Statistical Data Modeling.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Set some Pandas options\n",
    "pd.set_option('display.notebook_repr_html', False)\n",
    "pd.set_option('display.max_columns', 20)\n",
    "pd.set_option('display.max_rows', 25)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Estimation\n",
    "==========\n",
    "\n",
    "An recurring statistical problem is finding estimates of the relevant parameters that correspond to the distribution that best represents our data.\n",
    "\n",
    "In **parametric** inference, we specify *a priori* a suitable distribution, then choose the parameters that best fit the data.\n",
    "\n",
    "* e.g. $\\mu$ and $\\sigma^2$ in the case of the normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x104ed1f10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.array([ 1.00201077,  1.58251956,  0.94515919,  6.48778002,  1.47764604,\n",
    "        5.18847071,  4.21988095,  2.85971522,  3.40044437,  3.74907745,\n",
    "        1.18065796,  3.74748775,  3.27328568,  3.19374927,  8.0726155 ,\n",
    "        0.90326139,  2.34460034,  2.14199217,  3.27446744,  3.58872357,\n",
    "        1.20611533,  2.16594393,  5.56610242,  4.66479977,  2.3573932 ])\n",
    "_ = plt.hist(x, bins=8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fitting data to probability distributions\n",
    "\n",
    "We start with the problem of finding values for the parameters that provide the best fit between the model and the data, called point estimates. First, we need to define what we mean by ‘best fit’. There are two commonly used criteria:\n",
    "\n",
    "* **Method of moments** chooses the parameters so that the sample moments (typically the sample mean and variance) match the theoretical moments of our chosen distribution.\n",
    "* **Maximum likelihood** chooses the parameters to maximize the likelihood, which measures how likely it is to observe our given sample."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Discrete Random Variables\n",
    "\n",
    "$$X = \\{0,1\\}$$\n",
    "\n",
    "$$Y = \\{\\ldots,-2,-1,0,1,2,\\ldots\\}$$\n",
    "\n",
    "**Probability Mass Function**: \n",
    "\n",
    "For discrete $X$,\n",
    "\n",
    "$$Pr(X=x) = f(x|\\theta)$$\n",
    "\n",
    "![Discrete variable](http://upload.wikimedia.org/wikipedia/commons/1/16/Poisson_pmf.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***e.g. Poisson distribution***\n",
    "\n",
    "The Poisson distribution models unbounded counts:\n",
    "\n",
    "<div style=\"font-size: 150%;\">  \n",
    "$$Pr(X=x)=\\frac{e^{-\\lambda}\\lambda^x}{x!}$$\n",
    "</div>\n",
    "\n",
    "* $X=\\{0,1,2,\\ldots\\}$\n",
    "* $\\lambda > 0$\n",
    "\n",
    "$$E(X) = \\text{Var}(X) = \\lambda$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Continuous Random Variables\n",
    "\n",
    "$$X \\in [0,1]$$\n",
    "\n",
    "$$Y \\in (-\\infty, \\infty)$$\n",
    "\n",
    "**Probability Density Function**: \n",
    "\n",
    "For continuous $X$,\n",
    "\n",
    "$$Pr(x \\le X \\le x + dx) = f(x|\\theta)dx \\, \\text{ as } \\, dx \\rightarrow 0$$\n",
    "\n",
    "![Continuous variable](http://upload.wikimedia.org/wikipedia/commons/e/ec/Exponential_pdf.svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***e.g. normal distribution***\n",
    "\n",
    "<div style=\"font-size: 150%;\">  \n",
    "$$f(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]$$\n",
    "</div>\n",
    "\n",
    "* $X \\in \\mathbf{R}$\n",
    "* $\\mu \\in \\mathbf{R}$\n",
    "* $\\sigma>0$\n",
    "\n",
    "$$\\begin{align}E(X) &= \\mu \\cr\n",
    "\\text{Var}(X) &= \\sigma^2 \\end{align}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: Nashville Precipitation\n",
    "\n",
    "The dataset `nashville_precip.txt` contains [NOAA precipitation data for Nashville measured since 1871](http://bit.ly/nasvhville_precip_data). The gamma distribution is often a good fit to aggregated rainfall data, and will be our candidate distribution in this case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "       Jan   Feb   Mar    Apr   May   Jun   Jul   Aug   Sep   Oct   Nov   Dec\n",
       "Year                                                                         \n",
       "1871  2.76  4.58  5.01   4.13  3.30  2.98  1.58  2.36  0.95  1.31  2.13  1.65\n",
       "1872  2.32  2.11  3.14   5.91  3.09  5.17  6.10  1.65  4.50  1.58  2.25  2.38\n",
       "1873  2.96  7.14  4.11   3.59  6.31  4.20  4.63  2.36  1.81  4.28  4.36  5.94\n",
       "1874  5.22  9.23  5.36  11.84  1.49  2.87  2.65  3.52  3.12  2.63  6.12  4.19\n",
       "1875  6.15  3.06  8.14   4.22  1.73  5.63  8.12  1.60  3.79  1.25  5.46  4.30"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "precip = pd.read_table(\"data/nashville_precip.txt\", index_col=0, na_values='NA', delim_whitespace=True)\n",
    "precip.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mKD5JZ5IGP6whdeNNIjVOy0m5p0dJAybmRsRrK5SPTvRLW+9IItqYBHd8jCxF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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x104ef6710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "_ = precip.hist(sharex=True, sharey=True, grid=False)\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first step is recognixing what sort of distribution to fit our data to. A couple of observations:\n",
    "\n",
    "1. The data are skewed, with a longer tail to the right than to the left\n",
    "2. The data are positive-valued, since they are measuring rainfall\n",
    "3. The data are continuous\n",
    "\n",
    "There are a few possible choices, but one suitable alternative is the **gamma distribution**:\n",
    "\n",
    "<div style=\"font-size: 150%;\">  \n",
    "$$x \\sim \\text{Gamma}(\\alpha, \\beta) = \\frac{\\beta^{\\alpha}x^{\\alpha-1}e^{-\\beta x}}{\\Gamma(\\alpha)}$$\n",
    "</div>\n",
    "\n",
    "![gamma](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Gamma_distribution_pdf.svg/500px-Gamma_distribution_pdf.svg.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ***method of moments*** simply assigns the empirical mean and variance to their theoretical counterparts, so that we can solve for the parameters.\n",
    "\n",
    "So, for the gamma distribution, the mean and variance are:\n",
    "\n",
    "<div style=\"font-size: 150%;\">  \n",
    "$$ \\hat{\\mu} = \\bar{X} = \\alpha \\beta $$\n",
    "$$ \\hat{\\sigma}^2 = S^2 = \\alpha \\beta^2 $$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, if we solve for these parameters, we can use a gamma distribution to describe our data:\n",
    "\n",
    "<div style=\"font-size: 150%;\">  \n",
    "$$ \\alpha = \\frac{\\bar{X}^2}{S^2}, \\, \\beta = \\frac{S^2}{\\bar{X}} $$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's deal with the missing value in the October data. Given what we are trying to do, it is most sensible to fill in the missing value with the average of the available values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "        Jan    Feb   Mar    Apr    May   Jun   Jul   Aug    Sep   Oct   Nov  \\\n",
       "Year                                                                          \n",
       "1871   2.76   4.58  5.01   4.13   3.30  2.98  1.58  2.36   0.95  1.31  2.13   \n",
       "1872   2.32   2.11  3.14   5.91   3.09  5.17  6.10  1.65   4.50  1.58  2.25   \n",
       "1873   2.96   7.14  4.11   3.59   6.31  4.20  4.63  2.36   1.81  4.28  4.36   \n",
       "1874   5.22   9.23  5.36  11.84   1.49  2.87  2.65  3.52   3.12  2.63  6.12   \n",
       "1875   6.15   3.06  8.14   4.22   1.73  5.63  8.12  1.60   3.79  1.25  5.46   \n",
       "1876   6.41   2.22  5.28   3.62   3.40  5.65  7.15  5.77   2.52  2.68  1.26   \n",
       "1877   4.05   1.06  4.98   9.47   1.25  6.02  3.25  4.16   5.40  2.61  4.93   \n",
       "1878   3.34   2.10  3.48   6.88   2.33  3.28  9.43  5.02   1.28  2.17  3.20   \n",
       "1879   6.32   3.13  3.81   2.88   2.88  2.50  8.47  4.62   5.18  2.90  5.85   \n",
       "1880   3.74  12.37  8.16   5.26   4.13  3.97  5.69  2.22   5.39  7.24  5.77   \n",
       "1881   3.54   5.48  2.79   5.12   3.67  3.70  0.86  1.81   6.57  4.80  4.89   \n",
       "1882  14.51   8.61  9.38   3.59   7.38  2.54  4.06  5.54   1.61  1.11  3.60   \n",
       "...     ...    ...   ...    ...    ...   ...   ...   ...    ...   ...   ...   \n",
       "2000   3.52   3.75  3.34   6.23   7.66  1.74  2.25  1.95   1.90  0.26  6.39   \n",
       "2001   3.21   8.54  2.73   2.42   5.54  4.47  2.77  4.07   1.79  4.61  5.09   \n",
       "2002   4.93   1.99  9.40   4.31   3.98  3.76  5.64  3.13   6.29  4.48  2.91   \n",
       "2003   1.59   8.47  2.30   4.69  10.73  7.08  2.87  3.88   8.70  1.80  4.17   \n",
       "2004   3.60   5.77  4.81   6.69   6.90  3.39  3.19  4.24   4.55  4.90  5.21   \n",
       "2005   4.42   3.84  3.90   6.93   1.03  2.70  2.39  6.89   1.44  0.02  3.29   \n",
       "2006   6.57   2.69  2.90   4.14   4.95  2.19  2.64  5.20   4.00  2.98  4.05   \n",
       "2007   3.32   1.84  2.26   2.75   3.30  2.37  1.47  1.38   1.99  4.95  6.20   \n",
       "2008   4.76   2.53  5.56   7.20   5.54  2.21  4.32  1.67   0.88  5.03  1.75   \n",
