Kolmogorov-Smirnov Test.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Kolmogorov-Smirnov (K-S) test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The K-S test (Chakravart, Laha, and Roy, 1967) is used to decide if a sample comes from a population with a specific distribution. The K-S test is based on the empirical distribution function (ECDF). Given N ordered data points $Y_1$, $Y_2$, ..., $Y_N$, the ECDF is defined as:\n",
    "\n",
    "$EN=n(i)/N$\n",
    "\n",
    "where $n(i)$ is the number of points less than $Y_i$ and the $Y_i$ are ordered from smallest to largest value. This is a step function that increases by $1/N$ at the value of each ordered data point. The graph below is a plot of the empirical distribution function with a normal cumulative distribution function for 100 normal random numbers. The K-S test is based on the maximum distance between these two curves."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import Image\n",
    "import matplotlib.pyplot as plt\n",
    "import math\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from statsmodels.distributions.empirical_distribution import ECDF\n",
    "import yfinance as yf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(100)\n",
    "\n",
    "mu = 0\n",
    "sigma = 1\n",
    "n_bins = 50\n",
    "x = np.random.normal(mu, sigma, size=50)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(8, 4))\n",
    "\n",
    "# plot the cumulative histogram\n",
    "n, bins, patches = ax.hist(x, n_bins, density=True, histtype='step',\n",
    "                           cumulative=True, label='Empirical')\n",
    "\n",
    "# Add a line showing the expected distribution.\n",
    "y = ((1 / (np.sqrt(2 * np.pi) * sigma)) *\n",
    "     np.exp(-0.5 * (1 / sigma * (bins - mu))**2))\n",
    "y = y.cumsum()\n",
    "y /= y[-1]\n",
    "\n",
    "ax.plot(bins, y, 'k--', linewidth=1.5, label='Theoretical')\n",
    "\n",
    "# tidy up the figure\n",
    "ax.grid(True)\n",
    "ax.legend(loc='right')\n",
    "ax.set_title('Cumulative step histograms')\n",
    "ax.set_xlabel('Sample')\n",
    "ax.set_ylabel('Likelihood of occurrence')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Characteristics and Limitations of the K-S Test\t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An attractive feature of this test is that the distribution of the K-S test statistic itself does not depend on the underlying cumulative distribution function being tested. Another advantage is that it is an exact test (the chi-square goodness-of-fit test depends on an adequate sample size for the approximations to be valid). Despite these advantages, the K-S test has several important limitations:\n",
    "\n",
    "1. It only applies to continuous distributions.\n",
    "\n",
    "2. It tends to be more sensitive near the center of the distribution than at the tails.\n",
    "\n",
    "3. The distribution must be fully specified. That is, if location, scale, and shape parameters are estimated from the data, the critical region of the K-S test is no longer valid. It typically must be determined by simulation.\n",
    "\n",
    "Several goodness-of-fit tests, such as the Anderson-Darling test and the Cramer Von-Mises test, are refinements of the K-S test. As these refined tests are generally considered to be more powerful than the original K-S test, many analysts prefer them. Also, the advantage for the K-S test of having the critical values be indpendendent of the underlying distribution is not as much of an advantage as first appears. This is due to limitation 3 above (i.e., the distribution parameters are typically not known and have to be estimated from the data). So in practice, the critical values for the K-S test have to be determined by simulation just as for the Anderson-Darling and Cramer Von-Mises (and related) tests.\n",
    "\n",
