputs": [{"output_type": "stream", "name": "stdout", "text": "<class '__main__.ZeroInflatedPoisson'>\n"}], "metadata": {"collapsed": false, "trusted": true}}, {"execution_count": 14, "cell_type": "code", "source": "fig, (pi_ax, lambda_ax) = plt.subplots(ncols=2, figsize=(16, 6))\n\nsns.boxplot(boot_pis, ax=pi_ax, names=['$\\pi$'], color=palette[0]);\nsns.boxplot(boot_lambdas, ax=lambda_ax, names=['$\\lambda$'], color=palette[1]);\n\nfig.suptitle('Boostrap distribution of maximum likelihood estimates');", "outputs": [{"output_type": "display_data", "data": {"image/png": 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Z/Y44CZu0ZQynUovx+qaSmsnMtVSXdwMgIh4FfgnswBhm4W/kTPpS\nc/39GwDo7V1buBJp+zfalzge1iu1mLa2NtraPLxX0sgiYlpEvLq+fRpwW2auo2EW/nr98cBNBUuV\nJOkljpxKktRiIuJq4DBgl4hYAXwB6ISXJi3cG/iHiBgAfgZ8ol63PiIGZ+FvBy7LTCdDkiRtFwyn\nkiS1mMyc22T93UCMsO4Vs/BLkrQ98LBeSZIkSVJxhlNJkiRJUnGGU0mSJElScYZTSZIkSVJxbQMD\nm3w97q2mt3ft9lOMJKnldXdP9bpLW8jP5onvzjtv5447bitdRkt7/PHHANh997cUrqS1HXzwYV7P\nfRIY7bPZ2XolSZKkLTBt2rTSJUgTgiOnkqQJy5HTLednsyRpPI322ew5p5IkSZKk4gynkiRJkqTi\nDKeSJEmSpOIMp5IkSZKk4gynkiRJkqTiDKeSJEmSpOIMp5IkSZKk4gynkiRJkqTiDKeSJEmSpOIM\np5IkSZKk4gynkiRJkqTiDKeSJEmSpOIMp5IkSZKk4gynkiRJkqTiDKeSJEmSpOIMp5IkSZKk4gyn\nkiRJkqTiDKeSJEmSpOIMp5IkSZKk4gynkiRJkqTiDKeSJEnSFhgYGGBgYKB0GVLLM5xKkiRJW+DO\nO2/nrrsWlS5DankdpQuQJEmSWlVf33Nce+3VAMyc+S66uroKVyS1LkdOJUmSpM3WVroAacJw5FSS\nJEnaTF1dXRx33Fza2tocNZW2kOFUkiRJ2gKzZx9augRpQmgaTiNiDjAfaAcuzcx5Q9Z/FPgs1TEN\na4EzMvPBhvXtwGLgicz84DjWLknSpBQR3wTeD/wmM/cdZv0uwJXAG6k+6/9rZv5DvW458CywAejP\nzFnbpmpp4mpr89BeaTyMes5pHSwvAuYAewNzI2KvId0eAQ7NzP2A84AFQ9afBTwMOL+2JEnj43Kq\nz+aRnAksycx3AIcDX42IwS+kB4DDM3OmwVSStD1pNiHSLGBZZi7PzH7gGuCYxg6ZeXdmrqkX7wF2\nG1wXEbsBRwOX4tnikiSNi8xcBKwapcuvgJ3q2zsBz2Tm+ob1fiZLkrY7zcLprsCKhuUn6raRfAK4\nuWH5a8DZwMbNqk6SJG2ObwD7RMSTwANURzENGgBujYjFEXFakeokSRpGs3NOx3wobkQcAZwKzK6X\nP0B1LsySiDh8LPvo7p7qN7mSJG25c4D7M/PwiNgD+EFE7J+Za4HZmfmriOiu239Rj8QOy89mSdK2\n0mzkdCXQ07DcQzV6+jIRsR/Vt7QfyszBw4zeA3woIh4FrgbeGxFXbHnJkiSpifcA3wXIzF8CjwJR\nL/+q/tkL3EB1Co8kScU1C6eLgT0jYkZEvBo4HripsUNE7A5cD5yUmcsG2zPznMzsycy3AicAP8zM\nk8e3fEmSNIxfAEcCRMQbqILpIxHRFRFT6/YdgaOAh4pVKUlSg1EP683M9RFxJnAL1aVkLsvMpRFx\ner3+EuDzwHTg4oiAkaeld7ZeSZLGQURcDRwG7BIRK4AvAJ3w0mfz3wKXR8QDVF9EfzYzfxsRbwOu\nrz+vO4CrMvP7JR6DJElDtQ0MmBklSZIkSWU1O6xXkiRJkqStznAqSZIkSSrOcCpJkiRJKs5wKkmS\nJEkqznAqSZIkSSrOcCpJkiRtpoiYHhHfjojXla5FanVeSkZqERGxK/AVYE9gA/AMcFN9TUNJklRI\nRHwSeFVmLihdi9TKOkoXIGnM3pKZJ0bEicBAZl5duiBJkgTA94BvAoZTaQt4WK/UIjLzrogIYA3Q\nXboeSZJUycxfAztGxE6la5FameFUai0fBe4C9ogIj3yQJGk7EBGvBdYB7y9di9TKDKdSa+nJzFXA\nb4A9ShcjSdJkFxHtwBeBc4EPl61Gam1OiCRJkiRtpoiYD3wjM38eEXcDh2Xmi6XrklqRI6eSJEnS\nZoiI44CfZubP66b/ARxdsCSppTlyKkmSJEkqzpFTSZIkSVJxhlNJkiRJUnGGU0mSJElScYZTSZIk\nSVJxhlNJkiRJUnGGU0mSJElScYZTSZIkSVJx/wcVS32yEo37SgAAAABJRU5ErkJggg==\n", "text/plain": "<matplotlib.figure.Figure at 0x7faf758dbb50>"}, "metadata": {}}], "metadata": {"collapsed": false, "trusted": true}}, {"source": "The next time you are fitting a model using maximum likelihood, try integrating with `statsmodels` to take advantage of the significant amount of work that has gone into its ecosystem.", "cell_type": "markdown", "metadata": {}}, {"execution_count": null, "cell_type": "code", "source": "", "outputs": [], "metadata": {"collapsed": true, "trusted": true}}], "nbformat": 4, "metadata": {"kernelspec": {"display_name": "Python 2", "name": "python2", "language": "python"}, "language_info": {"mimetype": "text/x-python", "nbconvert_exporter": "python", "version": "2.7.6", "name": "python", "file_extension": ".py", "pygments_lexer": "ipython2", "codemirror_mode": {"version": 2, "name": "ipython"}}}}
@jeffjose
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jeffjose commented Mar 30, 2017

This looks like a jupyter notebook.

@mtzl
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mtzl commented Jul 17, 2017

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