       "2009   4.59   2.85  2.92   4.13   8.45  4.53  6.03  2.14  11.08  6.49  0.67   \n",
       "2010   4.13   2.77  3.52   3.48  16.43  4.96  5.86  6.99   1.17  2.49  5.41   \n",
       "2011   2.31   5.54  4.59   7.51   4.38  5.04  3.46  1.78   6.20  0.93  6.15   \n",
       "\n",
       "       Dec  \n",
       "Year        \n",
       "1871  1.65  \n",
       "1872  2.38  \n",
       "1873  5.94  \n",
       "1874  4.19  \n",
       "1875  4.30  \n",
       "1876  0.95  \n",
       "1877  2.49  \n",
       "1878  6.04  \n",
       "1879  9.15  \n",
       "1880  3.32  \n",
       "1881  4.85  \n",
       "1882  1.52  \n",
       "...    ...  \n",
       "2000  3.44  \n",
       "2001  3.32  \n",
       "2002  5.81  \n",
       "2003  3.19  \n",
       "2004  5.93  \n",
       "2005  2.46  \n",
       "2006  3.41  \n",
       "2007  3.83  \n",
       "2008  6.72  \n",
       "2009  3.99  \n",
       "2010  1.87  \n",
       "2011  4.25  \n",
       "\n",
       "[141 rows x 12 columns]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "precip.fillna(value={'Oct': precip.Oct.mean()}, inplace=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's calculate the sample moments of interest, the means and variances by month:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Jan    4.523688\n",
       "Feb    4.097801\n",
       "Mar    4.977589\n",
       "Apr    4.204468\n",
       "May    4.325674\n",
       "Jun    3.873475\n",
       "Jul    3.895461\n",
       "Aug    3.367305\n",
       "Sep    3.377660\n",
       "Oct    2.610500\n",
       "Nov    3.685887\n",
       "Dec    4.176241\n",
       "dtype: float64"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "precip_mean = precip.mean()\n",
    "precip_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Jan    6.928862\n",
       "Feb    5.516660\n",
       "Mar    5.365444\n",
       "Apr    4.117096\n",
       "May    5.306409\n",
       "Jun    5.033206\n",
       "Jul    3.777012\n",
       "Aug    3.779876\n",
       "Sep    4.940099\n",
       "Oct    2.741659\n",
       "Nov    3.679274\n",
       "Dec    5.418022\n",
       "dtype: float64"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "precip_var = precip.var()\n",
    "precip_var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then use these moments to estimate $\\alpha$ and $\\beta$ for each month:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "alpha_mom = precip_mean ** 2 / precip_var\n",
    "beta_mom = precip_var / precip_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Jan    2.953407\n",
       " Feb    3.043866\n",
       " Mar    4.617770\n",
       " Apr    4.293694\n",
       " May    3.526199\n",
       " Jun    2.980965\n",
       " Jul    4.017624\n",
       " Aug    2.999766\n",
       " Sep    2.309383\n",
       " Oct    2.485616\n",
       " Nov    3.692511\n",
       " Dec    3.219070\n",
       " dtype: float64, Jan    1.531684\n",
       " Feb    1.346249\n",
       " Mar    1.077920\n",
       " Apr    0.979219\n",
       " May    1.226724\n",
       " Jun    1.299403\n",
       " Jul    0.969593\n",
       " Aug    1.122522\n",
       " Sep    1.462581\n",
       " Oct    1.050243\n",
       " Nov    0.998206\n",
       " Dec    1.297344\n",
       " dtype: float64)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha_mom, beta_mom"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the `gamma.pdf` function in `scipy.stats.distributions` to plot the ditribtuions implied by the calculated alphas and betas. For example, here is January:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x105244050>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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DqmI3H4cUwdFr4ctOGc0Tj+ecVbj21GuZtmIaB4oOpB8oDj9yZkMYcoYhI1hO1+wVtH7L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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1050d3350>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats.distributions import gamma\n",
    "\n",
    "precip.Jan.hist(normed=True, bins=20)\n",
    "plt.plot(np.linspace(0, 10), gamma.pdf(np.linspace(0, 10), alpha_mom[0], beta_mom[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looping over all months, we can create a grid of plots for the distribution of rainfall, using the gamma distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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XFCpUiHXr1t2w/9ChQ1StWjXtc5UqVfj1118BOHbsGOXLlwegfPnyHDt2LIe/Kc+Sr+oU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\nkiRJA3GAJUmSJEkDcYAlSZIkSQP5BlLlMILFYYlLAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x104f71110>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "axs = precip.hist(normed=True, figsize=(12, 8), sharex=True, sharey=True, bins=15, grid=False)\n",
    "\n",
    "for ax in axs.ravel():\n",
    "    \n",
    "    # Get month\n",
    "    m = ax.get_title()\n",
    "    \n",
    "    # Plot fitted distribution\n",
    "    x = np.linspace(*ax.get_xlim())\n",
    "    ax.plot(x, gamma.pdf(x, alpha_mom[m], beta_mom[m]))\n",
    "    \n",
    "    # Annotate with parameter estimates\n",
    "    label = 'alpha = {0:.2f}\\nbeta = {1:.2f}'.format(alpha_mom[m], beta_mom[m])\n",
    "    ax.annotate(label, xy=(10, 0.2))\n",
    "    \n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Maximum Likelihood\n",
    "==================\n",
    "\n",
    "**Maximum likelihood** (ML) fitting is usually more work than the method of moments, but it is preferred as the resulting estimator is known to have good theoretical properties. \n",
    "\n",
    "There is a ton of theory regarding ML. We will restrict ourselves to the mechanics here.\n",
    "\n",
    "Say we have some data $y = y_1,y_2,\\ldots,y_n$ that is distributed according to some distribution:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$Pr(Y_i=y_i | \\theta)$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, for example, is a **Poisson distribution** that describes the distribution of some discrete variables, typically *counts*: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1062f4510>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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wDb6Lfl3nApur6vYk1wFvB353eOXTTnvJgcczM0cxM7N0b+p//dd/ydNP712y7UnSOPR6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\ng0KS1MqgkCS1+v97xKpdAe+O7AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1050d3050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = np.random.poisson(5, size=100)\n",
    "plt.hist(y, bins=12, normed=True)\n",
    "plt.xlabel('y'); plt.ylabel('Pr(y)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The product $\\prod_{i=1}^n Pr(y_i | \\theta)$ gives us a measure of how **likely** it is to observe values $y_1,\\ldots,y_n$ given the parameters $\\theta$. Maximum likelihood fitting consists of choosing the appropriate function $l= Pr(Y|\\theta)$ to maximize for a given set of observations. We call this function the *likelihood function*, because it is a measure of how likely the observations are if the model is true.\n",
    "\n",
    "> Given these data, how likely is this model?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the above model, the data were drawn from a Poisson distribution with parameter $\\lambda =5$.\n",
    "\n",
    "$$L(y|\\lambda=5) = \\frac{e^{-5} 5^y}{y!}$$\n",
    "\n",
    "So, for any given value of $y$, we can calculate its likelihood:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.041303093412337726"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "poisson_like = lambda x, lam: np.exp(-lam) * (lam**x) / (np.arange(x)+1).prod()\n",
    "\n",
    "lam = 6\n",
    "value = 10\n",
    "poisson_like(value, lam)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.338402687045475"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(poisson_like(yi, lam) for yi in y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.6625623857972949"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lam = 8\n",
    "np.sum(poisson_like(yi, lam) for yi in y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can plot the likelihood function for any value of the parameter(s):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x106a2f190>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hZn3NbKqZTTezYfVcc2368bfMbLtajzUzs0lmVhLvmx5+GA46KHYKKTc//zmc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\nKrYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106a4c050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lambdas = np.linspace(0,15)\n",
    "x = 5\n",
    "plt.plot(lambdas, [poisson_like(x, l) for l in lambdas])\n",
    "plt.xlabel('$\\lambda$')\n",
    "plt.ylabel('L($\\lambda$|x={0})'.format(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How is the likelihood function different than the probability distribution function (PDF)? The likelihood is a function of the parameter(s) *given the data*, whereas the PDF returns the probability of data given a particular parameter value. Here is the PDF of the Poisson for $\\lambda=5$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x107124a10>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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IiBh0nZx5tD7TKsNVERHR0ZnHhZJ+1rJ8csuybZ8+A3VFRMQA6+QO85Gp2kRExNzS0U2C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\nIiIq+//a9cozRz9JawAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106a422d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lam = 5\n",