    "Note that although the K-S test is typically developed in the context of continuous distributions for uncensored and ungrouped data, the test has in fact been extended to discrete distributions and to censored and grouped data."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example: Goldman Sachs\n",
    "The goal is to determine whether or not the simple return of the stock of the company goldman sachs is a normal distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "pandas_type = \"pandas.core.frame.DataFrame\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def download_stock_data(ticker: str) -> (str, pandas_type):\n",
    "    \"\"\"\n",
    "    Downloads and reshapes stock data based on the yfinance library.\n",
    "    \"\"\"\n",
    "    stock_data = yf.download(ticker)\n",
    "    columns_to_remove = [col for col in stock_data.columns if col != \"Adj Close\"] #lists all column not 'Adj Close'\n",
    "    stock = stock_data.drop(columns = columns_to_remove)\n",
    "    stock = stock[-260:] #260 is the expected number of business days in 1 year (conventional time frame to compute VaR)\n",
    "    return ticker, stock"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculate_stock_return(stock: pandas_type) -> pandas_type:\n",
    "    \"\"\"\n",
    "    Calculates daily returns over the chosen timeframe.\n",
    "    Plots stock returns.\n",
    "    \"\"\"\n",
    "    stock = stock[1:].values / stock[:-1] - 1 #computes simple stock returns\n",
    "    stock.dropna() #drops all NAs\n",
    "    return stock"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def standard_deviation(stock: pandas_type) -> float:\n",
    "    \"\"\"\n",
    "    Calculates standard deviation of stock over the chosen timeframe.\n",
    "    \"\"\"\n",
    "    mean = stock.mean()\n",
    "    stock = (stock - mean)**2\n",
    "    return math.sqrt(stock.sum()/(stock.count()-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cumulative_normal_distribution(ecdf, mu, sigma):\n",
    "    \"\"\"\n",
    "    Returns the selected parent distribution (i.e. normal distribution)\n",
    "    to be compared to the build empirical distribution function EDF\n",
    "    \"\"\"\n",
    "    y = ((1 / (np.sqrt(2 * np.pi) * sigma)) *\n",
    "    np.exp(-0.5 * (1 / sigma * (ecdf - mu[0]))**2))\n",
    "    y = y.cumsum()\n",
    "    y /= y[-1]\n",
    "    return y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[*********************100%***********************]  1 of 1 downloaded\n"
     ]
    }
   ],
   "source": [
    "stock = download_stock_data(\"GS\")\n",
    "stock_return = calculate_stock_return(stock[1])\n",
    "stock_standard_deviation = standard_deviation(stock_return)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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3FbjAzD4n2Fv9I3cvi9d1A1YmaRwRERGpBy98nc+d730HwIMXDuOogR12riN3mP9x8Cw5QLN2sN8F0LZvsAp7p30gvUmSohYREdk1hN1m7cKa2rj7YyG6ugV4BTgfKAOOSag7HpgYJh4RERFpeBsKy7jhlam4w++O67/zyXnhWnj+h0GCDsEM+TUTILNZ8oIVERHZBYV9cOvRKsoTV3avMUF397fNbAAwFPjG3ecmVH9MsGCciIiINEJjZ65gc2mUQZ1b8OMRvWvfwfgH4Yv7YW3Cx3//E+GEO5Sci4iIED5B71lJWS5wInAewersobj7fGB+JeUPhO1DREREGs6m4jLGfLuMW16fAcDxe3cKf3HhWpj4GMwaA4u/DMrSs6DzftBrJBx2HaTV8rl1ERGR3VTYZ9AXVlK8EJhoZgb8kiBRr5GZpQHDgTwgq5KxwtwqLyIiIg3gzndm8cDH8ygpjwHw/SGduSLs7PnmNfDkGbA0/gRbWnqQkB96LUQy6iliERGRXVcy9ib5hCBBr5GZDQReBnoTbN1WkRPiVnkzOxa4B4gAD7n7bRXq8whWoG8Vb/Nbdx8TJkYREREJzFlZwN/fnwPA8J5tOGNoV874XlfS0kKsor74K3j8FCgtCM6Pux2GnANZLesxYhERkV1bMhL0A4CCkG3/GR/zLOBboKS2g5lZBLgPOArIB74ys1fdfXpCs+uB59z9/vgfBcYAPWo7loiIyJ5q2tINXPP0JACOHtiB0RcOq10H7/5fkJznHQhH/xG61vJ6ERGRPVDYVdxvqKQ4ExgMnAD8I+R4Q4GL3P3FkO0rMxyY4+7z4rE9A5wMJCboDrSIH7cEltZhPBERkT1CeTTGh7NW8fgXC/nou1UAtMzO4MqRtVwQbvWc+PPmBmc+Cjkdkx6riIjI7ijsDPpNlZSVEDyHfivw55D9rAZKQ7atShdgccJ5PrB/hTY3Ae+Y2TVAM+DIOo4pIiKy2youi/LYuAU8+Ml8Vm0Kbm7Lzohw7vA8rjliL1o3ywzXUeFaWPAJfHo3eAz2OUfJuYiISC2EXSQuLUnj3QX8xMzedPfoTvZR1bPric4FHnX3O8zsQOBxMxvs7rHtOjK7HLgcIC8vbyfDERER2XVFY87ZD4xjcv4GAHq3a8aZw7px9rBu4RPz4o3w+i9g6gts/Uhu3jFYEE5ERERCS8Yz6LXRDugHTDezd4G1Ferd3W+soY98oFvCeVd2vIX9UuDYeIfjzCwLaAusrDDYaGA0wLBhwyom+SIiIru1kvIof3x9xtbk/N8X/w8j+