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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "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 are \"significantly different \"or effects are \"statistically significant\", a concept that is poorly understood, and hence, often misused by practioners. Even when interpreted *correctly*, statistical significance (as characterized by a small p-value) is a questionable goal for statistical inference, as it is not a measure of evidence in any statistical sense.\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. These estimates can then be passed on to individuals with domain expertise who can then appraise them for \"real-world\" significance."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import pandas as pd\n",
      "import matplotlib.pyplot as plt\n",
      "import statsmodels.api as sm\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 = np.array([ 1.00201077,  1.58251956,  0.94515919,  6.48778002,  1.47764604,\n",
      "        5.18847071,  4.21988095,  2.85971522,  3.40044437,  3.74907745,\n",
      "        1.18065796,  3.74748775,  3.27328568,  3.19374927,  8.0726155 ,\n",
      "        0.90326139,  2.34460034,  2.14199217,  3.27446744,  3.58872357,\n",
      "        1.20611533,  2.16594393,  5.56610242,  4.66479977,  2.3573932 ])\n",
      "_ = plt.hist(x, bins=10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAECCAYAAADXf53UAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nqbj/B4FiJaZ5ASuTAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107da3550>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "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",
      "</div>\n",
      "\n",
      "* $X=\\{0,1,2,\\ldots\\}$\n",
      "* $\\lambda > 0$\n",
      "\n",
      "$$E(X) = \\text{Var}(X) = \\lambda$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Continuous Random Variables\n",
      "\n",
      "$$X \\in [0,1]$$\n",
      "\n",
      "$$Y \\in (-\\infty, \\infty)$$\n",
      "\n",
      "**Probability Density Function**: \n",
      "\n",
      "For continuous $X$,\n",
      "\n",
      "$$Pr(x \\le X \\le x + dx) = f(x|\\theta)dx \\, \\text{ as } \\, dx \\rightarrow 0$$\n",
      "\n",
      "![Continuous variable](https://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/500px-Normal_Distribution_PDF.svg.png)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "***e.g. normal distribution***\n",
      "\n",
      "<div style=\"font-size: 150%;\">  \n",
      "$$f(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]$$\n",
      "</div>\n",
      "\n",
      "* $X \\in \\mathbf{R}$\n",
      "* $\\mu \\in \\mathbf{R}$\n",
      "* $\\sigma>0$\n",
      "\n",
      "$$\\begin{align}E(X) &= \\mu \\cr\n",
      "\\text{Var}(X) &= \\sigma^2 \\end{align}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Example: Nashville Precipitation\n",
      "\n",
      "The dataset `nashville_precip.txt` contains [NOAA precipitation data for Nashville measured since 1871](http://bit.ly/nasvhville_precip_data). "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "precip = pd.read_table(\"data/nashville_precip.txt\", index_col=0, \n",
      "                       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\n",
        "\n",
        "[5 rows x 12 columns]"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "_ = precip.hist(sharex=True, sharey=True, grid=False, figsize=(10,6))\n",
      "plt.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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AdRFxG/AZ6m2mgTOAV0TECuDlwAeaDVOSJEnqj3Zl3ja1zfQwcGhTQUmSJEmD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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107dcd050>"
       ]
      }
     ],
     "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",
      "The **gamma distribution** is often a good fit to aggregated rainfall data, and will be our candidate distribution in this case.\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": "code",
     "collapsed": false,
     "input": [
      "from IPython.display import IFrame\n",
      "IFrame('http://en.wikipedia.org/wiki/Gamma_distribution?useformat=mobile', width=600, height=350)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "\n",
        "        <iframe\n",
        "            width=\"600\"\n",
        "            height=350\"\n",
        "            src=\"http://en.wikipedia.org/wiki/Gamma_distribution?useformat=mobile\"\n",
        "            frameborder=\"0\"\n",
        "            allowfullscreen\n",
        "        ></iframe>\n",
        "        "
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "<IPython.lib.display.IFrame at 0x1080d1450>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "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>\n",
      "\n",
      "Let's deal with the missing value in the October data."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "precip.Oct.ix[1960:1970]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "Year\n",
        "1960    1.38\n",
        "1961    1.12\n",
        "1962    2.29\n",
        "1963     NaN\n",
        "1964    1.83\n",
        "1965    0.57\n",
        "1966    2.50\n",
        "1967    1.57\n",
        "1968    3.92\n",
        "1969    2.01\n",
        "1970    2.94\n",
        "Name: Oct, dtype: float64"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "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": 7,
       "text": [
        "        Jan    Feb    Mar    Apr   May   Jun   Jul   Aug   Sep   Oct   Nov  \\\n",
        "Year                                                                         \n",
        "1871   2.76   4.58   5.01   4.13  3.30  2.98  1.58  2.36  0.95  1.31  2.13   \n",
        "1872   2.32   2.11   3.14   5.91  3.09  5.17  6.10  1.65  4.50  1.58  2.25   \n",
        "1873   2.96   7.14   4.11   3.59  6.31  4.20  4.63  2.36  1.81  4.28  4.36   \n",
        "1874   5.22   9.23   5.36  11.84  1.49  2.87  2.65  3.52  3.12  2.63  6.12   \n",
        "1875   6.15   3.06   8.14   4.22  1.73  5.63  8.12  1.60  3.79  1.25  5.46   \n",
        "1876   6.41   2.22   5.28   3.62  3.40  5.65  7.15  5.77  2.52  2.68  1.26   \n",
        "1877   4.05   1.06   4.98   9.47  1.25  6.02  3.25  4.16  5.40  2.61  4.93   \n",
        "1878   3.34   2.10   3.48   6.88  2.33  3.28  9.43  5.02  1.28  2.17  3.20   \n",
        "1879   6.32   3.13   3.81   2.88  2.88  2.50  8.47  4.62  5.18  2.90  5.85   \n",
        "1880   3.74  12.37   8.16   5.26  4.13  3.97  5.69  2.22  5.39  7.24  5.77   \n",
        "1881   3.54   5.48   2.79   5.12  3.67  3.70  0.86  1.81  6.57  4.80  4.89   \n",
        "1882  14.51   8.61   9.38   3.59  7.38  2.54  4.06  5.54  1.61  1.11  3.60   \n",
        "1883   3.76   7.90   3.98   9.12  4.82  3.82  4.94  4.47  2.23  5.27  3.11   \n",
        "1884   7.20   8.18   8.89   3.51  3.58  6.53  3.18  2.81  2.36  2.43  1.57   \n",
        "1885   6.29   2.00   2.33   3.75  4.36  3.72  5.26  1.02  5.60  2.99  2.73   \n",
        "1886   5.18   3.82   4.76   2.36  2.10  7.69  1.90  5.50  3.68  0.51  5.76   \n",
        "1887   5.13   8.47   3.36   2.67  3.43  2.31  3.77  2.89  6.85  1.92  2.29   \n",
        "1888   6.29   3.78   6.46   4.18  2.97  4.68  2.36  7.03  3.82  2.82  4.33   \n",
        "1889   3.83   1.84   2.47   2.83  5.30  5.33  2.74  1.57  6.81  1.54  6.88   \n",
        "1890   8.10  10.95   8.64   3.84  4.16  2.23  0.46  6.59  5.86  3.01  2.01   \n",
        "1891   6.15   6.96  10.31   2.24  2.39  6.50  1.49  3.72  1.25  0.84  6.71   \n",
        "1892   2.81   2.73   4.10   7.45  4.03  5.01  5.13  3.39  4.78  0.25  3.91   \n",
        "1893   1.27   4.88   3.37   4.11  7.31  4.74  2.12  1.92  6.43  3.68  2.97   \n",
        "1894   4.28   8.65   2.69   4.05  2.53  3.55  5.45  2.43  3.07  0.53  1.92   \n",
        "1895   5.71   0.98   5.09   3.07  2.05  2.90  7.14  1.40  6.69  1.57  2.14   \n",
        "        ...    ...    ...    ...   ...   ...   ...   ...   ...   ...   ...   \n",
        "\n",
        "       Dec  \n",
        "Year        \n",
        "1871  1.65  \n",
        "1872  2.38  \n",
        "1873  5.94  \n",
        "1874  4.19  \n",
        "1875  4.30  \n",
        "1876  0.95  \n",
        "1877  2.49  \n",
        "1878  6.04  \n",
        "1879  9.15  \n",
        "1880  3.32  \n",
        "1881  4.85  \n",
        "1882  1.52  \n",
        "1883  4.97  \n",
        "1884  3.78  \n",
        "1885  2.90  \n",
        "1886  1.48  \n",
        "1887  5.31  \n",
        "1888  1.77  \n",
        "1889  1.17  \n",
        "1890  4.12  \n",
        "1891  4.26  \n",
        "1892  6.43  \n",
        "1893  3.50  \n",
        "1894  2.81  \n",
        "1895  4.09  \n",
        "       ...  \n",
        "\n",
        "[141 rows x 12 columns]"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "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": 8,
       "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": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "precip_var = precip.var()\n",
      "precip_var"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The history saving thread hit an unexpected error (OperationalError('disk I/O error',)).History will not be written to the database.\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "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": 9
    },
    {
     "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": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "estimates = pd.DataFrame({'alpha_mom': alpha_mom, 'beta_mom': beta_mom})\n",
      "estimates"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "     alpha_mom  beta_mom\n",
        "Jan   2.953407  1.531684\n",
        "Feb   3.043866  1.346249\n",
        "Mar   4.617770  1.077920\n",
        "Apr   4.293694  0.979219\n",
        "May   3.526199  1.226724\n",
        "Jun   2.980965  1.299403\n",
        "Jul   4.017624  0.969593\n",
        "Aug   2.999766  1.122522\n",
        "Sep   2.309383  1.462581\n",
        "Oct   2.485616  1.050243\n",
        "Nov   3.692511  0.998206\n",
        "Dec   3.219070  1.297344\n",
        "\n",
        "[12 rows x 2 columns]"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "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 import gamma\n",
      "\n",
      "xvals = np.linspace(0, 10)\n",
      "yvals = gamma.pdf(xvals, alpha_mom[0], beta_mom[0])\n",
      "precip.Jan.hist(normed=True, bins=20)\n",
      "plt.plot(xvals, yvals)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "[<matplotlib.lines.Line2D at 0x105d98fd0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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qFHAB+IXWerfRUIk1aK17Il/3AA0mw6To08APTYeIRSl1N+EW6QHT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\n9IUQYgaRoi+EEDOIFH0hhJhB/j8LG/RFX2J0FQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107f98e50>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "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 = np.linspace(*ax.get_xlim())\n",
      "    ax.plot(x, gamma.pdf(x, alpha_mom[m], beta_mom[m]))\n",
      "    \n",
      "    # Annotate with parameter estimates\n",
      "    label = 'alpha = {0:.2f}\\nbeta = {1:.2f}'.format(alpha_mom[m], beta_mom[m])\n",
      "    ax.annotate(label, xy=(10, 0.2))\n",
      "    \n",