    "xvals = np.arange(15)\n",
    "plt.bar(xvals, [poisson_like(x, lam) for x in xvals])\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('Pr(X|$\\lambda$=5)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Why are we interested in the likelihood function? \n",
    "\n",
    "A reasonable estimate of the true, unknown value for the parameter is one which **maximizes the likelihood function**. So, inference is reduced to an optimization problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Going back to the rainfall data, if we are using a gamma distribution we need to maximize:\n",
    "\n",
    "$$\\begin{align}l(\\alpha,\\beta) &= \\sum_{i=1}^n \\log[\\beta^{\\alpha} x^{\\alpha-1} e^{-x/\\beta}\\Gamma(\\alpha)^{-1}] \\cr \n",
    "&= n[(\\alpha-1)\\overline{\\log(x)} - \\bar{x}\\beta + \\alpha\\log(\\beta) - \\log\\Gamma(\\alpha)]\\end{align}$$\n",
    "\n",
    "(*Its usually easier to work in the log scale*)\n",
    "\n",
    "where $n = 2012 − 1871 = 141$ and the bar indicates an average over all *i*. We choose $\\alpha$ and $\\beta$ to maximize $l(\\alpha,\\beta)$.\n",
    "\n",
    "Notice $l$ is infinite if any $x$ is zero. We do not have any zeros, but we do have an NA value for one of the October data, which we dealt with above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Finding the MLE\n",
    "\n",
    "To find the maximum of any function, we typically take the *derivative* with respect to the variable to be maximized, set it to zero and solve for that variable. \n",
    "\n",
    "$$\\frac{\\partial l(\\alpha,\\beta)}{\\partial \\beta} = n\\left(\\frac{\\alpha}{\\beta} - \\bar{x}\\right) = 0$$\n",
    "\n",
    "Which can be solved as $\\beta = \\alpha/\\bar{x}$. However, plugging this into the derivative with respect to $\\alpha$ yields:\n",
    "\n",
    "$$\\frac{\\partial l(\\alpha,\\beta)}{\\partial \\alpha} = \\log(\\alpha) + \\overline{\\log(x)} - \\log(\\bar{x}) - \\frac{\\Gamma(\\alpha)'}{\\Gamma(\\alpha)} = 0$$\n",
    "\n",
    "This has no closed form solution. We must use ***numerical optimization***!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Numerical optimization alogarithms take an initial \"guess\" at the solution, and iteratively improve the guess until it gets \"close enough\" to the answer.\n",
    "\n",
    "Here, we will use Newton-Raphson algorithm:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$x_{n+1} = x_n - \\frac{f(x_n)}{f'(x_n)}$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which is available to us via SciPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.optimize import newton"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a graphical example of how Newtone-Raphson converges on a solution, using an arbitrary function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x107202750>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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M2ERy5wHzq9a6SRbjx8MLL8CoURXa3+5AV/POOTp37kydOnUAWLZsGU8//TQ1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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106b5c4d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# some function\n",
    "func = lambda x: 3./(1 + 400*np.exp(-2*x)) - 1\n",
    "xvals = np.linspace(0, 6)\n",
    "plt.plot(xvals, func(xvals))\n",
    "plt.text(5.3, 2.1, '$f(x)$', fontsize=16)\n",
    "# zero line\n",
    "plt.plot([0,6], [0,0], 'k-')\n",
    "# value at step n\n",
    "plt.plot([4,4], [0,func(4)], 'k:')\n",
    "plt.text(4, -.2, '$x_n$', fontsize=16)\n",
    "# tangent line\n",
    "tanline = lambda x: -0.858 + 0.626*x\n",
    "plt.plot(xvals, tanline(xvals), 'r--')\n",
    "# point at step n+1\n",
    "xprime = 0.858/0.626\n",
    "plt.plot([xprime, xprime], [tanline(xprime), func(xprime)], 'k:')\n",
    "plt.text(xprime+.1, -.2, '$x_{n+1}$', fontsize=16)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To apply the Newton-Raphson algorithm, we need a function that returns a vector containing the **first and second derivatives** of the function with respect to the variable of interest. In our case, this is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.special import psi, polygamma\n",
    "\n",
    "dlgamma = lambda m, log_mean, mean_log: np.log(m) - psi(m) - log_mean + mean_log\n",
    "dl2gamma = lambda m, *args: 1./m - polygamma(1, m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where `log_mean` and `mean_log` are $\\log{\\bar{x}}$ and $\\overline{\\log(x)}$, respectively. `psi` and `polygamma` are complex functions of the Gamma function that result when you take first and second derivatives of that function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Calculate statistics\n",
    "log_mean = precip.mean().apply(np.log)\n",
    "mean_log = precip.apply(np.log).mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Time to optimize!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.5189679152399647"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Alpha MLE for December\n",
    "alpha_mle = newton(dlgamma, 2, dl2gamma, args=(log_mean[-1], mean_log[-1]))\n",
    "alpha_mle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now plug this back into the solution for beta:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$ \\beta  = \\frac{\\alpha}{\\bar{X}} $$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.84261607548413797"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta_mle = alpha_mle/precip.mean()[-1]\n",
    "beta_mle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compare the fit of the estimates derived from MLE to those from the method of moments:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x107115c90>]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hP8r/W07N1hqLClNKNadhH0SHU8EGUqiPpC/7yJEtmmK6xdD/d/3Z/oT27pUKFRGUOqGo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\nGvZKKRUFNOyVUioKaNgrpVQU+P9w9eKwJ/eRRAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107115b90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dec = precip.Dec\n",
    "dec.hist(normed=True, bins=10, grid=False)\n",
    "x = np.linspace(0, dec.max())\n",
    "plt.plot(x, gamma.pdf(x, alpha_mom[-1], beta_mom[-1]), 'm-')\n",
    "plt.plot(x, gamma.pdf(x, alpha_mle, beta_mle), 'r--')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For some common distributions, SciPy includes methods for fitting via MLE:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2.2427517753152308, 0.65494604470188622, 1.570073932063466)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.stats import gamma\n",
    "\n",
    "gamma.fit(precip.Dec)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This fit is not directly comparable to our estimates, however, because SciPy's `gamma.fit` method fits an odd 3-parameter version of the gamma distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: truncated distribution\n",
    "\n",
    "Suppose that we observe $Y$ truncated below at $a$ (where $a$ is known). If $X$ is the distribution of our observation, then:\n",
    "\n",
    "$$ P(X \\le x) = P(Y \\le x|Y \\gt a) = \\frac{P(a \\lt Y \\le x)}{P(Y \\gt a)}$$\n",
    "\n",
    "(so, $Y$ is the original variable and $X$ is the truncated variable) \n",
    "\n",
    "Then X has the density:\n",
    "\n",
    "$$f_X(x) = \\frac{f_Y (x)}{1−F_Y (a)} \\, \\text{for} \\, x \\gt a$$ \n",
    "\n",
    "Suppose $Y \\sim N(\\mu, \\sigma^2)$ and $x_1,\\ldots,x_n$ are independent observations of $X$. We can use maximum likelihood to find $\\mu$ and $\\sigma$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we can simulate a truncated distribution using a `while` statement to eliminate samples that are outside the support of the truncated distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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cqUszwTKNxmOoS1JDLL9IU8Iyi7aCoS5NDX+4Wptn+UWSGuJIXZpS/qC1NsKRujS1nPGi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\nqEtSQwx1SWrI/wPeC0gQwKxCfgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106a4ced0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.random.normal(size=10000)\n",
    "a = -1\n",
    "x_small = x < a\n",
    "while x_small.sum():\n",
    "    x[x_small] = np.random.normal(size=x_small.sum())\n",
    "    x_small = x < a\n",
    "    \n",
    "_ = plt.hist(x, bins=100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can construct a log likelihood for this function using the conditional form:\n",
    "\n",
    "$$f_X(x) = \\frac{f_Y (x)}{1−F_Y (a)} \\, \\text{for} \\, x \\gt a$$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats.distributions import norm\n",
    "\n",
    "trunc_norm = lambda theta, a, x: -(np.log(norm.pdf(x, theta[0], theta[1])) - \n",
    "                                      np.log(1 - norm.cdf(a, theta[0], theta[1]))).sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For this example, we will use another optimization algorithm, the **Nelder-Mead simplex algorithm**. It has a couple of advantages: \n",
