bitcoin_ml_analysis.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "banned-reply",
   "metadata": {},
   "source": [
    "# Bitcoin Analysis: Maximum Likelihood\n",
    "\n",
    "## Introduction\n",
    "\n",
    "In this notebook, we are going to analyze the distribution of Bitcoin returns and try to calibrate a distribution using maximum likelihood. The file ```cryptocurrency_prices.txt``` contains the daily price time series of the following seven cryptocurrencies (in this order): \n",
    "\n",
    "- Bitcoin, \n",
    "- Dash, \n",
    "- Ethereum, \n",
    "- Litecoin, \n",
    "- Monero, \n",
    "- Nem, \n",
    "- Ripple. \n",
    "\n",
    "There are 929 daily prices and the time series ends in February 2018.\n",
    "\n",
    "First, let's import the libraries and the file and get the Bitcoin time series."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "allied-minute",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import norm\n",
    "from scipy.special import erf\n",
    "from matplotlib.ticker import ScalarFormatter\n",
    "from matplotlib.ticker import FuncFormatter\n",
    "from scipy.optimize import fmin\n",
    "\n",
    "# Import and format data\n",
    "data = pd.read_csv('cryptocurrency_prices.txt', header=None)\n",
    "btc = data.iloc[0, 0].split('\t')\n",
    "btc = list(filter(None, btc))\n",
    "btc = pd.DataFrame([float(i) for i in btc])\n",
    "btc.columns = ['prices']"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "constant-spice",
   "metadata": {},
   "source": [
    "Let's compute the daily log-returns, $r_t$, as \n",
    "\n",
    "\\begin{equation}\n",
    "    r_t = \\log \\left(\\frac{P_t}{P_{t - 1}} \\right)\n",
    "\\end{equation}\n",
    "\n",
    "where $P_t$ is the price at time (i.e., day) $t$.\n",
    "\n",
    "Note that in this notebook (and all my notebooks), $\\log$ denotes the natural logarithm unless stated otherwise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "conventional-portrait",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>prices</th>\n",
       "      <th>log_return</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>279.58</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>261.00</td>\n",
       "      <td>-0.068768</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>265.08</td>\n",
       "      <td>0.015511</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>264.47</td>\n",
       "      <td>-0.002304</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>270.39</td>\n",
       "      <td>0.022138</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   prices  log_return\n",
       "0  279.58         NaN\n",
       "1  261.00   -0.068768\n",
       "2  265.08    0.015511\n",
       "3  264.47   -0.002304\n",
       "4  270.39    0.022138"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculate returns and log returns\n",
    "btc['log_return'] = np.log(1 + btc.prices.pct_change())\n",
    "\n",
    "btc.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "transparent-monkey",
   "metadata": {},
   "source": [
    "We now have a set of returns $\\{r_t\\}_{1 \\leq t \\leq T} = (r_1, r_2, ..., r_T)$ with $T = 928$. \n",
    "\n",
    "## Analysis of the distribution\n",
    "\n",
    "Let's compute the four first moments of the distribution: the mean $\\mu$, the variance $\\sigma^2$, the skewness $\\gamma$ and the kurtosis $\\kappa$, which are defined as\n",
    "\n",
    "\\begin{equation} \\label{eq:mean}\n",
