4. Statistical Data Modeling.ipynb

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  "name": "4. Statistical Data Modeling"
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     "source": [
      "# Statistical Data Modeling\n",
      "\n",
      "Some or most of you have probably taken some undergraduate- or graduate-level statistics courses. Unfortunately, the curricula for most introductory statisics courses are mostly focused on conducting **statistical hypothesis tests** as the primary means for interest: t-tests, chi-squared tests, analysis of variance, etc. Such tests seek to esimate whether groups or effects are \"statistically significant\", a concept that is poorly understood, and hence often misused, by most practioners. Even when interpreted correctly, statistical significance is a questionable goal for statistical inference, as it is of limited utility.\n",
      "\n",
      "A far more powerful approach to statistical analysis involves building flexible **models** with the overarching aim of *estimating* quantities of interest. This section of the tutorial illustrates how to use Python to build statistical models of low to moderate difficulty from scratch, and use them to extract estimates and associated measures of uncertainty."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import pandas as pd\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)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "x = 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",
      "_ = hist(x, bins=8)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x107b54a50>"
       ]
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     "prompt_number": 2
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     "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 \u2018best fit\u2019. 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",
      "\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",
      "\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",
     "collapsed": false,
     "input": [
      "precip = pd.read_table(\"data/nashville_precip.txt\", index_col=0, na_values='NA', delim_whitespace=True)\n",
      "precip.head()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "       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"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "_ = precip.hist(sharex=True, sharey=True, grid=False)\n",
      "tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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8/fbbCA4OxoABA+Dq6opRo0Y1V6xkIWpfN+QFXiIyBZ2DJKRSqXrEnqenJzp1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\nIrFAERGRRWKBIiIii8QCRUREFun/A0YTXjj9DG0yAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107f97c90>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "precip.fillna(value={'Oct': precip.Oct.mean()}, inplace=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "<class 'pandas.core.frame.DataFrame'>\n",
        "Int64Index: 141 entries, 1871 to 2011\n",
        "Data columns (total 12 columns):\n",
        "Jan    141  non-null values\n",
        "Feb    141  non-null values\n",
        "Mar    141  non-null values\n",
        "Apr    141  non-null values\n",
        "May    141  non-null values\n",
        "Jun    141  non-null values\n",
        "Jul    141  non-null values\n",
        "Aug    141  non-null values\n",
        "Sep    141  non-null values\n",
        "Oct    141  non-null values\n",
        "Nov    141  non-null values\n",
        "Dec    141  non-null values\n",
        "dtypes: float64(12)"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, let's calculate the sample moments of interest, the means and variances by month:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "precip_mean = precip.mean()\n",
      "precip_mean"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "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"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "precip_var = precip.var()\n",
      "precip_var"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "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"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We then use these moments to estimate $\\alpha$ and $\\beta$ for each month:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha_mom = precip_mean ** 2 / precip_var\n",
      "beta_mom = precip_var / precip_mean"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha_mom, beta_mom"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "(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,\n",
        " 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)"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "from scipy.stats.distributions import gamma\n",
      "\n",
      "hist(precip.Jan, normed=True, bins=20)\n",
      "plot(linspace(0, 10), gamma.pdf(linspace(0, 10), alpha_mom[0], beta_mom[0]))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10814a110>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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POXwtg94Jt+iOxfsSARGM6VdWVuLVV18d9b36+vpRjx955BE88sgjo75XWFiI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\nQ5+IKI0w9ImI0ghDn4gojfx/x/DcAaaZhjIAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10814a150>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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 = 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",
      "tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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xbNmyPsfkCViW7XWCi/ZShoXhwtlylxg+5gyeVnZs1TvtucrwaYZ1kV/avn37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\nIyva5MyKt2hgBcml8BYJUX6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QLxtYARo9rshcc/5Vezw+D2MnxeL491epgUUcTtuig9ZF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PEq2dJARI0El5czxvC9J9kSzfQfnCwVBXuw8Ms8Opm7xoM7zEawAJJSrOj1Oo4frgh1KEK0KRwL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\nbrXsTa0AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10834a4d0>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "y = np.random.poisson(5, size=100)\n",
      "plt.hist(y, bins=12, normed=True)\n",
      "xlabel('y'); ylabel('Pr(y)')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "<matplotlib.text.Text at 0x1083496d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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AABgjPAAAxggPAIAxwgMAYIzwAAAYIzwAAMYIDwCAMcIDAGCM8AAAGCM8AADGCA8AgDHC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\nAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10825b3d0>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "0.041303093412337726"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.sum(poisson_like(yi, lam) for yi in y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "11.711976556065167"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lam = 8\n",