    "\n",
    "- it does not require derivatives\n",
    "- it can optimize (minimize) a vector of parameters\n",
    "\n",
    "SciPy implements this algorithm in its `fmin` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 11077.807912\n",
      "         Iterations: 44\n",
      "         Function evaluations: 82\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 0.02244612,  0.99710875])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.optimize import fmin\n",
    "\n",
    "fmin(trunc_norm, np.array([1,2]), args=(-1, x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In general, simulating data is a terrific way of testing your model before using it with real data."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kernel density estimates\n",
    "\n",
    "In some instances, we may not be interested in the parameters of a particular distribution of data, but just a smoothed representation of the data at hand. In this case, we can estimate the disribution *non-parametrically* (i.e. making no assumptions about the form of the underlying distribution) using kernel density estimation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.62402604,  1.06204307,  7.27542973,  5.91978919,  9.95408103,\n",
       "        9.51695422,  9.54608936,  4.3288413 ,  4.15275767,  2.47654605,\n",
       "        3.74949496,  4.9779126 ,  0.68937206,  8.82221055,  2.87731832])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Some random data\n",
    "y = np.random.random(15) * 10\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x106b3ed10>]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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v8JH5xx/zfLPz5wP//jcwYACwYQOwYAFPeP355+6R/3bmdrGnA/JG6qsBDLT3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\nUw8EAoEyIph6IBAIlBH/AwODy39retVbAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10733cb50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0, 10, 100)\n",
    "# Smoothing parameter\n",
    "s = 0.4\n",
    "# Calculate the kernels\n",
    "kernels = np.transpose([norm.pdf(x, yi, s) for yi in y])\n",
    "plt.plot(x, kernels, 'k:')\n",
    "plt.plot(x, kernels.sum(1))\n",
    "plt.plot(y, np.zeros(len(y)), 'ro', ms=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SciPy implements a Gaussian KDE that automatically chooses an appropriate bandwidth. Let's create a bi-modal distribution of data that is not easily summarized by a parametric distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a bi-modal distribution with a mixture of Normals.\n",
    "x1 = np.random.normal(0, 3, 50)\n",
    "x2 = np.random.normal(4, 1, 50)\n",
    "\n",
    "# Append by row\n",
    "x = np.r_[x1, x2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 0.00557232,  0.02228927,  0.04457855,  0.08358477,  0.0612955 ,\n",
       "         0.10030173,  0.1838865 ,  0.05572318]),\n",
       " array([-7.66367033, -5.86908508, -4.07449983, -2.27991459, -0.48532934,\n",
       "         1.30925591,  3.10384116,  4.89842641,  6.69301165]),\n",
       " <a list of 8 Patch objects>)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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DVbW927yB8dO4XkzJ+SZgbzduf/ci5g9X1TfXLCBLZ0zyHMZHIbd3F1ucD3wuydaq+sYa\nRgQmfy8BkryG8dP2X1mTQLOb5QK/QUjyROAfgA9X1Uf7zrOEnwde1r0x4ZOAs5J8qKpe3XOuhe5l\n/Ox3f7d9A+OCH5IXAP/2SOck2cv4+7towQ9lieZwkhd1H/8y4xfdhuijjPOR5CeBTWtd7pNU1Req\n6qlVdWFVXcj4B/b5fZT7NN1bSb8NuLiqjvWdZ4FHL/DL+D2rL2F8Qd+gZPwofi1wV1W9u+88S6mq\nt1fV5u5n8hXApwZY7lTV14B7ut9tgO3AF3uMtJhDwLYk39/9/29n/ML1oobyBz/eCPxVd7rP/3Xb\nQ3QdcF2SO4GHgMH9kC4w5KWGvwQ2ATd3zzZuqarf7jfS2Dq6SO+FwKuAO5Lc1t22q6r+ucdMsxjy\nz+Wbgeu7B/b/ortwcyiq6vYkH2J8EHIC+DzwN0uN90InSWrUUJZoJEkrzIKXpEZZ8JLUKAtekhpl\nwUtSoyx4SWqUBS9JjbLgJalR/w9fIE+fvZA+6wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106b46dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(x, bins=8, normed=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x107360190>]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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d9Ntjj0C/VhCrKRqAn36KLl0wdy7svHNmY+WQpmhEQrR2LfToAePGweTJ0eVsJXe22gr+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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1073600d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import kde\n",
    "\n",
    "density = kde.gaussian_kde(x)\n",
    "xgrid = np.linspace(x.min(), x.max(), 100)\n",
    "plt.hist(x, bins=8, normed=True)\n",
    "plt.plot(xgrid, density(xgrid), 'r-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise: Cervical dystonia analysis\n",
    "\n",
    "Recall the cervical dystonia database, which is a clinical trial of botulinum toxin type B (BotB) for patients with cervical dystonia from nine U.S. sites. The response variable is measurements on the Toronto Western Spasmodic Torticollis Rating Scale (TWSTRS), measuring severity, pain, and disability of cervical dystonia (high scores mean more impairment). One way to check the efficacy of the treatment is to compare the distribution of TWSTRS for control and treatment patients at the end of the study.\n",
    "\n",
    "Use the method of moments or MLE to calculate the mean and variance of TWSTRS at week 16 for one of the treatments and the control group. Assume that the distribution of the `twstrs` variable is normal:\n",
    "\n",
    "$$f(x \\mid \\mu, \\sigma^2) = \\sqrt{\\frac{1}{2\\pi\\sigma^2}} \\exp\\left\\{ -\\frac{1}{2} \\frac{(x-\\mu)^2}{\\sigma^2} \\right\\}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[<matplotlib.axes._subplots.AxesSubplot object at 0x1074a7710>,\n",
       "        <matplotlib.axes._subplots.AxesSubplot object at 0x107439950>],\n",
       "       [<matplotlib.axes._subplots.AxesSubplot object at 0x1073ffc90>,\n",
       "        <matplotlib.axes._subplots.AxesSubplot object at 0x1075e8bd0>]], dtype=object)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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X+yU5N8n1SXYkOaqLgkmSJE2TUWak/wDwmar6hSQbgS0tl0mSJGnqLDplRJJ9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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1072ad250>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cdystonia = pd.read_csv(\"data/cdystonia.csv\")\n",
    "cdystonia[cdystonia.obs==6].hist(column='twstrs', by=cdystonia.treat, bins=8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Write your answer here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Regression models\n",
    "\n",
    "A general, primary goal of many statistical data analysis tasks is to relate the influence of one variable on another. For example, we may wish to know how different medical interventions influence the incidence or duration of disease, or perhaps a how baseball player's performance varies as a function of age."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1076a07d0>]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107678750>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.array([2.2, 4.3, 5.1, 5.8, 6.4, 8.0])\n",
    "y = np.array([0.4, 10.1, 14.0, 10.9, 15.4, 18.5])\n",
    "plt.plot(x,y,'ro')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can build a model to characterize the relationship between $X$ and $Y$, recognizing that additional factors other than $X$ (the ones we have measured or are interested in) may influence the response variable $Y$.\n",
    "\n",
    "<div style=\"font-size: 150%;\">  \n",
    "$y_i = f(x_i) + \\epsilon_i$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $f$ is some function, for example a linear function:\n",
    "\n",
    "<div style=\"font-size: 150%;\">  \n",
    "$y_i = \\beta_0 + \\beta_1 x_i + \\epsilon_i$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and $\\epsilon_i$ accounts for the difference between the observed response $y_i$ and its prediction from the model $\\hat{y_i} = \\beta_0 + \\beta_1 x_i$. This is sometimes referred to as **process uncertainty**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We would like to select $\\beta_0, \\beta_1$ so that the difference between the predictions and the observations is zero, but this is not usually possible. Instead, we choose a reasonable criterion: ***the smallest sum of the squared differences between $\\hat{y}$ and $y$***.\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$R^2 = \\sum_i (y_i - [\\beta_0 + \\beta_1 x_i])^2 = \\sum_i \\epsilon_i^2 $$  \n",
    "</div>\n",
    "\n",
    "Squaring serves two purposes: (1) to prevent positive and negative values from cancelling each other out and (2) to strongly penalize large deviations. Whether the latter is a good thing or not depends on the goals of the analysis.\n",
    "\n",
    "In other words, we will select the parameters that minimize the squared error of the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "ss = lambda theta, x, y: np.sum((y - theta[0] - theta[1]*x) ** 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "333.35000000000002"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ss([0,1],x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 21.375000\n",