    "    \\mu = \\mathbb{E}[r_t] = \\frac{1}{T} \\sum_{t = 1}^T r_t,\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation} \\label{eq:std}\n",
    "    \\sigma^2 = \\mathbb{E} \\left[(r_t - \\mu)^2 \\right] = \\frac{1}{T} \\sum_{t = 1}^T (r_t - \\mu)^2,\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation}\n",
    "    \\gamma = \\frac{1}{\\sigma^3}\\mathbb{E} \\left[(r_t - \\mu)^3 \\right] = \\frac{1}{\\sigma^3T} \\sum_{t = 1}^T (r_t - \\mu)^3,\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation}\n",
    "    \\kappa = \\frac{1}{\\sigma^4}\\mathbb{E} \\left[(r_t - \\mu)^4 \\right] = \\frac{1}{\\sigma^4T} \\sum_{t = 1}^T (r_t - \\mu)^4.\n",
    "\\end{equation}\n",
    "\n",
    "Instead of the variance, we compute the standard deviation, $\\sigma$, which is simply the square root of the variance. Also, we compute the excess kurtosis which is equal to $\\kappa - 3$.\n",
    "\n",
    "Note that the values returned below are slightly different from those computed with the above formulas. This is because we compute the unbiased estimates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "indoor-resort",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean:               0.0\n",
      "Standard deviation: 0.04\n",
      "Skewness:           -0.28\n",
      "Excess kurtosis:    5.37\n"
     ]
    }
   ],
   "source": [
    "# Compute statistics    \n",
    "mean = btc['log_return'].mean()\n",
    "sdev = btc['log_return'].std()\n",
    "skew = btc['log_return'].skew()\n",
    "kurt = btc['log_return'].kurt()\n",
    "\n",
    "print('Mean:               ' + str(round(mean, 2)))\n",
    "print('Standard deviation: ' + str(round(sdev, 2)))\n",
    "print('Skewness:           ' + str(round(skew, 2)))\n",
    "print('Excess kurtosis:    ' + str(round(kurt, 2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "brilliant-reviewer",
   "metadata": {},
   "source": [
    "In finance, the assumption that asset returns are normally distributed is often made (e.g., Black Scholes model). Obviously, the distribution of our returns is not normal. For the Gaussian distribution, skewness and excess kurtosis are both equal to zero. \n",
    "\n",
    "The Gaussian probability density function (PDF) is given by\n",
    "\n",
    "\\begin{equation}\n",
    "    f(x) = \\frac{1}{\\sigma \\sqrt{2\\pi}} \\exp \\left(- \\frac{(x - \\mu)^2}{2\\sigma^2} \\right).\n",
    "\\end{equation}\n",
    "\n",
    "Let's compare the distribution of our returns with a Gaussian distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "connected-emission",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 720x504 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Normal distribution\n",
    "x = np.linspace(min(btc['log_return'].dropna()), max(btc['log_return'].dropna()), 100)\n",
    "f = norm.pdf(x, loc=mean, scale=sdev)\n",
    "\n",
    "# Plot BTC histogram vs normal distribution on linear scale\n",
    "fig = plt.figure(figsize=(10, 7))\n",
    "grid = plt.GridSpec(2, 1, wspace=0, hspace=0.5)\n",
    "ax = fig.add_subplot(grid[0, 0])\n",
    "ax.hist(btc['log_return'], 50, density=True, ec='k', label='BTC')\n",
    "ax.plot(x, f, 'r', label='Normal')\n",
    "ax.title.set_text('PDF: BTC vs normal - Linear scale')\n",
    "ax.set_xlabel('r')\n",
    "ax.set_ylabel('f(r)')\n",
    "ax.legend()\n",
    "\n",
    "# Plot BTC histogram vs normal distribution on semilog scale\n",
    "ax2 = fig.add_subplot(grid[1, 0])\n",
    "ax2.hist(btc['log_return'], 50, density=True, ec='k', label='BTC')\n",
    "ax2.semilogy(x, f, 'r', label='Normal')\n",
    "ax2.set_ylim(0.03, 30)\n",
    "ax2.set_xlabel('r')\n",
    "ax2.set_ylabel('f(r)')\n",
    "ax2.title.set_text('PDF: BTC vs normal - Semilog scale')\n",
    "ax2.legend();\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "contemporary-enhancement",
   "metadata": {},
   "source": [
    "We can also compare the complementary cumulative distribution function (CCDF) of Bitcoin with the normal CCDF.\n",
    "\n",
    "The Gaussian cumulative distribution function (CDF) is given by\n",
    "\n",
    "\\begin{equation}\n",
    "    F(x) = \\frac{1}{2} \\left[ 1 + \\text{erf} \\left( \\frac{x - \\mu}{\\sigma \\sqrt{2}} \\right) \\right]\n",
    "\\end{equation}\n",
    "\n",
    "and the CCDF is given by\n",
    "\n",
    "\\begin{equation}\n",
    "    C(x) = 1 - F(x) = \\frac{1}{2} \\left[ 1 - \\text{erf} \\left( \\frac{x - \\mu}{\\sigma \\sqrt{2}} \\right) \\right]\n",
    "\\end{equation}\n",
    "\n",
    "where $\\text{erf}(\\,.)$ denotes the error function given by\n",
    "\n",
    "\\begin{equation}\n",
    "    \\text{erf}(z) = \\frac{2}{\\sqrt{\\pi}} \\int_0^z e^{-\\xi^2} d\\xi.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "magnetic-fundamental",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Calculation CCDF as rank-frequency plot\n",
    "r_sorted = np.array(btc['log_return'].sort_values().dropna())\n",
    "y = np.linspace(1, len(r_sorted), len(r_sorted))\n",
    "y = 1 - y/(len(r_sorted)+1)\n",
    "\n",
    "# Gaussian CCDF\n",
    "c = 0.5*(1 - erf((r_sorted-mean)/(sdev*np.sqrt(2)))) \n",
    "\n",
    "fig = plt.figure(figsize=(10, 5))\n",
    "ax = fig.add_subplot(111)\n",
    "ax.semilogy(r_sorted, y, label='BTC')\n",
    "ax.semilogy(r_sorted, c, 'r', label='Normal')\n",
    "ax.set_ylim(np.exp(10^(-4)), 1.1)\n",
    "ax.set_ylabel('C(r)')\n",
    "ax.set_xlabel('r')\n",
    "ax.set_title('CCDF')\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "silent-cruise",
   "metadata": {},
   "source": [
    "It is clear that our returns distribution is not Gaussian. In particular, the distribution has fatter tails. Before focusing on the analysis of the tails, let's review the concept of maximum likelihood estimation.\n",
    "\n",
    "## Maximum likelihood\n",
    "\n",
    "The maximum likelihood problem consists of:\n",
    "\n",
    "- Assuming a PDF to model the data,\n",
    "- Finding the PDF's parameters that maximize the probability of observing the data if they were generated by the chosen PDF.\n",
    "\n",
    "Let us consider a set of data points $\\{x\\} = \\{x_i\\}_{1 \\leq i \\leq N}$ and a PDF, $f(x, \\{\\theta\\})$, depending on a set of parameters $\\{\\theta\\} = \\{\\theta_j\\}_{1 \\leq  j \\leq M}$. The likelihood of observing the data with the chosen PDF is given by\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mathcal{L}(\\{x\\} \\mid \\{\\theta\\}) = \\prod_{i = 1}^N f(x_i \\mid \\{\\theta\\}).\n",
    "\\end{equation}\n",
    "\n",
    "The maximum likelihood problem consists of finding the set of parameters that maximize the likelihood, i.e.,\n",
    "\n",
    "\\begin{equation}\n",
    "    \\{\\theta^*\\} = \\arg \\max \\mathcal{L}(\\{x\\} \\mid \\{\\theta\\}).\n",
    "\\end{equation}\n",
    "\n",
    "In practice it is preferable to work with the log-likelihood, i.e., the logarithm of the likelihood function. Indeed, it is easier to work with sums rather than products. Moreover, using the log-likelihood may help avoiding numerical precision issues that arise when multiplying very small numbers together.\n",