      "np.sum(poisson_like(yi, lam) for yi in y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "7.2853069474261591"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can plot the likelihood function for any value of the parameter(s):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lambdas = np.linspace(0,15)\n",
      "x = 5\n",
      "plt.plot(lambdas, [poisson_like(x, l) for l in lambdas])\n",
      "xlabel('$\\lambda$')\n",
      "ylabel('L($\\lambda$|x={0})'.format(x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "<matplotlib.text.Text at 0x1080a3710>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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lZx2HkH7o3cgL0RIq/mmiPBjzJ8jwlpqWWSGex6UxkwsXLtx2FondTp+SRtq5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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108123c10>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "lam = 5\n",
      "xvals = arange(15)\n",
      "plt.bar(xvals, [poisson_like(x, lam) for x in xvals])\n",
      "xlabel('x')\n",
      "ylabel('Pr(X|$\\lambda$=5)')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "<matplotlib.text.Text at 0x10899ba90>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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d//RX50cffST+/ve/F3t6ekRRvPL76XpG7WWub76yJSkpSXrtSrzRarWYNGkS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\nMCEiItkYJkREJBvDhIiIZPt/5cwSS3FXK2gAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108991790>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "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 $n = 2012 \u2212 1871 = 141$ 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",
     "collapsed": false,
     "input": [
      "from scipy.optimize import newton"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "# some function\n",
      "func = lambda x: 3./(1 + 400*np.exp(-2*x)) - 1\n",
      "xvals = np.linspace(0, 6)\n",
      "plot(xvals, func(xvals))\n",
      "text(5.3, 2.1, '$f(x)$', fontsize=16)\n",
      "# zero line\n",
      "plot([0,6], [0,0], 'k-')\n",
      "# value at step n\n",
      "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",
      "plot(xvals, tanline(xvals), 'r--')\n",
      "# point at step n+1\n",
      "xprime = 0.858/0.626\n",
      "plot([xprime, xprime], [tanline(xprime), func(xprime)], 'k:')\n",
      "plt.text(xprime+.1, -.2, '$x_{n+1}$', fontsize=16)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "<matplotlib.text.Text at 0x108ae8e10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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w2Go8LN1awgu3R0cpR0QPeTeL4ElIaPLJXlEUFm45xuD2ybSPslKOzMOPfJLw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\nF0KIKCEJXwghosT/ByIk0e9o3HtaAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1089e6f10>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "# Calculate statistics\n",
      "log_mean = precip.mean().apply(log)\n",
      "mean_log = precip.apply(log).mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Time to optimize!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Alpha MLE for December\n",
      "alpha_mle = newton(dlgamma, 2, dl2gamma, args=(log_mean[-1], mean_log[-1]))\n",
      "alpha_mle"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 22,
       "text": [
        "3.5189679152399647"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And now plug this back into the solution for beta:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$ \\beta  = \\frac{\\alpha}{\\bar{X}} $$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "beta_mle = alpha_mle/precip.mean()[-1]\n",
      "beta_mle"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 23,
       "text": [
        "0.84261607548413797"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "dec = precip.Dec\n",
      "dec.hist(normed=True, bins=10, grid=False)\n",
      "x = linspace(0, dec.max())\n",
      "plot(x, gamma.pdf(x, alpha_mom[-1], beta_mom[-1]), 'm-')\n",
      "plot(x, gamma.pdf(x, alpha_mle, beta_mle), 'r--')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 24,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1089e4f10>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Uk8KQ3w/h0OxDRP8wGjVK7f0kIQY5t6EPMHbsWB566KEubVlZWa7fx8TEkJOT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\nQogAIqEvhBAB5P8BUL5B+4nuU50AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1089e4d50>"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For some common distributions, SciPy includes methods for fitting via MLE:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.stats import gamma\n",
      "\n",
      "gamma.fit(precip.Dec)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 25,
       "text": [
        "(2.2427517865962434, 0.65494603858755807, 1.5700739281147422)"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "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",
      "$$f_X(x) = \\frac{f_Y (x)}{1\u2212F_Y (a)} \\, \\text{for} \\, x \\gt a$$ \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",
     "collapsed": false,
     "input": [
      "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",
      "_ = hist(x, bins=100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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pqemhx4PBoILB4NKqAgAsCXeoAoCB2FsGOSN5k5Mktv8FloiVO3JG+k1ObP8LLA3hDgAG\nItwBwECEOwAYiHAHAAMR7gBgIEYhkZMYiwSWhnBHToqPTemXH/5dkvRmXSA1GknQA/NDWwY5j/l3\nYOEIdwAwEG0ZrFg3E+O0a4BZsHLHihW+PUG7BpgF4Q4ABiLcAcBAhDsAGIhwBwADMS2DFSX9ztWJ\nqWmbqwFyF+GOFSX9ztXjT261uRogd9GWAQADEe4AYCDCHQAMRLgDgIEIdwAwEOEOAAYi3AHAQMy5\nwyhsAwzcs6Rwf+mll7RmzRoVFBTI6XSqtbVVo6Oj6uzsVDgc1qZNm3T48GEVFhZaVS+QUXIbYElq\nCwZS4U7oI98seeV+4sQJrV+/PvV1d3e3duzYoaamJr3//vvq7u7W888/v9SPAZZkttAHTLXknvvM\nzMx9X1++fFl79uyRJNXU1Kivr2+pHwHMKbnnDPvNAPcsKdwdDodee+01NTU16cMPP5QkxeNxeb1e\nSZLH41E8Hl96lcAckg/RnpiamftkIA8sqS3T0tKijRs3KhQKqbW1VZs3b77vdYfDsaTiAACLs6Rw\n37hxoyTJ7/fr8ccf1+DgoDwej4aHh+X1ehWLxeTxeCwpFFio+WwPnLzQykVWmGbRbZnx8XGNjo5K\nkhKJhPr7+1VaWqrKykpduHBBknTx4kVVVVVZUiiwUMlWTaZ2TfJCa+R/J3T1xoiu3hjRzcT4MlcK\nWG/RK/d4PK62tjZJ0oYNG3TgwAF9+9vf1vbt29XZ2anGxsbUKCSQ69L3iWeaBiZYdLgXFxenwj3d\nmjVr1NTUtKSigGziaU7IB9yhirzD05yQD9hbBgAMRLgDgIEIdwAwEOEOAAYi3AHAQIQ7ABiIcAcA\nAzHnDnxF+k1O7DmDlYqVO/AV6XvSJJ/eBKw0hDsAGIhwBwADEe4AYCDCHQAMxLQMME/JpzZJD5+i\nmet1YDmxcgfmKfnUptmmaOZ6HVhOrNyBDHiwB1Yqwh3IYLYHe3CjE3IdbRlgEbjRCbmOcAeWKLmK\np22DXEK4A0uUXMVPTM089PWbiXFdvTGiqzdGdDMxvszVIV8R7kCWMUUDO3BBFciC+UzZMBePbCLc\ngSyYbcomXXJFL0ltwQDhDkvRlgEAA7FyB5bRQm6KSm/brF/l1O2JKUm0cDA/hDuwjOZq16QH+sTU\ntH7xn/+TOjf5Plo4mA/CHcgB6bPy6YEOLFZWeu4DAwNqbm5WY2Oj/vCHP2TjIwCjzDUrDyyU5Sv3\n6elpnT17Vq+++qp8Pp9+/vOf69FHH5Xf77f6o4C8xL42mA/Lw31wcFAlJSUqLi6WJFVXV+vy5cuE\nO2CR9L79m3UBhW9P3BfyC52fT57PLwqzWB7u0WhURUVFqa99Pp8GBwet/hgA+lfQJ0Neuv9CbPrx\n9Imb9H8nz0+/UDvbLwhuvFo5bL+g+u//tlml3kIlxibtLgVYsWabwvnq8dn+LT04pvmwXxDpxxcy\ntbPcvxT4JSQ5ZmZmLL2C87e//U3vvfeefvGLX0iSfve738nhcOjpp59+4NwrV65oeHjYyo8HAON5\nvV499thjGc+xfOW+bds2DQ0NKRwOy+fz6dNPP9VPfvKTh547V3EAgMWxfOUu3RuF/M1vfqOpqSk9\n8cQTCgaDVn8EACCDrIQ7AMBebBwGAAYi3AHAQLaOQv7xj3/Ue++9p3/+859qbW3VN7/5TTvLsczA\nwIDeeuut1DWH/fv3212SZc6cOaP+/n653W61t7fbXY7lbt26pa6uLsXjcbndbtXU1Kimpsbusiwz\nMTGhEydO6O7du1q1apV27dqluro6u8uy1PT0tI4dOyafz6djx47ZXY6lXnrpJa1Zs0YFBQVyOp1q\nbW2d9Vxbw720tFSNjY361a9+ZWcZljJ9+4Xa2lrt379fp0+ftruUrHC5XKqvr9eWLVuUSCR09OhR\nBQIBY/7/Vq1apePHj2v16tW6e/eujh07psrKSpWUlNhdmmV6enrk9/s1OjpqdylZceLECa1fv37O\n82xty2zevFnf+MY37CzBcunbL7hcrtT2C6aoqKjQunXr7C4ja7xer7Zs2SJJcrvd2rZtm2KxmL1F\nWWz16ns39IyNjWlqakoul+33MlomEomov79fe/fulamzIvP9ucz5X80RbL9gjqGhIYVCIZWVldld\niqWmp6fV3NysL7/8Ui+88IIeeeQRu0uyzFtvvaUf/vCHxq7aHQ6HXnvtNTkcDu3bt09PPvnkrOdm\nPdxbWloeehfqwYMHVVlZme2PBxZlbGxMHR0dqq+vV2Fhod3lWKqgoEBtbW0Kh8NqbW3Vjh07tHXr\nyt87/sqVK3K73dq6dav+8pe/2F1OVrS0tGjjxo0KhUJqbW3V5s2bVVFR8dBzsx7ur776arY/Iqf4\nfD5FIpHU15FIRD6fz8aKsFCTk5Nqb2/X7t27VVVVZXc5WVNcXKydO3dqYGDAiHC/fv26rly5ov7+\nft29e1ejo6M6ffq0fvzjH9tdmmU2btwoSfL7/Xr88cc1ODg4a7gzCmmx9O0XJicn9emnn/IXygoy\nMzOjc+fOye/368CBA3aXY7lEIqE7d+5IkkZGRvT555+rtLTU5qqs8dxzz+ns2bPq6urSkSNH9K1v\nfcuoYB8fH0+1mxKJhPr7+zP+39l6h+qf//xn/frXv1YikdDatWu1detWvfLKK3aVYxmTt1/o6OjQ\ntWvXNDIyIo/Ho2effVa1tbV2l2WZv/71rzp+/LhKS0vlcDgk3QuN73znOzZXZo1//OMf6urq0vT0\ntLxer3bt2qW9e/faXZblBgYG9Pvf/17Nzc12l2KZcDistrY2SdKGDRu0a9cufe9735v1fLYfAAAD\n0ZYBAAMR7gBgIMIdAAxEuAOAgQh3ADAQ4Q4ABiLcAcBAhDsAGOj/ALyjHIOQTw2lAAAAAElFTkSu\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108acb810>"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can construct a log likelihood for this function using the conditional form:\n",
      "\n",
      "$$f_X(x) = \\frac{f_Y (x)}{1\u2212F_Y (a)} \\, \\text{for} \\, x \\gt a$$ "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "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()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 27
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "from scipy.optimize import fmin\n",
      "\n",
      "fmin(trunc_norm, np.array([1,2]), args=(-1, x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 11102.552602\n",
        "         Iterations: 48\n",
        "         Function evaluations: 89\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 28,
       "text": [
        "array([ 0.03391135,  0.995191  ])"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "# Some random data\n",
      "y = np.random.random(15) * 10\n",
      "y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 29,
       "text": [
        "array([ 2.62168662,  9.57094631,  0.55470409,  5.5091919 ,  6.43860024,\n",
        "        7.15947092,  5.97391474,  8.64691181,  7.47841321,  2.30472374,\n",
        "        7.78811628,  1.11174965,  6.50428732,  4.01692277,  4.79648118])"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "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",
      "plot(x, kernels, 'k:')\n",
      "plot(x, kernels.sum(1))\n",
      "plot(y, np.zeros(len(y)), 'ro', ms=10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 30,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108c22e50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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eqboncj0MtIsrp7BOBb4nxylxy95eHugcLHRK3QpHZ6beibtvqt6oacSnBwsx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\nSUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108c06f90>"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "# 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]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(x, bins=8, normed=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 32,
       "text": [
        "(array([ 0.0106135 ,  0.021227  ,  0.03714725,  0.0742945 ,  0.063681  ,\n",
        "        0.0530675 ,  0.16450926,  0.10613501]),\n",
        " array([-8.73310783, -6.84871543, -4.96432303, -3.07993063, -1.19553823,\n",
        "        0.68885417,  2.57324657,  4.45763897,  6.34203137]),\n",
        " <a list of 8 Patch objects>)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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JjkFPRCQ4Bj0RkeAY9EREgvv/+s4TvgIQUcAAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1082334d0>"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "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-')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 33,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108cc7090>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ffoqmF15Ay0MP8QtORB2A3/GmDqEoKoJar0dQYyPqtm9Hy+9+x5In6iCc0ZNP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\n2rRWl7yksHv3bgiCgN69e0u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       "text": [
        "<matplotlib.figure.Figure at 0x108cdb590>"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "cdystonia = pd.read_csv(\"data/cdystonia.csv\")\n",
      "cdystonia[cdystonia.obs==6].hist(column='twstrs', by=cdystonia.treat, bins=8)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 34,
       "text": [
        "array([[<matplotlib.axes.AxesSubplot object at 0x108ceadd0>,\n",
        "        <matplotlib.axes.AxesSubplot object at 0x108cfe190>],\n",
        "       [<matplotlib.axes.AxesSubplot object at 0x108e965d0>,\n",
        "        <matplotlib.axes.AxesSubplot object at 0x108eb17d0>]], dtype=object)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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SEfUA74Ikou7weAbMZrNh1apVsNvtCAkJQXp6OubMmSMjNiIiv5aSkoKUlBRfh0FEfZDH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\nSUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108d17f10>"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 34
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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",
      "plot(x,y,'ro')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 35,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108f54f50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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UPfroo6qvr3d7lLC4dOmSzp8/r/z8fO3fv1/ffPONDhw44PZYYTd58mRJUkpKihYvXnxH\nL2SLJlOmTNGkSZO0cOFCxcXFadmyZfL7/W6PFXZ+v1/333+/vF6v26OEVUNDg2bNmqXp06crMTFR\nS5Ys+de2RCzgjuPo8OHDSklJUXZ2dqQ+xjVtbW3q6OiQJLW3t+vrr79Wamqqy1OFx4YNG3To0CFV\nVFRox44dmjNnjl555RW3xwqrrq6ugeOhtrY2+f3+MfP35/P5NH36dH3//ffq7+9XXV2dHnnkEbfH\nCrvTp09r2bJlbo8Rdg899JAuX76s3377TT09PfL7/Zo7d+6geyN2Bn7p0iXV1tYqNTVVBQUFkv4I\nw7x58yL1kaOqtbVVFRUV6u/vl8/nU05Ojh5++GG3x4oIj8fj9ghhFwgEVFJSIklKTExUdnb2v/5P\nYlF+fr4qKirU1tam1NRUbdy40e2RwioYDOrChQvKy8tze5SwmzBhgtavX6+SkhJ1d3dr7ty5mjNn\nzqB7eZQeAIziSUwAMIqAA4BRBBwAjCLgAGAUAQcAowg4ABhFwAHAKAIOAEb9P6PSghthTu6CAAAA\nAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108ec27d0>"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "ss = lambda theta, x, y: np.sum((y - theta[0] - theta[1]*x) ** 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ss([0,1],x,y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 37,
       "text": [
        "333.35000000000002"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b0,b1 = fmin(ss, [0,1], args=(x,y))\n",