      "         Iterations: 79\n",
      "         Function evaluations: 153\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(-4.3500136038870876, 3.0000002915386412)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b0,b1 = fmin(ss, [0,1], args=(x,y))\n",
    "b0,b1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x107671cd0>]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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6HL04gDQI8z5FLw6gHdQsfYheHEC7CPM+Qi8OoFPULH2AXhxAtwjzAtGLA8gKNUtB6MUB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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107724950>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'ro')\n",
    "plt.plot([0,10], [b0, b0+b1*10])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 20)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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rhJgfgLwpdcdNzA9AHpW242Y3PwB5VbqOm5gfgLwrVcfNRXsBFEEpCjcX7QVQJIUflRDz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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10743d310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, y, 'ro')\n",
    "plt.plot([0,10], [b0, b0+b1*10])\n",
    "for xi, yi in zip(x,y):\n",
    "    plt.plot([xi]*2, [yi, b0+b1*xi], 'k:')\n",
    "plt.xlim(2, 9); plt.ylim(0, 20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Minimizing the sum of squares is not the only criterion we can use; it is just a very popular (and successful) one. For example, we can try to minimize the sum of absolute differences:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 10.162463\n",
      "         Iterations: 39\n",
      "         Function evaluations: 77\n",
      "0.00157170444494 2.31231743181\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1077cd890>]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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VK1xdfvhwOOqoxv38cUVFjCkspElZGRVNm9KjoCApF2KW7KYQl4xTVgZ33QWFhWG/74IC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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1077cd350>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sabs = lambda theta, x, y: np.sum(np.abs(y - theta[0] - theta[1]*x))\n",
    "b0,b1 = fmin(sabs, [0,1], args=(x,y))\n",
    "print b0,b1\n",
    "plt.plot(x, y, 'ro')\n",
    "plt.plot([0,10], [b0, b0+b1*10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are not restricted to a straight-line regression model; we can represent a curved relationship between our variables by introducing **polynomial** terms. For example, a cubic model:\n",
    "\n",
    "<div style=\"font-size: 150%;\">  \n",
    "$y_i = \\beta_0 + \\beta_1 x_i + \\beta_2 x_i^2 + \\epsilon_i$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 14.001110\n",
      "         Iterations: 198\n",
      "         Function evaluations: 372\n",
      "-11.0748186039 6.0576975948 -0.302681057088\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10772f4d0>]"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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rHJhsZq+b2bkZPpek2cKFoWtmq63g5ZcV8CK5KNMt+aPc/TMz2wkoNbP57j59/YslJSUb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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107865dd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ss2 = lambda theta, x, y: np.sum((y - theta[0] - theta[1]*x - theta[2]*(x**2)) ** 2)\n",
    "b0,b1,b2 = fmin(ss2, [1,1,-1], args=(x,y))\n",
    "print b0,b1,b2\n",
    "plt.plot(x, y, 'ro')\n",
    "xvals = np.linspace(0, 10, 100)\n",
    "plt.plot(xvals, b0 + b1*xvals + b2*(xvals**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Although polynomial model characterizes a nonlinear relationship, it is a linear problem in terms of estimation. That is, the regression model $f(y | x)$ is linear in the parameters.\n",
    "\n",
    "For some data, it may be reasonable to consider polynomials of order>2. For example, consider the relationship between the number of home runs a baseball player hits and the number of runs batted in (RBI) they accumulate; clearly, the relationship is positive, but we may not expect a linear relationship."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 4274.128398\n",
      "         Iterations: 230\n",
      "         Function evaluations: 407\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x107865190>]"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Ulmicc5vN7EpgMlAfeELJXUQkGBlvVSAiIpmR0Z2std0EFTQzW2Fm75jZQjOb\nF3Q8ZczsSTMrNrMlFZ5rZmYFZlZkZlPMrOrDbjOomjgHm9nn8Xu60MxOCTjGVmY2w8zeM7N3zex/\n4s+H6n7WEGfY7ue2ZvammS0ys6Vmdmf8+bDdz+riDNX9jMdUPx7L+Pj3db6XGRvBxzdBfQh0A74A\n5hPS+ryZfQwc6Zz7JuhYKjKz44ENwNPOuUPiz90NrHHO3R3/pbmTc25QCOO8FfjOOTc0yNjKmFkL\noIVzbpGZbQ+8BfQBLiRE97OGOM8iRPcTwMzynHM/xOfiZgPXA70J0f2sIc4TCd/9/CNwJNDEOdc7\nkf+tZ3IEX6tNUCFSp9nqTHDOzQLWVnq6NzAi/ngE/n/8gaomTgjRPXXOrXTOLYo/3gC8j9+zEar7\nWUOcEKL7CeCc+yH+sBF+Dm4tIbufUG2cEKL7aWZ7Aj2AxymPq873MpMJvlaboELCAVPNbIGZXRJ0\nMFvR3DlXHH9cDDQPMpituMrMFpvZE0H/qV6Rme0DHAG8SYjvZ4U434g/Far7aWb1zGwR/r7NcM69\nRwjvZzVxQrju533AQKC0wnN1vpeZTPDZNJt7rHPuCOBU4Ip4ySH0nK+3hfU+/wPYFzgc+Aq4N9hw\nvHjZ43ngaufcdxV/Fqb7GY/zOXycGwjh/XTOlTrnDgf2BE4wsy6Vfh6K+1lFnDFCdD/NrCewyjm3\nkGr+qqjtvcxkgv8CaFXh+1b4UXzoOOe+iv+7GngRX14Kq+J4nRYzawmsCjieKjnnVrk4/J+dgd9T\nM2uIT+7POOfGxZ8O3f2sEOe/yuIM4/0s45xbD7yKrx+H7n6WqRDnUSG7n52A3vG5wFFAVzN7hgTu\nZSYT/AKgjZntY2aNgLOBlzP4+bViZnlm1iT+eDvgJKCavgqh8DJwfvzx+cC4Gq4NTPw/yDKnE/A9\nNTMDngCWOufur/CjUN3P6uIM4f3cpaysYWaNge7AQsJ3P6uMsyxxxgV6P51zNznnWjnn9gXOAaY7\n584lkXvpnMvYF77k8SHwEXBjJj+7DjHuCyyKf70bpjjxv82/BDbh5zMuBJoBU4EiYArQNIRxDgCe\nBt4BFsf/w2wecIzH4eubi/CJaCG+zXWo7mc1cZ4awvt5CPB2PM53gIHx58N2P6uLM1T3s0K8nYGX\nE72X2ugkIhJRgR3ZJyIi6aUELyISUUrwIiIRpQQvIhJRSvAiIhGlBC8iElFK8CIiEaUELyISUf8P\nEx6LK076p9AAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1077f3b90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ss3 = lambda theta, x, y: np.sum((y - theta[0] - theta[1]*x - theta[2]*(x**2) \n",
    "                                  - theta[3]*(x**3)) ** 2)\n",
    "\n",
    "bb = pd.read_csv(\"data/baseball.csv\", index_col=0)\n",
    "plt.plot(bb.hr, bb.rbi, 'r.')\n",
    "b0,b1,b2,b3 = fmin(ss3, [0,1,-1,0], args=(bb.hr, bb.rbi))\n",
    "xvals = np.arange(40)\n",
    "plt.plot(xvals, b0 + b1*xvals + b2*(xvals**2) + b3*(xvals**3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, we need not fit least squares models by hand. The `statsmodels` package implements least squares models that allow for model fitting in a single line:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python2.7/site-packages/statsmodels/stats/stattools.py:72: UserWarning: omni_normtest is not valid with less than 8 observations; 6 samples were given.\n",
      "  \"samples were given.\" % int(n))\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.891</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.864</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   32.67</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 18 Jun 2014</td> <th>  Prob (F-statistic):</th>  <td>0.00463</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>20:56:03</td>     <th>  Log-Likelihood:    </th> <td> -12.325</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>     6</td>      <th>  AIC:               </th> <td>   28.65</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>     4</td>      <th>  BIC:               </th> <td>   28.23</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th> <th>[95.0% Conf. Int.]</th> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th> <td>   -4.3500</td> <td>    2.937</td> <td>   -1.481</td> <td> 0.213</td> <td>  -12.505     3.805</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>    3.0000</td> <td>    0.525</td> <td>    5.716</td> <td> 0.005</td> <td>    1.543     4.457</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>   nan</td> <th>  Durbin-Watson:     </th> <td>   2.387</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td>   nan</td> <th>  Jarque-Bera (JB):  </th> <td>   0.570</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.359</td> <th>  Prob(JB):          </th> <td>   0.752</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 1.671</td> <th>  Cond. No.          </th> <td>    17.9</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.891\n",
       "Model:                            OLS   Adj. R-squared:                  0.864\n",
       "Method:                 Least Squares   F-statistic:                     32.67\n",
       "Date:                Wed, 18 Jun 2014   Prob (F-statistic):            0.00463\n",
       "Time:                        20:56:03   Log-Likelihood:                -12.325\n",
       "No. Observations:                   6   AIC:                             28.65\n",
       "Df Residuals:                       4   BIC:                             28.23\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
       "------------------------------------------------------------------------------\n",
       "const         -4.3500      2.937     -1.481      0.213       -12.505     3.805\n",
       "x1             3.0000      0.525      5.716      0.005         1.543     4.457\n",
       "==============================================================================\n",