    "\n",
    "As an example, suppose we have a dataset which could be well described by a Gaussian distribution and we want to determine its mean and standard deviation. The likelihood function is given by\n",
    "\n",
    "\\begin{align}\n",
    "    \\mathcal{L}(\\{x\\} \\mid \\mu, \\sigma) &= \\prod_{i = 1}^N \\frac{1}{\\sigma \\sqrt{2\\pi}} \\exp \\left(- \\frac{(x_i - \\mu)^2}{2\\sigma^2} \\right) \\notag \\\\\n",
    "    & = \\frac{1}{\\left(2\\pi\\sigma^2\\right)^{\\frac{N}{2}}} \\exp \\left(- \\frac{1}{2} \\sum_{i = 1}^N \\frac{(x_i - \\mu)^2}{\\sigma^2} \\right)\n",
    "\\end{align}\n",
    "\n",
    "and the log-likelihood function is given by\n",
    "\n",
    "\\begin{equation}\n",
    "    \\ell(\\{x\\} \\mid \\mu, \\sigma) = \\log(\\mathcal{L}(\\{x\\} \\mid \\{\\mu, \\sigma\\})) = - \\frac{N}{2} \\log \\left(2 \\pi \\sigma^2 \\right) - \\frac{1}{2} \\sum_{i = 1}^N \\frac{(x_i - \\mu)^2}{\\sigma^2}.\n",
    "\\end{equation}\n",
    "\n",
    "In order to find the maximum log-likelihood, we need to set the partial derivatives with respect to the parameters to zero and solve the corresponding system of equations.\n",
    "\n",
    "Setting the partial derivative with respect to $\\mu$ to zero, we obtain\n",
    "\n",
    "\\begin{equation}\n",
    "    0 = \\frac{\\partial \\ell}{\\partial \\mu} = \\sum_{i=1}^N \\frac{x_i - \\mu}{\\sigma^2}\n",
    "\\end{equation}\n",
    "\n",
    "which implies that \n",
    "\n",
    "\\begin{equation}\n",
    "    \\mu^* = \\frac{1}{N} \\sum_{i=1}^N x_i.\n",
    "\\end{equation}\n",
    "\n",
    "Setting the partial derivative with respect to $\\sigma$, we obtain\n",
    "\n",
    "\\begin{equation}\n",
    "    0 = \\frac{\\partial \\ell}{\\partial \\sigma} = - \\frac{N}{\\sigma} + \\sum_{i=1}^N \\frac{(x_i - \\mu)^2}{\\sigma^3}\n",
    "\\end{equation}\n",
    "\n",
    "which implies that \n",
    "\n",
    "\\begin{equation}\n",
    "    \\sigma^* = \\sqrt{\\frac{1}{N} \\sum_{i=1}^N (x_i - \\mu)^2}.\n",
    "\\end{equation}\n",
    "\n",
    "Note that we recover the expressions for the mean and standard deviation obtained in equations \\eqref{eq:mean} and \\eqref{eq:std}.\n",
    "\n",
    "## Tail fitting\n",
    "\n",
    "Let's make the hypothesis that our data is well described by a power law in the tail region. We assume that our data follows a power law distribution for $|r| > r_{min}$.\n",
    "\n",
    "Let's calibrate a power law distribution on the tails of our data. The power law is given by\n",
    "\n",
    "\\begin{equation}\n",
    "    f(r) = \\frac{\\alpha}{r_{min}} \\left( \\frac{r}{r_{min}} \\right)^{-(\\alpha + 1)}\n",
    "\\end{equation}\n",
    "\n",
    "for $r \\in [r_{min}, \\infty)$.\n",
    "\n",
    "We want to find the maximum likelihood estimate of the tail exponent $\\alpha$. The likelihood function is \n",
    "\n",
    "\\begin{equation}\n",
    "    \\mathcal{L}(\\{r\\} \\mid \\alpha) = \\prod_{i=1}^N \\frac{\\alpha}{r_{min}} \\left( \\frac{r_i}{r_{min}} \\right)^{-(\\alpha + 1)}.\n",
    "\\end{equation}\n",
    "\n",
    "Taking the logarithm, we obtain the log-likelihood function,\n",
    "\n",
    "\\begin{align}\n",
    "    \\ell(\\{r\\} \\mid \\alpha) &= \\sum_{i=1}^N \\log \\left( \\frac{\\alpha}{r_{min}} \\left( \\frac{r_i}{r_{min}} \\right)^{-(\\alpha + 1)} \\right) \\notag \\\\\n",