      "b0,b1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 21.375000\n",
        "         Iterations: 79\n",
        "         Function evaluations: 153\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 38,
       "text": [
        "(-4.3500136038870876, 3.0000002915386412)"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(x, y, 'ro')\n",
      "plot([0,10], [b0, b0+b1*10])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 39,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108ebab50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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6d+6cNm3apG9961sqKSkxYzZYjN4ccJ6EQz0nJ0eSlJ+fr+985ztqbW0l1FMcvTngXAmF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\nAHAQQh0AHIRQBwAHIdQBwEH+HwEg8WHrbxoVAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108cf5550>"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(x, y, 'ro')\n",
      "plot([0,10], [b0, b0+b1*10])\n",
      "for xi, yi in zip(x,y):\n",
      "    plot([xi]*2, [yi, b0+b1*xi], 'k:')\n",
      "xlim(2, 9); ylim(0, 20)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 40,
       "text": [
        "(0, 20)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108f94310>"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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",
      "plot(x, y, 'ro')\n",
      "plot([0,10], [b0, b0+b1*10])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 10.162463\n",
        "         Iterations: 39\n",
        "         Function evaluations: 77\n",
        "0.00157170444494 2.31231743181\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 41,
       "text": [
        "[<matplotlib.lines.Line2D at 0x109062590>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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gZzLBEhiItOXLMTUjQ+xYRORmRC3zmhYDcktUsNgE/OaRBCRGBosZp9+VFhfj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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108d11510>"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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",
      "plot(x, y, 'ro')\n",
      "xvals = np.linspace(0, 10, 100)\n",
      "plot(xvals, b0 + b1*xvals + b2*(xvals**2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "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"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108fa6bd0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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sPRGRmFwi7ImISFwMeyIiL8CwJyLyAgx7IiIvwLAnIvICDHsiIi/AsCci8gIM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\nAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10896e2d0>"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "ss3 = lambda theta, x, y: np.sum((y - theta[0] - theta[1]*x - theta[2]*(x**2) - theta[3]*(x**3)) ** 2)\n",
      "\n",
      "bb = pd.read_csv(\"data/baseball.csv\", index_col=0)\n",
      "plot(bb.hr, bb.rbi, 'r.')\n",
      "b0,b1,b2,b3 = fmin(ss3, [0,1,-1,0], args=(bb.hr, bb.rbi))\n",
      "xvals = arange(40)\n",
      "plot(xvals, b0 + b1*xvals + b2*(xvals**2) + b3*(xvals**3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 4274.128398\n",
        "         Iterations: 230\n",
        "         Function evaluations: 407\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 43,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108ef1810>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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kEAiFgpOtROTxZDtGb89mprPNHfjw2Hn8en60Uw4sIyJyBZIGfeDy5VD0c0zC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\nT0Tk5hj0RERu7v8DLdrDpSm4gIwAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109066550>"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "import statsmodels.api as sm\n",
      "\n",
      "straight_line = sm.OLS(y, sm.add_constant(x)).fit()\n",
      "straight_line.summary()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "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, 03 Jul 2013</td> <th>  Prob (F-statistic):</th>  <td>0.00463</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Time:</th>                 <td>21:46:32</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",
        "</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>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 44,
       "text": [
        "<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, 03 Jul 2013   Prob (F-statistic):            0.00463\n",
        "Time:                        21:46:32   Log-Likelihood:                -12.325\n",
        "No. Observations:                   6   AIC:                             28.65\n",
        "Df Residuals:                       4   BIC:                             28.23\n",
        "Df Model:                           1                                         \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",
        "\"\"\""
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "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()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "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, 03 Jul 2013</td> <th>  Prob (F-statistic):</th>  <td>0.0191</td> \n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Time:</th>                 <td>21:46:32</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",
        "</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>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 45,
       "text": [
        "<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, 03 Jul 2013   Prob (F-statistic):             0.0191\n",
        "Time:                        21:46:32   Log-Likelihood:                -11.056\n",
        "No. Observations:                   6   AIC:                             28.11\n",
        "Df Residuals:                       3   BIC:                             27.49\n",
        "Df Model:                           2                                         \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",
        "\"\"\""
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exercise: Polynomial function\n",
      "\n",
      "Write a function that specified a polynomial of arbitrary degree."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 45
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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",
      "plot(x, y, 'ro')\n",
      "xvals = np.linspace(0, max(x), 100)\n",
      "plot(xvals, calc_poly(betas, xvals))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 7.015262\n",
        "         Iterations: 663\n",