       "Omnibus:                          nan   Durbin-Watson:                   2.387\n",
       "Prob(Omnibus):                    nan   Jarque-Bera (JB):                0.570\n",
       "Skew:                           0.359   Prob(JB):                        0.752\n",
       "Kurtosis:                       1.671   Cond. No.                         17.9\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "straight_line = sm.OLS(y, sm.add_constant(x)).fit()\n",
    "straight_line.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>            <td>y</td>        <th>  R-squared:         </th> <td>   0.929</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.881</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   19.50</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Wed, 18 Jun 2014</td> <th>  Prob (F-statistic):</th>  <td>0.0191</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>20:56:03</td>     <th>  Log-Likelihood:    </th> <td> -11.056</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>     6</td>      <th>  AIC:               </th> <td>   28.11</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>     3</td>      <th>  BIC:               </th> <td>   27.49</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th> <th>[95.0% Conf. Int.]</th> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>  -11.0748</td> <td>    6.013</td> <td>   -1.842</td> <td> 0.163</td> <td>  -30.211     8.062</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x</th>         <td>    6.0577</td> <td>    2.482</td> <td>    2.441</td> <td> 0.092</td> <td>   -1.840    13.955</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>I(x ** 2)</th> <td>   -0.3027</td> <td>    0.241</td> <td>   -1.257</td> <td> 0.298</td> <td>   -1.069     0.464</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td>   nan</td> <th>  Durbin-Watson:     </th> <td>   2.711</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td>   nan</td> <th>  Jarque-Bera (JB):  </th> <td>   0.655</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td>-0.809</td> <th>  Prob(JB):          </th> <td>   0.721</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 2.961</td> <th>  Cond. No.          </th> <td>    270.</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   R-squared:                       0.929\n",
       "Model:                            OLS   Adj. R-squared:                  0.881\n",
       "Method:                 Least Squares   F-statistic:                     19.50\n",
       "Date:                Wed, 18 Jun 2014   Prob (F-statistic):             0.0191\n",
       "Time:                        20:56:03   Log-Likelihood:                -11.056\n",
       "No. Observations:                   6   AIC:                             28.11\n",
       "Df Residuals:                       3   BIC:                             27.49\n",
       "Df Model:                           2                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
       "------------------------------------------------------------------------------\n",
       "Intercept    -11.0748      6.013     -1.842      0.163       -30.211     8.062\n",
       "x              6.0577      2.482      2.441      0.092        -1.840    13.955\n",
       "I(x ** 2)     -0.3027      0.241     -1.257      0.298        -1.069     0.464\n",
       "==============================================================================\n",
       "Omnibus:                          nan   Durbin-Watson:                   2.711\n",
       "Prob(Omnibus):                    nan   Jarque-Bera (JB):                0.655\n",
       "Skew:                          -0.809   Prob(JB):                        0.721\n",
       "Kurtosis:                       2.961   Cond. No.                         270.\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from statsmodels.formula.api import ols as OLS\n",
    "\n",
    "data = pd.DataFrame(dict(x=x, y=y))\n",
    "cubic_fit = OLS('y ~ x + I(x**2)', data).fit()\n",
    "\n",
    "cubic_fit.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise: Polynomial function\n",
    "\n",
    "Write a function that specified a polynomial of arbitrary degree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Write your answer here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model Selection\n",
    "\n",
    "How do we choose among competing models for a given dataset? More parameters are not necessarily better, from the standpoint of model fit. For example, fitting a 9-th order polynomial to the sample data from the above example certainly results in an overfit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 7.015262\n",
      "         Iterations: 663\n",
      "         Function evaluations: 983\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1074ad290>]"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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62D2QV4Ab3P2VqOuqzcyuA34A7OLuZ0RdT13MbCHwA3dfGXUtDTGz0cBL7v5Q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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10821e450>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def calc_poly(params, data):\n",
    "        x = np.c_[[data**i for i in range(len(params))]]\n",
    "        return np.dot(params, x)\n",
    "    \n",
    "ssp = lambda theta, x, y: np.sum((y - calc_poly(theta, x)) ** 2)\n",
    "betas = fmin(ssp, np.zeros(10), args=(x,y), maxiter=1e6)\n",
    "plt.plot(x, y, 'ro')\n",
    "xvals = np.linspace(0, max(x), 100)\n",
    "plt.plot(xvals, calc_poly(betas, xvals))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One approach is to use an information-theoretic criterion to select the most appropriate model. For example **Akaike's Information Criterion (AIC)** balances the fit of the model (in terms of the likelihood) with the number of parameters required to achieve that fit. We can easily calculate AIC as:\n",
    "\n",
    "$$AIC = n \\log(\\hat{\\sigma}^2) + 2p$$\n",
    "\n",
    "where $p$ is the number of parameters in the model and $\\hat{\\sigma}^2 = RSS/(n-p-1)$.\n",
    "\n",
    "Notice that as the number of parameters increase, the residual sum of squares goes down, but the second term (a penalty) increases.\n",
    "\n",
    "To apply AIC to model selection, we choose the model that has the **lowest** AIC value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 21.375000\n",
      "         Iterations: 79\n",
      "         Function evaluations: 153\n",
      "Optimization terminated successfully.\n",
      "         Current function value: 14.001110\n",
      "         Iterations: 198\n",
      "         Function evaluations: 372\n",
      "15.7816583572 17.6759368019\n"
     ]
    }
   ],
   "source": [
    "n = len(x)\n",
    "\n",
    "aic = lambda rss, p, n: n * np.log(rss/(n-p-1)) + 2*p\n",
    "\n",
    "RSS1 = ss(fmin(ss, [0,1], args=(x,y)), x, y)\n",
    "RSS2 = ss2(fmin(ss2, [1,1,-1], args=(x,y)), x, y)\n",
    "\n",
    "print aic(RSS1, 2, n), aic(RSS2, 3, n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hence, we would select the 2-parameter (linear) model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logistic Regression\n",
    "\n",
    "Fitting a line to the relationship between two variables using the least squares approach is sensible when the variable we are trying to predict is continuous, but what about when the data are dichotomous?\n",
    "\n",
    "- male/female\n",
    "- pass/fail\n",
    "- died/survived\n",
    "\n",
    "Let's consider the problem of predicting survival in the Titanic disaster, based on our available information. For example, lets say that we want to predict survival as a function of the fare paid for the journey."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0                      Allen, Miss. Elisabeth Walton\n",
       "1                     Allison, Master. Hudson Trevor\n",
       "2                       Allison, Miss. Helen Loraine\n",
       "3               Allison, Mr. Hudson Joshua Creighton\n",
       "4    Allison, Mrs. Hudson J C (Bessie Waldo Daniels)\n",
       "5                                Anderson, Mr. Harry\n",
       "6                  Andrews, Miss. Kornelia Theodosia\n",
       "7                             Andrews, Mr. Thomas Jr\n",
       "8      Appleton, Mrs. Edward Dale (Charlotte Lamson)\n",
       "9                            Artagaveytia, Mr. Ramon\n",
       "...\n",
       "1298                  Wittevrongel, Mr. Camille\n",
       "1299                        Yasbeck, Mr. Antoni\n",
       "1300    Yasbeck, Mrs. Antoni (Selini Alexander)\n",
       "1301                       Youseff, Mr. Gerious\n",
       "1302                          Yousif, Mr. Wazli\n",
       "1303                      Yousseff, Mr. Gerious\n",
       "1304                       Zabour, Miss. Hileni\n",
       "1305                      Zabour, Miss. Thamine\n",
       "1306                  Zakarian, Mr. Mapriededer\n",
       "1307                        Zakarian, Mr. Ortin\n",
       "1308                         Zimmerman, Mr. Leo\n",