      "plt.tight_layout()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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w71WbgqnOjGiD5QFfXwRL3Vm0aBHbtm3jww8/5IEHHuAnP/kJ0PvrlgCHw6Dm\nQgPp2X1fYLgryqQYWzCc4qPSi9WVYK874BzS+NZbb7F169b2BDHBSBpYvRThMJheVcf5lHi2jE5n\nR84wpq8u5PZvzSU6JpJXfruTixcaAh2mzzkcDu6//36mT5/O17/+dVpanJmsDhw4wLXXXsvs2bO5\n+eabuXjxIm+88QYHDhzgvvvuY/HixbS0tPD444+zbNkyrrnmGp588skBx7N3717y8vLIzc0lKiqK\nm2++mU2bNnUqt23bNq699loAZs2axalTp6ipqSE2NhbDMKivr6e5uZnm5ub2dR/CVXfZzlrtnbP7\nxdkc2EyKqG4WB46OUN1mS+tO/sR0Tg7CIRrBVncA5s6d6/H7PmzYMIqKioiMvPI7kJGRwZQpUwBI\nS0ujqKjI5xfmUFd/wUxWjncaWNm5Q7BZ7VRXmr1yvGAWqnXHfQHh+vp6YmKcPZe9vW4JMNe1EJ8Q\n3e8FhrviHCY4OB7yhVvdaRMKD5Y8NrA0TVuoado+TdMOaZp2fxfbb9c07aDr9aqmaZN7u2/IMAwm\n1tRjjo6kJDkO3CYERkVFsOLmyVy9II/XntvFyTCvtCdOnGDFihXs2rULh8PBO++8g9Vq5dvf/jYv\nvvgiu3btYvXq1bzyyiusWbOG6dOn89xzz/HBBx8QGxvLvffey+bNm3nnnXfYt28f//rXvzqd429/\n+1t7+lr315133tmpbEVFBSNGjGh/n52dTUVF52EzS5cuZf369TQ3N7Np0yaqq6spLy8nLi6OJ598\nkunTp1NYWMjs2bM7rXIebqrMrTy4sbjTq9Xe+Q9WYqsVc3T3y+XVNtu6PFaVubXbfbJGDqG5qZVL\nF8N/iIa7YKs73nT69GmOHTvGzJkzfXqeUOfNHiylFAVTszh2MPyHCYZy3fnHP/7BtGnTuP/++/n1\nr38N9P66JaCutomU1DivHjN3bBo1VQ00NnhedDfUhUvdeeaZZ9o/V0qxZs0aFi1axEsvvTSgc/hS\njwsNa5oWATwPLAPKgN2apm3Wdf2oW7HTwEJd1+s0Tfsa8AdgTi/3DQm59c0ktNr4JHvoFY0rd1Nm\njmR4ZhKvv7iHYRmJpKaF50rhycnJXH/99QB86UtfYsuWLYwbN47z58+3rxVlt9vJzc1t38f9ScP+\n/ft54YUXOHnyJPX19Rw7dowVK1ZccY5bbrnF40J3bXqb/eamm26ivLycG264gfz8fIqKioiIiKCm\npobvfe9DxurpAAAgAElEQVR77Ny5kyFDhnDnnXfyzjvvtK8DMdgltdowx3h3PXJlUu2LDg+mNNPB\nVne8paGhgXvuuYfHHnvsiqeO4krKMDDXNHo1PXTB1Cz+/so+FqwYH9aZwEK57lx//fVcf/31rF+/\nnjvuuIOtW7d6/Rzh7HJtE0O8NAe4TWSkidHjhnH6WDVTZo706rGDTbjUnbVr17bXnU2bNpGZmcnx\n48e57bbbGD9+fPvCx8HE053TLKBY1/WzAJqmvQasAdobSbqu73Qr/zbwWG/3DQWpza3kXW7kk+yh\nOEw9X8AyR6YwZ8lYNv31U778jVmYIgbHCEzDMEhNTe32wtF24TcMg4ceeogXXniBiRMn8sMf/rC9\nu9rdX//61ysm07fJy8vjxRdfvOKzrKwsysrK2t+XlZWRnZ3dad/4+HgefPBBHnzwQcC50veoUaPY\nsWMHM2fOZMyYMQCsWbOGHTt2SAPLJclioyox1uvHzZ+Ywa4PTg+qBlZXAll3vMFqtfK1r32NW2+9\nlVWrVnn9+OEksdVGXHIM0V58YDE8K4nIqAgqzl8mOzfVa8cNBaFWd26++Wa+//3vU19fT3Z2dq+u\nW8LVwPJSFlt3+YXpHD1UEfYNrK6Ect1JTk4mMzMTgAkTJrB69Wr27t0blA0sTy2AEcB5t/elrs+6\ncy/wRj/3DToOh8Hk6joOD0+muZfjf2fMHUVkVAS7t53xXDgE1dfX8/bbb2OxWFi/fj3Lli1j3Djn\ngplvvvkmhmFgtVo5duwYAImJiVRXVwNgsVhoaGggNzeX8vJyNm3a1OVT11tvvZWtW7d2enVVUYuK\nijh9+jQlJSW0trayYcMGVq5c2WXcra3OYWsvvfQS8+bNIzExkTlz5nDgwAEuXbqExWJh8+bNLFmy\nxFv/XCEvsdXW4xDB/sodM5TqSjNNDd0PJQw3wVZ3+qrjmHfDMHjggQcoKCjgW9/61oCPH+6SLVZS\n0r27MH3bMMGjYT5MMFTrzpkzZ9rrzbvvvktsbCzJycm9vm4JuHyxmZSh3h0iCJA3YTilZ2rDPkFZ\nuNWdpqYmzGbnvNOamho2b97MxIkT+30eX/LUwOr1LDJN05YAdwA/6uu+warmXC2WiAguxvc+pa4y\nKVZ+aTJ7PjrHhTBLRa2UYty4cWzatInZs2ejlGL58uVERUXxyiuv8Nvf/pZZs2axaNEidu/eDcDa\ntWt5/PHHWbx4MQDf+c53WLZsGXfffXd70omBiIyMZN26daxdu5YlS5Zw++23M2HCBABefPHF9gp+\n7Nix9rSeH3/8MU888QTg7D7/7ne/y9q1a1m1ahWTJk1iwYIFA44rHEQ4DGLtdpq8OLm4TWRUBKPy\n0zh9otrrxw5GwVh3AO655x5WrlzJqVOnmDx5Mn/605+AK+tOVVUVkydP5n/+53946qmnmDJlCg0N\nDezatQtd19m2bVv7mPtgzugUaCkWK8kZiV4/bsG0LI5/WokjTNeWC+W689ZbbzFv3jwWLVrEG2+8\n0V6mp+uWuFLdJd/0YMXGRZGVM4SzJ2u8fuxgEY5158KFC6xevZqFCxdyzz338K1vfYulS5d6JS5v\nUz1l4tA0bQ7wsK7rK13vfwA4dF3/ZYdyU4HXget0XS/uy75ttmzZYsyYMcMLv1LvVZotPU7G3/vW\nZ3zY5KA8qeunJ0+symdadtdPJI/sL2fX1tOs/T9zifTBDaoIffv27eOaa64Z8MSJvtadg+VmHtxY\n3Onzny3L4+ebP+95TWy1MbXqMjtyhnXa1t0+bXqqG20O7y3l1LFq1txe1OvYhQDv1R3w3rWnu3rV\nZm7pRa5ZM5GFRd4fCvbSuo9YurqQnDFDvX5sEV6Cse70ZN0jm7n7uwuJT4j2+rH37zxHRWkdq26d\n6vVji/DT17rjaezPHmCcpmmjgXLgNuAr7gU0TcvF2bi6o61x1dt9A60tm1pXYm12llY1UJnVvwtW\n4fQsTh6p4uMPTjP/2nEDCVOIgIi12WmJ9N3DgbwJw3n/7WPYbA4iIwfHfEUxOEU4DOKtNpKG+SYJ\nyPhJmZz4rFIaWCKsNDe1YhgQFx/lk+OPLUzno83F2O0OIgbJnHnhPz1+o3RdtwF3ARuAvcDzuq4f\n1TTtPk3T7nMV+ykwFPidpmn7NU37pKd9ffR7eN2I+maG5ad5TGzRHaUUS1YXcODjEhp7WBNIiGAV\na7PT7MMGVkJiDGnpiZSeqfXZOYQIBvFWG01Rkb1KfNTdOnU9rTM3fnIGJz+rwgjTYYJicKqrbWZI\nWrzPMmQmD4kjZWgcZWcv+eT4YnDzOHtd1/WtQFGHz551+/ke4J7e7hsKlGEwwtxMeuFY2Nf/da2S\nh8QxaUY2O98/xbIbg3MSnhDdibP6tgcLYGzBcE4dvcDoccN8eh4hAineaqepl3Wpp5EVbZ5YlU9m\n0udzg9PSE4mJjaL8/GVGjBpc2QRF6PE0PQMgIymayz5YA6uj/MJ0io9eIHdsmk/PIwYf76cHCyKe\nKnGr3dHl52lNrbRERpDghYmVsxeN5YVntjFz/mifTNQUwldibQ5q4n37J2JMQTobXt7L0hsKw3od\nHzG4xdl8kyzG3fjJGZw4XCkNLBH0evsQodkHa2B1lF+YwYZX9rFkdYFcg4RXhXUDy1Ml/tmyrtfg\nGWluojTZO09N4hOjKZo7ih2bi1mlyURKETp8PUQQYFhGIihFTVUDwzO9m8JaiGARb7VRF+ObeSRt\nxk/OZP3Le1m8Sm4URXioq20mc0SyT88xLNOZ2VOuQcLbwrqB1R8xNjupLVY+TR/itWPOnD+aPzz1\nIdWVZqnAImTE+TjJBTjnKo4tGM6pYxekboiwFW+1U5kYR3SE4mC5ucey3Y2s8GRYRiKRkSYqy+rJ\nGpnSr2MIEUwuX2yiYGqmT8+hlCK/IJ3iI3INEt4lDawORpibqUiMxd7P5BZdiY6JZPaiMWx/5wQ3\n/dtVXjuuEL6iDINouwOLH7L7jS1I56PNJ5mzeKzPzyVEIMRbnUMEa5ttXS5r4K67kRWeKKUYPzmT\nE4crpYElQl50hKK6ppEKm8HlLh5KJMdGUt/S/SLBGUnRV8xT7En+xHS2/vM4c5fKNUh4jzSwOshs\naOGz4d7vkp42K4c9H52l7NwlGSMvgl6MzYElwoThh6FGOXlDqa1upLHBQkJi7xf1FiIUmBwGUQ4H\nLX5IAz1+ciZvvrqfhSvGyzBBEdJqGlppMlv42bbzXV6HuluDsU3HRDA9GTE6lbraZsx1LSSlxPY7\nZiHcSeJ/NzE2OzF2h0/GykdGRTBn0Rh2fXDa68cWwtv8MTywTUSkiVH5aZw+Xu2X8wnhT/Ftcxn9\n0OBJz3IOcaqu6HkYohDBzmK20BIZ4ZeHfBERJvLGD+PUsQs+P5cYPKSB5WZocyu1cdE+uxBOmjGC\nqvJ6ufiJoBdrs9Pi46xn7sYWpHPqqFzcRPiJs9po9lNdUko5Fx0+XOmX8wnhKy31LT7PvOlurCtd\nuxDeIkME3aQ1t1ITF+2z40dGRXDVvFF8su00q7VpPjuPEAPlzCDov+cvYwqGs+Wto1itdqL8eFEV\nwteca2D571I7fnIGG/96iHnXjpNhgiJktdRbBpTF1lNCmY5zuGzJsZw/c4m9Zy8RGR3ZpzlcQnRF\nGlhtDIO05lZOpSb69DTTZuXyhye3crm2SdbFEkErzuagPsZ/fx7i4qPJyE7mXPFF8gvT/XZeIXwt\n3mqnIdp/dSlzZAp2m4OaygaGZ0lWNBGaLOaB9WB5SijT1RyuGZERPPP3Y1QlxvZpDpcQXZEhgi6J\nrTbsSvl8KEdMbCRTr85hz7azPj2PEAMR68c5WG3yJ6ZTfKTKr+cUwtfi/bDIsDulFOOnZHL80wq/\nnVMIb2upt/htaG2b6vgYhjdZ/HpOEb6kgeWS1tzKRR8OD3Q34wujOHaogkazVGQRnPyxyHBH+RMz\nOHX0Ao5+rgMkRDCKt9r82sACmDAli+OHKzEMw6/nFcJbWupbaPLzNehCQgzDmiwoqTfCC6SB5ZLW\n3MrFeP80sBKSYiiYmsW+nef8cj4h+sQwXFkE/fvnISU1jqQhcZSdu+zX8wrhK8owiLE5/N4bnDki\nGbvdoLpSEiqJEGQYWMz+78GyREbQHBVBanOrX88rwpM0sHCuUzKkxUptrH8aWAAzF4zm0CfnsfSw\nUJ4QgRDlMHAohd3k/z8P+YXpFB+VYYIiPMRZnT3B/kg17U4pxYTJmRz/VLIJitATbXdgijQF5BpU\nlRBLRqOMLhID53HmraZpC4FnXGV/r+v6ug7bC4AXgCLgR7quP+W27SxQD9gBq67rs7wXuvcMsbTS\nEB2JzQ8LQbafc2g8o/KHcWj3ea5ekOe38wrhSSCGB7YZNymDDf+7j8WrCiQDmgh58Va735/Ct5kw\nJZO3/3KQ+ZJNUISYeJud2OTALPhblRDDrPJLGA4ZJigGpscWhaZpEcDzwM3AVcDdmqYVdih2Ebgf\neLKLQxjAYl3Xi4K1cQWQ1uS/+VfuZi4Yzb4d57DLnBMRRPy5yHBHwzISUUoWShXhId7m//lXbTJG\nJONwGFKXRMiJs9qJTQ5MBr/mqEgsESYuV9QH5PwifHjqspkFFOu6flbXdSvwGrDGvYCu69W6ru8B\nrN0cI+gfnflz/pW7zBEpDEmLl2EcIqjE2hx+n3/VRilF/sQMTko2QREG4q12mqICsxrK59kE5foi\nQku81U5MgHqwwJnsourUxYCdX4QHT3dRI4Dzbu9LXZ/1lgG8p2nafk3TvtHX4Pwhyu4gzmqnLiYq\nIOe/ekEee7adkWxPImgEcoggwLiJGRQfvRCw8wvhLc4GVuDq0gRXA0uuLyKUxNnsxAZwDaqqhFgu\nnL4owwTFgHhqYA302zVP1/VpwFeBH2qatmCAx/O6tGYLl+Ki/D4JuU3e+GHY7QYlp2oDcn4hOgrk\nEEGA7NwhNNZbuFzbFLAYhPCGeKvd76mm3WVkJ2NgcEGGCYoQ4hwiGLgerMboSCKjIqgorQtYDCL0\neWpglQE5bu9zcPZi9Yqu6xWu/x4FNuAcchhUhjZbqY0L3JMSpRQzF4xm9/buVxwXwp8CsciwO5NJ\nMbYwneIj0oslQpcyDGdvcAB7sNqzCR6SRYdF6Ii12YlO9P+0DXfpY9M48ZkMrxX956mBtQcYp2na\naE3TooHbgDe7KXtFF5CmafGapiW5fh4OrAI+HWC8XpdisXI5NjDDA9sUTsumusIsa5aIoBBrc9Ac\noDlYbcZNyuDEYbm4idAVa7NjiTQFbHREm4JpWRw7VCHDnURoMAxi7A6iAzAv3l3G2DROHq6S4bWi\n33q8i9J13QbchbP3aS/wvK7rRzVNu0/TtPsANE3L1DTtPPAfwI81TSvRNC0RyAS2aZp2AGdyjKd1\nXX/Hl79MX9lb7cRZ7ZijAzMJuU1kpIkZc3PZs/1sQOMQwuQwiHQ4aPXjkgVdGTU2jUs1jdRfbg5o\nHEL0VyATXLgbnplEdEwkpecuBToUITyKsTuwRpgwBfgalDgsARQyvFb0m8e//rqub8W5xpX7Z8+6\n/VzJlcMI2zQA0wcaoC81VDdgjo4M+BNGgGmzc/nDkx9irmshKSVwY4/F4BZrs2OJiIAB1onoCMXB\n8q4vTMmxkdR3scB2RlI0ma6JzRGRJsZNyuDYoUpmLZR14kToCfT8qzZKKQqnZ3P0QDk5eUMDHY4Q\nPYqxOWgJcOMKXFk4J2Vy8nAlGdnJgQ5HhKDAP14LIPOFBuoCPDywTWxcFJNmZLNvxzkWXTch0OGI\nQcpbGQRrm238fHPX8wp/tiyvy21PrMpvb2ABFEzNYuum49LAEiEp0BkE3RVOy+LldTtYen0hkUES\nkxBdibXbsQTBgwmA8ZMz2PjXQ8yTxbpFPwT+MUEANVQ1BCw9e1eumjeaT/eU0tLc3ZJiQvhWXADX\nwOpoZN5QGswWamsaAx2KEH0WyEWGO0oeEsfwrCROH68OdChC9CgmiK5BmSNTMAyoKpNFh0XfBce3\nOBAMA/MFc8ATXLhLHhLHmAnDOfjJec+FhfCBQGcQdGcyKcZPzpAMaCIk+XIOVtsQ3O5elWZLp30m\nFmVzZH+5T+IRwluC6RqklGLidKk3on8G7RDBOJsdU4RpQF3RPc0zgSvnlPTW1Qvz+NsLe7jqC6Nk\nKIfwu1ibnUuxgc3e5K5gahb/Wn+YOUvGyhANETraUrT76EaxpyG40Hm4LcD4SRm8/49jNDe1Ehfg\nDG1CdCfW7qAhwInH3BVOz+LPv9vFolUTiAiCuWEidATPt9jPUixWEtMTB3SM/lzkPBmemUR6VhJH\nDpQz9equcocI4TtxNjsVQTI8A5yLDtusdmqqGhiemRTocITolRi7A5vJhMMUPA8FYmKjyBs/jOOH\nKpk+JzfQ4QjRpZgg6sECSE1LIGVoHOeKLzJmwvBAhyNCSPDcSflZSouVpAE2sHzl6oV57N52RtYt\nEX4Xa3PQEkQ9p0opJkx1ruMjRKhwDnMKvsvrxKJsjhyQ4U4ieMXaHFiCrO5MdGXhFKIvgutb7EdD\nLFYSM4KzgZWTN5SY2CiKj14IdChiEDEMgxi7nZaI4GlggXOY4LFDFbLgowgZcTaHz4YHDsToccO4\ndLGJy7VNgQ5FiM6C9Bo0YUoWp49X02rpvLyIEN0ZlA0sk8MgsdVGwrDgbGAppZi1MI9PPjwtN5XC\nb2wtNuxKBdWwJoD0rCRMJkVlaV2gQxGiV4Jpor67iAgTBVMy+WxfWaBDEaKTKIcRlNeg+MRoRoxO\n5eSRqkCHIkLIoGxgJbdaaYiOJCLIuqHd5U/MwNJio+TUxUCHIgaJ1sbWoFl/xJ1SioKpWTK0SYSM\nOKvvElwM1JSrR3J4bxkOuyPQoQhxhVhb8KyB1ZEMExR9FbwtDB9KabEG1fpXXTGZFHMWj2XHllPS\niyX8orWxlZYgzZI0+aoRHDtYgc1qD3QoYhA6WdPEZ1UN3b7O1DZfUT5Ye7AA0rOSSUyO4cyJmkCH\nIsQVYm2OoL0GjS1Ip+J8HQ31LYEORYSIQZlFMMVi5UJ837L7BULB1Ex2vFfM+TO15I5JC3Q4IswF\naw8WQEpqPOnZyZz8rIrC6dmBDkcMMi/vrWDX+e4XG72hMI2Feant7+NsdpoDOELC0xIiQ8cPZ+f2\nszSkxPV4nP4sNSJEf8XYg/fBRFR0BPkTMzh2qIKZ8/MCHY4IAYOygTWkxcrJocE5/8qdKcLEnMVj\n+Pi9U9LAEj5naWrFEqRPDwGmXp3DgV0l0sASwc0wiLPZA5qN09MSIiaHwcqKen7yxvEe4+zPUiNC\n9FcwZhB0N6komy1vHeWqeaNlXUbhUfB+k30kxmbHZBhBOz6+o8Lp2dRdaqb07KVAhyLCXGtja1Cm\nlm4ztjCdmqoGLl1sDHQoQnQrymHgUAqbKXjrksOkGDZuGCPNzZ4LC+EnwbYGVkc5Y4ZiOAzK5H5M\n9ILHHixN0xYCz7jK/l7X9XUdthcALwBFwI90XX+qt/sGQkqLlbrYKAiRpw8RESZmLx7Dx+8Xc8ud\nVwc6HBHGgnmIIEBkpImJRdl8uqeUhSsmBDocIboUawveBBfuMgrTyT56gVOpCRghcj0U4S3WHrxz\nsMCZcGna7BwO7DrPyLyhgQ5HBLkev8mapkUAzwM3A1cBd2uaVtih2EXgfuDJfuzrdymW4E9w0dGk\nohFcrG6k4vzlQIciQkSDxUZZXUuXr+4Ec5KLNlNnjuSzfeXYJQOaCFJx1uB+Ct8mPjWe5qgIhjdZ\nAh2KEEBwZxFsM2nGCM6cqKaxQeqN6JmnHqxZQLGu62cBNE17DVgDHG0roOt6NVCtadrqvu4bCCkW\nK2eGJAQyhD6LiDQxe+EYPtpSzC1fnxnocEQIqG22cc/fOle1wvR47pzZ9Rym1sZWLMnBPTcxLT2R\nIUPjOHO8mvyJGYEOR4hOAp3goi/OJ8cxsr6ZCwmxgQ5FDHaGQYzNEdTD1AFi46IYNymDw3vLmL1o\nTKDDEUHM0zd5BHDe7X2p67PeGMi+vmEYJFls1IdYDxbA5JkjuVTdyPnTtYEORYQhk8PAYXdgDbIF\nHrsyZeZIDu0pDXQYQnQpmFO0d1SVEEuyxUqc1RboUMQgF+lwLkdjD+K5i22mz87l4CfncThkCR3R\nPU89WAP59gTdNy/BascaYcIa5MOguhIZaWLeteP48F/H+eo350gGG+FVsXY70fHRITE3cfyUTD7Y\neBxzXQtJKfLkXQSXOJuDS3HRgQ6jVwylKE2OI7euiePDkgMdjggT5y41s6e0+2UCcoZ0zkwZaw/+\n3qs2mSNTiIuP4uzJGsZMGB7ocESQ8tTAKgNy3N7n4OyJ6o2B7OsTzvlXoZuZvnBqFru3neHkZ1WM\nn5wZ6HBEGImxOYhOCI2bwujoSAqmZnFwVwnzl48PdDhCXCFUkly0KUmOZ17pRU6nJobkw0cRfI5X\nN/HsrrJut//82s7rSAV7BsGOps/O5cCuEmlgiW55+mu6BxinadpoTdOigduAN7sp2/HRd1/29Ytk\nizUkhwe2USbFwhXj2f7OSRwyyV94UazNHjINLICr5o/i4CfnabXI0CYRXOJC7EaxNTKCqoRYcuua\nAh2KGMRibfagXoexo4KpWZSfu0zdJak3oms9dufoum7TNO0uYAOfp1o/qmnafa7tz2qalgnsBpIB\nh6Zp3wEm6rre0NW+vvxlPEmxWKlM9N+QougIxcHy7rvJM5Ki+7yI4+hxw0hIiuHwvjKmXp3jeQch\neiHG7urBagiNhntqWgIj84ZyeG8pM74wOtDhCAFApMOByTBCYi6ju7Mp8cwqr+XskATsIRa7CA8x\ndkdIPZiIio5g0oxsDnx8nkXXybIhojOP4+V0Xd+Kc40r98+edfu5kiuHAva4b6AowyCx1YY52n89\nWLXNNn6++Uy3259Yld/nBpZSioUrx/PGn/ZTOC2bqOjQ+YMkgleMzU5MQjQ0dJ/GPdjMWpjHW38+\nwPTZuZhC6MmnCF+xVtfwwBCYy+iuKTqSS3HRjDA3U5ISH+hwxCAUa3OE3BSOq+aN5uV1O5i1KI+4\n+NAZASL8Y9DclSS12miKigyLp3NZOUPIyhnC3h1nAx2KCBOxITQHq01WzhCSh8Rx/HBloEMRAnAm\nuAil+VfuzqQkMKquEWUEXX4qMQiEwhpYHSUPiSN/Yjr7d5YEOhQRhAZNAyulJbQTXHS0aOUE9m4/\nS/3l5kCHIsJAjN3hzCIYYq5ekMfubWcx5KZQBIFQStHeUX1sFE2RkWSGUC+2CB8xNkfQL3TfldmL\nxrB/5zmZDyw6Cb1vcz+FYoKLSrOFg+XmLl/nLHbyZ4zg/bePBTpMEQZibXaiE0OvgTVmwnBsVjsl\np2R9OBF4cTY7zVGh2cACODsknrzLjSAPLISfxdpD8+FE6rAERuWncWCX9GKJK4VPl44HKRZryI0t\nrzK38uDG4m63/9fyPPbqBzlzopq88ZIqVPSTYRBtdxAVH1oPIMCZWdPZi3WGUflp7Z9Xmi1UmVs7\nle9PYhkheivWZg+5B3nuLsZF41CKjEYLVX5MCCUGtwhXchhbiE7hmL14LH97YQ9Fc0cRFcIPWIR3\nDYoGVoTDQZzNTkN0eP26EZERLL2+kPfeOsrXvpNGZIgs0ieCS4zdgTXChMkUmt+fwunZfLT5JFVl\ndWSMSAG6fzjRn8QyQvRWnM1Ocyj/HVaKE0MTKawxcyFB6onwD+fwwNBLDtNmeGYSWSNT+HR3KTO+\nMCrQ4YggEcJXgt5LtjizBxohWnl7MrYgnaHpiezZ3n22QiF6EmNzhNT6Ix1FRpqYs3gsW/95QuZi\niYCKtYVWqumu1MbH0BwVwch6md8r/CPWbscSyg8mgNlLxrJ72xnsttBY6kT4Xmh/o3spxRJeCS46\nWnp9AXu3n5UF70S/xNjttIT4xW3K1SMx1zVz9mRNoEMRg5TdaifSEdoPK9qcGJrImMuNWGXivvCD\ncHgwkTUyhbT0RA7tKQ10KCJIhP6VoBdCMcFFX6SkxjNz/mj+tf4zDIc8wRd9E2tzhFx63I4iIkws\nXDGBD/95AofUAREALQ2WkB7m5K4hJoqauGjO7SsLdChiEIixhf5DPoCFK8ezc0sxLc1Wrxyvp0Rn\nB8vNVJotXjmP8I3w7dZxk2KxUpyaGOgwfOrqBXkUH73A/o9LZAyw6JMYuz0o0uNGRygOlps7fd7b\nxBT5E9PZve0MRw6UQ2ayL0IUolvN9RZawmiCe/HQREYfrsS8LJ+kFEl4IXwn1uYIizny6VnJjJuU\nwY4txSy9vnDAx/OU6EzmFAe3wN9V+ViU3UGkw6ApjC58XTFFmFh161R2vlfMxQsNgQ5HhJBg6cGq\nbbbx4MbiTq+usgF2RSnFousm8NG7J7Hb7D6OVogrtZgtoZ3gogNLZAQjJmXw0eaTgQ5FhLlQX97A\n3bxl4zh6oFzuw0To92B1l44ZoNXuYEiLa3hgEA7b6O6JfZtWe98mS6YOS2DesnFs/OshvvrNOUQE\nQa+ECH6husBjV0aMSiVzZAolBysCHYoYZJrNlpCfR9LR6KtGsvu1A5SXXCY7d0igwxFhKt5qpylM\n6k58YjSzF4/l/beP8aWvX4UKwntP4R8h38DqqQv1Z8vyGNLSyqXY4Jx/Vdts4+ebu8/+97NleX0+\n5rTZORQfu8CuD07zhWvyBxKeGCScGZzC4+IGsGDFeP73tzuJTh9Caxj9XiK4tZhbaA6z71tUTCRL\nVhfwz9c/5d++/QUiw6SXQQQPZRjE2IOvB8vTA3Dofvh60ZxcDn1yntPHqxlbkO6rEEWQC/kGliep\nLVZODg3v+VfulFKsvHkyL6/bwaj8NEaMSg10SCKYGYYzTXsYDW0aOiyBkZMzmXikmgMZKUHZey3C\nT3O9hZbI8LukTpiSyfFDlex8/xQLlo8PdDgizMTa7LRGmIJuGR1PD8Ch+zlQEZEmFq8u4P1/HGV0\n/tplhJ4AACAASURBVDAiwuj6KnrP49VA07SFwDOusr/XdX1dF2X+H7AaaAK+ruv6MdfnZ4F6wA5Y\ndV2f5b3QPbNb7SS22qgL4wyCXUlMjmXFlybz1p8PcMe/zyUxWSYoi65FutaNsgXZxW2gxlydw6GD\nlWQ2tFCZFBfocESYMwyDxkvNNGaG3wMtpRTL1kzkpf/+iHGTMsh0LeYthDfEW+00RYXfg4kxE4Zz\n8JPz7HivWB5MDFI9Nqs1TYsAngduBq4C7tY0rbBDmVXANF3XpwLfAV5022wAi3VdL+pP46rSbOGt\nI9Xdvs5fbulxf3NVA+aYSBym8Lp57I2xBelMm5XLG3/aj00WvhPdaO+9CrMGlinCxGfpyUyobSBa\nEl4IX7PYUQpaw/Rak5AUw6JVE/jX64dlIVXhVeE0/6qj5V+cxOG9ZZSeqQ10KCIAPPVbzgKKdV0/\nq+u6FXgNWNOhzI3ASwC6ru8ChmialuG2vd9XnKZWB+t2lHb7qm7sObtYfWU9l2Kj+3v6kDdn8RgS\nk2PZ8uYRDEPWBhKdxdjszrV7wlB9TBRlSXFMrDGDfP+FDxkNFhKGxofdgwp3E6dnk5QSy66tpwMd\niggjcTZ72GZ5TkiKYfkXJ7Hxb59iafHO2lgidHhqYI0Azru9L3V91tsyBvCepmn7NU37xkAC7Q9z\nRX3QJrjwB2VSXHfLFMpLLnPwk/OedxCDTqw9vOZfdXQqNYF4q43Mxp57u4UYCKPBQkJqeA9FVUpx\n7RcncWBXCSWnLgY6HBEm4q22oEtw4U1jC9MZnZ/GlreOBjoU4Wee7qx6+9i3u8d283RdnwZ8Ffih\npmkLeh3ZACnDoKG6kcuDuIEFEB0TyRfXFrFjczFnTlQHOhwRZGJsdlrCuIFlKMXh4SlMuNhAnNUW\n6HBEmHKYLSQOjQ90GD6XlBLLam0qb+uHqLvUHOhwRBhwzsEK3wYWwOLVBVSUXOb4p5WBDkX4kaeZ\nhWVAjtv7HJw9VD2VGen6DF3XK1z/Papp2gacQw63DSTg3kqxWIkbEofdFL43j57SiLalEE1NS2DN\nHUX8/ZX93PjV6eTkDfVjlCKYxdocNEQH9wTjnr7n3aXJdVcfG8Wp1ASKKi9ja+2+kdXdmnq9OYcY\n3NqHCJY1BToUnxuVP4yZ80fz5p/28+X7ZhMV5jfHwocMw7nIcBhm33QXHR3JKm0qG17eR3p2Eqlp\nCYEOSfiBp2/1HmCcpmmjgXLgNuArHcq8CXwbeE3TtDnAZV3XqzRNiwcidF03a5o2HFgFPODV6HuQ\n2mwleWQSNIbv3AtPaUTdU4iOGJXK9bdN481XD3Dz164ia6RkghIQY7dTExnc8xR7+p53lya3o9Lk\neJIsNj595wQz7pmF6iIZQXdr6vX2HGLwMhoGRw9Wm5nzR3OhvJ53N3zGdbdOkcVURb/E2B1YTSbs\nYZoc5oqHdhERjLo6h1f/sJtZt0wlOi5KHt6FuR67d3RdtwF3ARuAvcDzrt6o+zRNu89VZiNwSNO0\nT3Gmc7/TtXsmsE3TtAM4k2M8rev6Oz76PTpJbWklOSvZX6cLCaPy01h582Q2vLyX6sqeF9ATg0O8\n1U5LmGZw6ujYsCRsrXa2bz4Z6FBEGImyO8BhEB0/eIajK6VYftNkaqrM7Nl+NtDhiBAVbw2+BYa9\nqe2hXdvr1yUNHMHE31/Zz/f/cbLLERMifHjsl9V1fStQ1OGzZzu8fwh4qMNnp4HpXoixz9T/z959\nx8dVngnf/53p6r3bkuUm915xkQ02piWmHkIwGxJIeF4Skt0k7IZsNgm7STZPIEt2yfImmBZCEnJC\nCaY4wTZgsDHGHYMt3C1bxbKK1TX1PH+MJMaqI2tG067v5zNIM+c+59yD5vI9d9d1UjucJOUmwbHG\nUGQhbI2bnM3lzsm88PRubvyHOeTIniYxy+jxEOdyh/0QwUDRFYWZV09i30sHycpJYtLMvFBnSUSB\nRIcLJckac704ZouRtevm8Pz6nVisJmYuGD34SUL4iHe6onaJ9v4cTU9kRk0jU883yurOUS4qJygl\nOVx0mAyYY3yBi/5MmpHHqs9P4YVn9sjCFzEsye6ixWJCj6EvhpY4Mzesm8Nbrx3myMcy4VgMX4LT\nhZIYm8N8UtLiuPWuBXzw9nEO7u45PVuIgUXrJsMD6lx4yebycGzHaalkRbGo/GSntTtoiAvveSWh\nNmFqDvGJFl75wz6WXTmR6fNGhTpLYoSl2J00WmOvESIrL4mbvjyPF5/ZjcejM2mG9GSJS5focKFk\nxc78q55SM+JR75qP9uQuFAWmzZWyRPgn3uWmOiH2yiCPQWF/biqFpxv4y18OMnHJmD57wB1u2dQ7\nkkVnBavDSXWiLdTZCHsFRWl84WsLefGZ3TTWt3HZqgkYonSyqegt2e6kLj6yW977W2FwsIIpJz+Z\nW748nxee2Y3HrTNldn6wsiiiXILDjSHGJ6qnZSZwy1fmoT25C49HZ8Z8GS4oBhcXA0u098dpNDDp\nuilsev4AO47Vcygruddokh+tKg5R7kQgRN8QQV0ntcMR0xsMD0V6ZgJfvGcRFacv8Jcnd9HcKBuy\nxopkuyvie7Dq210XTSLuejjcgw+7yMpL4pavzOPdv3/KgZ3lI5BbEY0SY3iIoK/0rETUuxew692T\nvPXaYTzS+i4GouveRS5ibA6WL5PVxJ68NKxuDzPPNWLwyHDBaBJ1FazUDicOowF7DAftUCUkWbnl\nrvkUjsvg9//7PscO14Q6SyLITG4PVreH1hhtPeySmZOEevcC9rx/mk+2HJUCTgyJye3B6NHBFpWD\nQYYsPTOB2+9dTP35Vl54ejftbbJKmuibxe3Bo4DLGHVfQ4fEbVDYl5uKW4G5VQ1YXe5QZ0kESNR9\nskc1t1ORFBfqbEQcg0Fh8eXjWHv7bN569RCb/voJHe3OUGdLBEmy3Umz1QQxtMBFf9IzE1h372Lc\nTjcLKuuJc0oBJ/yT6PQuFBNrKwgOxBZn5sYvzSWnIIXn/ncH1WdlJV/RW7wr8he46Bqi3t/D3zlU\nuqJwMDuF8wlWFlXUk9UqI4miQWR/unsweTxktdk5kpEU6qxErIKiNP7hviW89+YRnnrkPZZdOZFp\ncwr63JhVRK6UKBgeGEgWq4npa0rY+vxHLKispywjiXMJVqmAigElOFwx3wt80WaqPlJn5lMYb0Z7\nejfjZ+ax+uoSzMP8f9XfvbrIxq2RIz4K5l/Vt7t4cPPJfo8PaQ6VonAqNYEGm5kZNY1ktDvwuGSY\nbSSLqgpWbksHdXEWHDHe5Txctjgzq9dOZca8UWzecIiPdp1h5bWTyC9MC3XWRIAk252ck4VgLqIo\nCuUpCTRazUw530xBczuHM6WxRvQv0RE7+8j1p2sz1f5YslJIq2vjd/+znStvmErh2Iyg3euha8ZL\nBStCxMX4/Kv+NNos7CjIYEptEwde/IjMOBu1Eb4YVayKqppIQZMMDwyknIIUvnjPImYuGM1rzx9A\ne3IX5SfqZN+GKJBsd9Joje0vhv1ptFn4YFQ6dXEWFlbUc/zDcpwOGTYoektwumiN8QrWYBwmIzPW\nlLDimkls/MtBNvxxH7Xneq/8KWJLNPRgBYvLaOCjnFTGLC6ipK6ZOVUNJDhcoc6WGKKoKRmS7E4s\nbg91sv9Vt/6WsAZItplo6ug/YLuGWigGhWlzRzF5Vj6H91ey6eVPiE+0MHfJGMZNysZo6ruOLkM5\nwpe9zYFJ16X1cAC6onA6NYHqRBuj69tY/9BW5iwpYtbCQmxxMrRSeCU6XLRE+DySkTJ+cjaF49I5\nsPMM2hO7KByXzuLLx5ORnRjqrIkQiHe5aDNLg/hA0grTeH9UBqMb25hfWU9NgpXTKQnSqBMhouav\nVNDcTmVSnMyZ8DHQ+OAfrSoecOxwz6EWRqOBaXNHMWV2AUc+rmbfjnI2v3KIKXPymT53VK9CUoZy\nhK/mmhaarGaJFT/YTUZmXDWeAqPCh++e4ImH32XG/FHMWDCa1PTY3VxWeOf8mjw6Hf00MkWDgRrp\nugxlM1SLxcT8ZcXMXDCafR+U8/zjO8kZlcLM+aMZOykLY5gM7x+sgRCkkXC4vD1YUfMVNGh0RaE8\nNYGqpDhGN7Uxr6qBJouJ06kJMpoozEXFp9vg0clt6eCDgksf2y38YzAoTJqRx6QZeTTUtnJw91m0\nJ3cRn2Bh/JRsJkzJIStP5q2Es8ZzLTI8cIgycxK55pYZXKhvY+/20/zhsR1kZCcydU4BE6flYpVl\numNOgsNFq8UY1Q0Vg03ih0vbDNViNbGwdCxzLiviyMfV7N52is0bDjF1dj4TpuWSW5Ac0pUZB2sg\nBGkkHA5nhwt0cMriWX5zGg2cSEvkVEoCeS3tTKptYvvv99IyJ59JM/PIzJHvXeEmKr4VZLd10GQ1\n0yHjeUdUWmYCy68qYemVE6k83cDRwzW88od96LpOYl4yec0dNMRZ6JChaGGluwcrBg3UIt9fS3zP\nczLmjuKyWfm0Vjaxb38lm189TEpuEplFqWQWpRGfGkduslW+fEW5RIebVmmBHxaz2cjU2QVMnV1A\n7bkWPtlXwRvaAZwON+MmZzNuUjYFRWnSgBFlOpo6aDdHd+NEsHgMChXJ8VQkxfFv8/NwVTXywtO7\nscWbGTsxi6LxmRQUpWKS78MhN+i/WqqqLgd+1Zl2vaZpj/aR5j+Ba4E24E5N08r8PXe4dI/O6KZ2\nypNluE6oGAwKo4rTGVWczoqrS6iraeGDfVVklTdSUteM26DQZDXTZDHTZDXRbDFJ13ao6DpNNS00\nZqWEOichMdiw2aGc86NVxTx3sBZjQToZ7Q4yD5wjc0c5ugJjitOZPCGT/MIUsvOSpbCLQsl2Z8yv\nIOgvf4Ya5iRZKL2qhNKrvGXIscM1fPjuCc5VNJGWmcCoojScSVYSHC7azEZ0+XIesTqaOmSBi+FS\nFDLzknBkJ7JoRj6N1c2cL2/g0zfKaKlrJSUniZScJHJGpWBOi8OW2HeD32Dz8f2dry96G7B0UFXV\nCDwFrAIqgF2qqm7WNO2wT5prgJmaps1QVXUh8AywyJ9zh8ve6uDQpqN4UKhJkD9wOFAUhcycJEbP\ngI/OtoKuE+90k+xwkmx3UXyhlUSHi3ee2Mkn2YmkZSaQlhFPclocyaneR2KStd/FM8TwGB1uUMAe\nJnMdooHbYKAmwUZNgq378z6vKI0LdW0c3l9JXU0LSSk2MrITychOJC0znpQ072c+KdmKQf4WEUXR\ndUrqmklvd7A3Vbau8Ic/Qw19h9x1xcrC0rG4XB5qKhs5e6qBsiN1zK6+gNXtpsVsotVios1s9D5M\nJuxtDnSPLvs2hrFEu5Oz+yplH8YA6DOu4uMx2WyktjtJOd7A3JoWaioa0RWFVrORVp+46TAZ+cdV\nY/nJu+X93mOo8/XFZwZrflsAHNM07RSAqqrPA2sB30rS54HfAWiatlNV1VRVVXOBYj/OvWTp7XZ2\naQcYNTWXPRbZEDRsKQptFhNtFhPVPutg/HRlEXlGhfrzLVyob+Pkp7U0XWin6UI7bS0OrDYTCUlW\nEpKsxMWbscVbvD/jzFhtJqw270+L1YTZYvzsYTZiNBlCOn4/nFla7SRnJUq8BEvn5z1/UjYz871j\n4t1uDw21bdTVtFBX00L58XoaGyo6P+t24hOtxCdavJ/3RCtxCWbi4i3Y4szY4s1YrSYsnZ91i8WI\n2WLCbDZIxSwErC43M8414jQq7CxIxyV/g6AzmQzkF6aRX5iGdXwWz71xDKPHQ6LDRYLDTbzLRXar\nnXhnGzv+uI/tTjcJSVYSk22fxVOChbh4C9Y4EzZbVxliwuwbUxYjBqmYBY2i6xRfaKWwsY385cX8\n9XRLqLMUtVwGA7UJVmoTrKxbVYy26QRWt4cEpzdmEpwuMtvs2Fxudj+3h5UeHbvRgN1oxGE0dD4U\n73WO1ZLRZsdlUHAbDLgUBZdBwaMoeCRcBjRYBasAOOPz/Cyw0I80BUC+H+deZPvmoxc9v9Du4jJn\n50o+eud/dDA43Vgb2ph53WRSC1JgkMmoIvxY4syMyk9i1JjeLcAej057m4O2ZgctzR20tznpaHPS\n3ubgQl0bdrsLR4cLe4cLh92F0+HG4fD+dDnduD06JqMBU2dly2hUOn96HwajgsHQ9dP3YUAxeHvh\nFMX7mu9zRfH+jgIK3ufdv/t+z1IUFLzHvD+6zuk67v2RmBW4/589Y6epwyd2fCQ1Okgelw61A6+Q\nJQLHaDSQmZNIZk7v5ajdLg+tLXZam72PlmY7HW1OWprt1J5roaPdiaPzc971uXc6vZ9zxaB0NygY\nTQZMnT+NRgMGg9L5WTdc9BlXun4O9vnu/Ax/9rwzwz6fbeWzD3jvz7vP8y69yuIBKvmD1f/j0gc+\nPlQ946ennHPNXOZwElfTRFtuCvZRqUzxyWRBimzaPZLcBgONNguNPf63P3TNeKZmxdPSbKelsYO2\nVgftrQ7aWp00XWjHXu3E3u6io6Mzrhxub/lh98aVwaBgMhnAYGCZ041HUdCh88ukgq7Q/Xzfq4c4\nEW/+LLYUpXf5MUCZ0R1XinJR2aD4FBa9Ys0nnc+PITWYxQd4LbC+Yqexyd6r/LHWt+ExG7gwp5Cc\nkhwobw1sRkT/FAW7yYjdZKS+x8r4P7xiDP/55gmsLjcWtwer24PF7cHs8WBzuKg/1UBRYxsmj47J\n48Go65g8OgZdx6DDlt/Uss3yWTlkNHrLIoPR+92r+7uWT1nkjQ9QuuJmKPHS+X4uio0e8eL9Mfxy\nqOfhoZY7ykBzYVRVvQm4StO0r3Y+Xwcs1DTtPp80rwI/1zRte+fzzcC/AGMGO9fXli1bZFKOiDlX\nXHHFsNuAJHZELApE7IDEj4g9EjtCXJqhxM5gPVgVwGif56Px9kQNlGZUZxqzH+d2C1TACxFrJHaE\nuHQSP0JcGokdIfo3WAVrNzBBVdUxQCVwK3BbjzQbgG8Az6uqugi4oGnaOVVV6/w4VwghhBBCCCGi\nxoAzdDVNcwFfAV4G9gBPaZp2WFXVe1RVvaczzRvAR6qqHsS7JPuXBzo3aO9ECCGEEEIIIUJswDlY\nQgghhBBCCCH8J2vMCiGEEEIIIUSASAVLCCGEEEIIIQJEKlhCCCGEEEIIESBSwRJCCCGEEEKIAJEK\nlhBCCCGEEEIEiFSwhBBCCCGEECJApIIlhBBCCCGEEAEiFSwhhBBCCCGECBCpYAkhhBBCCCFEgEgF\nSwghhBBCCCECRCpYQgghhBBCCBEgUsESQogAUhTlTkVRnKHOhxBCCCFCQypYUU5RlGcURdkU6nwI\nEakkhoQYGokZIfzTGSseRVFe7OPY2s5j0mAXgaSCFf30zocQ4tJIDAkxNBIzQvhHB8qBaxVFye5x\n7B7gNMOIJUVRzMPImxgGqWBFP6XzgaIocxRF2agoyjlFUZoVRflQUZQ1FyVWlFOKovxAUZSfKopy\nUlGUakVR/ktRFGNIci9E6Cndv/TRMq8oyjpFUTwjny0hwtaQYkZRlB8rinJUUZSViqLsVBSlUVGU\ntxVFGT+SmRYiRI4CO4E7u15QFKUQWAU8zWff4dIURXlOUZTTiqK0KYpSpijKt30v1BVvnTG2F2hV\nFMU6Yu9EdJMKVmzoav1IAv4ErACuAPYCGxRFmdAj/T8CKcAtwM+BbwFfGpGcChH+pGVeiKHxJ2Zy\ngR/jLXNUIAt4Koh5EiIcdDVGPA7c7fP63cBmvD1YXSzAQWAtMKfznJ8rinJnj2vOA67H+71tNiBD\nDEPAFOoMiBGhAOi6vvWiF72tG9firUj9zOdQla7r3+j8fXdn8K5CCjshwKd1XgjhF39iJgH4tq7r\newAURZkM/FJRFIuu646g5k6I0NKBF4D/VhSlFNgGfBm4D0jtTqTr54D/63NemaIoC4EvAs/4vJ4C\n/KOu62eDnG8xAOnBiiGKomQpivKYoiiHFUVpABrwthoW+iTTgdd6nLoXyBmhbAoR7qQHS4ih8Sdm\naroqV5324a2Y9ZyXIkTU0XXdDvwe+Brehm8T8KpvGkVRDIqifE9RlP2KopxXFKUZb29WYY/LHZLK\nVehJD1Zs6CrcngGmA/cCJ4EOYAPebmdfzT2ee5DKuBDgjYWe/24mhSIjQkQIf2OmZ7nj7vwpZY+I\ndr7DBPcCo4GndF13K8pFnb/fAX4C/B9gF96Y+QGwssf1KoOaW+EX+YcrtiwHfqPr+mu6rn+Cd1zu\npBDnSYhIUgPMVC4u9T6H9GoJ0R+JGSH8oOv6YbwVp8uAJ/pIshx4S9f1J3RdP6Dr+onO13rGkgxj\nDwNSwYotZcAtiqIsVRRlFvBroJGLg1ECU4j+bQLSgJ90rsr5M2AWEjdC9EdiRgj/rQEydV0/2cex\nMmCxoijXK4pSoijKT/HO0ZJYCkNSwYp+BsDV+fs/AAeAl/EuWPFHYA8Xt3701aooe5qIWNYdQ7qu\nb8E7JONLwBtARufznvEh8SJi2VBjpr8yRuJIRLuLPvu6rrfrun6hjzQA/wH8BngEeL3ztV/hXyyJ\nEabo+sB/B1VVl+P9A5qA9ZqmPdrj+Frg3/H+QSuAH2uatqvz2CmgCe9YaqemaQsC/QbEwBRFeRM4\no+v6XaHOixCRSGJIiKGRmBFCxLoBF7lQVdWIt6djFd7K0y5VVTdrmnbYJ9lmTdNe6UxfCvwS75hQ\n8Fa6VmiaVh/wnIsBde4IfjPev8Wdoc2NEJFHYkiIoZGYEUIIr8FWEVwAHNM07RSAqqrP410SsruC\npWlaq0/6FLwr0/mSsaGh8WdgPN4KrxbivAgRiSSGhBgaiRkhhGDwClYBcMbn+VlgYc9EqqregHdM\naCLeHaS76MBbqqp6gMc0TVs/vOwKf+m63nPZTiHEEEgMCTE0EjNCCOE1WAXLr4lymqa9DLysquqt\neBdQmN15aImmaVWqqk4G3lBVtUzTtPf6usaWLVtkUp6IOVdcccWwe3gldkQsCkTsgMSPiD0SO0Jc\nmqHEzmAVrAq8G551GY23F6tPmqb9WVXVX6uqmqJpWqOmaVWdrx9WVfVlvEMO+6xgAcyZM8fffAsR\n8fbu3Ruwa0nsiFgSyNgBiR8ROyR2hLg0Q42dwSpYu4EJqqqOwbsz9K3Abb4JVFUdB5zQNE1XVfUa\noF3TtEZVVeMBo6ZpzaqqZgHXAN8cUu6EEEIIIYQQIoIMuA+Wpmku4Ct4h/3tAZ7q7I26R1XVezqT\n3QQcVFV1H97Vg9Z2vp4LvKeq6n7geeARTdPeDMabEEIIIYQQQohwMFgPFpqmbeWzOVVdr/3W5/df\nAL/o47wTeHdrF0IIIYQQQoiYMGAPlhBCCCGEEEII/0kFSwghhBBCCCECRCpYQgghhBBCCBEgUsES\nQgghhBBCiACRCpYQQgghhBBCBIhUsIQQQgghhBAiQKSCJYQQQgghhBABIhUsIYQQQgghhAgQqWAJ\nIYQQQgghRIBIBUsIIYQQQgghAkQqWEIIIYQQQggRIFLBEkIIIYQQQogAkQqWEEIIIYQQQgSIabAE\nqqouB37VmXa9pmmP9ji+Fvh3QAcqgB9rmrbLn3OFEEIIIYQQIpoM2IOlqqoReAq4EZgL3KWq6uQe\nyTZrmjZT07RZwC+AXw7hXCGEEEIIIYSIGoMNEVwAHNM07ZSmaU7geWCtbwJN01p9nqYAHf6eK4QQ\nQgghhBDRZLAhggXAGZ/nZ4GFPROpqnoD8AiQiLe3yu9zhRBCCCGEECJaDNaDpftzEU3TXtY0bQzw\ndeCvw82UEEIIIYQQQkSiwSpYFcBon+ej8fZE9UnTtD8Do1RVTelM5/e5QgghhBBCCBHpBhsiuBuY\noKrqGKASuBW4zTeBqqrjgBOapumqql4DtGua1qiq6qDnCiGEEEIIIUQ0GbAHS9M0F/AV4GVgD/CU\npmmHVVW9R1XVezqT3QQcVFV1H3AznQtZ9HducN6GEEIIIYQQQoTeoPtgaZq2FZjd47Xf+vz+C7zL\ns/t1rhBCCCGEEEJEq8HmYAkhhBBCCCGE8JNUsIQQQgghhBAiQKSCJYQQQgghhBABIhUsIYQQQggh\nhAgQqWAJIYQQQgghRIBIBUsIIYQQQgghAkQqWEIIIUQI6bqO7tFDnQ0hhBABMug+WLFm++ajnPj0\nPOjeQs9oMnDtrTNJTY8PddaEEEJEGY9H528vHORCfRs3f3keFqsUy0IIEemkB8tHa4udve+f5vLr\nJrP6+qmsuXEao8aks3Xjp6HOmhBCiCjj8ej87cWDtDTbychO5KVn9+B0uEOdLSGEEMMkFSwfh/ZV\nMn5KNgVFaeSOSiGnIIXLVo3nXEUjZ07Uhzp7QgghooTu0fn7Sx/TfKGDG+6Yw+rrp5KUYmPDH/fh\ndnlCnT0hhBDDIBWsTrquc3DXWabPG33R62azkeVXlfD264fxyBh5IYQQw6R7dN786yc01rdxw5fm\nYLYYMRgUrr5pOiaTkdf+fACPWypZQggRqaSC1aniVAMoUFCU2utYyfRczBYjn+ytCEHOhBBCRJMT\nR85TdfYCN35pLhbLZ3OuDEYD135hJg67i13vnQxhDoUQQgyHVLA6fbT7LNPnjUJRlF7HFEVh5bWT\n2bbpKPYOVwhyJ4QQIlqcOlrL5Jn5fS5oYTIZWHH1JPZ9UI5berGEECIiDbpckaqqy4FfdaZdr2na\noz2O3w78c+fTT4CfaZr2ceexU0AT4AacmqYtCFzWA6ej3cnxwzWsuHpSv2lyR6UwZkIGO7ceZ/ma\nkhHMnRBCiGhSfryeq2+e3u/xrLwkUtPjOXaohpLpuSOYMyGEEIEwYA+WqqpG4CngRmAucJeqqpN7\nJDsBLNc0bSbwd+AJn2M6sELTtNnhWrkCOHygiqLxmcQnWgZMt+zKiRzYeYaOducI5UwIIUQ0aWnq\noLXZTnZ+8oDpZi0qZP8H5SOUKyGEEIE02BDBBcAxTdNOaZrmBJ4H1vom0DRth6ZpjZ1PXwdGkG1Y\nOgAAIABJREFU9bhG7zF3Yebg7rPMmN8z270lJtsoGp/BkY+rRyBXQgghok358XpGF6djMAxcNE6Y\nmkNDXSvnq5tHKGdCCCECZbAKVgFwxuf52c7X+vM14BWf5zrwlqqq+1RV/eqlZTG4qisa6Wh3UjQu\nw6/0k2flc2h/ZZBzJYQQIhqdPl5H4fjByxuj0cCM+aOlF0sIISLQYBUsv9clV1V1JbAO+Fefl5d0\nDh38IvB9VVWXDT2LwVV2oIqps/NRBmlN7FI8MYu6cy00XWgPcs6EEEJEE13XKT9eR9G4dL/Sz5g/\nirKPqmRYuhBCRJjBKlgVgO/GUKPx9mJdRFXVGcDjwOc1TbvQ9bqmaVWdPw8DL+MdchhWaqqayS/s\nvTR7f0wmAxOm5nD4QFUQcyWEECLaXKhrQ9d10jIT/EqfmGyjeGKmbBEihBARZrAK1m5ggqqqY1RV\ntQC3Aht8E6iqWgi8CKzTNO2Yz+vxqqomdf6eBVwDHAxk5gOhtrqZzJykIZ0zZVY+h/ZVouuy8bAQ\nQgj/nD5WR9H4jD63A+nP7MVF7P+gHF02uhdCiIgxYAVL0zQX8BW8vU97gKc0TTusquo9qqre05ns\nh0A68JvOuVYfdr6eC7ynqup+vItjPKJp2ptBeReXqLXFjsejk5hsHdJ5BUVpOBwumXzsY+bMmTQ0\nNAw7TaD88Ic/ZOHChaxYsYIHHniApqamPtOtX7+e2bNns3jxYp599tnu13/+858zdepUSktLKS0t\nZfPmzSOSbxF7wi12uvz6178mIyOj3/u+//77rFixgqVLl/L44493v+5v7MWi08frKPRzvm+X/MJU\nTBYj5Sfqg5SryBVusTPccqesrIwvfOELLF++nNtuu41PP/10RPItYk+4xU6Xwcqd1tZW7r33XkpL\nS1m0aBG7d+8GwrPcGXQfLE3TtgKze7z2W5/f7wbu7uO8E8CsAOQxaGqrW8jMSRxSayKAYlCYMjOf\nw/uryM4beKndWOHP/0NFUUas12/lypX8+Mc/BuCf/umfeOSRR/jRj350UZqmpibWr1/P22+/jclk\n4vrrr2fZsmUUFxejKAr33nsvX//610ckvyJ2hVvsAJw9e5Z33nmH0aNH93nc7XZz33338fLLL5OX\nl8cVV1xBaWkpJSUlfsVeLPJ4dM6cqOeKz/Xc6WRgiqIweWYeRz6upsiPxTFiSbjFznDLnYceeohb\nb72VG264gRdffJGHHnqIJ554oo87CTE84RY7MHi5A3D//fdz2WWX8dhjj+FyuWhrawP8i72RNtgQ\nwahWe66ZzNyhDQ/sMnlWHocPVOKJsWEbd9xxBytXrmTt2rX89a9/7XW8vLycRYsWcd999zFr1izu\nvPNOOjo6uo8/++yzLF26lC984QucPn0agD179rBmzRpKS0u59957OXny5LDzuXLlSgwGAwaDgcsv\nv5zKyt4rP+7cuZMZM2aQmppKYmIiS5cu5dVXX+0+LkNARSBFSuwA/OAHP+DBBx/s9/iePXsoLi6m\nsLAQs9nMjTfeyMaNGwH/Yi8W1VQ1EZ9oITHZNuRzJ07N5eihczFX3nSJlNgZbrmTnJxMQ0MDHo+H\n+vp6UlP9nx8uRF8iJXZg8HKnqamJHTt2sG7dOgBMJhPJyd5OjnAsd2K6gnW+upmsnMRLOjczJ4n4\nBAtnT8bWsI1HH32Ut99+m9///vf88Ic/5MKFC73SHD16lDVr1rBz5048Hg9vvvnZyND29na2bdvG\n/Pnz+fOf/wzAxIkTeeONN9i6dSurV6/uboXw1dzc3D1cz/exYsUKjhw5MmCen332Wa6++uper192\n2WXs3buX06dPU11dzaZNmy4KyvXr13f/w9PY2NjrfCGGIlJi54033iA/P5+pU6f2+16qqqooKPhs\nx478/Hyqqnov/NNf7EWj6mY7Byqb+33sP1hNQm5yv8erm+39Xjs1I56EJCsVp0d2uE64iJTY8XUp\n5c6DDz7Ib3/7W4qLi3niiSdC3gIvIl+kxI4/5c7p06fJyMjg61//Opdddhnf+ta3aG/vvaJ3uJQ7\ngw4RjGa151qYPu+zDYarm+2ca3YMel5OkoXcJGv3nlhDHVMfyV566SU2bNjA+fPnaWpq4vjx48yd\nO/eiNMnJyVx33XUA3HTTTWzZsoXPf/7zAHzhC18AYNmyZTz00EMAdHR08NOf/pTt27ej6zpOZ+8l\niZOSkti6deuQ8/vLX/6SxMRErr/++l7HEhIS+NnPfsb9999PU1MTixcvxmDwtjl85Stf4Z//+Z9p\nbm7mhz/8IT/4wQ949NFHh3x/IbpEQuy0tbXxX//1X7z00kvdr/XVk+vP8JKBYi8anWt2cP8bx/o9\nfoujg3ec8GQ/aR66Zjy5Sf3PB544NYcjH1czuti/Jd6jSSTEjq+hljtGoxGA++67j69+9avceeed\nPPHEE3zzm9/k6aefHvL9hegSCbHjb7njcrnYt28f3/3ud3n44Yf59re/zSuvvNKdRwivcidmK1i6\nR6eupuWiFQQHKyC7dBWEk2bk8bv/2Y7neg8GY/R3Bp46dYqnnnqKV199lbS0NEpLS7Hb+2917eL7\nZaxryIPZbO4+98knnyQtLY233nqLsrIy7rjjjl7XaG5u5tprr+3zi93jjz9OSUlJr9f/+Mc/smnT\npj67xbtcddVVXHXVVQD8x3/8B6NGeSvcWVlZgPcfnrvvvpt77rmn32sIMZhIiZ1Tp05x5swZli9f\nDkBlZSUrV65k8+bN3TEBkJeXR0XFZ0uHV1RUkJ+f3/3cn9iLJYqu03yumYb8S2+MmzA1lxee3sXl\n1072e9/GaBApsdNlOOXOzp07+c1vfoPJZGLdunX86le/GvR9CtGfSIkdf8ud/Px80tPTu2Pnxhtv\n5M9//nN3BSvcyp2YrWBdaGjDFm/Garv0/wVJKTZS0uOoOH2B0WOjv1WxurqazMxM0tLS+OCDD/j4\n44/7TNfU1MTrr7/OqlWreOmll7j55psHvG5VVRWLFy8GuGhFJV9JSUm8++67fud18+bNPProo7z+\n+uvYbP3PeTh//jxZWVmcPXuW1157jU2bNgHe95qbm4vL5eKFF15gypQpft9biJ4iJXamTJly0cpl\ns2bN4u233yYtLe2idLNnz+bEiROUl5eTm5vLyy+/zPr16wH/Yy+WJDhdWBOsuIbREJeZk4jFaqLq\nbOOQ9m6MdJESOzD8cmfp0qVs3Lixe07jihUr/L63ED1FSuz4W+7k5ORQXFzM7t27mTNnDps2baK0\ntBQIz3In+rtd+lF7roWsIe5/1Zdxk7I5VlYTgByFv0WLFjFq1CgWLlzIb37zm+4Pdk8TJkxg48aN\nLFy4EEVRuPLKK4GLW0UURel+/rWvfY1nnnmGyy+/nIKCgiGv6tiX733ve7S2tnLDDTdQWlrKd7/7\nXcD7D8Ott97ane7OO+9k6dKl3HXXXTz22GPdEyYffPBBli5dypVXXonT6eSnP/3psPMkYlckxY4v\n3+v5xo7JZOLRRx/tnkB9++23d7dI9hd7sSze6caWMvxCf+LUHI58Uh2AHEWOSIqd4ZY73/nOd3jj\njTdYtmwZmzZt4tvf/vaw8yRiVyTFjq/+yh2Axx57jAceeID58+dTVVXFjTfeCIRnuaOEy0ppW7Zs\n0efMmTNi99vx1jGcTjfL13zWTXmgstnvIYIz872Vs5rKJjb8cT93fWdZwD9kkai8vJzbbruN7du3\nhzorYW/v3r1cccUVw/7QjHTsdOlvzmLXHMUuukfnxJHzfPpRNdPmFsTUnMWhkNjxX6BiBwIXPwOV\nH2MutFJakMTvWj39nu9brvSnprKJV/6wj7u/u1zKGx8SO/4Lx9gRoSOx47+hxk7MDhE8X93ChCnZ\nw75OVl4SbreH+vOtZGRf2oqE0UYK/tjQ35zFrjmKDruLj/dUsG/HaSxWExOn57LxhYOMLcmi9OoS\nLNaY/eenXxI70am7B6u1bVjXycpLAgVqqprJyZc9GH1J7IhLMdjiZj0bDKORxE5wxOw3nNpzzSy+\nfNywr6MoineY4OEaqWABhYWFbNu2LdTZECHm8eg8//hOUtLjuerm6eQXpqIoCjMXjOadN8p45n+2\ns+aGabJxqg+JnegV73QRl2KDyuFVsBRF8e6J9XG1VLB8SOyISzXY4maDre4Z6SR2gicm52C5nG6a\nGtpJz0wIyPXGTc7i+OHYmIclhD8+3nMWi9XE5784i4KitO4WMlucmatums7qtVN4/c8HqDrTe08O\nIaJNoOZgAUyclsORT84F5FpCCCGCIyYrWHXnW0nNiMdoCszbHz02g7qaFlpbBl/+MpKVl5ezZMmS\nIZ3zpz/9ierq4E7K/sY3vkFJScmAedu2bRtFRUXdm909/PDDgHc/h1WrVrF8+XJWr17NY489FtS8\nxgKXw8X2zcdYce2kfoceFE/M4orPT2HjCwdxOt0jnMORF66x8/7777NixQqWLl3K448/3mea5uZm\n1q1bx5w5c7jyyis5efIk4N2c0ncDyaKiIn77298GNb+RyOjRMXs8WBIsAblebkEKDruLupqWgFwv\n3IVr7PhT7nQtWLF8+XJuvfVW9u7d233Mn9gTYjgiOXYA/v3f/52lS5eyevXqizYn/t3vfseaNWtY\nuXIlDzzwQFDzOhwxWcGqrW6+aP+r4TKZDBSNz+DEp+cDds1oMRLBevvtt/OXv/xl0HRLlixh69at\nbN26tXuFGZvNxoYNG3j33Xd57bXXeO655zhx4kRQ8xuuqpvtHKhs7vWobh5aw8GpPRUUjc8gtyBl\nwHQl03PJyk1i26ajw8l21Ap27Ljdbu677z6effZZ3n77bZ577rmLlsrt8uKLLzJq1Cj27t3LT37y\nE77xjW8A3pWnuuLp7bffJj4+nmuvvTZo+Y1UcU4X7SZjwOY5KAbvsPTjMbJ67aUIl3KntLSU9957\nj3fffZdvfvOb/Nu//Rvgf+wJMdLCJXY2bdrExx9/zLZt2/jP//xPvv71rwPQ0NDAI488wksvvcSW\nLVs4fvw4W7ZsCWp+L1VMVrDOn2smMzew86XGTc6OiWGCHo+H++67j1mzZnHnnXfS0dEBwP79+1m9\nejULFy7kxhtvpK6ujldeeYX9+/dzzz33sGLFCjo6OvjFL37BqlWruOKKK7p7kYZr8eLF3ZvhDaS/\nFTPj4+MBaG1txe12Y7EEpqU50nSNRe/5GGgCcE82p5uzH1ez7MqJfqVftXYKn35UxZmT9Zea7YgR\nbrGzZ88eiouLKSwsxGw2d++909N7773H6tWrAViwYAHHjx+ntrb2ojTvvPMOY8aM6d4wVXwm3umm\nzRzY6c7jJmdzoix2GvTCLXbAv3InIeGzaQhNTU1Yrd65PP7GnhDDFamxs3Hjxu4NhOfNm0dTUxM1\nNTXYbDZ0XaepqYn29nba29v9+v4XCoNWsFRVXa6q6l5VVT9SVfW+Po7frqrqgc7HH1VVnebvuaFS\nWx2YPbB8jS3Jovx4Pa4oH+505MgR1qxZw86dO/F4PLz55ps4nU6+8Y1v8Mwzz7Bz506uvfZannvu\nOdauXcusWbN4/PHHeeedd7DZbHzta19j8+bNvPnmm+zdu5e///3vve7xwgsvXDT0qOvx5S9/+ZLz\nrSgKH374IYsXL0ZVVcrKyrqPeTweli1bRklJCXfffbd8SRyGCfUtjJ6RR5Kf803i4i2sXjuVv71w\nEIfdFeTchVa4xU5VVRUFBQXdz/Pz86mqquqV7vLLL+ell16ivb2djRs3cv78eSorKy9K48/mlLEq\n3uWmzWwM6DULx6ZTU9VEe5v/jR+RLNxiZyhee+01Zs6cyX333cd///d/A/7HnhDDFamx01+MxMXF\n8fDDDzNr1iwmT57MwoULmTt37iXfJ5gGbFZTVdUIPAWsAiqAXaqqbtY07bBPshPAck3TGlVV/RLw\nBLDIz3NDojYIPVhx8Ray85IoP1HP2JKsgF47nCQnJ3PdddcBcNNNN7FlyxYmTJjAmTNn+OIXvwh4\nhz8UFhZ2n+Pbc7Rv3z6efvppjh49SlNTE2VlZaxZs+aie9x8880B/7I2Y8YMDh48iNls5k9/+hO3\n3347e/bsAcBgMPDee+9RXl6OqqosXLiQGTNmBPT+sSClw0Fah4MxcwoGT+xj3ORsjnxyjq1/+5TV\na6cGKXehF26x4++QtRtuuIHKyko+97nPMX78eGbPno3R+FmFweFw8Pe//50f//jHfl0v1sQ7XTRa\nzQG9pslsZPTYDE4eqWXKrPyAXjschVvsDMV1113Hddddx0svvcS6devYunVrwO8hhq69zcGZE/Uc\nLTtPkt1Jc4BjNFxEcuz0NeqotraW7373u+zYsYPU1FS+/OUv8+abb3ZvjhxOBhu3sAA4pmnaKQBV\nVZ8H1gLdlSRN03b4pH8d+Km/54ZCe5sDh91FcmpcwK89bnI2xw6di+oKVl90XSctLa3fgqPri5yu\n63zve9/j6aefZsqUKXz/+9/v7q729Ze//IVf//rXvV4vLi7mmWeeuaQ8JiV91mN5xx138OCDD9LQ\n0EBaWlr364WFhaxevZrt27dLBesSFDW2cTI1AeMltNavvHYST/7yXeYuGROw1T0jQShjJy8vj4qK\niu7nFRUV5Of3/rIeHx/P/fffz/333w/ArFmzKCoq6j6+efNmZs6cSWZm5uBvOAYlON1UJQahvJnk\nXb02FipYfYmEcsfXjTfeyL/8y7/Q1NREfn6+X7EnBjfUfaz27TjNx3sqaKhrpaAoDRKtzKm+wPl4\nK8fSEnCYAtvbHI4iIXZ6lk+VlZXk5eWxd+9e5s2bx9ixYwFYu3Yt77//fkRWsAqAMz7PzwILB0j/\nNeCVSzx3RNSeayEzJykoG6uNm5SF9uQpdF2P2o3bmpqaeP3111m1alX3sKAJEyYAsGHDBj73uc/h\ncrk4fvw4kyZNIjExkfPnvXMF7HY7LS0tFBYWUllZycaNG7ntttt63eOWW27hlltuCWi+a2pqyMrK\nQlEU/va3v2Gz2UhLS6Ourg6TyURKSgr19fVs3ryZn//85wG9dyxQdJ2MdgdlGZc29NYWZ2bOZWP4\n4O3jXHNLdFZuwy12Zs+ezYkTJygvLyc3N5eXX36Z9evX95lvm82GxWLhd7/7HUuWLCEx8bMRAC++\n+CI33XTTpfwviQneOViB/9I2tiSLrRs/xe32YDRG93TqcIsdf508eZIxY8agKAqbNm3CZrORnJzs\nd+yJwQ1lH6uyA1Xs2naKa26eTt7oVIwmAwcqm3mqzsHYC61cdraOk6kJnE6Jhyj5DhepsXP11Vez\nfv16brrpJnbt2kVycjLZ2dksWrSIBx54gIaGBuLj49m8eTP33HNPQO8dKIP9q9z3qgB9UFV1JbAO\n+NehnjuS6mpagrYhcFpmAiazkfNVzUG5fqgpisKECRPYuHEjCxcuRFEUrrzySsxmM8899xyPPfYY\nCxYsoLS0lF27dgHe3qJf/OIXrFixAoBvfetbrFq1irvuuqt74vxw3X333Vx11VUcP36cadOm8Yc/\n/AGAZ555prv1ZMOGDSxZsoTly5ezYcOG7jTnzp1j7dq1LFu2jK9+9avce++9lJaWBiRfsSS93UGL\n2TSs1r85lxVy8tPz1Ne2BjBn4SEcY8dkMvHoo49yxx13sHLlSm6//XZKSkqAi2OnrKyMpUuXUlpa\nygcffMBDDz3UfY3W1la2bt3aPQRFXMzo8WDyeLAHoQKUmGwjNSOeilMNAb92OAnH2AH/yp1XX32V\nJUuWUFpayiuvvNKdZqDYE8FRU9XEllcPcf262YwqTr9omx6X0cCRjCQ+zE8nv6WDosbhbQgeLiI5\ndlavXs3UqVNZsmQJ3//+97t7yJKTk/nOd77DHXfcwTXXXMPUqVNZtmxZQPIVaEp/K6sBqKq6CPix\npmlXdT5/APBomvZ/e6SbAbwIXK1p2rGhnNtly5Yt+pw5cwLwli7Ws/v46I7TGE0Gxs4f3Sutw+3h\nX/8++BLdD10znpn5fbfUv/36YWxxFhZfPu7SMy1iwt69e7niiiuG3UwWqNg5UNncZ0tgf5933/ST\nzzfRbjZyKjVhwPgYzI63jtNQ1xq1vVgiMAIVOxDc+EmyO5lW08SO0Rn8aFUxD24+2e/5lxI37285\nhr3DycprJ19SnkXsCcfYGY7+yq0uD10znompVp773x0su3Iik2bmDXi+zelmQWU9h7KSqY23Dqs8\nE9FlqLEz2BDB3cAEVVXHAJXArcBF/YOqqhbirVyt66pc+XvuSOjZfTytppG6OAtV53vv7fOjVcXD\nvt/Ykiy2bToqFSwRO3SdrDY7u/PSBk87iDmXFfLEw+9SX9saU3OxRHQK1vDALuMmZ/Pqn/az4pr+\nN/UWIpZ5PDqvPX+ACdNyelWu+tJhNnIgJ4XZ1RfYlZ8+AjkU0WrAcQuaprmArwAvA3uApzRNO6yq\n6j2qqnYNevwhkA78RlXVfaqqfjjQuUF6H36zudx0BHES46gx6dSfb6W1ZWibswoRqZLtLlwGhTbL\n8Pf6sdo+m4slRKQLdgUrOy8Jl9NNQxQOqxUiEE7tOQvAcj/3ZgRotFk4kpHE7OoLONqdwcqaiHKD\nfiPSNG0rMLvHa7/1+f1u4G5/zw01m8tDhyl4E4KNJgOF47zL504b4nLVQkSi7LYOahKsgyf0k/Ri\niWgR73RxwRa85Z8VRWHcpGyOl50nPSs4c4uFiFQGj075R1X8w/+3CMMQ50FWJsWR6HDx8ZtHmPd/\nFkoPsRiy6F56qCddx+ZyYzcGdxnOsZOyOFFWE9R7CBEuslvtnI/3b2Nhf3T1Yu18R3qxRGTzbjI8\n/J7dgYztXK5dCHGxvJYOUrITSbvEhrqj6Yk42p0cPiCbQIuhi6kKlsXtwWk04DEEtyVi7MQsTh+r\nw+3yBPU+QoRavNOFyaPTaA3sl8hZi0Zz7FANLU2999wQIlIEe4ggQOG4DGqqmmhv638vICFijq5T\n2NRGoR/zrvq9hKIwqXQsWzd+ir1DhgqKoYmpCpbN5aFjBPYLSUiykpaZQMXp6F4+V4isVjvn460B\n3zMkLt7CpJl57N95ZvDEQoShYC7R7stsNjJ6rHdYuhDCK63DiaLrpI9OHdZ1UvOSGVuSxfZN/a9U\nKERfYqyCFdwFLnyNLcniuAwTFFEuu83O+QDOv/I1Z3ERH314BpfTHZTrCxFM3t4r04hsWDp+crYM\nExTCR2FjG+Up8QGZO7VszUQOf1RFTWVTAHImYkUMVrBG5i2Pm5TFiU/Pj8i9hAgFR5uDRIeLujhL\nUK6fkZ1ITkGyjH8XESne6aZtBBv0Th2tlWHpQgBxTjdpHQ6qEuMCcr34BAvLrpzA5g2H0D397x0r\nhK/gzr4NMza3Z8R6sLLzk3HY3bISmoh4FqPCgcrmXq9Xn6ynLs6C3qOFsL/0AMk2E00drl6v5yRZ\nyE3q3RM2d8kY3nmjjGlzC2QVJxFR4p2uoM+/6pKQZCU9K4EzJ+sZMyFzRO4pRLga3dRGZVIc7gDO\nt58+dxQHd5/l470VTJ83KmDXFdErtipYLjeN1uAtmetLURTGlmRxouw86UulgiUiV327iwc3n+z1\n+jqLToOtd+9Vf+nBu5l3X8ceumZ8nxWsovEZ6DqUH6+naHzGJeReiNCId7q50Ed8BMv4ydkcL6uR\nCpaIaUaPh/zmdj4oCGx5oRgULv/cFF55bi+TZuRhtoxM44mIXDE1RDBuBIcIAoyTcfEiijWfaw7q\nHj/gbaiYu6SIPe+fCup9hAi0kVhB0NfYSd7yRtdlCJOIXbktHTTYLHQEIfbyRqWQNzqVfR+UB/za\nIvrEVAXL6hq5IYLgbX0/VynL54roY/R4sDfbabEEvxN88qx8qs40Ul/bGvR7CREo3j2wRq68ycxJ\nRFEUaqtbRuyeQoSb7FY71YmB25expyWrJrDrvZOybLsYVMxUsBRdx+IO/pK5vsxmI4Xj0jlRJotd\niOiS0uEkITOh1/yrYDCbjcyYP4q9758O+r2ECASDR8c8wuWNoiiMm5zNMRk1IWKU0aOT1uGkLj54\nQ3MzcxIZW5LJ7m2ngnYPER1ipoJldXmwmwwjsmSur/FS4IkolGJ3kpSdOGL3m7lgNGUHqnDYey+Q\nIUS4sbrdIStvZHsQEavS2+002sy4DJ99te1adKm/h8M99JU3F18+nn07ymlrkdFJon8xs8hFnMtN\nh3HkJyWOnZTNW6+V4XS6MY/gcBEhgim1w0liThK0NI7I/ZJT4xg9Np1P9lUye1HhiNxTiEtldXlC\nUt4UjEnjQl0bLU0dJCYHb5iUEOEou2vjex8DLboE3oWXhio1PZ5JM/PY+e4JVl4zacjni9gQQz1Y\nI7vARZf4BAvZeUmUH68b8XsLERS67u3Byhm5HiyA2YsK2bfjtEziF2HP5u4cMTHCjEYDYyZkyB6M\nIvboOpntDs4HcXigr0UrxvLJngqaGztG5H4i8gzag6Wq6nLgV51p12ua9miP45OAp4HZwL9qmvZL\nn2OngCbADTg1TVsQuKwPzUjugdXT+CnZHDtUw7hJ2SG5vxCBFO904zYoWEaoIOsyemw6iqJw5kQ9\nheNkyXYRvqwuN/YQ9GCBd/Xaw/urmDF/dEjuL0QopNidOA0K7eaRGZiVmGxj+vxRfPDOcVavnToi\n9xSRZcAmNlVVjcBTwI3AXOAuVVUn90hWB9wHPNzHJXRghaZps0NZuYKuJdpDVMGanMPxshrZAVxE\nhRS7kwsjtJ+cL0VROnuxZIlcEd5sLk9IRkwAjC3J4uypeux9bOgtRLTKarNTk9B7L8Vgmr+0mE8/\nqpZeLNGnwUqABcAxTdNOaZrmBJ4H1vom0DTtvKZpu4H+1qwc2Vm+/bCFaIggQGpGPHHxFqrOXgjJ\n/YUIpNQOJ41B3v+qP1Nm53PmZD1NF9pDcn8h/OFd5CLwDXrVzfYBJ+xXN9ux2swUjEl+GXUwAAAg\nAElEQVTnxKey2IWIHVmtjl7zr4ItPtHCtHkFfLj1xIjeV0SGwfpSC4AzPs/PAguHcH0deEtVVQ/w\nmKZp64eYv4AZ6T2weho/JZujh2rIL0wLWR6ECIQUu5PKpLiQ3NtiNTFlVj4Hdp5h2ZqJIcmDEIPx\nLnIR+Aa9c80O7n/jWL/HH7pmPLlJViZOy+HIx+eYPDM/4HkQItzEOd1YPB4aQzCyYv7SYp7+1TYW\nrhgrC8uIiwxWAgx3TNsSTdNmAl8Evq+q6rJhXu+ShXKIIMD4KTkcPyQtiiKyGT0eEpwumqyhW4B0\n1qLRHNx9FpfTHbI8CDEQW5B6sPw1fnI2p4/V4XDIMEER/bLa7NTGWUZ8WwSAhCQrU+fk8+G7/a9U\nKGLTYBWsCsB3puxovL1YftE0rarz52HgZbxDDkec0aNj0HWchtCNVszNT8bhcFF/viVkeRBiuJLt\nLpot5hHZYLg/6VmJZOcnUXawOmR5EKJfut45YiJ0i/TGxVvIL0zh5Ke1IcuDECMlFPOvfM1fVsyh\nfZW0NMlcLPGZwUqA3cAEVVXHqKpqAW4FNvST9qJvXKqqxquqmtT5exZwDXBwmPm9JNau3qsQfilU\nDArjJ+dw5JNzIcuDEMOV2uHkQojmX/mavbhIlmwXYcni9uAyKCFthACYOC2XIx9LI4SIbk67i5QO\nJ/VxoatgJSbbmDIrn13bToUsDyL8DFjB0jTNBXwFb+/THuApTdMOq6p6j6qq9wCoqpqrquoZ4J+A\nH6iqWq6qaiKQC7ynqup+vItjPKJp2pvBfDP9sYV4eGCXkum5fPqRFHgicqXYnSEZ597T2IlZ2Ntd\nVJbLwjEivFhDuCWIr/GTczh1tBanDKUVUaz+bCMXbGbcIRyhBDB/eTGf7Kmgtdke0nyI8DHoRApN\n07bi3ePK97Xf+vxezcXDCLu0ALOGm8FAiAvShOOhKhiTRnubg7qaFjKyR3aTViGGTddJ7XBwODMp\n1DlBMSjMXlzI3vdPU1AkC8eI8GFzuUOyyXBP8YkWsvOTOXW0lglTckKdHSGCoqGikfq4kd2TsS9J\nKTYmzchj97ZTlF5dEursiDAQ+lJgBIRLD5bBoDBxWi6fytwREYHiXG48ioI9DBorAKbNLeD0sTrZ\ng0SEFavbQ0eINhnuSYYJimjXUNFIgy30FSyABaXFHNx9lrYWR6izIsJAeHxTCjKrO3R7YPU0aUYu\nZR9VydwREXGS7S6arOaQzmX0ZbWZmTwzjwM7ZeNhET5sLk9Y9GABTJiSzYmy87hcnlBnRYiAa29z\n0N5kD+mqtr6SU+MomZ7Lnu2nQp0VEQbCoxQIsrgQ74HlK290Ki6nm9pqWU1QRJYkhzNsCrIusxcX\n8tEuWbJdhA+ryx02PViJyTaycpM4fUxWExTR5+ypBlLzkkK+oIyvBaVjOfDhGdrbpBcr1sVEBStc\nhggCKIpCyfQ8yg5WhTorQgxJkt1FsyW8KljpWYlkFyRT9pHEkwgPNnf49GABTJyWI8PSRVQ6c6Ke\ntPyUUGfjIilpcUyYmsOe7adDnRURYuFTCgSJruudFazweaslM7yrCcowQRFJkhwumkd4BcHqZjsH\nKpt7Pap9Vmqas7iIvTvKJZ5EWLCGUYMeQMn0PI4frpFNh0VE6q8MOFDZzNEjtSTmhX7RpZ4WrhjL\ngZ3ldLQ7Q50VEULh1RwdBM4OFx5FwW0InwpWTn4yKHCuooncUeHV+iJEXywuN0ZdH/HVOM81O7j/\njWO9Xn/omvHkJnn3PSmekMnbrx/m7MkGRo9NH9H8CdGTzeW5pIVgLEaFA5XNA6ZxuIc+lyohyUp+\nURrHPqlhyuz8IZ8vRCj1VwaY3B6W17VhzYgPQa4Glpoez7jJ2ezZfoolqyaEOjsiRKK+gmVvsYdV\nayJ4hwlOmp5L2cEqqWCJiJDk6BweGEZj3bsoBoV5S8awa9tJqWCJkDJ6PCjouC5hT576dhcPbj45\nYJofrSq+pHxNnZ3Pwd1npYIlokZah4MLNjOGMGo897VoxTj+8P/vYO6SMdjiQr93pBh54fnJDKCO\nVkdY7IHVU8mMPD49KMMERWRIcnSuIBimpswpoPpsI7XnZPEYETrWrgWVwqwhYtzkbM5VNMmWBiJq\npHU4abCFb5mUmhHP2EnZsqJgDAu/mkeA2Vsc2MOsBwsgKzcJi8VExekLoc6KEIMKxwUufJnNRmYv\nKmT3toF7AIQIJpvLHTb7xPkym41MmJrD4QOVoc6KEBcZaI7VgcrmfofFprc7aAiDDYYHsvjycez/\noFxWFIxR4fuNKUDsrY6wLPDAu1Hqwd1nGTUmLdRZEWJASQ4nJ1PDb6y7r1mLCnnyl+/RsrqDxGRb\nqLMjokx1s51zzf1/UXK4Pdjcod0SZKB5XNbCVA68e4L5y4pRwqyHTcSu/uZYdelrWKzJ7SHe6aYx\njEdVgHcu1sRpuXz47klKryoJdXbECIvYCtZghV2XthZ7WC2Z62vK7Hye+q/3sHdMxmqL2D+FiHJu\nl5s4l5vWMO7BAoiLtzB5Vh573z/NcinMRID580XQO0QwdOXNgPO4dJ21DjfnKpvILZC5vyJypXY4\nabSZw2r/q/4sWjmO3/3PduYtGUNC58JMIjaE9zemAQxW2HVRnY6w2fSxp4REK4XjMij7qIqZC0aH\nOjtC9Kmtvo1WsykiCrO5S8bw3P/uYOGKcdJoIUacze2mxRymnztFIa8km0N7K6WCJSJaeocjrOdf\n+UpKsTF1Tj473znB5Z+bHOrsiBEUnl07AeRoc4RtDxbA9HmjOLj7bKizIUS/2mrbwnr+la/U9HiK\nxmfw0a4zoc6KiEFWV3htMtxT3qQsDn9UhfsSlnsXIlykdThosI3M/KuuYbf9PXz3ZOzPguVjObS/\nkqYL7SOQYxEuBv3WpKrqcuBXnWnXa5r2aI/jk4CngdnAv2qa9kt/zx0JjlYH9oTAzh3xZ78SgGSb\niaaOgTd31BOsNDV2cL6qmaww3DBPiNa61hHfYHg45i8v5pXn9jFncRHGMP6yK6KP1eUO2xETAPEp\ncaRnxnPySC3jJ2eHOjtCDJnR4yHBMXLzrwbbPsF3T8b+JCRZmTF/1P9j777j46rOhI//zsxoZtRG\nzZJlSZYtW6644m6wZWxjihMcILlAAruEQEgChN28yb7Ju6m7pPeFJRASQksCl1ANOBhT3DDGHfde\nVVwkq2s07b5/zEgeyyojaaQper6fj5Dm3nNHj4yO7j3tOXz0/mGW3Dgh3CGKKNVpA0vTNDPwJLAY\nKAU2aZq2Stf1vUHFKoEHgM/04No+pQwDr8uLK8xJLkLZrwT8c/JDKXfv2Gx2bj4lw8eiz3W2drGj\nbE0NlY3UWWOngZWbn0ZWTgq7tpxi8qzCSIcjBhC7N7pHsAAmTCvgk49PSgNLxKR0p5tamwVfD/aa\ni6QZ84t48jdrmT6viMxByZEOR/SDru4EM4FDuq4f03XdDTwPLAsuoOv6WV3XNwPu7l7b12weHwmJ\nCVG3J0lb+eP86XM9bm+kQxFxrmXtYnsfLm87e7IZBo1VjdTZYmOKYIu5i4r56IMjeDwyFUr0D5/P\nR4LXF7VZa1uMnTSE8pPVVFc2RjoUIbot3emmup+mB4ZTYpKV6fOKWPvPA5EORfSTru4E+UDwYoZT\ngWOh6M21YWHzerEmR39FTHTYyclzcHDP6UiHIsRFktxeEhIT8Jii+6GxrbzCdAYNTmGXrG8U/cTd\n6PbPlojyDr0Eq5kJ0wrY9tHxSIciRLdlxFCCi7YunzuMirIaTh2tinQooh909dTUTpd2yHpzbVjY\nPD6sSbFREf3JLkojHYYQF0l1eUjOis3pDHMXFbNx9REZGRb9wtXgiugeWN0xZfZQdm8tw9Xc+Rph\nIaKJMgwczR5qYrSBlZBgZt6S0XywYj+GL+KPyKKPddXAKgWC84cPxT8SFYreXBsWdq+PhKToH8EC\nKB4/mHMVdVSeqY90KEK0SnW5SR4U3RsMd2TI0HSyc1P5REaxRD9wRfGm9m2lZSRRMDyDPdvLIh2K\nECFzNHtoSjDH3IyKYOMmDcEwDPbvrIh0KKKPdbWwYjMwStO04UAZcAtwWwdl286L6M61fcLm8WJL\ntoKz6w2JI81iMTF51lC2rD8mWWZE1HA0e0jKSoaq6j79Ph1l5uwo8UZnmTwHp1pbszrNXVTMq89t\nZdL0AiwJsTG6IGKTfwQrdh78ps4Zxntv7GXyzKGoKJ/WKARAegxPD2yhTIqS68bwz5d2UTw+R+5L\ncazTBpau6x5N0+4CXuFCqvW9mqbdGzj/uKZpucAmwAH4NE17EBiv63p9e9f25Q/Tls3rIyHZCpXR\n38ACmDp7GE/+di1XLB4lO36LqOCfIpgEB/u2gdVRZs4fLC7qVnm4OG1ubkEag/McfLLpJJfPHR62\neIVoq7nBRXOMTBEEKByZCcDJI1UUjsyKcDRCdC3d6eZ0Suw/GxWOyCI7N5VtH51gxrz273Ei9nWZ\nGkzX9dX497gKPvZ40NcVXDwVsNNr+5N/DVZsTBEESEqxMmZSLts2HOfKJaMjHY4Y4KweL8owYiJR\nTGfmLirm5We2MmFaAdYYy4YoYoerwYUzRqYIAiilmDq7kK0bjksDS0Q/wyDD6WLfoPjYL7Tk2tH8\n/fGNjJs8hBSHPdLhiD4QO3eDHrB7vViTY2s4efoVw9nx8UlZfCwizuHyUGdLiPnpQ4Pz0ygcmcnH\nq49EOhQRx1wxNoIFMH5qHqeOnqfmfFOkQxGiU0luL16Tirk61pHM7BQmzijgg7f2RzoU0UfiuoEV\nayNYABmDkikoymTXFskoKCIrtdlNrTU+RnzmLRnN9o0n5UFS9BlXfXNMrcECsNosTJxewKa17U+3\nFSJaZMTo/ledmXNVMWUnznP80LlIhyL6QGzdDbrB7POhMDBbY6+3Y8a8IjavP4avgwX+QvQHR7N/\nBCseONITmTqnkLVvS2+h6AOGgavRHTNp2oNNnzecfTvKqa2WzgcRveIhwUVbCVYziz49nlWv78Hj\n6dnzXkVdMzvK6jr8qKhrDnPUIlTx0T3dDpvHh9NsjsnpTXmF6aQ67BzYfZqxk4ZEOhwxQKW6PByM\nkxEsgBnzi3jyN2spPX6e/GEZkQ5HxBGb14fFbsGIwftNcoqNiTMK2Lj6CBMXFnO6ruOkUMEZOoXo\nT+lON8fTY3PLkM6MHJfDzs2n2LTmCHMWFnf7+tN1Lr711qEOzwcnfRL9K36entqweX00x8h0jfZS\nTg+6bDCr3z2EMyu5tZHosFuodXa9NktugqK3LF4fVq+PxjhKIWu1Wpi3ZDTvv7mPL3xlNsoUew/D\nIjrZPV5sMZzdbMY8f+dD6tjBfH/tyQ7LycOaiASrx4vV56M+IT4fWRd+ehzPPvIhYycPISMrOdLh\niDCJz99W/De8ZnNsPBy2m3LaMJhd08yvX9zN6RR/hpkfLC7qMDV1MLkJit7yJ7iwQAz2yHdm/JQ8\ntm44zt5Pyhk/JS/S4Yg4kejxYkuJ3fUhSclWJs0s4OjmjhtXQkRKutNNtS0h7u5HLRzpicyYX8Q7\nr+7hc1+cLp1/cSJuG1ixNILVLqU4kJXK+LO1nEm2xeTUExG74inBRTBlUiz81DiW/307I8ZkY0+M\nrzn9IjLsHh/WFBvUx+662Rnzitj2yzXYc9JxxtHItYh90ZzgorNN76HrGUUVdc2crnORMGIQ1dvK\neGPlQQqDlobIjKTYFX9PUAE2j4+mWG5gAVWJVhoTzBTUNnEyLf7mHou+U1brpMl96cOe22eEdL3D\n5aEyMTpvaL2VPyyD4vGDef/NfVz32YmRDkfEgdYpgvWxmygiMclKwYRcRhyoZE+2I9LhCNEqvdnF\n/szo3P+qs03voesZRcFrqJISbMxcd5Q/H6mhMdDBKTOSYlf8NrC8XqrjIOPMgcwUplVUU54qG9GJ\