        "         Function evaluations: 983\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 46,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108d0f950>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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6dfhNRUXHsdU1DTj25Ot4/b5bhsSzLgdKjwX+1ltvobi4GFqtFps2bQIAvP/++9i3bx+0\nWi0AYMmSJZgwYcLAJiWiIeng9u03lDcAhN+6ABEVJ5AUMVmkVNLQY4HPnTsXCxcuxJYtWzqOyWQy\n5OTkICcnZ0DDEdHQp7TZbvj869TJOJo6Gbcc/gOAp8QJJRE93hnIyMhASEjITccFQRiQQETkX5yB\n/39pfGtgMF59cDX+84NNkCk5dNITj2/t7t69G7/4xS+wdetWmM1mb2YiIj8ye8UK5CUlAQAKFi3H\ntPPHsdPRhFnLl4uczPd5dBMzOzsbDz74ICwWC/70pz/h3XffxcqVK72djYj8wO3Z2QCA5R8U4cyt\nszDtyNuYu3Fjx3HqmkcFHhbWPq0nODgYCxYsQEFBgVdDEZF/uXX2PGxtjsOLM+Ix9bm/iB1HMjwa\nQtHr9QAAl8uFL7/8EgkJCV4NRUT+wy0IyD9wETMSwzB1JOd890WPV+CbN2/GmTNnYDQasXLlSjz0\n0EMoKSlBZWUllEolMjIysGzZssHISkRD0Hvf1sFkc+HJqXFiR5GcHgv8+eefv+nYvHnzBiQMEfmX\nk1dM+Pv3DSi4Lx1KOWed9BVXYhJJSGdLzqV6s6/RbMfG/ZX4jzmjMCyE+3x7ggVOJBGdLTnPq6wE\nAMmVuMnmxC93l+OBsdGYHK8VO45kcYsvIok4uH07Nv5gyfnGigoc2rFDpESesTnd+O/PLmDiCA0e\nuo3bxPYHC5xIIn645PwahdU6yEk853IL2PBFJYaFqLDiR3HcqKqfWOBEEnH9kvPrudTqQU7iGZdb\nwO8PXYLF4caa2QmQs7z7jQVOJBHXLzm/Zm1ioiSWnNucbry8twLNbQ6svzMJAQpWjzfwJiaRRFy7\nUZm3YwcUVitcajVmL1/u8zcwzXYX/qvoAiKClXhx3miWtxexwIkk5PbsbJ8v7OvVmGz41WcVuDU2\nBE9Pj+ewiZexwIloQBys0KPgcDUenRCD+8cO4w3LAcACJyKvsjvd2H70Mr6pNuLXC0YjfdjNzxMg\n72CBE5FXCIKAIxcN2HH0MtKGBWPr/WMQolKIHWtIY4ETUb9V6i3Y9vVlNJodeG7GSK6uHCQscCLy\niCAIKL5iws7TDTjb0IYlE2Jwzy3DuCnVIGKBE1GfVLVYcfhiC/aV6SEAeGDsMLw4PwmBSk4PHGws\ncCLqkiAIqGu142x9G0obzDhWbYLJ7sTto3R4eno8xg8P5ewSEbHAiXxIm92FWpMdda126C0OGKxO\nGK1OtNpdsLsE2JxuOFwCBAgAABlkUMiBAIUcAXIZVAo51AFyBCrbP9QKWce/qxRyKBUyBMhlUMhl\nEARAgACXu32lpNnhgtnugsHiRF1re4YrRhsCFDKMGRaC9GHBeGF2AtKHBXM+t49ggROJQBAEVBts\nKG1oQ1lTG8qbLKjUW2F1uhGrUSE2VIWI4ABo1UpEBgcgITwIgYr2gg5QyHCtPwUBcAkCHK72D5vL\nDbvTDevVD7PNDZtTgPXqcadb6PiQy9r/ByCTAWqlHCEqBUJUCmjVCqRE6TpyhAcHiPuLRV1igRMN\nkuY2B45eMqD4ignf1bRCIZfhlpgQpEQG45HxWiRFBCEiSMkhCeo1FjjRAGow23GgXI8vKw242GJF\nZrwGk+O0eDxzBGI1KpY19QsLnMjLHC43vr5kxO7SJpxtMGPGKB2WTIzBhBEaqLiRE3kRC5zIS4xW\nJz4524i/lzQgTqvGXekReOmOJKg5vY4GCAucqJ8azXa892099pU340cJYXhtQQpGRwaJHYv8AAuc\nyEMGqxPvfVuHPeeasCAtEtsfGIMoPl2dBhELnKiP7E43PjhVj7+drsec0eHY8UAGIkOG7lS7I0VF\nOLh9O5Q2G5yBgZi9YoWk9iQfyljgRL10bbe97UcvIyUyCAX3pmO4tvPnVA4VR4qKcGjdOvymoqLj\nWF5lJQCwxH0AC5yoF2pNNrx5uAoNrQ48P3MkJsX5x257B7dvv6G8AWBjRQXyduxggfsAFjhRN1xu\nAf8oacBfimvx0G0xWDwu2q9221PabJ0eV1itg5yEOsMCJ+pCtcGK/AMXoZTLsfnHaYgPU4sdadA5\nAzsfInKp/e/Xwhf1WOBvvfUWiouLodVqsWnTJgCAxWJBQUEB6uvrERMTg2effRZq/gelIUIQBHxy\ntgl/PHYFP500HPfcEuW3mzfNXrECeZWV2HjdMMraxETMXr5cxFR0TY8FPnfuXCxcuBBbtmzpOPbh\nhx8iPT0dubm5+Oijj/Dhhx9i6dKlAxqUaDC0WBx4/VAVGsx2/D4nDQnh7Rcm/joT49r3mLdjBxRW\nK1xqNWYvX+4X37sU9FjgGRkZqK+vv+HYsWPHsH79egBAVlYW1q9fzwInyTtV24oNX1Ri7uhwvDg/\nEQFXl737+0yM27Oz/eL7lCKP1vgaDAbodDoAQFhYGAwGg1dDEQ0mQRDw/rd1eOXzCjw/cySemhbX\nUd5A+0yMjZ3MxDi0Y8dgRyW6Qb9vYnI3NZIys92F3+6/iGaLA1vuS0d06M0rKTkTg3yVR1fgYWFh\naGlpAQDo9XqEhYV5NRTRYKhqseLZv5ciMiQAv89J7bS8Ac7EIN/lUYFnZmZi//79AIADBw5gypQp\n3sxENOD+VWXACx+fx4PjovHcjJE3DJn80OwVK5CXlHTDsbWJiZjFmRgksh6HUDZv3owzZ87AZDJh\n5cqVePjhh7F48WIUFBRgzZo1HdMIiaRAEAR8eKoeH5yux/o7kjA2NrTH13AmBvkqmSAIwkC+wd69\nezFp0qSBfAuiXnG6BRQeqUJJnRmvLEjucsiEyBecOHEC8+fP7/YcrsQkv2C2u/DK3gooZDK8fk8a\nglUKsSMR9RsLnIa8+lY7/nNPOSYMD8XPfhQPhR/tZUJDGwuchrSKZgte3FOO+8cOw4O3xYgdh8ir\nWOA0ZH17xYRf76vE09PjMDc5Quw4RF7HAqch6cuKFrxxuAq/nJuIiXEaseMQDQgWOA05n55txDsn\narDhrmSkRAWLHYdowLDAaUh579s6fHymEZsWpSLOD/fvJv/CAqchQRAE/J9vruBolRGv35PKp8OT\nX2CBk+S5BQFbDlfjfFMbNi1KhVbN39bkH/g7nSTN5RaQf+AiGswObFyYghAu0CE/wgInybK73Hht\nXyUcLgGv3pUMtdKjvdmIJIu/40mSbE431n92AXIZsP7OJJY3+SVegZPkWBwu/FfRBUQEByB3zigu\njSe/xQInSTHbXXhxTzniwwLx/MwEljf5NRY4SYbJ5sQvd5cjLSoYq26Ph5yP8yM/xwInSTBancj7\ntAzjhofiZ9Pi+CxWIrDASQL0FgfydpVh6kgtnpgyguVNdBULnHxaU5sDa3eVYVaSDv8+KZblTXQd\nFjj5rEazHbm7yjAvJQI/mRgrdhwin8MCJ59U32pH7q7zWJgehUfG80EMRJ1hgZPPqTXZkLurDD++\nZRgeHBctdhwin8UCJ59yxWjD2l1lWDwuGveNHSZ2HCKfxgInn1FtsGLtrjIsmRiLRWOixI5D5PNY\n4OQTLumtyPu0DMsyh2NBWqTYcYgkgQVOoitvasN/7inHk1PicEcqHz5M1FsscBLVuYY2vFRUjlXT\n4zF7dLjYcYgkhQVOoimpM+O/P7uAF2YlYPqoMLHjEEkOC5xEUXzFhNf2VSJ3zihMGakVOw6RJPWr\nwFetWoWgoCDI5XIoFAps2LDBW7loCDt6yYDfHbyEl+Yn4rbhGrHjEElWv6/A169fj9DQUG9kIT9w\n8IIeW45U4+Xs0ciIDhE7DpGk9bvABUHwRg7yA3vONeF/jl3BhoXJSI4MFjsOkeT1q8BlMhlefvll\nyGQyZGdn44477vBWLhpiPjxVj53f1yP/7lSM1KnFjkM0JPSrwF955RWEh4ejuroaGzZsQFxcHDIy\nMryVjYYAQRDwzvEaHKxowe9z0hAdqhI7EtGQ0a9HeYeHt8/bjY+Px9SpU1FWVuaVUDQ0uNwCCr+q\nxr+qjNiUk8ryJvIyjwvcZrPBYrEAAIxGI4qLi5GQkOC1YCRtdpcbG76oxEW9FfmLUhEeFCB2JKIh\nx+MhFIPBgPz8fACARqPBokWLMH78eK8FI+ky21341ecXoAlU4tUFyVAp+/UXPSLqgscFHh0d3VHg\nRNc0tTnw0p5yjIkOwarp8VDI+Qg0ooHClZjkNRf1Fry45wLuSo/EkgkxfH4l0QBjgZNXfFdjwq/3\nVmL5NO4oSDRYWODUb3vLmrH968tYNy8RE0dwaTzRYGGBk8fcgoB3j9dgb5kev12UgsTwILEjEfkV\nFjh5xOp0I//ARTSZHXjz3jROEyQSAed3UZ/Vt9qx+uNzUClk+O3dKSxvIpHwCpz65LuaVrz2RQUe\nuDUaD42L5kwTIhGxwKlXBEHAP8804s8napGbNQqZ8XwIA5HYWODUI4vDhTcPV6G8yYLX70lDXFig\n2JGICBwDpx5c0lvx3D/OQSaT4c1701neRD6EV+DUpb1lzdj29WU8kTkcd6VHcrybyMewwOkmZrsL\nW45U4VxDGzby6TlEPotDKHSDM/VmPL3zLNRKOQrvH8PyJvJhvAInAO37d//lRC0+LW3CczNGYmaS\nTuxIRNQDFjihrLEN+QcuIlYbiG0PjEFEMBfmEEkBC9yP2Zxu/LW4FrtKm7BiWhzmp4TzRiWRhLDA\n/dSJy0a8ebgKqZHB2PbAGETyqptIcljgfqbBbMfb/7qCkjoznp0Rj6kjw8SOREQeYoH7CZvTjf89\nVY+dp+txT0YUnp85EkEBCrFjEVE/sMCHOJdbwBflerxzvAapUcHYcl86hmu4mpJoKGCBD1GCIODr\nS0b8z7ErCA5Q4D/mjMJtw0PFjkVEXsQCH2LcgoCjl4z468la2JxuPJ45Aj9K0HJ2CdEQxAIfIpxu\nAYcqWvDet7WQy2RYMiEWtyeGQc7iJhqyWOASZ7Q6sau0Ef8oacQITSCemDICU+J5xU3kD1jgEiQI\nAk7XmbG7tAlfXTTg9lFheCV7NPctIfIzLHAJqTHZsL9cj8/ON0Muk+Gu9Eg8NXUEdHwmJZFfYoH7\nuBqTDV9dNGB/uR41JjtmJeqwenYCbokO4TAJkZ9jgfsYh8uNM/VtOF5txFeXDGixODEtQYufThqO\niXEaKOVdl/aRoiIc3L4dSpsNzsBAzF6xArdnZw9ieiIaTB4XeElJCd555x24XC7Mnz8fCxcu9GYu\nv2F3unG+sQ3f15lxssaEkjoz4sPUmBSnwfMzE5A+LBiKbkr7miNFRTi0bh1+U1HRcSyvshIAWOJE\nQ5RHBe52u7F161a89NJLiIiIwLp16zBu3DjEx8d7O9+QYnO6cVFvRVlTG8qaLChrbEOF3ooEXSBu\niQ7BwvQo5GUlQqvu+3+Wg9u331DeALCxogJ5O3awwImGKI8KvKysDLGxsYiOjgYAzJgxA8eOHWOB\nA2izu9BodqDGZEOtyY4akw3VBhsutVjR3OZAnDYQyVHBSIkMQtZoHVKjgr2yJ4nSZuv0uMJq7ffX\nJiLf5FGBNzc3IzIysuPziIgIlJWVeS2UWARBgNMtwO4SYHe54XAJsDrcsDhdaHO4YXG40GpzwWx3\nwWRzwWhzwmBxosXqRHObA01tDrgEYFhIAGI1KsSGBiJWo8Jtw0ORoFNjuCawV8MhnnAGdr6/iUut\nHpD3IyLxDcpNzF/uLoPL3b7MWxDaf3QLgICrP149JqD934H28wS0f7T/4+rngnDD177xs+uOC9d+\n7rqvdd17X/vR5RbgEgQ4XQIcbgFyGRColEOlkCNAIUOQUo6gAAWCAuRQK+XQBCoQolJCE6hAfJga\nY2OU0KmV0AUpMSwkACEqhSizQ2avWIG8ykpsvG4YZW1iImYvXz7oWYhocHhU4BEREWhqaur4vKmp\nCREREZ2eq9Pp8CBaPEsnGtfVjx7Y2j8EA6BH+4dY1FFRuPMPf8De645dG/k+ceKEGJGIqB90up6f\nS+tRgScnJ6O2thb19fWIiIjAkSNH8POf/7zTcydPnuzJWxARUQ9kwg/HJHqppKQEf/zjHzumEd59\n993ezkZERN3wuMCJiEhccrEDEBGRZ1jgREQSNaDTCKWw3P6tt95CcXExtFotNm3aJHacLjU2NqKw\nsBAGgwGe3HxKAAAE3UlEQVRarRZZWVnIysoSO9ZN7HY71q9fD4fDAZVKhenTpyMnJ0fsWJ1yu93I\ny8tDREQE8vLyxI7TqVWrViEoKAhyuRwKhQIbNmwQO1KnrFYr3n77bVy6dAkOhwMrV65EWlqa2LFu\ncOXKFWzevLnj87q6OjzyyCM+ef/u888/x/79++FwOJCRkYHHHnus8xOFAeJyuYRnnnlGqKurExwO\nh7BmzRqhqqpqoN7OYyUlJcKFCxeEF154Qewo3dLr9UJFRYUgCIJgMBiEJ5980id/PQVBEKxWqyAI\ngmC324UXXnhBqKmpETlR5/75z38Kb7zxhrBx40axo3Tp6aefFkwmk9gxelRQUCDs3btXEARBcDqd\ngtlsFjlR91wul/DUU08JDQ0NYke5iclkEp5++mnBYrEILpdLeO2114Ti4uJOzx2wIZTrl9srlcqO\n5fa+JiMjAyEhIWLH6JFOp0NiYiIAQKvVIjk5GXq9mDPPuxZ4dVWo1WqFy+WCUul7m142NTWhuLgY\n8+bNu2lxmK/x9XxtbW04e/Ys5s2bBwBQKBQIDvbth4ucOnUKMTExiIqKEjvKTVQqFYD2X1e73Q6b\nzYbQ0M4fSD5gf7KG6nJ7X1BbW4vq6mqkpqaKHaVTbrcba9euRVVVFR577DGf/EPyzjvv4Cc/+Qks\nFovYUbolk8nw8ssvQyaTITs7G3fccYfYkW5SX18PrVaLwsJCXLhwAampqXjiiSc6isgXHT58GDNn\nzhQ7RqdUKhWefPJJrFq1CgEBAVi4cCFSUlI6PZc3MSXGarVi8+bNWLZsGdQ+us+JXC5Hfn4+3nzz\nTezZswcVP9glUWzHjx+HVqtFUlKSz1/dvvLKK8jPz8dzzz2HnTt34syZM2JHuonL5UJ5eTmmTZuG\nDRs2wOl04quvvhI7VpecTieOHz+O6dOnix2lU0ajEW+//TZef/11FBYW4ty5c12uph6wAu/Lcnvq\nHafTiU2bNmHWrFmYMmWK2HF6FB0djYkTJ6KkpETsKDcoLS3F8ePHsWrVKrzxxhv4/vvvsWXLFrFj\ndSo8PBwAEB8fj6lTp/rk32IjIyMRGhqKzMxMqFQqzJgxA8XFxWLH6lJxcTFGjx4NrVYrdpROlZWV\nITU1FbGxsdBoNJg+fXqXf4YGrMCvX27vdDpx5MgRZGZmDtTbDXmCIGDbtm2Ij4/HokWLxI7TJaPR\nCLPZDAAwmUw4efIkEhISRE51oyVLlmDr1q0oLCzE888/j7Fjx+KZZ54RO9ZNbDZbxxCP0WhEcXGx\nz/1aAu33Z2JjY3H+/Hm43W6cOHECt912m9ixunT48GHMmDFD7BhdGjNmDMrLy9Ha2gqHw4Hi4mKM\nHz++03MHbAxcoVBg5cqV+N3vftcxjdAX9wvfvHkzzpw5A5PJhJUrV+Lhhx/G3LlzxY51k9LSUhw6\ndAgJCQnIzc0F0F5EEyZMEDnZjVpaWlBYWAi32w2dToecnByMGzdO7Fjd8tVnixoMBuTn5wMANBoN\nFi1a1OUfZLGtWrUKhYWFMBqNSEhIwNKlS8WO1Cmr1YpTp05hxYoVYkfpUnBwMBYvXoz8/HzY7XaM\nHz8eY8eO7fRcLqUnIpIo3sQkIpIoFjgRkUSxwImIJIoFTkQkUSxwIiKJYoETEUkUC5yISKJY4ERE\nEvX/AC/sps0J0nhyAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1091ad410>"