       "Name: name, Length: 1309, dtype: object"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "titanic = pd.read_excel(\"data/titanic.xls\", \"titanic\")\n",
    "titanic.name"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1082b5510>"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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4/PE9rFlTwz/+42+Ymxtb+IRm+pK+Xk5xcTHl5UbMZtvCKaTVVFQ8hqJU09r6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\nTAqycDfGmBRk4W6MMSno/wEqCIW8zQZzKAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1082a5610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "jitter = np.random.normal(scale=0.02, size=len(titanic))\n",
    "plt.scatter(np.log(titanic.fare), titanic.survived + jitter, alpha=0.3)\n",
    "plt.yticks([0,1])\n",
    "plt.ylabel(\"survived\")\n",
    "plt.xlabel(\"log(fare)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I have added random jitter on the y-axis to help visualize the density of the points, and have plotted fare on the log scale.\n",
    "\n",
    "Clearly, fitting a line through this data makes little sense, for several reasons. First, for most values of the predictor variable, the line would predict values that are not zero or one. Second, it would seem odd to choose least squares (or similar) as a criterion for selecting the best line."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 277.621917\n",
      "         Iterations: 55\n",
      "         Function evaluations: 103\n"
     ]
    }
   ],
   "source": [
    "x = np.log(titanic.fare[titanic.fare>0])\n",
    "y = titanic.survived[titanic.fare>0]\n",
    "betas_titanic = fmin(ss, [1,1], args=(x,y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1083988d0>]"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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2f5SjR90UFHgoKcnh+edriMXmo2m5wFGgGqhG1/vRNDuRiIbTaSUjI52ZM0tPXQPQ3AzR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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1083989d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "jitter = np.random.normal(scale=0.02, size=len(titanic))\n",
    "plt.scatter(np.log(titanic.fare), titanic.survived + jitter, alpha=0.3)\n",
    "plt.yticks([0,1])\n",
    "plt.ylabel(\"survived\")\n",
    "plt.xlabel(\"log(fare)\")\n",
    "plt.plot([0,7], [betas_titanic[0], betas_titanic[0] + betas_titanic[1]*7.])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we look at this data, we can see that for most values of `fare`, there are some individuals that survived and some that did not. However, notice that the cloud of points is denser on the \"survived\" (y=1) side for larger values of fare than on the \"died\" (y=0) side."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Stochastic model\n",
    "\n",
    "Rather than model the binary outcome explicitly, it makes sense instead to model the *probability* of death or survival in a **stochastic** model. Probabilities are measured on a continuous [0,1] scale, which may be more amenable for prediction using a regression line. We need to consider a different probability model for this exerciese however; let's consider the **Bernoulli** distribution as a generative model for our data:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$f(y|p) = p^{y} (1-p)^{1-y}$$ \n",
    "</div>  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $y = \\{0,1\\}$ and $p \\in [0,1]$. So, this model predicts whether $y$ is zero or one as a function of the probability $p$. Notice that when $y=1$, the $1-p$ term disappears, and when $y=0$, the $p$ term disappears.\n",
    "\n",
    "So, the model we want to fit should look something like this:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$p_i = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, since $p$ is constrained to be between zero and one, it is easy to see where a linear (or polynomial) model might predict values outside of this range. We can modify this model sligtly by using a **link function** to transform the probability to have an unbounded range on a new scale. Specifically, we can use a **logit transformation** as our link function:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$\\text{logit}(p) = \\log\\left[\\frac{p}{1-p}\\right] = x$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a plot of $p/(1-p)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x108432650>]"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x108232850>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "logit = lambda p: np.log(p/(1.-p))\n",
    "unit_interval = np.linspace(0,1)\n",
    "plt.plot(unit_interval/(1-unit_interval), unit_interval)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's the logit function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x108464390>]"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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qs8jNrBkwCKgA9gZ6mNletVx3K/A0oLGQFNSqVXD66TBwIOyzT+w0IoWXbUTeHljs7m+7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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1083f2a50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(logit(unit_interval), unit_interval)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inverse of the logit transformation is:\n",
    "\n",
    "<div style=\"font-size: 150%;\">  \n",
    "$$p = \\frac{1}{1 + \\exp(-x)}$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, now our model is:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$\\text{logit}(p_i) = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can fit this model using maximum likelihood. Our likelihood, again based on the Bernoulli model is:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$L(y|p) = \\prod_{i=1}^n p_i^{y_i} (1-p_i)^{1-y_i}$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which, on the log scale is:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$l(y|p) = \\sum_{i=1}^n y_i \\log(p_i) + (1-y_i)\\log(1-p_i)$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can easily implement this in Python, keeping in mind that `fmin` minimizes, rather than maximizes functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "invlogit = lambda x: 1. / (1 + np.exp(-x))\n",
    "\n",
    "def logistic_like(theta, x, y):\n",
    "    p = invlogit(theta[0] + theta[1] * x)\n",
    "    # Return negative of log-likelihood\n",
    "    return -np.sum(y * np.log(p) + (1-y) * np.log(1 - p))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remove null values from variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = titanic[titanic.fare.notnull()][['fare', 'survived']].values.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "... and fit the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 827.015955\n",
      "         Iterations: 47\n",
      "         Function evaluations: 93\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(-0.88238984528338194, 0.012452067664164127)"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b0,b1 = fmin(logistic_like, [0.5,0], args=(x,y))\n",
    "b0, b1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10865bc50>]"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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fuzKLiCShxAj/hQshOxseeQRWrfJ5mZnQty+8+CK88YZvAM46y1vt+fke6CUl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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1083c9a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "jitter = np.random.normal(scale=0.01, size=len(x))\n",
    "plt.plot(x, y+jitter, 'r.', alpha=0.3)\n",
    "plt.yticks([0,.25,.5,.75,1])\n",
    "xvals = np.linspace(0, 600)\n",
    "plt.plot(xvals, invlogit(b0+b1*xvals))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As with our least squares model, we can easily fit logistic regression models in `statsmodels`, in this case using the `GLM` (generalized linear model) class with a binomial error distribution specified."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Generalized Linear Model Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>          <td>y</td>        <th>  No. Observations:  </th>  <td>  1308</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                 <td>GLM</td>       <th>  Df Residuals:      </th>  <td>  1306</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model Family:</th>       <td>Binomial</td>     <th>  Df Model:          </th>  <td>     1</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Link Function:</th>        <td>logit</td>      <th>  Scale:             </th>    <td>1.0</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>IRLS</td>       <th>  Log-Likelihood:    </th> <td> -827.02</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>           <td>Wed, 18 Jun 2014</td> <th>  Deviance:          </th> <td>  1654.0</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>               <td>20:56:06</td>     <th>  Pearson chi2:      </th> <td>1.33e+03</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Iterations:</th>         <td>6</td>        <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th> <th>[95.0% Conf. Int.]</th> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th> <td>   -0.8824</td> <td>    0.076</td> <td>  -11.684</td> <td> 0.000</td> <td>   -1.030    -0.734</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>x1</th>    <td>    0.0125</td> <td>    0.002</td> <td>    7.762</td> <td> 0.000</td> <td>    0.009     0.016</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                 Generalized Linear Model Regression Results                  \n",