n0L13qJpntpZfcvxHV4e2a3xqs5ujcdyon3/NaJ76/TqOHjhL0ejsSIcjYtyFKYKx28ACGDY1j/3b\nSjmWnkRjnK53EbHF4/KQ7PJSGycZbTvTaLVwOCOFiWdr+DgvU2YuxbjYHuLphM0T41MEA+ptCZxL\ntDK8ujHSoYgBwuzzkejx0hCHUwRbWG0Wltw4gZWv7qY5hMQxQnTG7vHFdJKLFgn2BI6lJzP2XB0Y\noY12C9GXasvrqLEl4Bsg65JOOhJxm0yMON8Q6VBEL8V+C6QDdq+XZnN8/HiHMlMoqG2kuV72MxB9\nL8Xlod4amymnu2P4qEEMG5nFGtkbS/SS3eP1r8GKA8fTkrB7vOQ0yP1GRF5taQ1VA2mtrFLsznZQ\nUNeEw+mOdDSiF+Kzi9ow/CNYMZJFsCvNFjOnHImc3HwSiI+fSUQvR7OHWuvAuKEtuH4sT/1+HWMm\n5lI4IivS4YgIKa1x8thHpR2en12YRkFa+w0oi8+HMsBii4+/zYZS7B3kYOKZGiqTrHhNFzoqu1rQ\nD7IoX4RXTVktVYkD6/ep2WJmb1Yqk89U42qSRlasissGVoLPwGNScTWkfDQ9mbFlNWSlJFGZNLD+\n2Ij+5Wh2UxMH6xdDYU9MYMmNE1jx4k7uuG8uSTGcalv0nAFsPFnb4fncVGuHDSy7x4vTYorJTe07\ncj7RSlWilZHnGziQdSG5QFcL+kEW5YvwsXp9NNc3U5uWEulQ+t2ZFDtpzW52vr2faffOwhQnM7IG\nki4bWJqmzQd+Fyj7hK7rD7dT5qfAUqARuFPX9X2B48eAWsALuHVdnxm+0Dtmi6E9sELlNZkYOX8E\nNSsPsKEgC49JKpvoG6kuDycdsZvgoqNe9o426h6c52DclCG8qe/g5junY4qjjhnR9+weH05LfN1v\nwJ9gae6pSspSE6mP4/WYInplNrlw5Drifrp6Rw5mpjDN42LtyoOUXDcm0uGIbur0r6amaWbgSWAx\nUAps0jRtla7re4PKXA9M1nV9kqZps4CngNmB0wawQNf1qr4IviP2WN8DqwPpBelUJtoYXVkvaXRF\nn1CGQbLbE9MPVB31sne0Ufcvry/mysWj0J/cxIb3DnHF4lH9EaaIE3aPl6Y4bGC5LGYOZaYw7lwt\nm4ZkxO0mryJ6ZTa5SBuZA6edkQ4lMpRi4pLRbHt5J4PzHYwN2h9LRL+uWiEzgUO6rh/Tdd0NPA8s\na1PmBuBpAF3XNwLpmqYNDjrf73+V/SNY8dfAAtiflUJWUzNZjbIAWYRfistDY4IlrqbXhsJkNvHp\nW6ewc/Mpjh44G+lwRAxpmSIYj06lJoIBw2ski63of5lNLhz5aZEOI6KsiQl85gtTeff1PZwp63ga\ns4g+Xd0V8oGTQa9PBY6FWsYA3tM0bZumaff0JtDusHl9NMdhjyL4pwruGeRg/LlaLL5LN5IVojcc\nzW7qYnj0qjeSU20svWUyK/6xk5rzsb2fkeg/ie74nCIIgFJ8MjiNYTWNpDtdkY5GDCB2txezYZCU\nkRjpUCIuJ8/B4mWX8fIzW6iulM6OWNFVAyvUjTA66u6+Qtf1ycDngf+nadq8kCPrBZvHF7c9igCV\nSTYqE22yV4kIO0ezZ0Bs6NiRoUWZzJw/gpef3kJTozxQiq75R7DitIGFP6PZ7mwHE0/X4Ja00aKf\nZDW5qLInxFXymN4YMzGX2VeN5MUnN1FfO0CnTMaYrlohpcDQoNdD8Y9QdVamIHAMXdfLA5/3Aq/g\nn3LY5+JpD6yO7MtKJcXlYZhM3RBhlOryUDtAR7BaTL9yOEVjBvHKM1txuWQTYtG5eF2DFexcko3T\nKXYOvX9YOvVEv8hsclGVKFldg02ZVcjEGQW8+ORm6QCMAV21QjYDozRNG65pmhW4BXi9TZnXgX8B\n0DRtNlCt6/ppTdOSNE1LDRzPBq4HdoY1+g7YPPE7RbCFz6TYlpvOsJpGBsl6LBEGXo+PFJeHOtvA\nbmABlFwzhvSsJJb/bTter0zFFe1ThhGYkh7fHXrgz2jmafbIeizR9wyDTKc0sNozq2QERaMH8fLT\nW3A1SwdgNOv0rqDruge4C//o0xbgSV3X92qadq+mafcGyrwFfKJp2k786dy/GLg8F1iradp2/Mkx\nfqvr+so++jkuYvP64n4EC/xTN3YMTmPCmRqSpadd9FJdRS11VstFG4sOVMqkuOamCSilePulXRg+\n6bUXl7J5fLjMpgGRRtpQitGLRlFY08jgepmiJPpOstuLT0FTgnT2taWUouS6MeQMcaD/eZNsRBzF\nuvzt1XV9NTC1zbHH27z+NvDtNseOAFPCEGO3KMMgweu/6Q0ENXYrB7JSmVJRzcb8TDwD5OcW4Vdz\nqkZ6DIOYzSY+fdsU/vGXzfzz5Z0suXECZqlfIshAmB4YzJZqY1tuOtPKz+M2Kapk03vRB7KaXFTK\nvahDSikWLxvPuncOsumlndhTk3EmDJy/Q7Ei7p4WEt1emi0Do0exRVlqImeSbUyrOI9FpjOJHqou\nraEySW5qwRKsZm7+4jQa61289tw23C5vpEMSUSSeU7R3pM6WwPbcdCadqSFNMguKTlQ1uDlS2dTh\nx9n69n9/ZP1V15RSzFsymqETc5lZViWzmKJQ3I2/pjvdVNsGXsU8mJnC6Kp6ppef79aQcUVdM6fr\nur5JDk61kpsqvZXxyurx0lzXTI0jJdKh9DurWbGjrK7dcw67hVqnhxGLRrHnvUM89dhHTP3UOAqy\nk6U+CBI9cZyivRPVdiu7ctKYcrqGLUPSqbcO3MyjomMV9c382/KDHZ5/6JqR2MwXd4ZbvD4ymlzs\nznb0dXj9orP7C4Crl53ihZPzeHF/JdPLz7Mr20GljCpHjfhrYDW7qLYPwD/2SnEgM4Xi8w1sfmUX\n4++dRXIID4Cn61x8661DXZb75fXF8kAZxzKdLhxDHANq5LdFVZOHH6062u65HywuunDOUIxu8lL6\n9BauuWUyueNz+jFKEY3sHi/1AzTr5rkkG/uzUphWXs2OwWlU2wdex6YIv8ENTqoSrbjjZDp2Z/cX\n8N9jeqsiJRGn2cykMzWUpiZyOCMZBuC9PNrEx29wkAyne2A2sACU4lBmCrmjBvH8ExupOS/ZnkRo\nshpdpBekRTqM6KYUB7JSOelI4uOXdrLvk/JIRyQibKCtwWqrIiWRXdkOplRUk1svm3OL3surd1KW\nao90GDGnOtHKR/mZpDvdTCuvxuqR6eyRFlcNrASvD6vHR90A7VFsMWLGUKbOLuSvj37Ekf1nIx2O\niHaGQVaTi7R8aWCF4mRaEpd/ejzrVh7kndd243HLjWygShyAa7DaqkyysXlIBqOq6ik6X48h+2SJ\nHkp0e0h2eTgn09x6xGUxs2VIOtX2BGaXVpFT75T6GEFx1RJJd7qosSfI0Chw+dzh5Axx8MYLO5gw\nrYC5i4oxmeTfRVwq2e3FUGBPk17DUDlyUrjj/jm8/fJu/vrYRyy5cQJDZARwYDEM7DG2Bquv1oPU\n2xLYmJfJ1Ipqdr69nzG3TiZREuaIbsqrc1KeYh+QU9U70u06qxSHM1OoTLIy/mwtO97ax4jPTcSR\nntjHkYq24qyBNYCnB7ajoCiTO+6byxvP7+ClpzZzzU0TpJKJS/hT4tpQclPrFps9gU/fNpk928t4\n9dmtFI/PYd6S0dgT5W/QQGDxGRgQU1tj9OV6EJfFzKa8TCYmW3j6f9Zz9WcuY+RYWacoQmQYDKl3\n8slg6agK1tM6W223sqEgi0nZNp595ENmLRjJlNmFWAb4iHt/iroG1rGqJg5Wdj2XO91+aejpTjeH\nM5P7IqyYlZxq43N3TWfj6iM8+8iHTJ9XxPQrhmOOs0oWajZEkIyIbWU1NVOWIg3vnlBKcdnUfEaO\nzWHdyoP85XfruPLqUVw2NQ9TDD14i+7zp2iPndGr/uAzKcbMKyJzZgH/fGknB3ad5qqlY6XTQXQp\nvdmNT0HtAF/iEU6GUoycWUjJ3GF8sGI/Wz48xtyFxXJ/6idR95v88cla/rSprMty/71kxEWvTT4D\nh8tNjU3+kLdlMpuYs7CYcVPyeG/5XnZvKWXRDeMYVjwo0qGFTajZEEEyIgZThkF6k5td2dJr2Bv2\nxAQWLxvPhGn5fLBiHx99cJjZC0Yyfkpe3HVmCD9/inb5f9ueoUWZ/OsDV7Dm7QP8+ddrmD6viMvn\nDCPBKg1S0b68OidlqYmyxKMPZOWkcPO/TqP0+HnWrTzIx6uPMOuqkYydNERGtPpQ1DWwesrhclOf\nYMFrkl+WjqRnJnHjv1zO4X1neee1PSSnWMmeOAQMQ/6oDVDpTjeNVnPcpMSNtNyCNG69Zxanjlax\n4f3DbHjvENOvHM74qfnSix9nZASrfcFrRrKnDyVpZBYHNp7g43XHKJpewNSZQ8nP6LsR81BmM8gs\nhuhi8hkMbnDyYUFWpEOJa/nDMtDunsGJw5VsWnuUNSv2M3FGAVNmFZIqa7DDLm4aWLL+KjRKKYrH\n5TBiTDYHdlWwetUh5tQ2czwtidPJdrySCGNAyWxyUZkoi9HDraAok88VZVJ2opqtHx5j/apDjByb\nw8TpBRQUZch6tzhg93hpSpAGVlvtrxmxkJqewqmPT3H445NMmVHA5JlDSc9MCvv3D2U2g8xiiC7Z\njc3U2BJolg6LPqeUYljxIIYVD6LyTD3bN57g6f9ZT96wdMZOGkLxuBystrhpGkRU3PwrpjtdlMs6\nkpCZTIqxk4bgzErm5y/tobCmkTGVdZxNslGWYqcq0SqjWnHO5DPIbXCye5Aj0qHErbzCdPIKp9DY\n4GLv9jJWvb4Hl8vDqHGDKR6fQ8HwDJkLH4PMPoOchmb2DkqNdCgxo86WwLYhGSybm4/rWBV/fXQD\ngwvSGDtpCCPHZkvWwQHK4/Iy8nw9hzNSIh3KgJOVk8KiT49n3pLRHNpzhn2flLPqtT0MK86ieFwO\nw4qzSHHIyFZPxUcDyzBId7rZKw+K3aaUojLJRmWSDavHS26Dk9FV9dg9Xs4l2TT8/f8AACAASURB\nVKhMtMoIR5waW1lHjS1BRn57oKPUuR1NPUpKtjLtiuFcPncYlWfqObTnDKv/uZ+aqiYKR2YytCiT\ngqJMBuWkoGQUOeqNqqqjxp5AlezX023J6YnMvX4sV1w9ioO7T3Nw12neW76X3HwHxeMHM3SE1IOB\nwjAMdr97kGp7AqeTpS5FitVmYfzUPMZPzcPZ5Obg7tMc3neG99/cR0qajeHFg8gfnkHe0HSSZeQ3\nZF02sDRNmw/8LlD2CV3XH26nzE+BpUAjcKeu6/tCvTYcktxevCYlw8u95LKYOZGWzIm0ZOxuL1lN\nzeQ0NDO2so4P/7aV8qJMhhSkMWRoOlk5KbJ4P4YNrneS4XTxUX6mjFT2QEepc7uaeqSUYtDgVAYN\nTmX2VSOpq3Fy4nAlJ49WsXn9MVxOD4ML0hg8xEFOnoOcvFTSMpJkD7sokhn4uyjrRXrmos6JnFSG\nL0xl6LwiKk9Ws+9wJRvXHcPldJOe5yA910FqdjKpg5KxBtYwyvqp+HFyWynO+mb2ZTnkPtRHutpH\nq219sicmMHF6AROnF+Dz+th98BwH953l6Jqj1Jyux2I1k5abSmpWMilZSQwvTGNkfpp0iLSj0waW\npmlm4ElgMVAKbNI0bZWu63uDylwPTNZ1fZKmabOAp4DZoVwbLhlOF+ftMsrSoqsKFayjjSWdCWZK\nE5IodSShDIMfzc7jXFktu/af48M1R2mqdZLosJOckUhyRhKJaXbsqTaGDkllRJ6jy8ZXqGnV5WYa\nfoluD+PO1bJlSIYkhYmw1DQ7l12ez2WX5wNQV+PkdFktp0tr2LO9jA/eqqWpwUVaZhKZ2clkZCXh\nSE/EkZFIapqd1DQ7NrtF1nT1E4vPx2Vna9md7Yip/a+iSef7+ih+oE3mZ/88REa9m7Rdp0l1eUh1\nefAqRb3VwpTiTEYVppMeqAtSB2LToMZmynadZ9bnJvGPNScjHU7c6mofrc46BU1mEz5HIo+dc4HF\nBnlWktxe0qpdpJ5pJMXlId8EPreX9Mwk0jOTSMtKJC0jyX9/cthISbOTlGQdkA2wrkawZgKHdF0/\nBqBp2vPAMiC4kXQD8DSArusbNU1L1zQtFygK4dqwkAQXF+uqQgULZWNJQym8qXb+52Q5oCDDgUpP\nJdntIbneS0rVeeweL4keL3kJJlyNLmw2C8mpNpJSbCSnWLEnJWBPvPBR1uDmz9sq8JhMeE0Kr1L4\nTAqvAq9Srb1Zshg5vHxeH5PO1HAkI4U62dIg6rQ0morHXdig1eXyUH2ukapzDZw/18iZ8loO7TtD\n7fkm6mudeL0GSSlWklNsJCZbSWypZ0kJ2OwWrHYLNpsFq81CgtXs/0gwY0kwY0kwYbGYMVtMMkoW\ngjHn6jiX6J9SLfpOs8VMRYqZipTA+g/DwO7xkeL2cEV6ItWVjRw9cI7a6ibqa5vx+XykOOwkp1hx\nmU2MP9uEy6zwmEy4Tf7PHpPCa1J4lKKxuon6lIRAHTBjNitpoPUTs89HmtPNZWdrGffp8dhTpC7F\nDKVotFpotFooDxy67fpixmbaqa5spLqqieqqRs6W13J0/1nqap3U1zhpdnpITLaSlGwlKcXa+hxo\ns1uwJSb47092/z3KGrhHWRIu3KvMFhMWiwmzxRRT9bSrBlY+ENy1cAqYFUKZfCAvhGsvsvKVXXga\n3Fzrbu4iLDi4+jDjzwZSwTb6s+CJ/mMoRb01gXprAqeDjv/y+mIm5qbQ1Oiisd5FQ10zjfUunE0u\nmhrdVFc10tzk4WyNk6LqRiyGD7PPwGwYrZ9NBhiAoeD9Jyr50GrGZFKYzSZMZuX/MPkfCJXyJ+xo\ndPuYVuXECNQ9A0CBgWrzGj755z5OJCYE2nD+80r5v1T+/7S6tC6rDo7T7YqfPbxbxTu18pVdF8dS\nH1SPDAO8Bnh97Hx1N81mEycckhAmVlitlsB0wfbXmLrdXhrrm2moc9HU6MLZ6MbZ5G6tb65mL65m\nD81ODx63F7cr8OH24vX48Li9eDw+TGZ/HTObTa0NLpNZYTaZUCblfwg1Kf9xk/+BVCn/MaUIfA58\nrRSKQJ0IVIuW4y1fg2qtd4FXLV+0ulClLq53OV33C3XL5rf3d3rfSTl4hp0HTpPhdLOhIDO831x0\nTSmcCWacCWYKJ+cxOe/i5CKuZg91NU5/MplTtayuLSXB68Pq9ZHkNrD4fFh8BpbAPWbr63vY5vXh\n8fjwerx4fQaWlt97s/9hruX332QK3Hfa/u63fs2FuqAu/M63vG79PaeT3/tLfucvvpd0dWvpzr0n\nnPcduPTeE6y22XuhXrm9+Gqd0OxBpdoxTxzCsKIMmtztz6IRscNmT2BwfhqD89vfT9Pr8bU+EzY2\nXLhHOZ1umhpd1LTcp1weXM0ePG5f6z3K4/LibamrXh9mk8LUzn3KXzcDdfWi+qlQJn8daamvqI7v\nVZ3V2SGjuvfvogzD6PCkpmk3A9fqun5P4PXtwCxd1x8IKrMc+Jmu6+sDr1cB/xcY3tW1wd59992O\nAxEiTi1atKjX3TFSd8RAFI66A1J/xMAjdUeInulO3elqBKsUGBr0eij+kajOyhQEyiSEcG2rcFV4\nIQYaqTtC9JzUHyF6RuqOEB3rqoG1GRiladpwoAy4BbitTZnXgfuB5zVNmw1U67p+WtO0yhCuFUII\nIYQQQoi40WkaJF3XPcBdwCvAFuBJXdf3app2r6Zp9wbKvAV8omnaTvwp2b/Y2bV99pMIIYQQQggh\nRIR1ugZLCCGEEEIIIUToZCMPIYQQQgghhAgTaWAJIYQQQgghRJhIA0sIIYQQQgghwkQaWEIIIYQQ\nQggRJtLAEkIIIYQQQogwkQaWEEIIIYQQQoSJNLCEEEIIIYQQIkykgSWEEEIIIYQQYSINLCGEEEII\nIYQIE2lgCSGEEEIIIUSYSANLCCGEEEIIIcJEGlhCCCGEEEIIESbSwIoTSqmnlFI+pdRL7ZxbFjjn\njkRsQkS7oPrz8zbHCwLH50cqNiFiiVIqWSn1c6XUFqVUnVJqv1LqOaXUsG68x5+UUu/3ZZxCRAOl\nVKJS6r+VUgeVUo1KqUql1MdKqQciHZvoHWlgxQ8DOAEsVUrltDl3L3A8UEYIcSkDcAJfV0oVRjoY\nIWKRUiob2Al8GvhvYArwFcAB7JKOCiEu8Qfgq8DPgWnAAuB/gbQIxiTCQBpY8eUgsBG4s+VA4GFx\nMfAXQAWOZQR6FI8Hekz2KaW+EXTNAqWURylVEPzmSql/UUpVK6US++OHEaKffQjsAH7SUQGl1FCl\n1EtKqYrAh66UGhI45wjUp9vaXJMXqE8L+zZ8ISLu14AdmGkYxquGYRw2DON9wzBuANYCTyilrABK\nqZlKqVVKqbOB+8oHSqkRSqkfAncBJYHRY59S6l8i9hMJ0bc+A/yPYRh/Mgxjr2EYOw3DeNowjIeC\nCymlrldKfRioK0eVUr9WSiUFnf9AKfWEUuo7gdGwMqXUb5RSln7/iQQgDax4ogKf/wjcHXT8bmAV\n/hGsFlb8vYzLgMsD1/xMKXUngGEYH+BvrN3V5nvcA/zVMIymMMcuRKQp/KNY3wRuU0pNu6SAv2Nh\nHf7e+OuA64FsYI1SKsEwjFrgFeCONpfeDpQahvFeH8YvREQFGk63AY8YhlHfTpGfAKOAuUqpmfg7\nND4BFgIz8XcCWoBfAn8LnM8NfOh9/gMIERllwCKlVEZHBZRSN+GvAy8D84DvAp8CHmtT9CZgeODc\nl4EvAL8If8giFMowZNZYPFBKPQXk4Z+aUQrcjP9h8BjwAJAOPGEYRkIH178AZBiGsSTw+t+BB4Ei\nwzAMpdRYYA8w1TCMHX370wjRv1rqj2EYS5RSL+OvC1cFRnFP4J+2MQp4AhhnGMb+wHUT8Y96/ath\nGM8qpa4B3gAKDMM4HSizE3jdMIz/7O+fS4j+opS6DH/H3Y2GYbzWzvlM4BxwP/4HwcbAyFZ77/Un\nYKRhGFf1YchCRJxSai7+DoUCYDfwEfBWcB1SSu0HXjQM47tBx74APIv/XlWjlPoAf0dFhmEYzYEy\nDwH/B8iUjvH+JyNYcSZQsZ7F33uxFH+P4PLgMkopk1Lq20qp7YHpGXX4R7OC1548DeQA1wRe3w1s\nlsaViGMto8D/F7hCKfXpNucuA461NK4ADMPYib9DY3zg0CrgDPB5AKXU5YHrnunb0IWIKZOBlZEO\nQohIMwzjQ2Ak/pGpp4FxwCtKqdfBnzQGf+fevweSxtQFntkexz/rojjo7da3NK4C3gFsgfcX/Uwa\nWPEleJrgTfinOz1pGIa3Tbn/AzwEPIJ/fdZk/D0otpYChmFUAf8A7lFKJQD/EnhfIeKaYRgH8d+8\nfg6Y25xWl15x0bVe4K/46wuBzx8HN8qEiFMHAS8wsYPzlwU+7+mfcISIDYZheA3D2GAYxm8Mw5gP\nfAP4lFJqHhee07+O/1mt5WMS/obXrqC36vT+JPqXNLDikGEYe4FNwFzgT+0UmQ+8F1hUucMwjCOB\nY23niz6Of8rhV/AvXP5730UtRMQF//7/CP+U23uDzu0ChgWmywKglJoE5HPxTe4ZYLJSagr+NSky\neiXinmEYLuB54H6lVGo7Rb4DHAA2ANu5MDuiPS4u7dwQYqA4GPicYhhGHXAIGGsYxpF2PoJHrOYq\npWxBr6/Gnx33cD/FLYJIAyt+XQMMMgzjaDvn9gFzlFKfUUqNUUr9GP8arYt6PwzDWA/sx7/o+O+G\nYTT0ddBCRFDr779hGOeAnwH/FnT+b/jXYz2ilJoaSITxe/w3vxeCrt0FbMO/aN+BdEyIgeMbQBPw\nsVLqRqVUsVLqKqXUcuBK4J7AA+F3geuUUr9VSk0K3IfuVEqNDrzPEWCsUmqOUmpQS+ZBIeKNUmq1\nUupepdR0pdRopdTt+BNTlOFP9AL+WUdfV0r9P6XUhEB9+YxSqm2Siybg90qpsUqppfiXdjwm668i\nQxpY8cMgqAfeMIwmwzCq2ykD/v1JHgN+C7wZOPY72t8n60/4sw7K9EARzy6qPwG/Bc62HDcMw4n/\nIbEa+Cf+unMGKDEMw9Pm2qfxT+N4yzCM830YtxBRwzCMs/inCL4BfA9/R8Pj+OvMRMMw1gbKbcS/\n5mQy8D7+7UXuwD9yBfBn/LMwVuCvY7f2308hRL96C3+2vzfxJ0z6MbAFWGgYRg2AYRjL8XeaL8Zf\nXz4GfgCcCnofA3gJOBl4zz/h79z7j375KcQluswiqGnafPwP3xbgCV3XH25zfhnwX/j/55YCP9R1\nfVPg3DGgFv+8bLeu6zPD/QOIvqWU+gWwyDCMS9JWCyGEEEKIyFJKvQ8cNAzjy5GORfh1ugGZpmlm\n4En8reZSYJOmaat0Xd8bVGyVruuvBcqX4N9osGW3dgNYoOt6VdgjF31KKZWGv3fxHvxp3oUQQggh\nRPRRSJKLqNLVDs8zgUO6rh8D0DTtefzpvFsbWLquB6/LScO/oC6Y/A+PTa9xIb30XyMcixBCCCGE\naF9709xFBHXVwMrHP5+zxSlgVttCmqbdiH+9QgowPeiUAbynaZoPeFTX9Sd6F67oL4ZhLIh0DEII\nIYQQonOyKXf06aqBFVJrWNf1V4BXNE27BXgFmBo4dYWu6+Wapo0D3tI0bZ+u62vbe493331XWt5i\nwFm0aFGvR3il7oiBKBx1B6T+iIFH6o4QPdOdutNVA6sUGBr0eigXZy25iK7rL2ia9oimaWm6rtfo\nul4eOL5X07RX8E85bLeBBXD55ZeHGrcQMW/r1q1hey+pO2IgCWfdAak/YuCQuiNEz3S37nTVwNoM\njNI0bTj+nPy34N84s5WmaSOBI7quG5qmXQ806bpeo2laEmDWdb1O07Rs4Hr8O1ELIYQQQgghRFzq\ndB8sXdc9wF34p/1tAZ4MjEbdq2navYFiNwM7NU3bBnwWfxIMgFxgraZp2/Hv7v5bXddX9sUPIYQQ\nQgghhBDRoKsRLHRdX82FNVUtxx4P+voX+HedbnvdEWBKGGIUQgghhBBCiJjQ6QiWEEIIIYQQQojQ\nSQNLCCGEEEIIIcJEGlhCCCGEEEIIESbSwBJCCCGEEEKIMJEGlhBCCCGEEEKEiTSwhBBCCCFE1PN4\nfNRWN0U6DCG61GWadiGEEEIIISLhfGUDh/ee5dihc5QdP49hwFe+fRU2uzzCiuglI1hCCCGEECLq\n+Lw+nvvfDVSdrWfS9AK+/B8LGDYyiwO7KyIdmhCdkgaWEEIIIYSIOlXnGklMtrLkxgmMnpCLPTGB\n8VPz2LO1LNKhCdEpaWAJIYQQQoioc7ailuzc1IuOjRibw7nTddScl7VYInpJA0sIIYQQQkSdM+V1\n5AxxXHTMYjExemIue7fLKJaIXtLAEkIIIYQQUedseR05Q1IvOX7Z1Hx2byvFMIwIRCVE16SBJYQQ\nQvShhrpmVr2+B6/XF+lQhIgpZyvqyG6ngTVkaBoYUHGqJgJRCdG1LnNcapo2H/hdoOwTuq4/3Ob8\nMuC/AAMoBX6o6/qmUK4VQggh4plhGLzz2m5OHK7CkZ7IzPlFkQ5JiJjQUN+Mx+0lNc1+yTmlFOOn\n5rN7axlDhqZHIDohOtfpCJamaWbgSeAmYBrwJU3TxrUptkrX9cm6rk8BfgH8uhvXCiGEEHFr345y\nzp9r5PNfmc2mNUeormqMdEhCxISz5f7RK6VUu+fHTx3C/p3leD0yMiyiT1dTBGcCh3RdP6bruht4\nHlgWXEDX9Yagl2mAM9RrhRBCiHjVUNfM+2/u47rPTmTQ4BSmXzmcVa/vkXUjQoTgbEUdObmODs+n\nZSSRNTiFIwfO9mNUQoSmqwZWPnAy6PWpwLGLaJp2o6Zpx/CPWN3TnWuFEEKIeGMYBu+8upuJ0wvI\nLUgDYPq8IuprnOz/RDZJFaIrZ8pryc67dP1VsMum5rNnm2QTFNGnqwZWSN1suq6/ouv6cOA+4NXe\nBiWEEELEsn07yjlf2cicRcWtx8xmE0tuvIz339qHs8kdweiEiH5ny+vIye28gTViTDYnj1Rh+GRU\nWESXrhpYpcDQoNdD8Y9EtUvX9ReAAk3T0gLlQr5WCCGEiAeGYfD+W/u49uYJWCwX32bzCjMoHp/D\nupUHIxSdENHP4/FRXdlIVk5Kp+WSU20kJiVw7kx9P0UmRGi6amBtBkZpmjZc0zQrcAvwenABTdNG\napqmAl9fDzTpul4TyrVCCCFEvGmoa8bwGR1mN5tz1Uj27iiTxflCdKDyTD1pmUlYEsxdls0fnsGp\no1X9EJUQoeu0gaXruge4C3gF2AI8qev6Xk3T7tU07d5AsZuBnZqmbQM+SyCRRUfX9s2PIYQQQkRG\nRV0zO8rqWj8+3nsWW3oiFXXN7ZZPcdjJGJTMqWPn+zlSIWLD2fLadjcYbk/B8AypSyLqdLkPlq7r\nq4GpbY49HvT1L/CnZw/pWiGEECKenK5z8a23DrW+HlrTSIrLw4w6F7mptnavGTk2h8P7zjCsOKu/\nwhQiZnS0wXB7CooyWffOQQzD6DCluxD9raspgkIIIYTohhS3hwZr5/2XI8dlc3jfGUnZLkQ7zpTV\nkTOk4xTtwdIyEgGoqWrqy5CE6BZpYAkhhBBhlOzyUN/F2pHs3FR8XoPKMw2dlhNioDEMwz+C1UUG\nwRZKqcA0QVmHJaKHNLCEEEKIMEp2e7scwVJKtU4TFEJcUFfjxGRWJHcwvbY9+cMzZR2WiCpdrsES\nQgghhD+Zxek61yXHXd4L2QAtXh9mn0Gzuev+y5Hjstnw3mFmlYwIa5xCxLKzFXUhJ7hoUTA8gy3r\nj/VNQEL0gDSwhBBCiBC0TWbR4geLi1q/9q+/MkMIi+2Hjsjijed30FjvIinFGtZYhYhVZ8vryM4N\nbf1Vi0E5KTgb3TTUNXdr5EuIviJTBIUQQogw8a+/Cq3v0mIxUTgyiyMHzvZxVELEjjPloWcQbKFM\nivxh6TJNUEQNaWAJIYQQYRLK+qtgI8dmc3ivrMMSokV1ZQOZ2cndvi5/eKZsOCyihjSwhBBCiDBJ\ncXlo6CKDYLCiMdmcOFyJx+PrurAQA0BTo5uk5O5PmR1alMGp4zKCJaKDNLCEEEKIMEl2e6jvxghW\ncoqNrJwU6XkXAn+K9qYGF4lJ3W9g5eQ5qK5sxNnk7oPIhOgeaWAJIYQQYWD2GVi9PposoY9gQWCa\noKRrFwK32wsKEqzdq0MAZrOJIUPTKZVRLBEFpIElhBBChEGS20NjgiWkDILBCkdmcfKIjGAJ0dTg\n7tHoVYuC4RmUSqILEQWkgSWEEEKEQYrLQ3031l+1GJznoLa6icaGS/fYEmIgaWp0kdiD9Vct8grT\nKTtRHcaIhOgZaWAJIYQQYdDdDIItTGYTeYXS8y6Ef/1VQo+vzy1I43RZLT6vJI0RkdXlnUDTtPnA\n7wJln9B1/eE2578A/Efg5W7gJ7qu7wqcOwbUAl7Arev6zPCFLoQQQkSPZJeHihR7j64dWpTByaNV\njLpscJijEiJ2NDW6e9XAsicmkJpm59zpenLyurdZsRDh1OkIlqZpZuBJ4CZgGvAlTdPGtSl2BJiv\n6/pk4G3gT0HnDGCBrutTpXElhBAinqW4PT0awQIoKMqUTVLFgOds7FkGwWBDhqZTflKmCYrI6upO\nMBM4pOv6MQBN054HlgF7Wwrour4hqPybwI/bvEf3VvsKIYQQMUYZBnaPt1t7YAXLzU/j/LkGnE1u\n7Ik978EXIpY1Nrh7tQYLYMjQNMpO1jB5VpiCEqIHulqDlQ+cDHp9KnCsI18GXgt6bQDvaZq2TdO0\ne3oWohBCCBHdktxenBYzRjczCLYwWyTFtBBNjb1bgwWQJyNYIgp01cAyQn0jTdOuAm4H/jPo8BWB\nqYOfB/6fpmnzuh+iEEIIEd2SXR4aEno2PbBFyzosIQaqpjCMYA0anEJdjVM2HBYR1VUDqxQYGvR6\nKP5RrItomjYJ+CNwg67rrd0Guq6XBz7vBV7BP+VQCCGEiCspbg/1PdgcNVhBUSanjsoIlhi4msKw\nBstkNjE4z0HFqZowRSVE93XV3bYZGKVp2nCgDLgFuC24gKZphcBLwO26rh8KOp4EmHVdr9M0LRu4\nHvh6GGPvF5Vn6klNs2O19a5nUgghRPxKdnk519vF+QVpVJ6px9XskXuOGJD8+2D1fg2iP9FFDcNH\nDQpDVEJ0X6cjWLque4C78I8+bQGe1HV9r6Zp92qadm+g2PeBTOCxwFqrjwPHc4G1mqZtB54Hfqvr\n+so++Sn6yJH9Z/nbYx/x+M8/4J8v7eTUsfMYRsizJgeUyZMnc/585z2voZQJlx//+MfMmzeP+fPn\n85WvfIWqqvan3UyePJkrr7ySkpISFi9e3O3rheitWK07DQ0NfO1rX6OkpITZs2ezadMmAPbu3cvC\nhQuZPn06X/rSl2hqauqXuCMtyeOhsYcJLlpYEswMznfIOqwQRVvd+f73v8+sWbNYsGAB3/nOd6it\nre2wrNfrpaSkhNtuu9BnPVDrTrCmBnevR7DAn+hC1mF1LBbrjtPpZPHixcyfP5+rr76aRx99tPXc\nq6++ypw5cxg0aBA7duzol5i70uVGw7qurw6kWZ+o6/r/BI49ruv644Gv79Z1PStQpjUdu67rR3Rd\nnxL4WNRSPlacrahjxT92cvOd07jr3+eRlZPCyld28czDH8q83naoEBZ2K6X6rYH69a9/nbVr17Jm\nzRpGjBjBY4891mFMy5cvZ/Xq1axatarb1wvRW7Fad771rW8xd+5cVq9ezbp16xgzZgwAv/nNb/j6\n17/O5s2bmThxIr/97W/7Je5Is3p9uMy9a2ABFAzPlHVYIYq2unPVVVexYcMG3nvvPRobGzv93X/s\nsccYM2bMRT/DQK07LQzDoKnRhb2XSS7gQqp26RRvXyzWHbvdzuuvv86aNWt44403eO655zhy5AgA\n48eP59lnn2Xu3Ln9Em8oumxgDUQNdc288swWFn5qLHmFGSSn2pgxr4gv/tuV5BWms3rF/kiHGDF3\n3HEHV111FcuWLePVV1+95PyJEyeYPXs2DzzwAFOmTOHOO+/E6XS2nn/mmWe48sorufXWWzl+/DgA\nW7Zs4ZprrqGkpISvfe1rHD16tNdxpqamAuDxeGhsbMRu73jzz/b+gHTnetG+6qpGmp2eSIcRNeKp\n7tTW1rJhwwZuv/12ACwWCw6Hf1PP9evXc+211wJw7bXXsnz58l7HFAv8Daze70oyVNZhXSJW6s5V\nV12FyWTCZDKxcOFCysrK2i1XWlrKqlWruOOOOy66/wzUutPC7fJiNptI6OVIMEBqmh2zxUTN+YE3\nChgs3upOUlIS4J9B4fV6sVr9o52jR4+muLi413GEkzSw2nC7vbz63FYuuzyfcZPzLjqnlKLkujEc\nO3SOYwfPRSjCyHr44Yd5//33efbZZ/n+979PdfWlQ/AHDx7kmmuuYePGjfh8PlauvDAztKmpiXXr\n1jFjxgxeeOEFwF8x3nrrLVavXs3VV1/ND3/4w0ves66ujpKSkks+FixYwIEDB9qN9aGHHmLs2LF8\n9NFH3H///e2WUUqxbNkySkpKePrpp7t9vWjfjo0nePaRD/nzb9aw/aMT+Ly+SIcUcfFUd44fP05W\nVhb33Xcfc+fO5cEHH2ydzrRgwQL+/ve/09zczPPPP09paWlP/rliitft9X/uYYr2YHmF6Zwpr8Pl\nks6JFrFUd1o888wzXHfdde2e+8///E9+9KMfYTJd/Ag2EOtOsMaG3qdoDzZkaDrlJwb2NMF4qzs+\nn4958+YxZswY7r77bgoKCrrxr9G/ZBVtGx+8tY+0jETmLmq/JWy1WVjymctY+cou7nzwygG3EPnl\nl1/m9ddf5+zZs9TW1nL48GGmTZt2URmHw8GnPvUpAG6++WbeffddbrjhBgBuvfVWAObNm8cvf/lL\nwD+v9sc//jHr16/HMAzc7kunYKamprJ69epuxfrd736Xb3zjGzz00EP8OQa56AAAIABJREFU8Ic/\n5Cc/+cklZVasWEFubi779+/nlltuYfTo0cyZMyfk68XFvB4f772xl5NHq/jC1+bgdnlZ/dY+tm44\nzoLrxzJiTHakQ4yYeKo7Ho+Hbdu28c1vfpNf/epXfOMb3+C1117j1ltv5dvf/jaPPPIIS5YsYenS\npa09jPHM7XTjMpsgDA2sBKuZnCGplJ+oZlixLNCH2Ko7AL/+9a9JSUnhM5/5zCXn3n77bbKzs5k0\naRLr1q276NxArDvBmhp7n6I9WEuii3FT8rouHKfiqe4AmEwm1q5dy4kTJ9A0jVmzZjFp0qRuf5/+\nMLBaB13wuL3s21HOXd+Y1+n81KLR2RSOzGLN2wdYfMP4fowwso4dO8aTTz7J8uXLycjIoKSkhObm\n5i6vC/63TE9PByAhIaH12j//+c9kZGTw3nvvsW/fPu64445L3qOuro6lS5e2+//lj3/8Y+v6j7aS\nkpK4/fbb+cpXvtLu+dzcXADGjBnD0qVL2bJlS2sDK5TrxQUNdc28/rdt2JOsfOGrc7DZ/X9ePvel\nGRzZf5Z3Xt3NVUvHMnpCboQj7X/xVnfy8vLIzMxsnc5000038cILL3DrrbdSWFjIL37xCwAOHTp0\nUW9ovHI3eXCZwjchZGhRJiePnpcGFrFXd/72t7/xzjvvtDsdC+Djjz9mxYoVvPPOOzQ3N1NXV8dX\nv/pV/vCHPwzIuhOsqaH3KdqDDRmaxupdFWF7v1gTb3UnWGFhIVdffTXr16+XBlYsOHaokuwhqSSn\n2Losu+D6sTz1+3WMnZhLQVFmP0QXeRUVFQwaNIiMjAw++ugjdu3a1W652tpa3nzzTRYvXszLL7/M\nZz/72U7ft7y8vLVR88wzz7RbJjU1lTVr1oQc6+HDhxk5ciQej4eXXnqJ8eMvbQg3Njbi9XpJTU3l\n3LlzrFq1ip/+9KchXy8utur1PeQWpLHgurEo04U/qkopRo7NwWaz8MYLOygcmYU9MXzTQGJBvNWd\nwYMHU1RUxObNm7n88st55513KCkpAeDcuXMMGjQIn8/Hr3/9a+66666Qv3es8rSMYIVJ/vAMNq3p\n/bqGeBBLdWfVqlU8/PDDvPnmmx2u2/3e977H9773PcC/5uqRRx7hD3/4AzAw606wcKVobzE438G5\n0/V4PD4sloG3Iibe6k5lZSUWi4W0tDSqqqpYtWoVP/vZzy4pFy2JTQbeb1wnDu6uYPRlofWu2xMT\nWHTDeFa+uhvDFx3/M/va7NmzKSgoYNasWTz22GOtD1RtjRo1ihUrVjBr1iyUUixZsgS4uFdEKdX6\n+stf/jJPPfUUCxcuJD8/P6TsNl35r//6L6644gquvfZaPB4PDz30EOD/w3DLLbcAcObMGZYuXcr8\n+fO5++67+epXv8rChQs7vV60r77WyYnDlVyxeNRFjatgBUWZFI8bzAdv7evn6CIv3uoOwKOPPsp3\nvvMdZsyYQXl5OTfddBMAL730EjNnzmThwoWMGjWKz3/+872OKdq5m9y4w9nAGpZORWkNXo+sXYyl\nuvPtb3+bhoYGbrzxRkpKSvjmN78JXFp3ggV/34FYd4KFK0V7C6vVQuagJM6UdZwuP57FW92pqKhg\n2bJlzJs3j3vuuad1mxCAN954gwkTJrB582ZuvfVWPve5z/U6pt5S0dLSe/fdd43LL788Yt/f6/Xx\nh5+8z79+/QpS00LLGGcYBs/97wbmLi5m5NicPo4wNpw4cYLbbruN9evXRzqUqLd161YWLVrU679M\nka47ABveO0xdTRNLbpzQablmp4enfr+Oa2+eyLDirH6KLjZI3QlduOoOdK/+7Cir41tvHbrk+N2D\nbKzZf44DWakXHf/l9cVMzku9pHwonnl4PYtuGE/+sIweXT+QSN0JXaTqTijWrjyAxWJizsLwZYN7\n59XdZGYnM+2K4WF7z3gidSd03a07MoIVcPJIFRmDkkJuXIG/RT/tiuFsWX+8DyOLPeHozRCxw+cz\n+GTzSSbPKuyyrM1uYfGy8ax8ZZdkSWuH1J3Y5A7zFEHwTxM8dUzStYdK6k7sC/caLIAhhemUDfBM\ngl2RutM3pIEVcGBXBaNCnB4YbMzEXKrO1nOmfGAOQbdVWFh4SWYkEd+OHjhLcoqNwXmOkMqPHJtD\nXmE661ddOhIwkEndiV2epvA3sAqGZ1IqDayQSN2JD+HOIgj+bQ/KTkoDqyNSd/qONLDw98Af2nOG\n0RMGd/tas8XE1NmFMoolBqxPPj7J5JlDu3XNVUvHsWdrKdVVjX0UlRD9x+3sXhbBirpmdpTVXfJR\nUXchw1fB8AxKj58fMGt8hWhqDO8+WAAZWUl4XF7qapxdFxYijKSBBZQeO0+Kw0Z6ZlKPrp80cyiH\n9pymvja+K/CJEye44oorunXN3//+dyoq+jZN6v3338+YMWM6je3FF19k3rx5rYsj9+zZA/j3c1i8\neDHz58/n6quv5tFHH+3TWONNbXUTpcerGTOpe6O/SSlWJk4vYMv6Y30TWJSJ5boD/sQXV155JVdf\nffVFm0R++OGHLFiwgCuvvJI//vGPfRprNOvuFMHTdS6+9dahSz5O17layySn2khKtnLudH1fhBwz\n4rXuTJ48mSuvvJKSkhIWL17cp7FGs+DOhvM1zZxocHfY6dATSqkBO00w1usO+Nc9ZWdns3z58tZj\nDQ0NrQkuZs+ezaZNm/oy3B6TBhZwYHfPpge2SEyyMnbyELZvPBnGqOJDf1TWL3zhC7z44oudlhk2\nbBhvvvkma9euZeHChTz44IMA2O12Xn/9ddasWcMbb7zBc889x5EjR/o03nhRUdfMqvePkF2cxd5z\nTd2+IV4+dxh7t5fT1OjquvAAFC1155133mHXrl2sW7eOn/70p9x3330AeL1eHnjgAZ555hnef/99\nnnvuOfbv39+n8UarcGcRbOFfh1UV9veNd9Fed8D/4L98+XJWr17NqlWr+jTWaBbc2VBV4+RnH57q\nsNOhp2SaYOiipe6A/x7zox/9iEWLFl2Uev1b3/oWc+fOZfXq1axbt67DvRwjbcA3sAyfwcHdp0Oe\nHtjR1I7k4my2fHSCrSeqw9LrEq18Ph8PPPAAU6ZM4c4778Tp9I/abd++nauvvppZs2Zx0003UVlZ\nyWuvvcb27du59957WbBgAU6nk1/84hcsXryYRYsW8atf/SosMc2ZM6d1M7yOzJw5E4fDv0ZoyZIl\nlJWVtZ5LSvKPXDY0NOD1erFawzsHPF6V1zSzZ2spL5x39eiGmOKwUzw+h+0fnejjSKNDrNadFStW\ncOuttwIwffp0amtrOX36NFu2bKGoqIjCwkISEhK46aabWLFiRVjiijWeJg8uc/gXihdIogsgvurO\nmTNnWs9HSxbnqGAYJHh9fdJRkTc0nfIBOIIFsVt3wL8h8Q033EBW1oWMw7W1tWzYsIHbb78dAIvF\n0vpsF226/E3WNG2+pmlbNU37RNO0B9o5/wVN03YEPv6madqEUK+NBhWlNVhtFrJyUkIq39HUjh9u\nKKVUKR7+x+6w9bpEowMHDnDNNdewceNGfD4fK1euxO12c//99/PUU0+xceNGli5dynPPPceyZcuY\nMmUKf/zjH/nggw+w2+18+ctfZtWqVaxcuZKtW7fy9ttvX/I9/vGPf1BSUnLJxxe/+MWw/AxPP/00\n119/fetrn8/HvHnzGDNmDHfffTcFBQVh+T7x7tzxKposZuqtPZ8zP2NeEds2nMDt9oYxsugUq3Wn\nvLyc/Pz81td5eXmUl5dTUVHR7vGBxhxYI+Xtg0xcBcMz/euwBviDeLzVHfCPYC1btoySkhKefvrp\nHn+PeGHxGXiVwuiDepRbkMaZ8jo8A3BfuVitO2VlZaxYsaJ1s+2WTIfHjx8nKyuL++67j7lz5/Lg\ngw/S1NTU4+/TlyydndQ0zQw8CSwGSoFNmqat0nV9b1CxI8B8XddrNE37V+BPwOwQr424Q3vOMOqy\n7ie3aM8JRxKjq+opdfRsLVcscDgcfOpTnwLg5ptv5t1332XUqFGcPHmydVNEr9dLYeGFlN3BDwfb\ntm3jL3/5CwcPHqS2tpZ9+/ZxzTXXXPQ9PvvZz3a5k3hPrV27Fl3XL/ojYTKZWLt2LSdOnOD/t3ff\n0XFd96Hvv2caymDQAaKSAAvYu1jUSHVTkmUlsrVlOVbc5Ovr+Ob5OolzYyXPulkv79mxnWfnydG1\nlxxJlpu8nViWbEm2RdlXklUskmKvYicBAgQJEB2Ycs77YwbQEESfcqb8PmvNAmZwZp8Ncn44Z7ff\nVkqxYcMGVqxYkZDzZ5KO05c4782JqYyyygKq64vY/04zq6aQ5j2dpXPsZPsN/kQ8IRNXngsScGNY\nVJqHZUFXxwDFZZl7XZlMJsbOiy++SFVVFYcPH+a+++6jqamJq6++Ou7nTxduMzGjVwCenPc2HK6Z\nPfnISSZJ19h56KGH+PKXvzzSsBquUzAYZOfOnfzN3/wN3/jGN/irv/ornn322ZGR4lQyYQMLWA8c\n1VqfBFBKPQ3cDYw0krTWb0Yd/zzwf0/1vang3JlLXHV9Y1zK6sjz4DZNfEOBuJSXLizLoqSkhFde\neWXMn0cHyN/93d/xxBNPsGTJEh566KGR4epoP/vZz/j2t799xeuNjY08+eSTM67n/v37+cIXvoDW\nmqKioit+Pnv2bG699VZef/11aWBNQee5Hi7lxj6dct31jfz65/tYsa4ehyO79uNIh9iprq6mubl5\n5HlLSws1NTUEAoHLXm9ubqampmZG50hn7pCJO3fsUVyP02B3S88Vr/tDU+tJNwwjMk2wI6sbWGNJ\n19iprq4GoKoqvO574cKF3HnnnezYsSOrG1iekEkgAdNshw0nusi2BtZY0iF2du/ezYMPPghAR0cH\nW7duxe12s2bNGkpLS9myZQsA99xzDz/96U/TsoFVC0RnbjgLbJjg+P8CPDvD9yadZVm0tXRPef+e\nSRkGLQV51PSk5nBlPHR3d/P8889zyy238POf/5wPfehDLFiwAIDnnnuOu+66i2AwyLFjx1i0aBEF\nBQW0t7cDMDQ0RG9vL7Nnzx4Z/r3//vuvOMe9997LvffeG9d6nz17lo997GN85zvfYe7cuSOvX7x4\nEZfLRVFR0UgQf/WrX43ruTPR0GCAga4Buuu8MZdV21BCXr6bowfaaFo282QzqS5dY+f222/nscce\n44Mf/CDbtm2jsLCQyspKSktLOX78OKdPn6aqqopnnnmGxx57LK7nTgce08SdN3YDq2MgyD9uPXHF\n6w/fMvVOveENh5etzd6py5kWO/39/YRCIXw+HxcuXGDr1q185Stfieu50407ZE1rq4PpqpldzLGD\n7QkrP1Wla+zs3Llz5PvPfe5zbNmyhdtvvx0IN9y2b9/OmjVreOmll9i8eXNczx0vk32apzwvRCl1\nI/BR4O+n+167dHUO4PY48fpim+YUrcWXR3XvIOYUeyjTiWEYLFiwgBdffJENGzZgGAa33XYbbreb\nH/7whzz66KOsX7+ezZs3j6TNfOCBB/ja177GDTfcAMDnP/95brnlFj71qU9x6623xqVeDz74IFu2\nbOHYsWMsW7aMH/3oRwA8+eSTI70nX/va1+js7OSv//qvL0uL29rayt133z2Svn049Wcmm8oePJNp\nPnWJwsqCuMyXNwyDddc3su21K29EM0U6x86tt97K0qVLufbaa3nooYdGeipdLhePPPIIDzzwADfe\neCN/9md/lrLZnBLJEzJx507WVzlz9Vm+4XAmxs758+e588472bRpEw8++CCf/exnuemmm+JSr3Tl\nSeAUQQg3sM5lWSbBdI6diTz66KN86UtfYt26dZw7d4577rknLvWKN2OiufVKqY3A/9Rab4k8/xJg\naq3/edRxK4D/BG7XWh+dznuHvfzyy9aaNWvi8CtN3eG9rRzY1cKfPjD18+5u6eGLLxyd8JirWjq4\n4Ya53LYpPlMPRWZ65513uPnmm2NuoUw3dsb7DH/9jvmsrPFNqYzXfnuE871+vttx5XTY6ZQzzAyZ\nPPaNV7nnz9dSUT2994rsE6/YgenFz1ix03Cpj811Pr7fe2Wn2sO3NI47gjXW62PFjmVafPufXuaT\nX7g+rp2BIjvZFTvjGY6pOZf6yAmZHCm7/PM/k+vJWCzL4tH/5/c88LmrKSzOi7k8kX2mGzuTdRds\nBxYopRqUUh7gPuC56AOUUrMJN64+Oty4mup77dbW0hW/6YFRmn15NB88P/mBQqSp5pOdlMQxdhxO\nB8vW1rJnm+wlJ9KLZ4I1WPFgOIyRaYJCZCq3aRJI4BRBwzDC+2Flabp2kXwTfpq11kHgk8AzwA7g\nca31QaXUZ5RSn4kc9mWgFPiOUmqnUurtid6boN9jRtqau5lVG/8GVps3l67Wbnq6rlwMKES6CwZN\n2lq6KaqK70jT8qvqOLj7HAF/5qdsF5nDEzJx5yVuiiBAfWMpp49fTOg5hLCTJ2QlZC+5aNk4TVDY\nZ9Krgtb6FWD1qNe+G/X9g8CDU31vqrAsK9zAqimktWdoyvtWTSX7k+kwmDWvnAO7Wtiwee6kxwuR\nTtqauygp9+LyxPemsrA4j5rZxRze18qyNbWTv0GIFDBRFsF4mT23lL3bzyb0HELYKVGbDEerqS/m\n1d8cTug5hBiW2G63FNZ9aRCny0FBYS7HprCuathUsz/VLK5k3yvHWL+pcSTlpRDpaHQHxMm9reSU\ne8ftbBgvNXVhrovuweAVr8/yeaiKrC1Zsb6et185Lg0skTbC+2AltoFVUV1If6+f3u5BCgpzE3ou\nIeJpsg7s4euIZ4ZTBCcqP/raAjCrrpD21l6CgRAut3Pa5xJiOrK2gdXWnJj1V8OKqnwYGLScvkTt\nnJKEnUeIRGvr8V/WAbGqtZNzBXn819DYCXImSk093sL+4Yvg3KZytj67n/bWHiriPAVRiETwmInN\nIgjgcIT3wzpzooPFK7NvrzGRvkZfP0Yb7rR2h0z8MxjBmqj86GsLgMfjoqzCS1tLt9yXiYRL7Hhs\nCjvfkpj1V8MMw2Dp2lr2v9M8+cFCpAvLongwQGeCpkSFk13UsXebTIcSacCyImuwEjuCBVA/t5Qz\nxzsSfh4h7OAJmQSSsNG8JIwRyZK1DazWeG4wPI4lq2o4sq+NYEAW7YvM4A2ECDoc+F2Jm16x/Kpa\nDu5uISBxI1KcM7LNicOV+Evp7LllnJYGlshEloXLtBK+BgvCCWPOnJA4EomXlQ2skQQXtUUJPY+v\nKJfKah/HDmXf7uEiM5UM+hM2ejWsqCSfWXVFvLuvLaHnESJW4cxnjqSssy2fVcDQQECy04qM4zIt\nQg4jLhvXT6a2oYSWU5cwp5CwTIhYZGUDq7d7CAMoKEz8po1LVtdwYFdLws8jRDIUDwa4lOAGFsCK\nq+pkTyyR8twhE38C9+6JZjgM6iRdu8hAniTGUb7XQ2FxLm0t3Uk5n8heWdnAamvuorK2MCm9jk3L\nqjh7ooP+vqmlgRfCLsPZ/0Y/orMFhkewPAmvy7xFlXS099F5oS/h5xJipjwzXJg/U7NlHZbIQG4z\n8Snao4WnCco6LJFYWZlFsK2lm6oEr78a5slx0dhUweE951h99ZyknFOImZgo+x9ATjCE07ToT0J6\nW6fLweLVNex7p5nrb2tK+PmEmAlPsm8M55ax/Q8nk3Y+IZLBEzIJJHiT4Wh1jaXsf6eZ9Zumtu2O\nEDORpSNY3VQmqYEFMk1QZIaioQBduW5I0r5uyyNZOE1z7HTwQtgt2SNYZZVeAv4QXZ0DSTunEInm\nDllJmyIIUNdYQvOpTrm2iITKzgZWS+ITXERrmF9GV+cAHTLdSaSxAn+QHk/yBr3LZ/nwFeVy8t0L\nSTunyE4d/YHLHn3+qWWwTHYDyzCMSLp2WYclMkeyR4K9BTl4fTm0n5N1WCJxsq6B1ds9iBkyKSzO\nTdo5HU4Hi1dWc2CnjGKJ9FXgD9KbxAYWwLK1dezbIXtiicT67DOHLnscvdg/pfclu4EF4f2wJF27\nyCQz3WQ4FpKuXSRa1jWw2lrC0wOTkeAi2pLVtRzY1YIlQ9IiTflsaGAtWlHFqaMXR5LEtPYMjZmI\no7VnKKn1EpmlcyB42cOcYgbnZGY/Gzac6MKy5FoiMoPbTM4mw9Hq50qiC5FYWZfk4kJbLxVVvqSf\nt7Lah9vtpPlUJ3WNpUk/vxCxcJgWucEQfe7k/snIyXUzb3ElB3e1sPbaBtp6/HzxhaNXHPf1O+ZT\n5Uv8tgtCRLOj572k3ItlWVy62E9JuTep5xYiEdymRTDJHRX1jaVsffYAlmlhJLlxJ7LDpHdLSqlN\nwLcixz6mtX5k1M8XAU8Aq4G/11r/S9TPTgLdQAgIaK3Xx6/qM9PR3kvtnJKkn9cwDJauqWH/zhZp\nYIm04w0E6Xe5krIR5GjL1tbyu18dZM01koVTpJZkrx2B8LWkYUE5J45ckAaWyAhO0yKYgEbO8NYj\n43HnuWhv66GyOnlJz0T2mLCBpZRyAo8DtwDNwDal1Fat9cGowy4Cfwn8yRhFWMANWuuUmeja0d7H\n8rV1tpx7yaoanvzX17np/YtxexKf6lqIeAmvv7LnM1vfUErAH6KtuRuS3MspxLgsK7IGK/mdDo1N\nFezbcVY6HURGcCWogTXe1iPDPlFVyNkTHdLAEgkx2d3KeuCo1vqk1joAPA3cHX2A1rpda70dCIxT\nRsqMvVqWReeFfkoq7On1KyjMpaq+iKMH2mw5vxAzFV5/5bbl3IbDYNmaOvZul2QXInnGWu8Xvem2\nM7IGKmTDqO6c+WU0n+okEJhatkMhUpnLNJM+RRCgpLaQM8dlHZZIjMk+0bXAmajnZyOvTZUF/E4p\ntVMp9enpVi7eBvrCbcB8r8e2OixbXcu+d5ptO78QM2FHBsFoy9bWcnhvKyG5oRRJMrzeL/rhD72X\nWMITssLrr2xoYOXmuamI9L4Lke4SNYI1mZLaIs6e7JDkYyIhJmtgxfqpu1ZrvRL4CPCQUur6GMuL\nSceFPkrK85OeQTDavCWVtDV309M1aFsdhJiuZO+BNZqvKJeaOcW0HZU9sURqsCODYLTGheWcOCzx\nINKfy7QIjdPAGl5HNd4jelR5unILcsjJc9PeNv46LSFmarI7pmagPup5PeFRrCnRWp+LfD2olHqG\n8JTD16ZbyXjpaO+l1KbpgcPcbicLl1dxYGczG26YZ2tdhJiK4FAQl2kx6LJ3/dOKdfX8/qV3IT/f\n1noIAfZkEIzW2FTBr36yC1hsWx2EiJVlWTgti+A4Hd+TraN6+JbGmM7fGEkYI+uwRLxNdnXYDixQ\nSjUopTzAfcBz4xx7WXQopfKVUr7I9xXAHcDeGOsbk44LfZSmQNalpWtq2P9Oi+xjItJCf0d/eHqg\njSO/AHObyhnoHsTrD9paDyHAngyC0Sqrffj9IS5NcVNkIVJRKBAKr2O06frSuLCCE4fbbTm3yGwT\njmBprYNKqU8Cz/BemvaDSqnPRH7+XaVUFbANKARMpdTngSVAJfBzpRSEMw1+U2v928T9Kpdr7Rmi\nrcd/2Wsnz3RRs3jWFWk7Yxlinonq+mIsLFrPdlFdX5zUcwsxXSMNLJs5nA5qF8+i7t0ODpcnfy87\nIaJ5bB7BGk7XfvxIO2uulmyCIj2F/CFb1l8Nq28s5VdP72JoMEBOrj2JnERmmvSuSWv9CuE9rqJf\n+27U961cPo1wWC+wKtYKztRYG5Je29zNs0EHfYcvzxoT6xDzdIX3xKpl345maWCJlNff0W/r+qto\nNUsqqd7ZzLulBZiyOaSwUbIaWGN1FgLM8nlobCpn/84WaWCJtBUK2NvAcnuc1Mwp4eS7F1m4vMq2\neojMkxp3TUlgWBa5wRD97tTYf2rJqhqeeuQNbrxzEa4UqZMQY+nvGEiJESyA/KI8enLcVPYP0VqQ\na3d1RBbzhEx63YmPi7E6CwG+fsd8muaX8Zuf7yMYCMl1RKSlkD80boKLZJnbVM6JI+3SwBJxlRp3\nTUmQFwgx5HRiJWme72Q7iAPkl3vZ+odTrLqqlipfTlLqJcS0WBb9nf30VpfaXZMRZ3151HUPSANL\n2MptmgRs2GQ4Wl6+h4oqH2dOdNDYVGFrXYSYifAIlr0JlBqbKnj71RNYlmVrlmmRWbKmgeUNBOlL\nYg/fZJlvAGYNWdS/doLqhRXSwBIpKSdk4nA6bF3MP9p5bw6LL3STFwgykIQRBCHG4gpZtt8YQvjm\n8MSRC9LAEmkp5A+Nm0EwWUrKvbjcTtpbeySboIgb+68OSeINhOhLkWlOw857c8L16pQsUCI1+fxB\n8ktTKy26ZRi0REaxhLBLKoxgQWQ/rCOSBU2kJ7vXYA1rbCqXbIIirrKmgZWf5BGsqbAMg2ZfLmf3\ntdldFSHGVJCCDSyAM4V51PYM4DBlqwNhD7dpEUiBEaxZ1YUE/CEunu+1uypCTJvdWQSHDY8ECxEv\n9l8dksTrD9GfgtOJmn15nDt8nmAgZHdVhLhCuIGVZ3c1rjDgdtGV46aqd9DuqohsZFm4TDMlGliG\nw6BpaRWH97baXRUhpi0UCBFKgTiqn1vK+XPdDA4E7K6KyBD2f6qTJNlrsKZqwO2isKKAI/tlFEuk\nnlQdwQI4U5TP7O5+kA27RZI5Ih850/6OdwAWrqjiyD5pYIn0E0yRESy320ntnBJOHb1od1VEhki9\nIZ0EcIdMDAtbN4WcSO2yKva8fYYlq2rsrooQIwzLwhsIkleSeiNYABfyPCw0LYqGpMdRJJfbNMMJ\nLlIk41hNfTGDAwEunu+lrLLA7uqILKL3tHHi4tjrYVfV+qgq8Ez4/lSZIgjD0wQlXbuIj6xoYOUH\ngvR7nClzMRytoqGEY384wYW2XspnycVRpIbhrQ2crtQb+QXAMDhTlMfsLkl2IZLLbZoEUuSmEN6b\nJnhkXytX3zTf7uqILLKzuYcdzWNvSZPrdkzewEqRJBcQThjzx1fY1JWnAAAgAElEQVSOY5kWRorU\nSaSv1BzSGUPItOjsD0zp0TMUvOy93kCIvhRcfzXM4XSwbG0te7efsbsqQowoCATp86Ro4yqipSCP\n8oEhhvr8dldFZBF3KDUSXERrWl7FYZkmKNJMOE17asRSSZmXPK+b5tOX7K6KyACp2+oYpc8f4h+3\nnqCtd/IbqU+vr6Es3z3y3OtPzfVX0ZZfVcePHn2T625twp3iN7UiO3j9IXpTuGMCIOh00OrN5ez+\nVtYvKLO7OiJLuFIkRXu02tnFDPYH6GjvpbRCZkKI9BBOcpE615lFy6s5tOccdQ0ldldFpLnU6DaY\noo6BABf7J38MBs3L3pcfSM0MgtGKS/OpmV3MgZ3NdldFCCCSGCbF9o4by5miPJr3txEKmZMfLEQc\nuM3U2GQ4muEwWLB0Fof3SsIkkT5SaQ0WRBLG7G3FlOuJiFFqXSESxJsGU50ArrqukR2vn8KSvX1E\nCkjVzJuj9Xrc5BfncXiPTI8SyeEOpUaK9tEWLpNsgiK9pNIaLAhPE/QV53LmRIfdVRFpbtLuaaXU\nJuBbkWMf01o/Murni4AngNXA32ut/2Wq700Gw7LIC4bod6V+T3xdYwluj5PjR9qZt6jS7uqIbGZZ\neP2pvXYxWsOaWt5+9TiLV1VjpGgyG5E53KaVclMEAWrnlNDf55dpgiJthEewUquzYtGKag7taWXO\n/HK7qyLS2ISfaqWUE3gcuAdYC3xKKbV41GEXgb8EvjGD9yZcXjCE3+nATKEekvEYhsHa6xrY/oeT\ndldFZLmckEnIYRBM0a0NRiubXYzDYXD8cLvdVRFZIFU2GR4tnE1QpgmKNGFZKTdFEGDh8ire3d9G\nKCjTBMXMTXaFWA8c1Vqf1FoHgKeBu6MP0Fq3a623A6M3o5n0vcmQn0a98BAO7M4LfbS1dNtdFZHF\nCtIgMUw0wzBYv2kub79ywu6qiCwQniKYWjeFwxYur+LQnnNYsgG3SHEOC3CAZdOsA4/TYHdLzxWP\nE/1Bcotz2bn3nC31EplhspZHLRCdO/wssGGKZcfy3rhJl3Ukw5xOB2uumcOO109yx70r7K6OyFLe\nQIjeNEhwEa1p2Sxee+kIZ092SgYokVBu00rI6O7wDV80/zQX29c2lBAKmZw700XN7OJ4Vk+IuHKZ\nJk4b7886BoL849axO+Xq/ZC3p5WrVtcmuVYiU0x2BxVLF1hKdJ/lp+GN4op19XzvG6/S0zWIryjX\n7uqILOT1B22Nm9aeIdp6rtySYaKbTYfTwbrrG3n71ePUNaxNZPVElkvURsNj3fA9fEvjtMowDIMV\n6+rZs+2MNLBESnNZFs4UTUDW5s2h/UQHgUAIdxp10ovUMVkXXDNQH/W8nvBI1FTE8t64SbcRLIDc\nPDeLV1Wz861TdldFZCm7M2+29fj54gtHr3j4QxP32yxbU0tbczftrT0THidELFwpuNFwtKVranh3\nfxtDg6Nn7guROpymhStF78/8LieFlQWckHW9YoYmu0JsBxYopRqUUh7gPuC5cY4d3Z03nfcmTH4g\nRH+ajWABrL2mgb3bzjI4IBdIkXzeQHqtXRzmcjtZc80ctr0qa7FE4rhNM6UTwHgLcmhYUM6BXbKG\nRKQul5m6I1gAsxaUc2iPxJCYmQmvEFrrIPBJ4BlgB/C41vqgUuozSqnPACilqpRSZ4AvAP+glDqt\nlCoY772J/GVGc5ombtNkMIUvhOMpLstn7qJKdrx+0u6qiCzjCpk4TIuhNIwbgFUb6jlxpJ3OC312\nV0VkIsvCZVopm+Ri2Ip19ex5+4wkuxApK7wGK3U78mbNK+fU0Yv09QzZXRWRhib9ZGutXyG8x1X0\na9+N+r6Vy6cCTvjeZMoPhOh3uyBN98W5+qZ5/OjRN1lzzRzy8j12V0dkiYJAkH6PM23jJifXzdpr\nG/jDS+9y1/2r7K6OyDAuy8I0DNsyn03V7LmlBPwhWs92UV0va7FE6hkZwUrRbOjuXBdNy6rYu/0s\nG2+cZ3d1RJpJ3a6DOEjH9VfRikvzWbB0FtteO8Gm9y20uzoizbzT3M2lgeBlr80qmLyh7vUH6U3h\nXsWxjM6+5plbxonXT7L/yAWWNslmkSJ+3KHU2GR4rIyDALN8Hqp8ORgOg+Xr6tj99hlpYImUNNLA\nGrS7JuNbvXE2z/zgHdZvasSRprM6hD3S6y5qmkZGsNLYxhvn8dQjb7D22ga8BTl2V0ekkZ/samP3\nud7LXrtlfgnvayqb8H3eQIi+NFu3OFb2tbr8XHwvH5UGloirVNlkeLwU01+/Yz5VvvC1YtmaWh7/\n5mvceOcicnLdya6iEBNyWlY4TftgyO6qjKuyppDC4lyOHjxP07Iqu6sj0oj9V4kE8vpDaT2CBVBY\nnMeildWyaF8kjTfNNhkeT7Mvj/7uQU4fu2h3VUQGcYdMgim+/mqY15fDnPnlHJRkFyIF2b0P1lSt\n3jiHnW+dtrsaIs1kdAMrPxBM+xEsgI03zGXfjmZ6u1N4HF1kjIJAMO1GsMZiGQbzN8zm1d8ckYX+\nIm7cZmqnaB9t1YZ6drxxEtOUGBCpJdWzCA5bsHQWHe19XGiT7T/E1KXPVWKaLMuKpJpO/eCdTEFh\nLkvX1PDH/33c7qqIDOcwLTwhkwFX+scNhNPsmqbFu/vb7K6KyBAu0ySQRmsx6ueWkpfv4cjeVrur\nIsRl0qWB5XQ5WLGuTkaxxLSkz1Vimvz9AUyDlN6rZDo2bJ7Hob2tsoGqSChvIEi/y5XyGdKmyjAM\nrr9tAa/95gjBYIqmqhJpxZ0GKdqjGYbBxhvn8ebvj2HJKJZIIS7TSospggAr19dzeE+rbN4tpiwz\nWh9j6Ls0kJYbpY4nv8DDNTfN43e/PCjTnUTCeANB+pLcozicDS364Q/FrzHU2FRBaWUB216VEWAR\nu/AarPS6dDY2leNyO3j3gIzkitThMk1caTCCBeGZRHPml7H/neaR11p7hq64dkU/WmX/rKyWOS2Q\nUfo7B+hPk56RqVq5vp49289yeE8ri1ZW210dkYHCiWGS+2dhrGxoD9/SGNdz3HzXYn7w7TdYtKKa\nknJvXMsW2cVtWgy402cEC8KjWFffNJ83Xj7KgqWzMDJkhFqkN2cajWABrL22gV8+vYsV62fjcjlo\n6/HzxReOjnt8dEZPkX3SqxtuGvovDWTEQv1oDqeDWz6whP/94iH8Q8HJ3yDENNkxgpUMhcV5rN88\nl63PHZARYBETd4qkaZ+ueYsqsCyL44fb7a6KEEB40+50WIM1rGZ2MRVVPva8LWuxxOQyqwUSpe9S\n+oxgjbdh5FhmleYze14Zb/7+GJu3yObDIr4K/EFOFGfmCM/aa+ZwYFcLh/acY/HKGrurI9KUO2Sm\n1RqsYYZhsPGGebz5u2PMXVgho1jCdumS5CLadbcu4D+e2M6ytXV2V0WkuIxtYPVfGqCvID1uFMfb\nMHIsX79jPpu3LOTJf/0Dy9bUUlZZkODaiWxhhkzygsmfIpgsDqeD2/5kKb/44U4amyrIzZONV8X0\nuUwrrbIIRmtaOos3Xj7KyXcv0NhUYXd1RJZL9X2wxu78NiisLuT53xyhbnWtLfUS6SEj76Qs02Sw\nZ4j+kkK7q5IQXl8OV980j98+s4/7HlyPI00v9iK1DHQOMOByYqZh7/xUVdcXM39xJa/++jC3/eky\nu6sj0pDbTL8kF8MMh8G1t8znlRcPM3teGU65dgibGJaFwwKHK3U/g+N1fucHLNa/fYaSpkobaiXS\nRep+smPg7/WT483JmFTTY1m9cQ5Op4M/vjK1kS8hJtN7oY/unMwf1dm0pYlTRy9yZJ/sCySmzx1K\nrzTtoy1YOouCwlzeeeOU3VURWcxpWgQdRlpOVe13u2jz5tKyu8XuqogUNukIllJqE/CtyLGPaa0f\nGeOYrwB3Av3Ax7XWhyKvnwS6gRAQ0Fqvj1/Vx+fvHiS/ODcZp7KN4TC4/d4V/ODbb9CwoIzq+mK7\nqyTSXN+FPrpzMnJQ+zI5uW7e/+GVPPPUO8yqLaKoJM/uKok0YZomTit8Y5iuDMPg5g8s5sf/6y0W\nrajGV5TZ10qRmlxmesfR8RIvjYfO45lVjN+VutMchX0mHMFSSjmBx4F7gLXAp5RSi0cdcwewUmu9\nAvg88GTUjy3gBq316mQ1rgD8XYPkF2f+TZOvKJebP7CE5/UeySooYtZ3sY8eT+aPYEF4quBV1zfy\n/E93Y8Zxzy2R2UJDofDoVRr2ukcrKfOyauNsfv/8QburIrKUyzIJGuk7iWrI5aRiYQVNHb12V0Wk\nqMk+3euBo1rrk1rrAPA0cPeoYz4AfB9Aa/1HoFgpNSvq50m/Eg11D+LNkl7phcurqGso4Xe/kgul\niIFl0X+xn54sGMEatu66Bjw5Lt54efx9TISIFhwKpu36q9HWb57L+ZYeThyRtO0i+dJ9BAugfm0d\nxYMByvtlQ2FxpcmuFLXAmajnZyOvTfUYC/idUmqnUurTsVR0OsJTBLOjgQVw0/sXc/ZEJ4d2n7O7\nKiJNeQMh3PnujLl5nIrwNNvl7HunmVNHL9hdHZEGgkNBAs70vikc5nY7uemuxbz83EGCgZDd1RFZ\nxmVahNK8geV0O9lfUciS9m5cMhNCjDLZ3dRUd+QcL0qu1VqvBD4CPKSUun7KNYuBv3sIbxY1sDw5\nLj7wkVW8/MsDtJy+ZHd1RBryDQUoKE+PbQ3iobVniN0tPRzt9rPwpgU8+5PdvL6vjdYe6YkU4wsO\nBdNyk+HxzF1YQWWNj1d/c8Tuqogs48yAESyAzjwP5705LLw4tb1MRfaY7ErRDNRHPa8nPEI10TF1\nkdfQWp+LfD0IPEN4ymFCOU2TUCBEToEn0adKKZU1hWz50HKe/dFOLnX0210dkWYK/UG8ZdnTwGrr\n8fPFF47yxReO8pU97ewoyON3ejenzslFUowvOLwGK4Pc9qfLOHbwPIf3SlZNkTyuNN7uYLR3Swso\nkamCYpTJPt3bgQVKqQallAe4D3hu1DHPAX8OoJTaCFzSWrcppfKVUr7I6xXAHcDeuNZ+DEVDQXJL\n8tIy9Wes5i2qZMPmRn7+/R0MDgTsro5II4VDAbxZNII12jlfHmd9+ez61UFJGCPGFRwKEsywvaNy\n89zc9ZFVbH12Px3tsmBfJEcmrMEaFnI4ZKqguMKEVwqtdRD4JOHRpx3A41rrg0qpzyilPhM55gVg\nj1JqL+F07p+IvL0KeE0ptYtwcoxvaq1/m6DfY0TRoJ+8ioJEnyZlrbmmgTnzynjux7sISaCLqbAs\nfP5gVjewAE4U51NU5eO5H++U2BFjyrQpgsOqaou47rYmnv3RLvx+6WAQiecyLUIZ1BHemeehzZvL\nivNdGNZUV9eITDZpyjCt9SvA6lGvfXfU878D/m7Ua8eBVXGo47QUDwbIz+IGFsCNdy7i2R/v4pc/\n3sX771+FK4V3Shf2ywuahAwDd152pGgfl2GwcNNcTv7+KL/6yW7u/PBKiR1xmdBQMOOmCA5bsa6O\n5lOdbP3FAW6/d3lWzgIRyeMyLQYz7O/rkbICVrVeYtGFHg6W++yujrBZZn26LYuioQB5FdndE+9w\nOvjA/aswHAa/+MEOAn7JECXG5xsKZFV69ok4HAZ33b8KwwHPPLVDpguKywT9qT+C5XEa7G7pueJx\nomNgzNeHE7sYhsGtdy+lva1Hti4QCee0zIyZIjjMMgz2zCqieCjA7C5ZC5/tMuquKj8QIuQwcOdn\nV4KLsThdDu768Ep+/Z/7+M8nt3PPx9bikZtoMYZCf5DuLNlgeCpcLgfv//AqXvrFfn72+Dbu+dha\n8uRvigCCg6GUT9PeMRDkH7eeuOL1h29pHPP1r98xnypfDgBuj5MPffwqfvq9t3E6HWy8cV7C6yuy\nU3gNVmp3VsxEyOFgZ1Ux65s7OH/8ItTISFa2yqhPd/FQgEs5mX2jOF7v5FiP8/0Bbv/QcsoqC9D/\nvo0+SUEtxuAbCtAtje/LOBwGt/3pUmrnlPDTx96m+9KA3VUSKSDoz5yNhsfj9eWgPrWO/Tub+eMr\nx+2ujshQmZTkYrRBl5NdVcUc+N1RTh+/aHd1hE0y6q6qeDBAV25m9zSP1zs5luGeyVvuXsLrW4/y\nw0ff5P0fXkntnJIE11Kkk0J/kJ4M75iYCcMw2Hz7Qry+HH746Jts+eBy5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Cml3jeDop4A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\nhBBCCCGEiJP/H3SisuAqZ+qTAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107deed50>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "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=np.sqrt(len(y)), normed=True)\n",
      "plt.xlabel('y'); plt.ylabel('Pr(y)')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "<matplotlib.text.Text at 0x1080809d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x107f98750>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "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 the set of 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": 15,
       "text": [
        "0.041303093412337726"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "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": 16,
       "text": [
        "12.681664407577044"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "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": 17,
       "text": [
        "8.058131431782682"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "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",
      "plt.xlabel('$\\lambda$')\n",
      "plt.ylabel('L($\\lambda$|x={0})'.format(x));"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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rW3k0lPlDqdfrZV11CyWTrPPdBCYjOYFlJxby61d3021zT0yIeLze6P5lW7Vq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\njBkxKybGGGNG7P8DOdAMqwWw/ioAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108284d90>"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "How is the likelihood function different than the probability distribution (or mass) function? 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 PMF of the Poisson for $\\lambda=5$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lam = 5\n",
      "xvals = np.arange(15)\n",
      "yvals = [poisson_like(x, lam) for x in xvals]\n",
      "plt.bar(xvals, yvals)\n",
      "plt.xlabel('x')\n",
      "plt.ylabel('Pr(X|$\\lambda$=5)');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x107f1c850>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "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": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here is a graphical example of how Newton-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",
      "plt.plot(xvals, func(xvals))\n",
      "plt.text(5.3, 2.1, '$f(x)$', fontsize=16)\n",
      "\n",
      "# zero line\n",
      "plt.plot([0,6], [0,0], 'k-')\n",
      "\n",
      "# value at step n\n",
      "plt.plot([4,4], [0,func(4)], 'k:')\n",
      "plt.text(4, -.2, '$x_n$', fontsize=16)\n",
      "\n",
      "# tangent line\n",
      "tanline = lambda x: -0.858 + 0.626*x\n",
      "plt.plot(xvals, tanline(xvals), 'r--')\n",
      "\n",
      "# point at step n+1\n",
      "xprime = 0.858/0.626\n",
      "plt.plot([xprime, xprime], [tanline(xprime), func(xprime)], 'k:')\n",
      "plt.text(xprime+.1, -.2, '$x_{n+1}$', fontsize=16)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": [
        "<matplotlib.text.Text at 0x103db3a10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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0YNp13WnXLMXsEMVJSMIXQpwRz3/Xs7VNN5764QM0d2Qw9rwcLu94FjaZVRP1JOELIU5L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\nIkFIwhdCiATx/wH2X5/1vJNLBwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107fcacd0>"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "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": 22
    },
    {
     "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(np.log)\n",
      "mean_log = precip.apply(np.log).mean()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 23
    },
    {
     "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": 24,
       "text": [
        "3.5189679152399647"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And now plug this back into the solution for beta:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$ \\beta  = \\frac{\\alpha}{\\bar{X}} $$\n",
      "</div>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "beta_mle = alpha_mle/precip.mean()[-1]\n",
      "beta_mle"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 25,
       "text": [
        "0.84261607548413797"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "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 = np.linspace(0, dec.max())\n",
      "plt.plot(x, gamma.pdf(x, alpha_mom[-1], beta_mom[-1]), 'm-')\n",
      "plt.plot(x, gamma.pdf(x, alpha_mle, beta_mle), 'r--')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 26,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1082c0e90>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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oeO/HdtDK5DtvGsRVWVyYEBaQ0O+lid3ciZtWJNM1g01+Fnx0PMxbaQzTFaKE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\ngA6+IhxszAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x107eb5c90>"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "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",
      "ahat, _, bhat = gamma.fit(precip.Dec, floc=0)\n",
      "ahat, 1./bhat"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 27,
       "text": [
        "(3.5189679152399753, 0.84261607548414119)"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that SciPy's `gamma.fit` method fits a slightly different version of the gamma distribution, which has to be transformed to match ours."