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "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"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "titanic = pd.read_excel(\"data/titanic.xls\", \"titanic\")\n",
      "titanic.name"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 48,
       "text": [
        "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"
       ]
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "jitter = np.random.normal(scale=0.02, size=len(titanic))\n",
      "plt.scatter(np.log(titanic.fare), titanic.survived + jitter, alpha=0.3)\n",
      "yticks([0,1])\n",
      "ylabel(\"survived\")\n",
      "xlabel(\"log(fare)\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 49,
       "text": [
        "<matplotlib.text.Text at 0x10913d710>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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9J6v1eDwgmQzf9+bH/Hw4dy5ibU0mhnc+vrhY7ttZXbWb9+47jhxxpFKPv2ul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\nSUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109360ed0>"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 277.621917\n",
        "         Iterations: 55\n",
        "         Function evaluations: 103\n"
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "jitter = np.random.normal(scale=0.02, size=len(titanic))\n",
      "plt.scatter(np.log(titanic.fare), titanic.survived + jitter, alpha=0.3)\n",
      "yticks([0,1])\n",
      "ylabel(\"survived\")\n",
      "xlabel(\"log(fare)\")\n",
      "plt.plot([0,7], [betas_titanic[0], betas_titanic[0] + betas_titanic[1]*7.])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 51,
       "text": [
        "[<matplotlib.lines.Line2D at 0x109445d50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Xe/iDm4p5b+3qqnDeTAJfXDPb5qL7rSx0QM1HaioL8ibkK1V1tUdWlofjmH7/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\nAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109421110>"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "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",
      "$$p_i = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$"
     ]
    },
    {
     "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",
      "$$\\text{logit}(p) = \\log\\left[\\frac{p}{1-p}\\right] = x$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here's a plot of $p/(1-p)$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "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)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 52,
       "text": [
        "[<matplotlib.lines.Line2D at 0x109579310>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Pe38wooIcO1/SAQBxZHlR/cNhGfms2gEgnpIg7hEZ7LcDQFxZH/eRsEqJOwDElfVxHw7L\nxbYMAMSV5XHvHYnImc/KHQDiyfK4942M3e4XABA/U8bd7XZrx44devrpp/Xmm29ed5zH49HXv/51\n/f73v5/RBMZW7sQdAOJp0rjHYjE1Nzdr+/btamxs1JEjR9Td3X3NcS+//LLuuusumaY5own0jURU\nyrYMAMTVpHH3eDwqLy9XWVmZHA6Hamtr1dbWdtW4N998U/fdd5+Ki4tnPIG+YETOPFbuABBPk8bd\n7/fL5XKNPzYMQ36//6oxbW1t2rhxoyTJZpv+V+VFY6YGRyMqJu4AEFdzPqH60ksvafPmzbLZbDJN\nc0bbMgOjERXl8t2pABBvky6ZDcOQz+cbf+zz+WQYxoQxJ0+e1P79+yVJg4OD+uCDD+RwOFRTUzPl\nm/cHIyph1Q4AcTdpWaurq9XT0yOv1yvDMNTa2qonnnhiwpgXX3xx/OempiatXr16WmGXxk6mEncA\niL9Jy2q329XQ0KC9e/cqGo2qrq5OlZWVOnTokCSpvr5+Tm8+wModABJiyrLefvvt2r1794Tnrhf1\nrVu3zujNx7Zl7DP6GwDA1Cz9hGr/aJQrZQAgASyNO9syAJAYlse9OJe4A0C8WRv30YiK2XMHgLiz\nNO6Do1EVsXIHgLhLgm0ZVu4AEG+s3AEgDVkW92jM1HA4qsIcVu4AEG+WxT0QGgs7Nw0DgPizLO6D\no1EtYNUOAAlhYdwj7LcDQIJYFvehUFQLuFIGABLC0m2ZIrZlACAhLD2hysodABLD0j13TqgCQGJY\nvHLnhCoAJIKlJ1T5ABMAJIZ1K/dR4g4AiWLtpZDEHQASwvLbDwAA4o+VOwCkIcviPhyKqiDH0jsO\nA0DaYlsGANKQJXGPxEyFY6byHKzcASARLKnr8OerdpuNe7kDQCJYEvdAKKqCbLZkACBRrFm5h6Mq\n5GQqACSMRXGPsXIHgASybM89n7gDQMJYti3DNe4AkDgWnVBlWwYAEsmSuI+EoyrIZuUOAIliUdxj\n7LkDQAJZt+fOyh0AEsa6lTv3lQGAhGHPHQDS0LS+odrtdqulpUXRaFR1dXXatGnThN8fPXpUr7/+\nuiSpsrJSjz76qKqqqq77eiPhmPIcrNwBIFGmXD7HYjE1Nzdr+/btamxs1JEjR9Td3T1hzOLFi/XD\nH/5Qe/bs0Ze+9CX9/Oc/n/Q1R8IxVu4AkEBTFtbj8ai8vFxlZWVyOByqra1VW1vbhDG33nqrCgoK\nJEmrVq2Sz+eb9DWHw3xCFQASacq4+/1+uVyu8ceGYcjv9193/DvvvKOamppJXzMYiXEvdwBIoLgW\n9tixYzp69Ki+8Y1vTDpu7GoZ4g4AiTJlYQ3DmLDN4vP5ZBjGVeO6urp08OBB7dixQ4WFhZO+5kg4\nqnxW7gCQMFMWtrq6Wj09PfJ6vYpEImptbb1q2+XSpUvat2+ftm3bpvLy8infNBiJKY89dwBImCkv\nhbTb7WpoaNDevXvHL4WsrKzUoUOHJEn19fV69dVXNTQ0pF/84hfjf7Nr167rv6bNJkcWX7EHAIli\nM03TnM83PHz4sP77Rw79n//y1/P5tgCQ0jo6OlRXVzft8ZZsfHOlDAAkFnEHgDRkTdz5dCoAJJRF\nK3eulAGARGJbBgDSENsyAJCGWLkDQBqypLK5xB0AEoqVOwCkIeIOAGmIuANAGmLPHQDSEHEHgDRE\n3AEgDbHnDgBpyJLK5tj5og4ASCRW7gCQhqxZuRN3AEgoTqgCQBqyJu524g4AiWTRyp0TqgCQSGzL\nAEAasuhSSOIOAIlkSWXtWWzLAEAisYQGgDRE3AEgDRF3AEhDxB0A0hBxB4A0RNwBIA0RdwBIQ8Qd\nANIQcQeANETcASANEXcASEOOqQa43W61tLQoGo2qrq5OmzZtumrMK6+8oo6ODuXm5mrr1q2qqKhI\nyGQBANMz6co9FoupublZ27dvV2Njo44cOaLu7u4JYzo6OtTV1aW9e/dqy5YtampqSuiEAQBTmzTu\nHo9H5eXlKisrk8PhUG1trdra2iaMaWtr0wMPPCBJWrZsmQKBgPr6+hI3YwDAlCaNu9/vl8vlGn9s\nGIb8fv+kY1wu11VjAADzKy4nVE3TjMfLAADiZNITqoZhyOfzjT/2+XwyDGPGY67kdDrV0dEx2/kC\nQEZyOp0zGj9p3Kurq9XT0yOv1yvDMNTa2qonnnhiwpiamhq9/fbbqq2t1SeffKLCwsJJJ7F69eoZ\nTRAAMHM2c4o9FbfbrZdeemn8UsiHHnpIhw4dkiTV19dLkl5++WV1dHQoLy9PDQ0NqqysTPzMAQDX\nNWXcAQCph0+oAkAaIu4AkIamvP1AvEznNgbpqqmpSZ2dnSouLta+ffskSSMjI/rpT38qr9erxYsX\na9u2bcrLy7N4pol36dIlHThwQP39/SouLtb69eu1fv36jDweoVBIzz//vMLhsHJycrR27Vo9/PDD\nGXksLovFYtq5c6cMw9DOnTsz9lh897vfVX5+vrKysmS327Vr166ZHwtzHkSjUfPxxx83L1y4YIbD\nYfPpp582z5w5Mx9vnRTcbrd58uRJ86mnnhp/7le/+pX5m9/8xjRN03zttdfMX//611ZNb1719vaa\nn332mWmaptnf329+5zvfMc+cOZOxxyMYDJqmaZqhUMh86qmnzHPnzmXssTBN03zjjTfMn/zkJ2Zj\nY6Npmpn778nWrVvNwcHBCc/N9FjMy7bMdG5jkM5WrFihwsLCCc9deduG9evX649//KMVU5t3TqdT\nN998sySpuLhY1dXV8vv9GXs8cnNzJUnBYFCxWEzZ2dkZeyx8Pp86Ozu1YcOG8Q9GZuqxkK7+cOhM\nj8W8bMtc6zYGHo9nPt46afX3949/HqCkpET9/f0Wz2j+9fT0qLu7W7feemvGHo9YLKYdO3bozJkz\n2rJlixYuXJixx6KlpUXf/OY3NTIyMv5cph4Lm82mH/3oR7LZbNq4caO+/OUvz/hYzNueO67PZrNZ\nPYV5FwwGtX//fn3rW9+6at8wk45HVlaW9uzZI6/Xq127dmn58uUTfp8px6K9vV3FxcW65ZZb9NFH\nH11zTKYcC0n68Y9/rNLSUnV3d2vXrl1X3UZ9OsdiXuI+01sUZIKSkhL19fXJ6XSqt7dXJSUlVk9p\n3kQiEe3bt0/333+/1qxZIymzj4cklZWV6e6775bb7c7IY3HixAm1t7ers7NT4XB4/ORhJh4LSSot\nLZUkVVZW6p577pHH45nxsZiXPfcrb2MQiUTU2tqqmpqa+XjrpFVTU6N3331XkvTee++NRy7dmaap\nn/3sZ6qsrNRXv/rV8ecz8XgMDAwoEAhIkgYHB/XBBx+oqqoqI4/F5s2b1dzcrAMHDujJJ5/UHXfc\noW3btmXksRgdHR3fmhoYGFBnZ+es/ncxb59QvdZtDDLF/v37dfz4cQ0ODqqkpERf+9rXdN9992Xk\nJV4ff/yxfvCDH6iqqmr8/1pu3rxZy5cvz7jjcfr0aR04cECxWExOp1Nr167Vhg0bMvbyv8vcbrfe\neOMN7dixIyOPhdfr1Z49eyRJRUVFWrt2rerr62d8LLj9AACkIT6hCgBpiLgDQBoi7gCQhog7AKQh\n4g4AaYi4A0AaIu4AkIaIOwCkof8PmaI/Q4spIzgAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1091d0b10>"