7Yj2KClGu4w7h+w4DNYOQ3WL9pW1+NQ6DUC9rsQcjrUY/QiIiK7n9AJenxxtguBAwluM18CfA48XovZ8OsTjvtUUu9ATQn6V0AfM+sZj+EcdtzibREwCnjUzAYQbOe2KmSMIiIiu73pSzfyy+e+YebyTaQZ3Hrq3hzer324i6e+AO8kfKRHMqH9ABh1I+w1qn4CFhER2QOEXSSuO/A20JdgBnsFsDfwI+A3ZnZsFXulbycZt8q7e7mZXR2PJwI84u7TzOxmYIK7vwpcCzxoZr8gSPovcnfNkIuIiABPj1/EDa9MpSzqdM9typ1nDeF73duEu3ju+/DKT4LjPkcHK7S36Q2Rhr4pT0REZPcT9tP0HwSroh/i7p9vKTSzg4HngXuBk6rrwMwygSuBse4+defCDcT3NB9ToeyGhOPpwMF1GUNERGR3NG9Vwdbk/AcHdOd3x/enaWbIrwPFG+G/l0J5MXzvIjjhLkhL1jI1IiIiEvZT9Qjgd4nJOYC7fwb8b7y+Wu5eCtwGhPwTvYiIiCTb7W/NoizqnDWsK7ecMjh8cg4w5VkoWgvt+is5FxERqQdhP5ULqLDAWoKVQGHIfmYAvYCPQ7YXERGRJFm3uZQPZgUf5784qm+4i8pLYP7HMONVmPREUDb0h0rORURE6kHYBP0J4ArgzUrqfgw8FrKfG4B7zOxrd/825DUiIiJSR8VlUa56ciIl5TEO7dOWTi2za74oWg6PHAtLJ24rO+w6OODK+gtURERkD1Zlgm5mlySczgbONLNvgRcIFonrAJwB5FB54l6Z3wDNgUlmtgBYxvZ7mLu7jwgdvYiIiITy8KfzGTdvDe1ymvDHUwbXfIE7fHx7kJxbBA77FQw8GToMqv9gRURE9lDVzaA/VElZV6CyT+b7gH+FGC8KTA/RTkRERJIkFnMeHxdstnLHmUPontus6sYFK2HaS/D1o7Ay/pF90NVw+P/Wf6AiIiJ7uOoS9J7JHszdRya7TxEREanep3NWs3xjMV1bZ3Non7Y7NoiWwbSXg6R84WdsvbmtWXs4+Gcw/LKGDFdERGSPVWWCHmZfcxEREWn8nv1qMQBnD+uGmW1fGS2Hh4/e9px5JBN6HQ77ngv9ToD0zAaOVkREZM9Vi71V6s7MDqupjbtrhXcREZEkKCwt589jZjJm6jLSDM4Y1nX7BsumwBu/3Jacf/8eGHQqZLVs+GBFRESk2kXi5gGnuvtkM5vP9ou5VeTu3jvEeB/W0A9AJEQ/IiIiUo1YzPnp05N4b8ZK0tOMXx3Tb9vK7QWrYOwf4tumOWS1gtNGQ99jUhqziIjInq66GfSPgI0JxzUl1mEcXklZLnAiMAK4OgljiIiI7PH+8cEc3puxkpbZGTx12f4M6hyfFS8pgH8fC2vmQFo6DL882DqtaZvUBiwiIiLVPoN+ccLxRckYzN0/qqLqRTO7C/g+4bdsExERkSq8MDEfgD+ftve25Bzg3f8LkvN2A+DsJ6DtXimKUERERCpKq6mBmWWa2UQzO7qeY3kDOKuexxAREdntrSkoYfHaQjIixmF9222rWLcAvnkqOD79ISXnIiIijUyNCbq7lxJsuVZez7H0A2L1PIaIiMhu753pK4g5HLxXW5o3id8st2YuPHQklBdD/xOh4+DUBikiIiI7CLuK+7vA0cD7dRnMzC6spDgTGAxcCrxYl/5FRET2dM9+tYi/vDUTgOMHd9pW8eZvYPMq6H4InPyPFEUnIiIi1QmboN8LPGFm6cDLwDIqLBrn7vNC9PNoFeUlwLPAz0LGIyIiIhV8m7+B37zwLQCH9mnLSft2DiqWfwtz3oX0bDjrMchuncIoRUREpCphE/Qti7v9EvhFFW3CbI/Ws5KyYndfETIOERERqcInc1YBcNp+XbjjrCGYGcSi8OFtQYNhF0Oz3BRGKCIiItUJm6BfQnK2WXNgmbuXVayIz853dvdFSRhHRERkj1JSHuX9GSsBOKxvuyA5XzIRXrwsWLU9kgkHXZPiKEVERKQ6oRJ0d380SePNBw4ExldSNyReHmYmXkREROLGz1/Lja9OY8ayjWRnRDhor1xYPB6eOB1KNkKr7nDsn6FF51SHKiIiItWocRV3ADN738z6V1HX18zCLh5n1dRloFXcRUREauWNKcs464FxzFi2kW5tsnn68gNon5MFY64LkvN+J8DVE6D/CakOVURERGoQ9hb3kUCLKupygBFVXWhmrYA2CUVdzKxXhWbZwA+B5SHjEREREeCZr4Inw04b2oVbT9mb7MwIFKyEFVODBifdC+mZKYxQREREwgqboEPVz6D3Bgqque5nwI3x6x34bxXtLN5OREREQvhg1ko+mb0agJ+N6hMk56Wb4YM/Qaw82O9ci8KJiIjsMqpM0M3sYuDi+KkDo81sU4Vm2QR7mI+tZoyXgQUECfgjwB+BuRXalADT3X1K6MhFRET2YOs2l3LVExMBOHW/LuS1aQqb18DokbAhvt7q/j9OXYAiIiJSa9XNoMeAaPzYKpxvsQa4H/hLVZ24+2RgMoCZOfCGu6/e2YBFREQEXpuylKKyKAf2yuXOLVuqTX8pSM7b9IZj/gQ9D0t1mCIiIlILVSbo7v4f4D8AZvYBcKW7z6zLYPE+MbO2wAFALvCau681syyg1N21UJyIiEgNvl64DoAT9ukUJOerZsFXDweVh14L/Y5NYXQiIiKyM8Jus3Z4ZeVmluvua2ozoJn9FbgGyCS4df5/gLXAK8CnwC216U9ERGRPs2hNIe9OXwFA/5xieOZ8mPkG4NC8A/Q7LrUBioiIyE4Ju83aZWZ2XcL53maWD6w0swlm1jFkP/8LXA3cDOzP9tuuvQacGDpyERGRPZC7c91/J1NYGuWEvTvxvdl3w8zXwdJg2CVwxWfQtE3NHYmIiEijEypBJ5jxLko4vxNYD/wcaEmQcIfxI+Bmd/8TMLFC3RyCFeFFRESkCp/NWcOX89fSMjuD2wfMwb55Kqi48GU48S5o3i61AYqIiMhOC5ug5wEzAcysJcG+579293sJtkY7JmQ/XYAvqqgrBZqF6cTMjjWzWWY2x8x+W0Wbs8xsuplNM7OnQsYnIiLSqN3x7iwArh1qNHv9yqBwwEnQ49AURiUiIiLJEHYf9AjBKu4AhxA8O/5h/Hwx0D5kP0sItmX7oJK6IcD8mjowswhwH3AUkA98ZWavuvv0hDZ9gN8BB7v7OjMLG5+IiEijtmRdcEPbKbGxwV7ne58Jpz0IZjVcKSIiIo1d2Bn02cAJ8eNzgM/dvTB+3plgkbcwngduMLODE8rczPoC1wLPhOhjODDH3ee5e2n8mpMrtLkMuM/d1wG4+8qQ8YmIiDRaXy9cx8pNJfTMWEfOzPhH5oCTlJyLiIjsJsIm6H8Dfm5mq4HzgHsT6g4HpoTs5yaCW+U/Jkj6IUjav42f3xaijy4Es/Zb5MfLEvUF+prZZ2b2hZlVuteMmV0eX+RuwqpVq0L+CCIiIqlx7/uziRDlmZx7sKJ10G1/6HN0qsMSERGRJAm7zdpTZraIYOX1r9z944TqFcCrIfspMrORBEn+MQQLw60h2FrtSXcvD9FNZdMEXuE8HegDjAS6Ap+Y2WB3X18hntHAaIBhw4ZV7ENERKTRiMacz+es4WfpL9Ch8DtolQfnPQsZWakOTURERJIk7DPouPunBPuUVyy/Mcz1ZpYBHA9McffHgcfDjl1BPtAt4bwrsLSSNl+4exkw38xmESTsX+3kmCIiIg2upDzKtKUbmbRoPRMWrKU0GuP0rHFB5TF/guzWqQ1QREREkqrKBN3M8oBl7l4WP66Wuy+qob7MzJ4DjiXEYnDV+AroY2Y9CRadO4dgRj7Ry8C5wKNm1pbglvd5dRhTRESkQc1bVcDJ933GpuJtN5cdmDaNLqyASCbsdWQKoxMREZH6UN0M+nzgQGA8sIAdbyOvKBJivHmEX/G9Uu5ebmZXA2/Hx3zE3aeZ2c3ABHd/NV53tJlNB6LAde6+pi7jioiINBR358FP5m1Nzs8a1pX98lpz6vg/wWrg4J9BRnZqgxQREZGkqy5BvwSYm3CcjGe0bwd+b2bvu/tOr8rm7mOAMRXKbkg4duCX8ZeIiMguIX9dIU+PX8Srk5eyeG0RGWnOy+fnMShjGawYC6unQno2HHZdqkMVERGRelBlgu7u/0k4fjRJ4x0BtCF4LvwLYBnbJ/7u7j9M0lgiIiK7jMVrCzn5vs9Yu7kUgKE563g67QaaPF/hBrCuwyC9SQoiFBERkfoWepG4JDkEKANWAb3jr0RaSV1ERPZIt7w+nbWbSxmVF+E3/VbSZ+4j2NJ4ct7zMGg3ANr3h/4npjZQERERqTfVLRL3SC36cXe/NESjnrXoU0REZI+wZH0Rn85YxP2Z/+K4lV/CynhFTme4eAy00ceniIjInqC6GfQj2H5GuxXQEign2Ls8N379BmBdfQUoIiKyu/t09ipuSX+E49K+BItAj0Og1wjY9wLI6ZDq8ERERKSBVPcMeo8tx2Z2CPA08GPgBXePmlkEOAP4G3BBPccpIiKy2ypePJkfRj6h3DJIv+Jj6DAw1SGJiIhICoR9Bv1O4M/u/tyWAnePAs/G9xm/GxheD/GJiIjs1gpKyln17VgAVucdS0cl5yIiInustJDt9gbmVFE3GxicnHBERET2LG9NnMOZ0TcA6DD4iBRHIyIiIqkUNkFfDpxVRd05wIrkhCMiIrLnKCgpZ91H/6J72krW5/TB9j031SGJiIhICoW9xf1u4C4z6wQ8T5CQdyBI2o8Bfl4/4YmIiOy+XnvqPi4u+g8YZB17C2RkpzokERERSaQvPk4AACAASURBVKFQCbq732NmBcCNwHEJVYuBy9w99JZsZmbA94HDCFaCv8ndF5rZCGC2uy8NHb2IiMgubNjiR0m3GMsHXUbHgcemOhwRERFJsbAz6Lj7w/G90bsCnYBlQL67e/VXbmNmrYExwP7ARiAHuBdYCFwGrAV+Gjp6ERGRXVRp4Sb6xOYRdaP1iTeCWapDEhERkRQL+ww6AB5Y7O7j4++hk/O4vwLdgIOBtkDit5H3gFG17E9ERGTXM+M1Sh4IFoRbbB1pkp2T4oBERESkMQg9g54kJwO/cvdx8X3UEy0iSN5FRER2W9EZY4g8ewE5wCpvyedD/kyPVAclIiIijUJDJ+jNgSVV1GWx/Yy6iIjI7mXFdGIv/IgI8Ez0cFYfcgtXHqWdSkVERCRQq1vck2AWcHQVdSOAbxswFhERkYZTupnY46eSUb6Zt6PDKDrqr1x9zN5E0vS3aREREQk09Az6fcB9ZrYBeCpe1srMLgauBi5v4HhERETq33dvw8tXkla4BoCx7S/iTwf3TnFQIiIi0tg0aILu7g+aWW/gD8DN8eJ3gRhwu7s/2ZDxiIiI1Lv5H8PT54DH+CbWm/vKT+bco44mPdLQN7GJiIhIYxc6QTezLsC1BPuXtwFOcvepZvZzYJy7fxmmH3f/rZndDxwFtAfWAO+6+7xaRy8iItJYlZfAuPvgw9vAYzwWPZobyy6kTbMsbu3SMtXRiYiISCMUKkE3s0HAJ0AUGAfsB2TGq7sDw4Hzwg7q7guBh2oVqYiIyK5i8Xh46QpYOxeAp6OjuLnsAk7/Xh6/P34ArZtl1tCBiIiI7InCzqDfAcwAjgGKgdKEus+Bv1R1oZnl1SYgd19Um/YiIiKNSulmeOY82LyKWG4fbvWLeXhpD04a0pm/nrEPZloUTkRERCoXNkE/BDjX3Qsq2b98BdCxmmsXAF6LmCr2LyIisutYNgU2r6K8RXdOj93O5GVFtM9pwm+P66/kXERERKoVNkGPVVPXFiiqpv4Sapegi4iI7HpiUZj4H3j/VgA+2dieycVF5LVpyiMXDaNzq+wUBygiIiKNXdgEfTxwMfBaJXVnAZ9VdaG7P1r7sERERHYxb/0Wxo8G4BsbwB+Lz2Jkv3bcc/Z+tGyakeLgREREZFcQNkG/BXjPzN4h2L/cgSPN7GfAqQQru9eKmXUGugBL3H1pba8XERFpFKLlMP4BGP8gAPkj7uSUtzvQuWU2D104TNupiYiISGihvjW4+0fAKUBP4BHAgNuAQ4FTwm6xBmBmF5rZfGAx8AWw2Mzmm9kFtQ1eREQk5V69Bt7+X8DhiP9jfMtjACMvt6mScxEREamV0Pugu/sbwBtmthfx/cvdfVZtBjOzq4G/A+8RzMqvADoA5wL/MbOW7n5fbfoUERFJiVg0uKV98lMA+Jn/4b4Vg7jz+ckADOnWKpXRiYiIyC4o1J/2zWyfLcfuPsfdP69tch53LfCoux/t7o+4+xvx96OAx4FfhYznWDObZWZzzOy31bQ7w8zczIbtRKwiIiJVG3Nd8Nw5wKgbuGfZQP72znc48JPDe3PtUf1SGp6IiIjsesLOoH9jZt8CjwFPufuynRyvI/BMFXVPESw4V634Nm/3AUcB+cBXZvaqu0+v0C4H+CkQ+vZ7ERGRam1cCovHw6IvYMLDQdnxf2PD4It47I4PAbj77H05ed8uqYtRREREdllhH447F1gI/BlYZGZvm9n5Zta0luN9C/Suoq4PMDVEH8OBOe4+z91LCRL+kytpdwtwO1BcyxhFRES2Fy2D5y6EOwfA8z+EL+8Pyof/mOX9fsCF/x7P2s2l7JfXipOGdE5trCIiIrLLCjWD7u7PAs+aWVuCZP0CglvSC8zsJeBxd38vRFc/A54xs9XAi+4ejc+Inw5cB5wToo8uBAvMbZEP7J/YwMz2A7q5++tmVuVt82Z2OXA5QF5eXoihRURkj/TlAzD9FcCg10joNhx6joAeB3PtQ18wefF6urTK5p6z98PMUhysiIiI7KpCLxIH4O6rgXuBe82sD/AD4EfA+SH7eg5oQTDrHTWzdUBrIAIUAM8lfLFxd+9eSR+VffPxrZVmacBdwEUhfp7RwGiAYcOGeQ3NRURkTxOLwbQX4dO7gvNznoT+J2yt/mrBWj6bswYzeO6KA+nSKjtFgYqIiMjuoFYJ+hbxW9uHx1/tgfKQl44lIZneSflAt4TzrkDiPuo5wGDgw3iy3xF41cxOcvcJdRxbRET2FEu/gZeugFUzgvO8A6HvcVur35+5gquenAjAZYf2UnIuIiIidRY6Qbcg2z2SYNb8FKA5MA74CfBsmD7c/aLah7iDr4A+ZtYTWEJwW/x5CWNsANomxP0h8Csl5yIiUiuv/zxIzlt0hRHXwb7nQ9q2pVtueX0GxWUxjhnUgeuO0YrtIiIiUnehEnQz+ytBEtwJmAvcQfDc+bx6jK1S7l4e30/9bYJb4x9x92lmdjMwwd1fbeiYRERkNzPvI1g6CSKZ8JMvoUlzADYUlvHR7FW8NXUZ81dvJj3NuP30IWREwq65KiIiIlK1sDPolxA8P/64u39elwHNrA1wAsFt6lkVqt3db6ypD3cfA4ypUHZDFW1H7lykIiKyR5r8LLx8RXA87JKtyfmfxszg4U/nE40FT2pF0ow7zhpCy6YZqYpUREREdjNhE/RO8S3N6sTMjgZeAJpV0cSBGhN0ERGRerFyJrxyFXgMBp0GR98KwMffrWL0x/OIpBkH9GrDqP4dOHpQB7rnVvVxJiIiIlJ7YbdZq3NyHncnMIngufWZ7l6WpH5FRETq7vVfQKwccvvAqQ9AJPiYfHr8IgB+cWQfrj6iTyojFBERkd1YlQm6mc0DTnX3yWY2n+pXX3d37x1ivB7AL9z929qFKSIiUs8mPwuL4k9xnfcspGdSHo0x+pN5vDt9BWZw+ve6pjZGERER2a1VN4P+EbAx4TgZ+4RPAjonoR8REZHkKS2EsX8Ijk+8C3J7s3DNZq55ehJT8jcAcMWI3nRqqa3UREREpP5UmaC7+8UJxxclabxfAo+a2XfuPi5JfYqIiOyceR/CN0/B7HehaC3kdIb9LgTg+penMiV/A51bZvHn0/dhRN92qY1VREREdntht1m7AXjI3ZdWUtcJuMzdbw7R1dfAWOBTM9sMrK9Q7+7ePUxMIiIidTLleXjxMrbeINZ+EJzxCETSicaciQvXAfDiVQfTsWXFTUdEREREki/sKu43Am8BOyToBLes3wiESdD/BlxNcKv7TCBZi8+JiIjULBaDyU/BhH/DkglBWZ9j4JhbIXcvMANg9Mfz2FwapXtuUyXnIiIi0mDCJuhWTV1roCRkPxcBt4TZ61xERCTpxt4En90THGe1hEN/BQdctXW1doDHxi3gL2/NBOB3x/Vv+BhFRERkj1XdKu4jgSMSin5sZidWaJYNnABMCzmeAx/XJkAREZE6WzIR3rsR5sc/gnoeBuc+C5lNt2tWVBrlnx/MBeCno/pw7OBODR2piIiI7MGqm0EfAVwfP3bg4kralALTgZ+GHO954DiC59BFRETq37LJ8PDRECsLZs0P+ikcdA2kN9mh6Z/fnMHyjcW0aZbJpYf0TEGwIiIisierbhX3PwB/ADCzGHCAu4+v43hvAneZWUuCZ9rXVTLu+3UcQ0REJFBWBK/9PEjO+58IJ/8DsltX2rS4LMrb05YDcMvJg2mZndGQkYqIiIiEewbd3dOSNN5L8fdL46+tQxA85+5AJEljiYjInu7DP8PSiZCWAaNurDI5B3j40/ms2FhCz7bNOHJg+wYMUkRERCQQdpG4rcysPbDDkrbuvijE5YfXdjwREZFai5bDxEfhi38F5yfdC+36Vtnc3Xnmq+Bj7KaTBtEkXX8rFhERkYYXdh/0NOCPwI+BVlU0q/HbjLt/FD40ERGRnfTO7+HLeHK+z9nBqxLuzrSlG3nks/ksXltETlY6B/XObcBARURERLYJO4P+c+AnwF8IEvVbgRhwfvz9tnqJTkREpLbyJ8D40WAROP0hGHTq1v3NExWXRbnk0a/4fO4aACJpxl/PGEJGJFlPdYmIiIjUTtgE/WLgZuBuggT9JXefaGZ/BN4B8qq60MzeB65y95nx4+q4u48KGZOIiMj2ykvgmfPAY3Dg1TD4tMqbRWPc+saMrcn5RQf14NzhefTrmNOQ0YqIiIhsJ2yC3guY4O5RMysn2P8cdy8zs7uBe4Gbqrg2cdoijWAhuKrsOMUhIiISRuFaeOJ0KFgBrfLgyJt2aOLuvDp5KX8fO5u5qzZjBvefP1T7nYuIiEijEDZB38C2heGWAv2AzxL6aFPVhe5+eMLxyNqHKCIiEsKLlwUrtrfoAqc/DJEdt0l7dfJSfvbMNwB0bZ3N7afvw0F7tW3oSEVEREQqFTZBnwQMBN6Ov/5gZkVAOcHz6BPrJzwREZEQ3vgVzHkv2E7tkrehVbcdmoydsYLfvvAtAAf0asPjl+6v581FRESkUQn7zeRuoDB+fCOwHHgSeBbIAK4O04mZnWxmFyecdzezcWa2ycz+a2bNw4cuIiICrJoFXz0IkSZw8n07JOexmPPr/07m0v9MoKgsyr7dWnHvuUOVnIuIiEijE2oG3d3fTThebmbDgd5AU2CGu5eFHO964PmE8zuBrsBo4AcEz7H/KmRfIiKypysvhTd/HRwPPh2G7Lid2mtTlvLchHwy09P49TH9uPjgnkTStOSJiIiIND5hb3Hfjrs7MGcnLu0NTAEws2zgeOBCd3/ezGYAv0MJuoiI1KRofTBrPv4hKFgelHUZukOz0vIYD30yH4BfHd2XHx3aqyGjFBEREamVKhN0MzusNh25+8chmmUBRfHjg+LjvxM/nwV0rs2YIiKyB5o5Bl69BgpXB+ftB8IBV8K+5+/Q9L4P5vDtkg1kZ0Q4Tiu1i4iISCNX3Qz6h1S/JdoWFm8XCdF2AXAI8BFwMvC1u2+I17UnWC1eRERke+sWwszXYf4n8N2bQVnegTDiN9BrJNiOt6xvLC7jxUn5ANzw/YF0a9O04eIVERER2QnVJeiHV1O3sx4A/mZmpwL7Alcm1B0ITK+HMUVEZFc2/VV48XIoL9pW9r2L4cS7Kk3MAaYu2cDlj01g6YZierVtxin7dmmgYEVERER2XpUJurt/lOzB3P0eM1sNHAD83d0fS6jOAf6d7DFFRGQXtmwKPPeD4LjnCNjnbOh5KLTKq/ISd+f3L09l6YZihnRtyd/P3Y/szDA3eYmIiIik1k4tElcX7v4kwRZtFct/HLYPMzsWuIfgtvqH3P22CvW/BH5EsE/7KuASd19Yl7hFRKQBrZkLb/0OZr8dnDfvAOf/F9Iza7x07IyVTF68ntxmmTx9+QE0zWzwjzoRERGRnRLqW4uZvV9DE3f3UUmIJ0wsEeA+4CggH/jKzF5198Tb4ycBw9y90MyuBG4Hdtx7R0REGp+Ny+CRY2DzKkjPhmGXwIjrQiXnsZhz/ctTAbhyZG8l5yIiIrJLCfvNJY0dF4zLBfoRzFB/l8ygajAcmOPu8wDM7BmCBee2Juju/kFC+y+ACxowPhERqYsJjwTJedfhcM5T0LxdqMvcnZtem8byjcV0bJHFxQf3rOdARURERJIrVILu7iMrKzez3sDLwJ+SGFNNugCLE87zgf2raX8p8Ga9RiQiIsmx6jsYd19wfMT1oZNzgIc/nc9j4xaSETGuP3EAkbTKF5ATERERaazqdO+fu881s9uAvwL7JSekGlX2javS7eDM7AJgGDCiivrLgcsB8vKqXnBIRETq2ebVsGoWfHIHlG2GQadBz8NCX766oIR7xs4G4J5z9uP4vbXnuYiIiOx6kvFw3iqgb1WVZvYi8Gt3n2NmFwJvuPuaOoyXD3RLOO8KLK1k3COB3wMj3L2kso7cfTQwGmDYsGFh9nwXEZFkKC+FWW/ApCdg6SQoTPhYSEuHo2+pcgu1iqIx57wHv2BTcTn9OuRw3OCO9RS0iIiISP2qU4JuZm2AXwJzq2l2MrBllfV/E+x3XpcE/Sugj5n1BJYA5wDnVYhrP4I9149195V1GEtERJKhtBDyv4KFn8PCzyB/wvb7mmfmQLt+0K4/DDgRWnYN3fWU/PV8t6KADi2a8PiPhmMhE3sRERGRxibsKu7z2fE28kygQ/z49GouX0GQlI8nuD29TjPV7l5uZlcDbxNss/aIu08zs5uBCe7+KsEt982B5+Nf1Ba5+0l1GVdERHbCyhnw+i8hfzzEyrev6zAYhv4Q+h8PLbqEnjGv6MkvFwFw5IAOtM/JqmvEIiIiIikTdgb9I3ZMrIuBhcDz7l7dDPpzwF1mdme8jy+qmd1wd68xJncfA4ypUHZDwvGRNfUhIiL1ZOk3MGsMLP4S5n24rbzTEMg7CLofBHkH1moBuMpMW7qBO975jvdnBjdK9chtVqf+RERERFIt7CruF9VhjF8AnwEDgRuBRwluTRcRkd1JtBxe/zlMenz78m4HwFmPQU6Hyq+rpYKScv40ZgZPxWfOm2ZGuOTgnvzgwO5J6V9EREQkVZKxSFy13N2B5wHM7CLgHnefXN/jiohIAylYBVOegW+egpXTIZIJQy+EniOg23DISd6ibSs3FXPpoxP4dskGMiNp/ODA7lw5sjdtmzdJ2hgiIiIiqRI6QTezAcAZBCuoV3zIz939hzX14e49axeeiIikRLQcSjZC8YZt78WJ51vK1sOM14IygMzmcPbj0PuIpIXi7jw9fjHPfLWIKfkbAGjdNIOnLz+A/h1bJG0cERERkVQLu0jchcAjBM+QrwRKKzQJvfCbmXUCriXYm7wNwYruHwJ3uvvysP2IiEiSlRTA+NEw4d+wYVHtru09Kpg13+tIaNI8qWHd8c53/OODOQA0SU/joN65/HRUHyXnIiIistsJO4P+f8ArwKXuvn5nBzOzvsCnQCuC59LnAB2BnwEXmtmh7j57Z/sXEZE6eP4imPNu/MSgSQvIaglZLbY/zmq5/Xlun2Dht3rY3uzLeWu2JufXnzCA8/fvTnZmJOnjiIiIiDQGYRP0jsAVdUnO4/4CbACGu/uCLYVm1h14J15/Wh3HEBGR2lo7b1tyfu6z0OcoSEtdIrxkfRE3vjKN92asAGB4jzZccnBP0tK0x7mIiIjsvsIm6J8BA4CxdRzvcIJEf0FiobsvNLObgH/WsX8REamtWCzYqxyCZ8f7HZvScNYXlnLO6HEsXltEdkaEyw/rxVWH91ZyLiIiIru9sAn61cCLZraGYKZ7XcUG7h4L0U8msKmKuk3xehERaSglBfDkmbDo82D19e//PdUR8cLEJSxeW8Sgzi3498X/Q/uciuuSioiIiOyewibo+cAk4Ikq6j1kX98A15jZm4kJvZkZcFW8XkRE6sIdyoqCFdaL1kHR+vhx/H3zKlg1C1bOgHXzg2uyWsJpD0KrbikNfXNJOY+NWwDAVSP3UnIuIiIie5SwCfqDwNnAy8BMdlzFPaybgdeBGWb2LLCM4Pn2M4E+wAk72a+IyJ5p3UKY9yEs+BSWT4HCtUESHg35n2mLQLt+cMKd0P3Aeg01jH9/Np+Fawrp26E5owa0T3U4IiIiIg0qbIJ+MnCdu99Tl8Hc/S0zOxH4I/B7wAhm378GTnT3d+rSv4jIHmX+J/DYSVDZE0aRJpDdGrJbQVarhPfWwSu3N7QfALl7QXqTho+dYH/zaUs3MmnxemYu28is5ZuYsiTY5/z/ThxIVoZWaxcREZE9S9gEfTMwPRkDuvtbwFtm1hRoDaxz98Jk9C0istsqKYDV3wWrra+dD2vnwsw3tiXnx/0Vug2H5h2CZDwjO7XxVsHdmb2ygC/mreGFr/OZnL9hhzYH75XLIXu1TUF0IiIiIqkVNkH/N3Ae8G5NDcOKJ+VKzEVEqhOLwey34aUrglvXK2o/EM57FlrlNXxsIc1ZuYnP567hi3lr+HLeWtZs3nb7faumGRzRvz0DO7VgQKcW9OuYQ9vmqZnRFxEREUm1sAn6QuBcM3sXeIvKV3F/JJmBiYjssdxh2WSY+gJMewk2LA7KczpB12HQplf81Ru67Q/pjW8DDHcnf10RD34yj8fGLdyurn1OEw7olcuhfdpy4j6dyc7UrewiIiIiED5Bvz/+3h0YVUm9A0rQRUSS4c1fw/jR285bdIF9z4MRv4FIRuriCmHFxmLufm82705fweqCkq3lo/q358iBHTigVy49cpsSbN4hIiIiIonCJug96zUKEZE9RXlJsNJ64ZrgVbTleG3wKlgezJoD7H0mDLs0mCVPS0tt3DWIxpyPvlvJz5/5ho3F5QC0bprB0LzWfH9IZ07Zr0uKIxQRERFp/EIl6O6+sOZWIiJSpZJN8MpPYPor4dofei2MuqF+Y9pJ7k5JeYxNxeVsLC7jha/zeXr8ItYVlgEwvEcb/nDyIPp3zNFMuYiIiEgthJ1BFxGRMNxh/UJYPhVWTIXl3wbv6xYkNDJo1x+a5kLTNvFXbvDKbgMdBkKnISkJf+Gazbzx7TKWrCuioKScguJyNhWXs6mknIKSsq3n5THf4druuU05dlBHrhnVh+ZN9PEiIiIiUluhvkGZ2XyC58yr5O69khKRiEhjFC2DonXBbehFa6FgJWxaDpuWBu8bE97LNu94fSQT2vWDPsfAEddDI5xZvuOdWdz7/pxQbTPT08hpkk7zrHR65Dbjp6P6MDSvlWbMRUREROog7BTHR+yYoOcCB8H/t3fnQXKV9RrHv890z0wmk4UsQIQQAgqiBBWJclOliCjIlauo4AULFRTlWmqp5RXF7cq19IpaluAVF0RZFEXBq6IUIkgiLihLBCQqJCQsQ4BAFiDLrP27f5zTkzOd7pnOTG/JPJ+qU6fPOe/7nrfnnbe7f33O+zabgZtqWSkzs5qIgIGt0L81CZr7t1T3uPep7ePEt22ArRuhb8ff666oey+Ytwj2XgTzDkvWcw9q6QneVq3bPBycn/iifThi/1lMn5JnWmd7us4Pr6dNydOZ98zrZmZmZrVW7Rj0M8rtl7QHyc+u3VjDOpmZVW/bJui5DZ68L1meuA82rE7GfA9sZYybf3aCoGtWcjt612zo3hNmPAumz4Pp+yTrGem6a1aNztkYq9Y9w1nfvwOAow7ekwtOPbzJNTIzMzObnCY0SDAiNkn6MvB54Ie1qZKZWQWFAjzdA0+uhPWr4ME/wb3XwVBf5Tz5LuiYCu3d0NGdPp4KHdMqPO6GKTOTILwYjE+dnexr2zWvGg8Vgs19g2ztH2RL3yBPbRukZ+NWejZu4/4nNvN/yx8BYN6MKZx/youaXFszMzOzyasWs/j0AvNrUI6ZWWKwLxnjvXFNMsnaY+lka+tXwmBvSWLBgiWw96Ew9+DkVvI5z0muYrdP3WWD6mrd9fAm7u7ZxKp1m3low9ZkYre+IbakAfnmvkF6BwpjlnPMIXtx3psOY3Z3RwNqbWZmZmbljDtAl5QHFgHnAitqVSEzm0RW3gD3L4XNj6fLumTdu6lynml7w5yDYM6zYa/nwfNeBzN3/+8II5Kr4Os397N+Sx9PPNPPL+9ay7V/e3TMvBJ0d+SZ2pEbHkO+z8wuFsyZyn6zp3Lg3G6WHDiHtjZP8GZmZmbWTNXO4l6g8kDOp4ETalYjM9u9DfTCQ7fA6mXwx/PLp2nLJxOtzdw3nWhtEcx7QTIL+pSZDa1uM617upcVa5/mrp5NXHnrwzz2dOndA9t94rWHsHBONzO62pnWmae7M093R47uzjxd7TkH32ZmZma7gGqvoH+WHQP0XuBB4LqI2Inpjc1stxQBA9uSWdCzM6P3b94+k3rvJlj2hWR29Kw3fju5Ml5cumZBW1tznkeTFApB32CBLf2D/PSOHi7+wxqeeGbk2PqOfBt7z+hkTncnc6d1MHdaJ688ZC+Ofd7eDsDNzMzMdgPVzuJ+bp3rYWatIiL5ze+hfigMJOPBNz2UzIy+YTVsWJNs929Og/AtaQC+hapnTM9PgcVnwv5L4MCjoXN6HZ9QcxQKwdO9A6zf0s+GdFn6z3WsfnILW/sH2dY/RO9AIXk8MFR2nHhXe44XzJ/JofvM5CULZ/GaQ+c5EDczMzPbjVUM0CW1kdy6viYi7qmQ5jBgYUT8sk71q1S344ELgBxwcUScV3K8E7gcOAJYD5wSEQ80so5WhQiIQrIm8zgKI7dHHCtuj5WukEk32rFCY8vP5qmm/MIQFAaTYHloIH08kATOQwOZx4PpuiTdUH9JnsHtgfdwnoGRZcbQ+Ns0PyWZBT07Y3rpducMOPxtsNch4z9PiYhgYCjoHRyib6BA3+AQQ4XYvkSM2C5EMDiU7C8UYLBQoBDBUCGZ8bx/qEDfwBB9gwV603Xf4PZ9fYNJQN2Xnq83s+4dKLBpaz8btw4wVNi5n3ib0t5GV3uOfWd1cdqR+3PK4v0ckJuZmZlNIqNdQX8r8A3gsFHSPAP8SNK7I+JHNa1ZBZJywIXAsUAPcJukayLi75lkZwIbI+I5kk4Fvgic0oj61dPQfTfALRey00FelUGnho8xMt0OeXfcVqVAtEJ9VLPfprZ6iLZ8urQTuQ4Gp+3LwMyFDMzYn4GZBzA4Yz4D7TMZzHcxmOtiKDeVgVwXQ7QRsT0ojmA4IC4GwIUIBh4r0Pdwz4gAuHTdlwm2h4PhcunSwDha8F9q+pQ8s7s7mN3dwZzuDmZN7WDP6Z0cv2geXe05ujpyw+speY8TNzMzM5vsxgrQL4mINZUSRMQDkr4LnA40JEAHXgqsiojVAJKuBE4EsgH6iSSzywNcDXxdkiJa8SN89Zb/bQUvWbO02dWoqUIkoXqBtuR7geG1KKDhNZntsdIHImLH9IzIV1252fTFJalz+XLLps+UX1pucU2m3NL6DJBngDyD5Bggx2DkGCRPP8k6uz+btrh/IIpptqcdID+cfkTadBtKAsWN8HdNjgAADvlJREFUwMPZHYMkN6e0jvac6Mzn6My30ZlvI5cT+bY22gS5NpFrayPXRrIe3qftx9J9bRId+TY68zmmtCfrzvakzCntxfIzx4r7M2n26Gpnj6kddOQn1zh6MzMzM5uY0QL0FwP/W0UZNwKn1aY6VdmXkaFCD3BkpTQRMSjpKWAO8GQ2kaSzgLMAFixYUK/61sy6vV/Gu+48p0zg1wblAkxBNgAtBqQ77FNbJqDcHnSWBr8gCtKI4LP8wnCdsueRRtYbJUFguWuG0o57y6er9Ncqk79M2p0pUxMqs/oro2XLLHue8mVWUycBHRKlv3i9E39OckqCW0lJ4KviY6WPtwfBbdJwoNwmkc+JKfkcncWAt70t3R4Z/O4QBJekLa47cm3kcw6GzczMzGzXNlqAPp3kutlYNqZpG6VcDFF6ZbyaNETERcBFAIsXL275q+snvGwxJ7xscbOrYWZmZmZmZnUw2iWnJ4H9qyhjASVXpuusB9gvsz0fWFspjaQ8MBPY0JDamZmZmZmZmY3DaAH6H0jGlo/ljDRto9wGHCTpAEkdwKnANSVprmF73U8GbtrVx5+bmZmZmZnZ7m20AP184FWSvpoGwiNIapd0AXAM8NV6VbBURAwC7weuB/4B/CQiVkj6rKTXp8m+C8yRtAr4MHBOo+pnZmZmZmZmNh4a7cKypA8BXyGZrvk3wIPpof1JfuZsDvCfEXFBnetZd4sXL47bb7+92dUwMzMzMzOz3ZykOyJihwnGRpskjog4X9JykivQbwS60kPbgGXAeRHx+xrX1czMzMzMzGzSGTVAB4iIm4GbJbUBc9Pd6yNiqK41MzMzMzMzM5tExgzQiyKiAKyrY13MzMzMzMzMJq1Rx6BPJpKeYPsY+1Y0l8b+nJ01l9t7cnF7Ty5u78nF7T25uL0nF7f35FLr9t4/IvYs3ekAfRch6fZykwjY7sntPbm4vScXt/fk4vaeXNzek4vbe3JpVHuP9jNrZmZmZmZmZtYgDtDNzMzMzMzMWoAD9F3HRc2ugDWU23tycXtPLm7vycXtPbm4vScXt/fk0pD29hh0MzMzMzMzsxbgK+hmZmZmZmZmLcABupmZmZmZmVkLcIDeZJJmS7pB0sp0PatCutPTNCslnZ7umy7pzszypKTz02NnSHoic+xdjXxeVt5E2jvdv0zSvZl23Svd3ynpx5JWSfqLpIWNeUY2mgn276mSrpX0T0krJJ2XSe/+3UIkHZ/2y1WSzilzvGL/lPTxdP+9kl5TbZnWPONtb0nHSrpD0t/S9TGZPGVf2635JtDeCyVty7TptzJ5jkj/D1ZJ+pokNe4ZWSUTaOvTSj6PFyS9KD3mvt2iqmjvoyQtlzQo6eSSY5U+p9emb0eElyYuwJeAc9LH5wBfLJNmNrA6Xc9KH88qk+4O4Kj08RnA15v9/LzUtr2BZcDiMnneC3wrfXwq8ONmP1cvE2tvYCrwyjRNB/B74F/TbffvFlmAHHA/cGDaTncBzy9JU7Z/As9P03cCB6Tl5Kop08su2d6HA/ukjxcBj2TylH1t97JLt/dC4J4K5d4KLAEEXFd8bfeya7Z1SZrDgNWZbfftFlyqbO+FwAuAy4GTM/tH+5xek77tK+jNdyJwWfr4MuANZdK8BrghIjZExEbgBuD4bAJJBwF7kXyIt9ZVk/Yeo9yrgVf5G/mWMO72joitEbEUICL6geXA/AbU2XbOS4FVEbE6bacrSdo9q1L/PBG4MiL6ImINsCotr5oyrTnG3d4R8deIWJvuXwFMkdTZkFrbeE2kf5cl6VnAjIi4JZJP9JdT/r3BGqtWbf0W4Ed1ranVwpjtHREPRMTdQKEkb9nPbbXs2w7Qm2/viHgUIF2Xu/VlX+DhzHZPui/rLSTf5GWn5T9J0t2Srpa0Xy0rbeNWi/a+JL1N6tOZN4bhPBExCDwFzKl15W2n1aR/S9oDeB3w28xu9+/WUM3rc6X+WSlvNWVac0ykvbNOAv4aEX2ZfeVe2625JtreB0j6q6TfSXp5Jn3PGGVa49Wqb5/CjgG6+3brmcj77Gjv3TXp2/nxZLKdI+lGYF6ZQ5+stogy+0p/H+9U4G2Z7V8CP4qIPknvIfnG7xis7urc3qdFxCOSpgM/JWnzy8fIY3VU7/4tKU/yZv+1iFid7nb/bh3V9L1KaSrtL/fluftza5hIeycHpUOBLwLHZY5Xem235ppIez8KLIiI9ZKOAH6etr3fr1tTLfr2kcDWiLgnc9x9uzVNpB/u7Hv6TnOA3gAR8epKxyQ9LulZEfFoemvEujLJeoCjM9vzSca0FMt4IZCPiDsy51yfSf8dkg8D1gD1bO+IeCRdPyPphyS36Fye5tkP6EkDupnAhok/GxtLvfs3cBGwMiLOz5zT/bt1FPte0XxgbYU0pf1ztLxjlWnNMZH2RtJ84GfA2yPi/mKGUV7brbnG3d7pHY19ABFxh6T7gYPT9NnhSu7frWFCfTt1KiVXz923W1Y17T1a3qNL8i6jhn3bt7g33zVAcfa/04FflElzPXCcpFlKZoE+Lt1XtMN4lzQYKHo98I+a1dgmYtztLSkvaS6ApHbg34Dit7TZck8GbioZ7mDNMaH+LelzJB8APpTN4P7dUm4DDpJ0gKQOkg9o15SkqdQ/rwFOTWcGPgA4iGSCmWrKtOYYd3unQ1WuBT4eEX8sJh7jtd2aayLtvaekHICkA0n69+p0uNMzkv4lvd357ZR/b7DGmshrOZLagDeTjGUm3ee+3bom8j5b9nNbTft2LWfE8zKuWQTnkIwrXZmuZ6f7FwMXZ9K9k2QCoVXAO0rKWA0cUrLvCyST0NwFLC097mXXa2+gm2Sm/rvTtr0AyKXHpgBXpelvBQ5s9nP1MuH2nk9ya9Q/gDvT5V3pMffvFlqA1wL3kcwI+8l032eB16ePK/ZPkqEQ9wP3kpnttVyZXlpjGW97A58CtmT6850k81JUfG330vxlAu19UuZ1ejnwukyZi0kCtfuBrwNq9vP0MuHX8qOBP5eU577dwksV7f0SkqviW4D1wIpM3rJxWa36ttLCzMzMzMzMzKyJfIu7mZmZmZmZWQtwgG5mZmZmZmbWAhygm5mZmZmZmbUAB+hmZmZmZmZmLcABupmZmZmZmVkLcIBuZmZWJ5LeIOlmSeskbZP0oKSfSzo+k+YMSSFpYRPqF5LObfR5K5F0rqRjml0PMzOzZnGAbmZmVgeSPgD8DFgJnAmcAHwuPZwNQq8FlgCPNrSCrekzjPzbmJmZTSr5ZlfAzMxsN/UR4OcRcWZm303AdyQNf0EeEU8ATzS6co0gqTMi+iZ7HczMzKrlK+hmZmb1MRt4rNyBiCgUH5e7xV3SA5J+IOltku5Nb4//vaSDJHVL+rak9ZIel/QVSflM3qPT8k6SdKmkjZKelnSFpDljVVrSCyVdk+bbJumPkl5eRb5z0/MuknS9pM3ATzLH3yTpz5K2Stok6SpJCzLHI334ybSc4dvvJS2TtKzMOR+QdGmZv+VRafmbgL+kxy6V1CPp8PRvuVXSSknvKSlznqTLJK2V1CfpUUm/krTXWH8DMzOziXKAbmZmVh+3AqdLOlvSwePIfxTwXuBjwOnAs4GfAlcAzwCnAhcBHwbOKpP/fCCAtwCfBF4PXD3aCSW9GPgTyZcL7wZOAtYDN0o6osp6/wL4XXq+r6blviet+9+Bk4H/ABYBv5M0Pc23JF1fmj5eAlxc5TlLXQGsSc91Tmb/DOCHwA+AE4HbgG9KemUmzffTc58NHAt8AOgBpo6zLmZmZlXzLe5mZmb18R6SgPhLwJckrQduAC6JiN9UkX8acHxEPAXJlV3gAuDWiPhImuYGSScAbwa+UZJ/RUS8I338a0kbgB9IelVE/LbCOb8MPAQcExH96XmvB+4BPg28oYp6fy0iLihuSJoGfJHkeb8zs/8vwH0k4/PPj4g/SwJ4JCL+XMV5RnN1RHy0zP7pwHsjYmlah5uB40i+xFiaplkCfCIirsjku2qC9TEzM6uKr6CbmZnVQUTcBxwOvAL4PHAn8EbgekmfqqKIW4rBeeqf6fr6knT/BPYrk/8nJdtXAQW2X6keQVJXWtergIKkfHrrvIAbSa7oV+NnJdtLSK5cX1EsMy23J617teXujNI6FG0tBucA6dj0lcCCTJrbgLMlfVDSYUq/NTAzM2sEB+hmZmZ1EhFDEXFzRHwqIl4NHAj8DfiMpFljZN9Yst0/yv4pZfI/XlKX/jTvvhXONxvIkVwpHyhZ3g/Myk5uN4rS2eiLY7dvLFPuYcCY4+LHodKM+KV/O4A+Rv79TgGuAT4K3A08Ium/qnzuZmZmE+Jb3M3MzBokItZKupjkVvWDSMap18ve2Q1JHcAs4JEK6TeRXGG/ELi8XILs5HajiJLt9en6DGBFmfTPVFFmL8lV+FKzq6xD1SJiHfA+4H2Snksy/v+/SWba/+Z4yzUzM6uGA3QzM7M6kLRfRDxc5tAh6brsDO819O/A9zLbbya5c+6WcokjYouk3wMvBJZXGYxX408kQfhzIuKyMdL2A11l9j8InCSpIzM2/iiSMeV1ExH3Ap9IJ7lbVM9zmZmZgQN0MzOzerlH0lKS8dBrSK4Av5Zk8rifRMRDdT7/oZIuAa4EDiYZB/+7USaIg2RG+JtJxsl/l+RW8bnAi4FcRJwzSt6yIuJpSWcDF0raE7gOeIrkVvtXAMsi4odp8r8DJ0j6Ncnt6GsjYm36HM4Cvpf+rNoBaV2fooYkzSS5Ff8KkvHxAySzvc8CqpnYz8zMbEI8nsrMzKw+PkbyPvtZkuDuxyQTpp0DvK0B5/8gyQRvPwb+B/gVyc+OVRQRy4GXkNyW/jWSel9AMlb85vFWJCK+TfKza88l+Rmz60huG8+TTJ5X9H5gC/BLksnazkrzLyX5YuPI9Ng7gLeS3JZfS73AcpKfmLua5MuVJcBpEfGLGp/LzMxsB4oY9zAtMzMzazGSjib5ybBjI+LGJlfHzMzMdoKvoJuZmZmZmZm1AAfoZmZmZmZmZi3At7ibmZmZmZmZtQBfQTczMzMzMzNrAQ7QzczMzMzMzFqAA3QzMzMzMzOzFuAA3czMzMzMzKwFOEA3MzMzMzMzawH/Dwt7+g8FRkaPAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 1008x252 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "edf = sorted(list(stock_return.iloc[:,0])) # ecdf.y[1:] == cum_dist_true_values should yield false\n",
    "len_edf = len(edf)\n",
    "cum_dist_edf = np.arange(1, len_edf+1) / len_edf\n",
    "\n",
    "mu = stock_return.mean()\n",
    "sigma = stock_standard_deviation\n",
    "\n",
    "y_edf = cumulative_normal_distribution(edf, mu, sigma)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(14, 3.5), tight_layout=True)\n",
    "ax.plot(edf, y_edf, linewidth = 2)\n",
    "ax.plot(edf, cum_dist_edf, linewidth = 2)\n",
    "ax.set_xlabel(\"Simple returns\", fontsize=16)\n",
    "ax.set_ylabel(\"Cumulative distribution\\nof simple returns\", fontsize=16)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### General Steps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The general steps to run the test are:\n",
    "\n",
    "1. Create an EDF for your sample data (see Empirical Distribution Function for steps),\n",
    "2. Specify a parent distribution (i.e. one that you want to compare your EDF to),\n",
    "3. Graph the two distributions together.\n",
    "4. Measure the greatest vertical distance between the two graphs.\n",
    "5. Calculate the test statistic.\n",
    "6. Find the critical value in the KS table.\n",
    "7. Compare to the critical value.\n",
    "\n",
    "#### Calculating the Test Statistic\n",
    "The K-S test statistic measures the largest distance between the EDF $F_{data}(x)$ and the theoretical function $F_0(x)$, measured in a vertical direction (Kolmogorov as cited in Stephens 1992). The test statistic is given by:\n",
    "\n",
    "$D = sup|F_{0}(x)-F_{data}(x)|$\n",
    "\n",
    "Where (for a two-tailed test):\n",
    "\n",
    "1. $F_0(x)$ = the cdf of the hypothesized distribution,\n",
    "2. $F_{data}(x)$ = the empirical distribution function of your observed data.\n",
    "\n",
    "For one-tailed test, omit the absolute values from the formula.\n",
    "If D is greater than the critical value, the null hypothesis is rejected. Critical values for D are found in the K-S Test P-Value Table below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='images/p-value-k-s-test.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Computing D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.08265250449733397\n"
     ]
    }
   ],
   "source": [
    "D = np.abs(np.array(y_edf) - np.array(cum_dist_edf))\n",
    "print(D.max())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a α of 0.05, the value that the maximum of D should not be greater than is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.08450628817616321\n"
     ]
    }
   ],
   "source": [
    "p_value = 1.36/math.sqrt(len(cum_dist_edf))\n",
    "print(p_value)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D.max() > p_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### The null hypothesis (that the distributions of the stock returns of Goldman Sachs and the normal distribution are the same) is accepted."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}