       "==============================================================================\n",
       "Dep. Variable:                      y   No. Observations:                 1308\n",
       "Model:                            GLM   Df Residuals:                     1306\n",
       "Model Family:                Binomial   Df Model:                            1\n",
       "Link Function:                  logit   Scale:                             1.0\n",
       "Method:                          IRLS   Log-Likelihood:                -827.02\n",
       "Date:                Wed, 18 Jun 2014   Deviance:                       1654.0\n",
       "Time:                        20:56:06   Pearson chi2:                 1.33e+03\n",
       "No. Iterations:                     6                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
       "------------------------------------------------------------------------------\n",
       "const         -0.8824      0.076    -11.684      0.000        -1.030    -0.734\n",
       "x1             0.0125      0.002      7.762      0.000         0.009     0.016\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logistic = sm.GLM(y, sm.add_constant(x), family=sm.families.Binomial()).fit()\n",
    "logistic.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise: multivariate logistic regression\n",
    "\n",
    "Which other variables might be relevant for predicting the probability of surviving the Titanic? Generalize the model likelihood to include 2 or 3 other covariates from the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Write your answer here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bootstrapping\n",
    "\n",
    "Parametric inference can be **non-robust**:\n",
    "\n",
    "* inaccurate if parametric assumptions are violated\n",
    "* if we rely on asymptotic results, we may not achieve an acceptable level of accuracy\n",
    "\n",
    "Parmetric inference can be **difficult**:\n",
    "\n",
    "* derivation of sampling distribution may not be possible\n",
    "\n",
    "An alternative is to estimate the sampling distribution of a statistic *empirically* without making assumptions about the form of the population.\n",
    "\n",
    "We have seen this already with the kernel density estimate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Non-parametric Bootstrap\n",
    "\n",
    "The bootstrap is a resampling method discovered by [Brad Efron](http://www.jstor.org/discover/10.2307/2958830?uid=3739568&uid=2&uid=4&uid=3739256&sid=21102342537691) that allows one to approximate the true sampling distribution of a dataset, and thereby obtain estimates of the mean and variance of the distribution.\n",
    "\n",
    "Bootstrap sample:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$S_1^* = \\{x_{11}^*, x_{12}^*, \\ldots, x_{1n}^*\\}$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$S_i^*$ is a sample of size $n$, **with** replacement.\n",
    "\n",
    "In Python, we have already seen the NumPy function `permutation` that can be used in conjunction with Pandas' `take` method to generate a random sample of some data without replacement:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([u'Meek, Mrs. Thomas (Annie Louise Rowley)',\n",
       "       u'Thorneycroft, Mr. Percival', u'Williams, Mr. Leslie',\n",
       "       u'Graham, Mr. George Edward', u'Petroff, Mr. Nedelio'], dtype=object)"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.permutation(titanic.name)[:5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, we can use the `random.randint` method to generate a sample *with* replacement, which we can use when bootstrapping."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41      Brown, Mrs. James Joseph (Margaret Tobin)\n",
       "1061              Nilsson, Miss. Helmina Josefina\n",
       "937                  Klasen, Miss. Gertrud Emilia\n",
       "426                            Hale, Mr. Reginald\n",
       "831                Goodwin, Mr. Charles Frederick\n",
       "Name: name, dtype: object"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random_ind = np.random.randint(0, len(titanic), 5)\n",
    "titanic.name[random_ind]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We regard S as an \"estimate\" of population P\n",
    "\n",
    "> population : sample :: sample : bootstrap sample\n",
    "\n",
    "The idea is to generate replicate bootstrap samples:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$S^* = \\{S_1^*, S_2^*, \\ldots, S_R^*\\}$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute statistic $t$ (estimate) for each bootstrap sample:\n",
    "\n",
    "<div style=\"font-size: 120%;\">  \n",
    "$$T_i^* = t(S^*)$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Jo2/rpbPbuuh2qFxQVaeTXAg8tMNrfKH794tJ/o7Rn/YthPw47/P2Nhd397Vmz3VRVV/b\nsvzhJO9McriqvjyjGhfFqmwTe+qzTcx6uGansaSPA89NcmmSpwGvBG6fXVkzcztw5tJVRxh9Cz9F\nkvOTPKNbfjrwUmDHow6WzDjv8+3AL8ETZ1A/vGWIqyV7roska+kmUknyfEaHPK9awMPqbBN76rNN\nHPhFQ5L8LPAO4FnAPyS5p6peluTZwLuq6qdrdU6iugW4NclrgVPAjQBb1wWjoZ6/7d7Hc4G/qKrj\n8yl3unZ6n5P8cvf4n1XVHUmuS3I/8HXgNXMs+cCMsy6AnwN+JcnjwDeAV82t4AOU5P3ANcCzkjwA\n3MzoiKOV2iZg73VBj23Ck6EkqWGLcnSNJOkAGPKS1DBDXpIaZshLUsMMeUlqmCEvSQ0z5CWpYYa8\nJDXs/wCVqXPsFDsT4gAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108656a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 10\n",
    "R = 1000\n",
    "# Original sample (n=10)\n",
    "x = np.random.normal(size=n)\n",
    "# 1000 bootstrap samples of size 10\n",
    "s = [x[np.random.randint(0,n,n)].mean() for i in range(R)]\n",
    "_ = plt.hist(s, bins=30)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bootstrap Estimates\n",
    "\n",
    "From our bootstrapped samples, we can extract *estimates* of the expectation and its variance:\n",
    "\n",
    "$$\\bar{T}^* = \\hat{E}(T^*) = \\frac{\\sum_i T_i^*}{R}$$\n",
    "\n",
    "$$\\hat{\\text{Var}}(T^*) = \\frac{\\sum_i (T_i^* - \\bar{T}^*)^2}{R-1}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.087385394806476724"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boot_mean = np.sum(s)/R\n",
    "boot_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10590407752057245"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boot_var = ((np.array(s) - boot_mean) ** 2).sum() / (R-1)\n",
    "boot_var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we have estimated the expectation of the bootstrapped statistics, we can estimate the **bias** of T:\n",
    "\n",
    "$$\\hat{B}^* = \\bar{T}^* - T$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.00528084680355842"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "boot_mean - np.mean(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bootstrap error\n",
    "\n",
    "There are two sources of error in bootstrap estimates:\n",
    "\n",
    "1. **Sampling error** from the selection of $S$.\n",
    "2. **Bootstrap error** from failing to enumerate all possible bootstrap samples.\n",
    "\n",
    "For the sake of accuracy, it is prudent to choose at least R=1000"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bootstrap Percentile Intervals\n",
    "\n",
    "An attractive feature of bootstrap statistics is the ease with which you can obtain an estimate of *uncertainty* for a given statistic. We simply use the empirical quantiles of the bootstrapped statistics to obtain percentiles corresponding to a confidence interval of interest.\n",
    "\n",
    "This employs the *ordered* bootstrap replicates:\n",
    "\n",
    "$$T_{(1)}^*, T_{(2)}^*, \\ldots, T_{(R)}^*$$\n",
    "\n",
    "Simply extract the $100(\\alpha/2)$ and $100(1-\\alpha/2)$ percentiles:\n",
    "\n",
    "$$T_{[(R+1)\\alpha/2]}^* \\lt \\theta \\lt T_{[(R+1)(1-\\alpha/2)]}^*$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.82890714, -0.77634577, -0.76588512, -0.76230089, -0.75578488,\n",
       "       -0.73850118, -0.72869116, -0.72862786, -0.72840095, -0.71831374])"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s_sorted = np.sort(s)\n",
    "s_sorted[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.81823418,  0.86179331,  0.92314175,  0.93496722,  0.9358216 ,\n",
       "        1.02058937,  1.03085586,  1.03121927,  1.22699691,  1.3599996 ])"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s_sorted[-10:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.5684053 ,  0.68682205])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha = 0.05\n",
    "s_sorted[[(R+1)*alpha/2, (R+1)*(1-alpha/2)]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise: Cervical dystonia bootstrap estimates\n",
    "\n",
    "Use bootstrapping to estimate the mean of one of the treatment groups, and calculate percentile intervals for the mean."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Write your answer here"
   ]
  }
 ],
 "metadata": {
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