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And here's the logit function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(logit(unit_interval), unit_interval)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 53,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1095a5550>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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lDAbDgGPCwsIuG0NERI43IicsCIIwEp+GiIiGYcA5dKVSCb1e3/tar9dDqVRe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\nD8FCJyLyEP8HxY2pL+qB/hIAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109586110>"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The inverse of the logit transformation is:\n",
      "\n",
      "<div style=\"font-size: 150%;\">  \n",
      "$$p = \\frac{1}{1 + \\exp(-x)}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So, now our model is:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$\\text{logit}(p_i) = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$"
     ]
    },
    {
     "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",
      "$$L(y|p) = \\prod_{i=1}^n p_i^{y_i} (1-p_i)^{1-y_i}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "which, on the log scale is:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$l(y|p) = \\sum_{i=1}^n y_i \\log(p_i) + (1-y_i)\\log(1-p_i)$$"
     ]
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 54
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Remove null values from variables"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x, y = titanic[titanic.fare.notnull()][['fare', 'survived']].values.T"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 55
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "... and fit the model."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b0,b1 = fmin(logistic_like, [0.5,0], args=(x,y))\n",
      "b0, b1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Optimization terminated successfully.\n",
        "         Current function value: 827.015955\n",
        "         Iterations: 47\n",
        "         Function evaluations: 93\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 56,
       "text": [
        "(-0.88238984528338194, 0.012452067664164127)"
       ]
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "jitter = np.random.normal(scale=0.01, size=len(x))\n",
      "plot(x, y+jitter, 'r.', alpha=0.3)\n",
      "yticks([0,.25,.5,.75,1])\n",
      "xvals = np.linspace(0, 600)\n",
      "plot(xvals, invlogit(b0+b1*xvals))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 57,
       "text": [
        "[<matplotlib.lines.Line2D at 0x109683150>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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kk9Szgz440MTvPz/OszdOJs0lVwUVQiSfpA2BXf52/mXzIVbJVFAhRBJLyuGg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\nrkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109674890>"
       ]
      }
     ],
     "prompt_number": 57
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "logistic = sm.GLM(y, sm.add_constant(x), family=sm.families.Binomial()).fit()\n",
      "logistic.summary()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "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, 03 Jul 2013</td> <th>  Deviance:          </th> <td>  1654.0</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Time:</th>               <td>21:46:37</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>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 58,
       "text": [
        "<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, 03 Jul 2013   Deviance:                       1654.0\n",
        "Time:                        21:46:37   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",
        "\"\"\""
       ]
      }
     ],
     "prompt_number": 58
    },
    {
     "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",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 58
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "np.random.permutation(titanic.name)[:5]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 59,
       "text": [
        "array([u'Swift, Mrs. Frederick Joel (Margaret Welles Barron)',\n",
        "       u'Uruchurtu, Don. Manuel E', u'Geiger, Miss. Amalie',\n",
        "       u'Padro y Manent, Mr. Julian',\n",
        "       u'Goldsmith, Mrs. Frank John (Emily Alice Brown)'], dtype=object)"
       ]
      }
     ],
     "prompt_number": 59
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "random_ind = np.random.randint(0, len(titanic), 5)\n",
      "titanic.name[random_ind]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 60,
       "text": [
        "470                  Keane, Mr. Daniel\n",
        "1253                  Torfa, Mr. Assad\n",
        "1013    Meanwell, Miss. (Marion Ogden)\n",
        "518       Nicholls, Mr. Joseph Charles\n",
        "665              Barbara, Miss. Saiide\n",
        "Name: name, dtype: object"
       ]
      }
     ],
     "prompt_number": 60
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "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",
      "_ = hist(s, bins=30)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAECCAYAAAAfE3cCAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAEdFJREFUeJzt3X9olfXfx/HX2WxuOncdt7tt6jCXW3OGCLllIrGTWuCK\n+xYGQVqY0D+pI8g11JB+UdPUMs5tFgglaRAylP6QbtQ6K9Du9stuTHPOgpqlh7adM6f75XbuP6J9\n6+t+XLvOtbMfn+cDBD377NrbD5cvP9fn+lyfyxOJRCICAExqcWNdAABg9BH2AGAAwh4ADEDYA4AB\nCHsAMABhDwAGmDLUF99//33V19crJSVFe/fulSR1dHTI7/crGAwqIyNDpaWlSkxMlCSdOHFCX375\npeLj47VhwwYtWLBg9P8GAIBhDTmyf+SRR7R9+/Z/fFZZWam8vDzt2bNHubm5qqyslCQ1NTXpq6++\n0s6dO7Vlyxbt379ffX19o1c5AMC2IcM+Pz9f06dP/8dnNTU1KioqkiT5fD5VV1dLkqqrq7V8+XJN\nmTJF6enpyszMVGNj4yiVDQAYiRHP2YfDYXm9XkmSZVkKh8OSpNbWVqWlpfW3S0tLU0tLi0tlAgCi\nEdUNWo/HE9XXAQCxMeQN2oFYlqVQKCSv16vW1lZZliVJSk1NVXNzc3+75uZmpaamDnqcb7/9Vjdv\n3nRQMgCYy+v1asmSJSP+vhGHfUFBgQKBgNasWaOqqioVFhb2f/7ee+/piSeeUEtLi65du6acnJxB\nj3Pz5k098MADIy4YAExWV1fn6Ps8Q+16uW/fPl28eFE3btyQZVl68skn9dBDDw259PL06dP9Sy/z\n8/MH/cGnT58m7AFghOrq6rRy5coRf9+QYT+aCHsAGDmnYc8TtABgAMIeAAxA2AOAAQh7ADAAYQ8A\nBiDsAcAAhD0AGICwBwADEPYAYADCHgAMQNgDgAFGvOslgMnj97YuBdu7bbVNT07QrJSpo1wRRgth\nDxgs2N6tl07Ye33o7uIcwn4CYxoHAAxA2AOAAQh7ADAAYQ8ABiDsAcAAhD0AGICwBwADEPYAYADC\nHgAMQNgDgAEIewAwAGEPAAZgIzRgErK7m2V3b18MqsF4QNgDk5Dd3SxfWZUdg2owHjCNAwAGIOwB\nwACEPQAYgLAHAAMQ9gBgAMIeAAxA2AOAAVhnD8CW+Djp+99uDNsuPTlBs1KmxqAijARhD8CWcGev\nXjv187DtdhfnEPbjkOOwP3XqlAKBgHp6epSfn69nn31WHR0d8vv9CgaDysjIUGlpqRITE92sF5gw\n7G5ZwEgYseAo7Nvb23Xs2DHt3btXCQkJ2rVrl86dO6fz588rLy9P5eXlOn78uCorK7Vu3Tq3awYm\nBLtbFjASRiw4ukGbkJAgSbp165a6u7vV1dWl6dOnq6amRkVFRZIkn8+n6upq9yoFADjmaGSfkJCg\n5557Tps2bdJdd92l1atXKzc3V+FwWF6vV5JkWZbC4bCrxQIAnHEU9m1tbTp48KDeffddTZ8+Xe+8\n845qa2v/0cbj8bhSIAAgeo6mcRobG5Wbm6vMzEzNmDFDy5Yt08WLF2VZlkKhkCSptbVVlmW5WiwA\nwBlHYb9gwQJduXJF7e3t6unpUX19vRYvXqyCggIFAgFJUlVVlQoLC92sFQDgkKNpnGnTpqmkpES7\nd+9Wd3e3Fi9erPvvv185OTny+/0qKyvrX3oJABh7jtfZ+3w++Xy+f3yWlJSk8vLyaGsCALiMJ2iB\nCYR3y8Ipwh6YQHi3LJxi10sAMABhDwAGIOwBwACEPQAYgBu0wBiz+1IQiVU2cI6wB8aY3ZeCSKyy\ngXNM4wCAAQh7ADAAYQ8ABiDsAcAAhD0AGICwBwADEPYAYADCHgAMQNgDgAEIewAwAGEPAAYg7AHA\nAIQ9ABiAsAcAAxD2AGAAwh4ADEDYA4ABCHsAMABhDwAGIOwBwACEPQAYYMpYFwDATL+3dSnY3m2r\nbXpygmalTB3liiY3wh7AmAi2d+ulE4222u4uziHso8Q0DgAYgLAHAAMQ9gBgAMIeAAzg+AZtZ2en\nDh48qF9++UU9PT3auHGjsrKy5Pf7FQwGlZGRodLSUiUmJrpZLwDAAcdhf/DgQS1cuFCbN29Wb2+v\nurq6VFlZqby8PJWXl+v48eOqrKzUunXr3KwXAOCAo2mcW7du6ccff9SKFSskSfHx8Zo2bZpqampU\nVFQkSfL5fKqurnavUgCAY45G9sFgUCkpKdq/f79++ukn5ebmasOGDQqHw/J6vZIky7IUDoddLRYA\n4IyjkX1vb6+uXLmipUuXqqKiQrdv39bZs2f/0cbj8bhSIAAgeo7CPi0tTcnJySooKFBCQoKWL1+u\nc+fOyev1KhQKSZJaW1tlWZarxQIAnHEU9l6vV5mZmbp8+bL6+vpUV1enRYsWacmSJQoEApKkqqoq\nFRYWulkrAMAhx6txNm3apP3796utrU1z587VunXrFIlE5Pf7VVZW1r/0EgAw9hyH/ezZs/Xmm2/e\n8Xl5eXlUBQEA3McTtABgAMIeAAxA2AOAAQh7ADAAYQ8ABuC1hMAI2X13andvXwyqAewh7IERsvvu\n1FdWZcegGsAewh6Aq+LjpO9/uzFsO658YouwB+CqcGevXjv187DtuPKJLW7QAoABCHsAMABhDwAG\nIOwBwACEPQAYgLAHAAMQ9gBgAMIeAAxA2AOAAQh7ADAAYQ8ABiDsAcAAhD0AGICwBwADEPYAYADC\nHgAMQNgDgAEIewAwAK8lxKT2e1uXgu3dttqmJydoVsrUUa4IGBuEPSa1YHu3XjrRaKvt7uIcwh6T\nFmEPYNyLj5O+/+3GsO24OhscYQ9g3At39uq1Uz8P246rs8FxgxYADEDYA4ABCHsAMABhDwAGIOwB\nwABRrcbp6+vT1q1blZqaqq1bt6qjo0N+v1/BYFAZGRkqLS1VYmKiW7UCAByKamR/4sQJZWVlyePx\nSJIqKyuVl5enPXv2KDc3V5WVla4UCQCIjuOwb25uVn19vVasWKFIJCJJqqmpUVFRkSTJ5/Opurra\nnSqBGPjrwZ3hfnX39o11qcCIOZ7GOXTokJ5++ml1dHT0fxYOh+X1eiVJlmUpHA5HXyEQI3Yf3Hll\nVXYMqgHc5WhkX1tbq5SUFGVnZ/eP6v/dX1M7AICx52hkf+nSJdXW1qq+vl49PT39N2Yty1IoFJLX\n61Vra6ssy3K7XgCAA47Cfu3atVq7dq0k6cKFC/r8889VWlqqw4cPKxAIaM2aNaqqqlJhYaGrxQIA\nnHFlnf1fUzYlJSVqaGhQWVmZLl++rJKSEjcODwCIUtS7Xi5cuFALFy6UJCUlJam8vDzqogAA7uIJ\nWgAwAGEPAAYg7AHAAIQ9ABiAsAcAAxD2AGAAwh4ADEDYA4ABCHsAMABhDwAGIOwBwACEPQAYgLAH\nAAMQ9gBgAMIeAAxA2AOAAQh7ADAAYQ8ABoj6tYQAMF7Ex0nf/3Zj2HbpyQmalTI1BhWNH4Q9gEkj\n3Nmr1079PGy73cU5xoU90zgAYADCHgAMwDQOJqTf27oUbO8etl13b18MqsFEY3duX5o88/uEPSak\nYHu3XjrROGy7V1Zlx6AaTDR25/alyTO/zzQOABiAkT3GFaZngNFB2GNcYXoGGB1M4wCAAQh7ADAA\nYQ8ABiDsAcAAhD0AGICwBwADEPYAYADCHgAMQNgDgAEcPUH7xx9/aP/+/QqHw0pJSZHP55PP51NH\nR4f8fr+CwaAyMjJUWlqqxMREt2sGAIyQo7CfMmWK1q9fr3nz5qmtrU1btmxRTk6OAoGA8vLyVF5e\nruPHj6uyslLr1q1zu2YAwAg5msbxer2aN2+eJCklJUXz589XS0uLampqVFRUJEny+Xyqrq52rVAA\ngHNRz9lfu3ZNTU1Nuu+++xQOh+X1eiVJlmUpHA5HXSAAIHpRhX1nZ6f27dun9evX3zE37/F4oioM\nAOAex2F/+/Zt7d27Vw8//LAKCwsl/TmaD4VCkqTW1lZZluVOlQCAqDgK+0gkog8++EBZWVl6/PHH\n+z8vKChQIBCQJFVVVfX/JwAAGFuOVuNcunRJ33zzjebOnavy8nJJ0tq1a1VSUiK/36+ysrL+pZcA\ngLHnKOwXLFigzz77bMCv/RX+AIDxg9cSYtTZfa+sxLtlgdFC2GPU2X2vrMS7ZYHRwt44AGAAwh4A\nDEDYA4ABCHsAMABhDwAGIOwBwACEPQAYgHX2uIPdh6DSkxM0K2VqDCoCEC3CHnew+xDU7uIcwh6Y\nIJjGAQADEPYAYADCHgAMQNgDgAEIewAwAGEPAAZg6SUAuGC8P59C2AOAC8b78ylM4wCAAQh7ADAA\nYQ8ABmDOHgCGEB8nff/bjWHbdff2xaAa5wh7ABhCuLNXr536edh2r6zKjkE1zhH2cGyyjHgAExD2\ncGyyjHgAExD2ABBDdq+IJXcfwCLsASCG7F4RS+4+gMXSSwAwACP7CW6878cBYHwg7Ce48b4fB4Dx\ngbAfp+yO2FnWCMAOwn6csjtiZ1kjADsIe0OMZLkXVwvA5EPYG2Iky724WgAmH9fD/sKFCzp06JB6\ne3u1cuVKrV692u0fETOsdAEwWbga9n19fTpw4IB27Nih1NRUbdu2TYsWLVJWVpabPyZmWOkCYLJw\nNewbGxuVmZmp9PR0SdLy5ctVU1MTk7C//Mct3e6LDNsuMzlBM6fdNer1AMB44mrYt7S0KC0trf/P\nqampamwcfmTshg//96r+7/f2Ydv993/dp19CnbaOafdGpd2bn8kJ8Wrv7nX1ZwOAHZPmBu1/5v+H\nls21hm0X7rytl//nJ1vHtHujciS7P3KTFMBY8EQikeHnPmxqaGjQ0aNH9fLLL0uSjh07Jo/HozVr\n1tzRtra2VqFQyK0fDQBG8Hq9WrJkyYi/z9WR/fz583Xt2jUFg0GlpqbqzJkzeuGFFwZs66RYAIAz\nro7spT+XXn788cf9Sy+Li4vdPDwAwAHXwx4AMP6wnz0AGICwBwADxGzp5dmzZ3X06FFdvXpVFRUV\nuvfeewdst2nTJiUlJSkuLk7x8fGqqKiIVYkTit3+nEzbV4ymjo4O+f1+BYNBZWRkqLS0VImJiXe0\n4/wcnJ1z7dNPP1VdXZ2mTp2qjRs3as6cOWNQ6cQwXH/+8MMPevvtt5WRkSFJWrp0qUpKSgY/YCRG\nmpqaIlevXo28+uqrkStXrgzabuPGjZEbN27EqqwJy05/9vb2RjZv3hy5fv16pKenJ1JWVhb59ddf\nY1zpxPDJJ59Ejh8/HolEIpFjx45FDh8+PGA7zs+B2TnXamtrI2+99VYkEolEGhoaItu3bx+LUicE\nO/15/vz5yM6dO20fM2bTOHPmzNHs2bNttY1wz3hYdvrz79tXTJkypX/7CtyppqZGRUVFkiSfz6fq\n6upB23J+3snOufb3Ps7NzdXNmzd51mYQdv/tjuRcHHdP0Ho8Hr3++uvyeDx67LHHtGrVqrEuacIa\ny+0rJppwOCyv1ytJsixL4XB4wHacnwOzc679e5u0tDS1tLT09zv+xU5/ejweNTQ06MUXX9Tdd9+t\nZ555Zsh9yFwN+zfeeGPA/6mfeuopFRQU2D7GzJkz1dTUpIqKCs2ZM0f5+fluljlhuNGf+Jeh+vPv\nPB7PkMfg/HSOqyL3ZGdn68CBA4qPj1dVVZV27dolv98/aHtXw37Hjh1RH2PmzJmSpKysLD344INq\nbGw09h9TtP2Zmpqq5ubm/j83NzcrNTU12rImrKH607IshUIheb1etba2yrIG3meJ83Ngds41zkf7\n7PRVUlJS/+9XrFihI0eOqL29XcnJyQMec1wtvezq6lJHR4ckqa2tTfX19Zo7d+4YVzVx/X37itu3\nb+vMmTNcEQyioKBAgUBAklRVVaXCwsI72nB+Ds7OuVZQUKCvv/5a0p/7aE2fPp0pnEHY6c9QKNR/\npVRbW6uEhIRBg16K4RO03333nT766CO1tbVp2rRpys7O1vbt29XS0qIPP/xQ27Zt0/Xr17Vnzx5J\n0owZM7Rs2TI9+uijsShvwrHTnxLbV9g12NJLzk/7BjrXTp48KUn9/XTkyBHV1dUpMTFRzz///IR9\nsVEsDNefX3zxhU6ePKm4uDjdc889Ki4uHnQJtsR2CQBghHE1jQMAGB2EPQAYgLAHAAMQ9gBgAMIe\nAAxA2AOAAQh7ADAAYQ8ABvh/rVKYia6jM90AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109684450>"
       ]
      }
     ],
     "prompt_number": 61
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "boot_mean = np.sum(s)/R\n",
      "boot_mean"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 62,
       "text": [
        "-0.22563001072303621"
       ]
      }
     ],
     "prompt_number": 62
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "boot_var = ((np.array(s) - boot_mean) ** 2).sum() / (R-1)\n",
      "boot_var"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 63,
       "text": [
        "0.084986794950926578"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "boot_mean - np.mean(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 64,
       "text": [
        "0.01515986676075437"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "s_sorted = np.sort(s)\n",
      "s_sorted[:10]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 65,
       "text": [
        "array([-1.38320881, -1.28007726, -1.25835439, -1.23663152, -1.17214697,\n",
        "       -1.09564706, -1.04243947, -1.03802018, -1.02332146, -1.02033221])"
       ]
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s_sorted[-10:]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 66,
       "text": [
        "array([ 0.37964688,  0.3867198 ,  0.38672838,  0.39383902,  0.41813411,\n",
        "        0.4314353 ,  0.43636116,  0.44595754,  0.44693849,  0.46676247])"
       ]
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha = 0.05\n",
      "s_sorted[[(R+1)*alpha/2, (R+1)*(1-alpha/2)]]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 67,
       "text": [
        "array([-0.85322216,  0.27618541])"
       ]
      }
     ],
     "prompt_number": 67
    },
    {
     "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",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 67
    }
   ],
   "metadata": {}
  }
 ]
}