     ]
    },
    {
     "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": 28,
       "text": [
        "array([ 3.72458625,  8.02504424,  2.5152199 ,  1.15249687,  3.69592197,\n",
        "        4.81974995,  7.21932052,  4.78104845,  2.67680421,  9.90455954,\n",
        "        8.162251  ,  7.12880674,  6.79736262,  5.98943549,  7.99250998])"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.stats import norm\n",
      "\n",
      "# Create an array of x-valuese``````````````````````````````\n",
      "x = np.linspace(y.min()-1, y.max()+1, 100)\n",
      "\n",
      "# Smoothing parameter\n",
      "s = 0.4\n",
      "\n",
      "# Calculate the kernels\n",
      "kernels = np.transpose([norm.pdf(x, yi, s) for yi in y])\n",
      "plt.plot(x, kernels, 'k:')\n",
      "plt.plot(x, kernels.sum(1))\n",
      "plt.plot(y, np.zeros(len(y)), 'ro', ms=10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 29,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1088d5150>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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X0xkYvHOgAA+PiiY5CRYIIW4Zj4KIuwshkTKuY2CkH86Xcyfu285XItRXjGkp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\nILpx19HR0emA6MZdR0dHpwPyf+CxfNeB3HhrAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1087de410>"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "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, 2, 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": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(x, bins=10, normed=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "(array([ 0.00940678,  0.02822033,  0.05644066,  0.13169488,  0.09406777,\n",
        "         0.1222881 ,  0.1222881 ,  0.23516943,  0.08466099,  0.05644066]),\n",
        " array([-4.36396175, -3.30089839, -2.23783504, -1.17477168, -0.11170832,\n",
        "         0.95135503,  2.01441839,  3.07748174,  4.1405451 ,  5.20360846,\n",
        "         6.26667181]),\n",
        " <a list of 10 Patch objects>)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x1087e1050>"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "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=10, normed=True)\n",
      "plt.plot(xgrid, density(xgrid), 'r-')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 32,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1088eb9d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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F1ttvU9qwodORRKQGUNFHgHfzZlIefJC47dvJfewxijt0cDqSiNQgWroJo7iv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\nQyLdAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1089013d0>"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "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",
      "plt.plot(x,y,'ro')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 33,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108a6b1d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXAAAAECCAYAAADn84z1AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nZhFnAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108a47490>"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "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>\n",
      "\n",
      "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": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ss([0,1],x,y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 35,
       "text": [
        "333.35000000000002"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.optimize import fmin\n",
      "\n",
      "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": 36,
       "text": [
        "(-4.3500136038870876, 3.0000002915386412)"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(x, y, 'ro')\n",
      "plt.plot([0,10], [b0, b0+b1*10])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 37,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108a6ff50>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108a6fa10>"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(x, y, 'ro')\n",
      "plt.plot([0,10], [b0, b0+b1*10])\n",
      "for xi, yi in zip(x,y):\n",
      "    plt.plot([xi]*2, [yi, b0+b1*xi], 'k:')\n",
      "plt.xlim(2, 9); plt.ylim(0, 20)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 38,
       "text": [
        "(0, 20)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108b45050>"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "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",
      "plt.plot(x, y, 'ro')\n",
      "plt.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": 39,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108a7a990>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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73XnwjpJ6lt06hbREzYyf7KN18SV5eUQ0N+ONiWH2okVaLxeRbutpmZcAo0/6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\nREKAylxEJAT8L0MvKusrK9b6AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108a7a890>"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "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",
      "plt.plot(x, y, 'ro')\n",
      "xvals = np.linspace(0, 10, 100)\n",
      "plt.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": 40,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108a3c290>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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h3WEvIiKhQzNoRUTCgMJeRCQMuDXOvk2MMVOBh1vqeNpa+5iX9XjFGJMCvAgM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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108c1e310>"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "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",
      "plt.plot(bb.hr, bb.rbi, 'r.')\n",
      "b0,b1,b2,b3 = fmin(ss3, [0,1,-1,0], args=(bb.hr, bb.rbi))\n",
      "xvals = np.arange(40)\n",
      "plt.plot(xvals, b0 + b1*xvals + b2*(xvals**2) + b3*(xvals**3))"
     ],
     "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": 41,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108b77690>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108b77890>"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "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>Sun, 23 Feb 2014</td> <th>  Prob (F-statistic):</th>  <td>0.00463</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Time:</th>                 <td>10:40:26</td>     <th>  Log-Likelihood:    </th> <td> -12.325</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>No. Observations:</th>      <td>     6</td>      <th>  AIC:               </th> <td>   28.65</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Df Residuals:</th>          <td>     4</td>      <th>  BIC:               </th> <td>   28.23</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>   \n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
        "</tr>\n",
        "</table>\n",
        "<table class=\"simpletable\">\n",
        "<tr>\n",
        "    <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th> <th>[95.0% Conf. Int.]</th> \n",
        "</tr>\n",
        "<tr>\n",
        "  <th>const</th> <td>   -4.3500</td> <td>    2.937</td> <td>   -1.481</td> <td> 0.213</td> <td>  -12.505     3.805</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>x1</th>    <td>    3.0000</td> <td>    0.525</td> <td>    5.716</td> <td> 0.005</td> <td>    1.543     4.457</td>\n",
        "</tr>\n",
        "</table>\n",
        "<table class=\"simpletable\">\n",
        "<tr>\n",
        "  <th>Omnibus:</th>       <td>   nan</td> <th>  Durbin-Watson:     </th> <td>   2.387</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Prob(Omnibus):</th> <td>   nan</td> <th>  Jarque-Bera (JB):  </th> <td>   0.570</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Skew:</th>          <td> 0.359</td> <th>  Prob(JB):          </th> <td>   0.752</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Kurtosis:</th>      <td> 1.671</td> <th>  Cond. No.          </th> <td>    17.9</td>\n",
        "</tr>\n",
        "</table>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "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:                Sun, 23 Feb 2014   Prob (F-statistic):            0.00463\n",
        "Time:                        10:40:26   Log-Likelihood:                -12.325\n",
        "No. Observations:                   6   AIC:                             28.65\n",
        "Df Residuals:                       4   BIC:                             28.23\n",
        "Df Model:                           1                                         \n",
        "Covariance Type:            nonrobust                                         \n",
        "==============================================================================\n",
        "                 coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
        "------------------------------------------------------------------------------\n",
        "const         -4.3500      2.937     -1.481      0.213       -12.505     3.805\n",
        "x1             3.0000      0.525      5.716      0.005         1.543     4.457\n",
        "==============================================================================\n",
        "Omnibus:                          nan   Durbin-Watson:                   2.387\n",
        "Prob(Omnibus):                    nan   Jarque-Bera (JB):                0.570\n",
        "Skew:                           0.359   Prob(JB):                        0.752\n",
        "Kurtosis:                       1.671   Cond. No.                         17.9\n",
        "==============================================================================\n",
        "\n",
        "Warnings:\n",
        "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
        "\"\"\""
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "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>Sun, 23 Feb 2014</td> <th>  Prob (F-statistic):</th>  <td>0.0191</td> \n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Time:</th>                 <td>10:40:27</td>     <th>  Log-Likelihood:    </th> <td> -11.056</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>No. Observations:</th>      <td>     6</td>      <th>  AIC:               </th> <td>   28.11</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Df Residuals:</th>          <td>     3</td>      <th>  BIC:               </th> <td>   27.49</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>   \n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
        "</tr>\n",
        "</table>\n",
        "<table class=\"simpletable\">\n",
        "<tr>\n",
        "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th> <th>[95.0% Conf. Int.]</th> \n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Intercept</th> <td>  -11.0748</td> <td>    6.013</td> <td>   -1.842</td> <td> 0.163</td> <td>  -30.211     8.062</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>x</th>         <td>    6.0577</td> <td>    2.482</td> <td>    2.441</td> <td> 0.092</td> <td>   -1.840    13.955</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>I(x ** 2)</th> <td>   -0.3027</td> <td>    0.241</td> <td>   -1.257</td> <td> 0.298</td> <td>   -1.069     0.464</td>\n",
        "</tr>\n",
        "</table>\n",
        "<table class=\"simpletable\">\n",
        "<tr>\n",
        "  <th>Omnibus:</th>       <td>   nan</td> <th>  Durbin-Watson:     </th> <td>   2.711</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Prob(Omnibus):</th> <td>   nan</td> <th>  Jarque-Bera (JB):  </th> <td>   0.655</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Skew:</th>          <td>-0.809</td> <th>  Prob(JB):          </th> <td>   0.721</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Kurtosis:</th>      <td> 2.961</td> <th>  Cond. No.          </th> <td>    270.</td>\n",
        "</tr>\n",
        "</table>"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 43,
       "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:                Sun, 23 Feb 2014   Prob (F-statistic):             0.0191\n",
        "Time:                        10:40:27   Log-Likelihood:                -11.056\n",
        "No. Observations:                   6   AIC:                             28.11\n",
        "Df Residuals:                       3   BIC:                             27.49\n",
        "Df Model:                           2                                         \n",
        "Covariance Type:            nonrobust                                         \n",
        "==============================================================================\n",
        "                 coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
        "------------------------------------------------------------------------------\n",
        "Intercept    -11.0748      6.013     -1.842      0.163       -30.211     8.062\n",
        "x              6.0577      2.482      2.441      0.092        -1.840    13.955\n",
        "I(x ** 2)     -0.3027      0.241     -1.257      0.298        -1.069     0.464\n",
        "==============================================================================\n",
        "Omnibus:                          nan   Durbin-Watson:                   2.711\n",
        "Prob(Omnibus):                    nan   Jarque-Bera (JB):                0.655\n",
        "Skew:                          -0.809   Prob(JB):                        0.721\n",
        "Kurtosis:                       2.961   Cond. No.                         270.\n",
        "==============================================================================\n",
        "\n",
        "Warnings:\n",
        "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
        "\"\"\""
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "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",
      "plt.plot(x, y, 'ro')\n",
      "xvals = np.linspace(0, max(x), 100)\n",
      "plt.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": 44,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108caa450>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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GGC/wCPA5YD/wpjHm99baD+NZ3HDjC4TY3dRFZWMHlY2dVDZ0UN3UxcRRWZxc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\nAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108c56bd0>"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "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": 45
    },
    {
     "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": 46,
       "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": 46
    },
    {
     "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",
      "plt.yticks([0,1])\n",
      "plt.ylabel(\"survived\")\n",
      "plt.xlabel(\"log(fare)\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 47,
       "text": [
        "<matplotlib.text.Text at 0x108f06050>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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44dI1TQghxGJa0Ajf9/3S6bX4X1vi9gghhFgiCy3pXPJ9/3Xg28D3giAYX7om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\nEQr4IiKJUMAXEUmEAr6ISCK+ABL0mrf1qUVxAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108ebaa10>"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "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": 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",
      "plt.yticks([0,1])\n",
      "plt.ylabel(\"survived\")\n",
      "plt.xlabel(\"log(fare)\")\n",
      "plt.plot([0,7], [betas_titanic[0], betas_titanic[0] + betas_titanic[1]*7.])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 49,
       "text": [
        "[<matplotlib.lines.Line2D at 0x108f09050>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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0WHY9sYoEnxXtkhY9CXwhxJxSSvHKxT6+cbSNX749j4dXpssJl3NEAl8IMWdG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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108f57b50>"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we look at this data, we can see that for most values of `fare`, there are some individuals that survived and some that did not. However, notice that the cloud of points is denser on the \"survived\" (y=1) side for larger values of fare than on the \"died\" (y=0) side."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Stochastic model\n",
      "\n",
      "Rather than model the binary outcome explicitly, it makes sense instead to model the *probability* of death or survival in a **stochastic** model. Probabilities are measured on a continuous [0,1] scale, which may be more amenable for prediction using a regression line. We need to consider a different probability model for this exerciese however; let's consider the **Bernoulli** distribution as a generative model for our data:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$f(y|p) = p^{y} (1-p)^{1-y}$$ \n",
      "</div>  "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "where $y = \\{0,1\\}$ and $p \\in [0,1]$. So, this model predicts whether $y$ is zero or one as a function of the probability $p$. Notice that when $y=1$, the $1-p$ term disappears, and when $y=0$, the $p$ term disappears.\n",
      "\n",
      "So, the model we want to fit should look something like this:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$p_i = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$\n",
      "</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "However, since $p$ is constrained to be between zero and one, it is easy to see where a linear (or polynomial) model might predict values outside of this range. We can modify this model sligtly by using a **link function** to transform the probability to have an unbounded range on a new scale. Specifically, we can use a **logit transformation** as our link function:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$\\text{logit}(p) = \\log\\left[\\frac{p}{1-p}\\right] = x$$\n",
      "</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here's a plot of $p/(1-p)$"
     ]
    },
    {
     "cell_type": "code",
     "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": 50,
       "text": [
        "[<matplotlib.lines.Line2D at 0x109000b90>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x108befd50>"
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "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": 51,
       "text": [
        "[<matplotlib.lines.Line2D at 0x109093950>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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A9/Fxj49bSy4PADe49LX9Yq2tBzDGDKerKFvcTdQ7a+3L1lqvtdYL/JsgvUvX\nWrvLWvuB2zn64M8RF66z1q4Hqt3O4Yu1ttxa+173zw8A7wIT3E3VO2ttU/dP4+h6d9bqYpxeGWMy\ngIuA3/Ppmzg/YdAL3RizhK63DVsH+2sfKWPMnUA5cBZwn8tx/PF14Bm3Q4Qgf464kKPQPeOdCbzl\ndpbeGGOGGWO2APuBh6y1Rb5+jwseBK4HvL4GDsgaujHmZWB8L790K11LAp/r8THXbjTqJ+ct1tp/\nWmtv7S71O4F7gB8OasBuvnJ2j7kVqLfW/m1Qw/XgT84gpb27A8AYE0fXu50fWmsb3c7Tm+53trO7\nb5583hjzhrV2s8uxDjHGLKLrmslmY8w5vsYPSKFba8/v7ePGmBOAycAWYwx0LQ9sMsbMtdZWDESW\n/vSV87AxTd0Xo/40CJH6ytBvTmPMl+l6S5YzKIH64M+fZ5Dy54gLOQLGmBHAU8CfrbVB/67RWltg\njHmerrX0oCl04Azg4u5jymOABGPMH621V/c2eFB3uVhrtwOHjtY1xuwD5gTpLpcsa+2H3WvoV9B1\nUS/oGGMuoOvt2PzuizyhINiOf/DniAvxkzEmAvgDsMNau9ztPH0xxiQDHdbaGmPMWOBC4Psux/oE\na+0twC1w6K79n/RV5uD+PvRgfqt7tzFmG7CBrv/4fuRynr6soOuCzivGmM3GmN+6Hag3xpj/MsYU\nAfOA54wxL7id6WPW2g7gGmA1sAlYGYzHVxhj/kLX9+M0Y0yRMeYrbmfqw5nAVcC53d+Tm7snHsEm\nDVjTvYb+f8AD1tpclzP50m9n6tZ/EZEw4fYMXUREAkSFLiISJlToIiJhQoUuIhImVOgiImFChS4i\nEiZU6CIiYUKFLiISJv4fddHY0FiKHEgAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108f74950>"
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The inverse of the logit transformation is:\n",
      "\n",
      "<div style=\"font-size: 150%;\">  \n",
      "$$p = \\frac{1}{1 + \\exp(-x)}$$\n",
      "</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So, now our model is:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$\\text{logit}(p_i) = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$\n",
      "</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can fit this model using maximum likelihood. Our likelihood, again based on the Bernoulli model is:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$L(y|p) = \\prod_{i=1}^n p_i^{y_i} (1-p_i)^{1-y_i}$$\n",
      "</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "which, on the log scale is:\n",
      "\n",
      "<div style=\"font-size: 120%;\">  \n",
      "$$l(y|p) = \\sum_{i=1}^n y_i \\log(p_i) + (1-y_i)\\log(1-p_i)$$\n",
      "</div>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can easily implement this in Python, keeping in mind that `fmin` minimizes, rather than maximizes functions:"
     ]
    },
    {
     "cell_type": "code",
     "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": 52
    },
    {
     "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": 53
    },
    {
     "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": 54,
       "text": [
        "(-0.88238984528338194, 0.012452067664164127)"
       ]
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "jitter = np.random.normal(scale=0.01, size=len(x))\n",
      "plt.plot(x, y+jitter, 'r.', alpha=0.3)\n",
      "plt.yticks([0,.25,.5,.75,1])\n",
      "plt.xvals = np.linspace(0, 600)\n",
      "plt.plot(xvals, invlogit(b0+b1*xvals))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 55,
       "text": [
        "[<matplotlib.lines.Line2D at 0x1090b9a10>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BwFq73xgzFbgO+MwQ1VtERIbAgCMBa20BWEp8Ff8KsMxau84Yc4cx5o6Soh8EnrbWdpZs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\nAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10909de50>"
       ]
      }
     ],
     "prompt_number": 55
    },
    {
     "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>Sun, 23 Feb 2014</td> <th>  Deviance:          </th> <td>  1654.0</td>\n",
        "</tr>\n",
        "<tr>\n",
        "  <th>Time:</th>               <td>10:40:31</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": 56,
       "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:                Sun, 23 Feb 2014   Deviance:                       1654.0\n",
        "Time:                        10:40:31   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": 56
    },
    {
     "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",
      "The NumPy function `permutation` can be used 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": 57,
       "text": [
        "array([u'Ahlin, Mrs. Johan (Johanna Persdotter Larsson)',\n",
        "       u'McCormack, Mr. Thomas Joseph', u'Reynaldo, Ms. Encarnacion',\n",
        "       u'Cumings, Mr. John Bradley',\n",
        "       u'Coutts, Master. Eden Leslie \"Neville\"'], dtype=object)"
       ]
      }
     ],
     "prompt_number": 57
    },
    {
     "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": 58,
       "text": [
        "308     White, Mrs. John Stuart (Ella Holmes)\n",
        "1004           McDermott, Miss. Brigdet Delia\n",
        "1126                  Petranec, Miss. Matilda\n",
        "773                          Dimic, Mr. Jovan\n",
        "945                              Lam, Mr. Ali\n",
        "Name: name, dtype: object"
       ]
      }
     ],
     "prompt_number": 58
    },
    {
     "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",
      "_ = plt.hist(s, bins=30)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x1090b9bd0>"
       ]
      }
     ],
     "prompt_number": 59
    },
    {
     "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": 60,
       "text": [
        "-0.26109460708189697"
       ]
      }
     ],
     "prompt_number": 60
    },
    {
     "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": 61,
       "text": [
        "0.089079064843898145"
       ]
      }
     ],
     "prompt_number": 61
    },
    {
     "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": 62,
       "text": [
        "-0.011697963201919293"
       ]
      }
     ],
     "prompt_number": 62
    },
    {
     "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": 63,
       "text": [
        "array([-1.21473328, -1.20296609, -1.14257574, -1.12029849, -1.09395259,\n",
        "       -1.03641168, -1.03146359, -1.01978558, -1.00359632, -0.94009612])"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s_sorted[-10:]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 64,
       "text": [
        "array([ 0.39140486,  0.39440803,  0.40575741,  0.40891146,  0.43260227,\n",
        "        0.43500618,  0.49686382,  0.54923955,  0.55645112,  0.59660475])"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "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": 65,
       "text": [
        "array([-0.84960042,  0.28246507])"
       ]
      }
     ],
     "prompt_number": 65
    }
   ],
   "metadata": {}
  }
 ]
}