    "    &= \\sum_{i=1}^N \\left[ \\log \\left( \\frac{\\alpha}{r_{min}} \\right) + \\log \\left( \\left( \\frac{r_i}{r_{min}} \\right)^{-(\\alpha + 1)} \\right) \\right] \\notag \\\\\n",
    "    &= \\sum_{i=1}^N \\left[ \\log \\left( \\frac{\\alpha}{r_{min}} \\right) - (\\alpha + 1) \\log \\left( \\frac{r_i}{r_{min}} \\right) \\right] \\notag \\\\\n",
    "    &= N \\log \\left( \\frac{\\alpha}{r_{min}} \\right) - \\sum_{i=1}^N (\\alpha + 1) \\log \\left( \\frac{r_i}{r_{min}} \\right).\n",
    "\\end{align}\n",
    "\n",
    "Setting the partial derivative with respect to $\\alpha$ to zero, we obtain\n",
    "\n",
    "\\begin{equation}\n",
    "    \\frac{\\partial \\ell}{\\partial \\alpha} = \\frac{N}{\\alpha} - \\sum_{i=1}^N \\log \\left( \\frac{r_i}{r_{min}} \\right) = 0.\n",
    "\\end{equation}\n",
    "\n",
    "This implies that \n",
    "\n",
    "\\begin{equation}\n",
    "    \\alpha^* = \\frac{N}{\\sum_{i=1}^N \\log \\left( \\frac{r_i}{r_{min}} \\right)}.\n",
    "\\end{equation}\n",
    "\n",
    "We can now fit a power law on the tails of our data. We assume that each tail contains 10% of the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "legislative-lawsuit",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tail exponent (right tail): 2.21\n",
      "tail exponent (left tail):  1.63\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x504 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Get tail returns\n",
    "p = 0.1 # Percentage of data in right tail\n",
    "r_right = r_sorted[round((1-p)*len(r_sorted))-1:]\n",
    "r_left = np.abs(r_sorted[:round((p)*len(r_sorted))])\n",
    "\n",
    "# Maximum likelihood estimate for tail exponents\n",
    "alpha_right = len(r_right)/np.sum(np.log(r_right/min(r_right)))\n",
    "alpha_left = len(r_left)/np.sum(np.log(r_left/min(r_left)))\n",
    "x_right = np.linspace(min(r_right), max(r_right), 100)\n",
    "x_left = np.linspace(min(r_left), max(r_left), 100)\n",
    "y_right = alpha_right*(x_right/min(r_right))**(-alpha_right-1)/min(r_right)\n",
    "y_left = alpha_left*(x_left/min(r_left))**(-alpha_left-1)/min(r_left)\n",
    "\n",
    "# Get histogram data\n",
    "fig = plt.figure(figsize=(10, 7))\n",
    "r_hist = plt.hist(r_right, 20, density=True)\n",
    "b_right = r_hist[0] # Value (count)\n",
    "a_right = r_hist[1] # Bin Edges\n",
    "a_right = np.mean(np.vstack([a_right[0:-1],a_right[1:]]), axis=0) # Bin Centers\n",
    "r_hist = plt.hist(r_left, 20, density=True)\n",
    "plt.clf()\n",
    "b_left = r_hist[0] # Value (count)\n",
    "a_left = r_hist[1] # Bin Edges\n",
    "a_left = np.mean(np.vstack([a_left[0:-1],a_left[1:]]), axis=0) # Bin Centers\n",
    "\n",
    "# Plot of the right tail\n",
    "ax = fig.add_subplot(grid[0, 0])\n",
    "ax.loglog(x_right, y_right, 'r', label='fitted power law')\n",
    "ax.scatter(a_right, b_right, label='empirical distribution')\n",
    "ax.set_yscale('log')\n",
    "ax.set_xscale('log')\n",
    "ax.set_ylim(0.1, 100)\n",
    "ax.set_xlabel('r')\n",
    "ax.set_ylabel('f(r)')\n",
    "ax.set_title('Right tail')\n",
    "ax.legend(loc='best')\n",
    "\n",
    "# Format axis in scalar form\n",
    "formatter = FuncFormatter(lambda y, _: '{:.16g}'.format(y))\n",
    "ax.xaxis.set_major_formatter(formatter)\n",
    "for tick in ax.xaxis.get_minor_ticks():\n",
    "    tick.label.set_fontsize(8) \n",
    "ax.xaxis.set_minor_formatter(ScalarFormatter())\n",
    "\n",
    "# Plot of the left tail\n",
    "ax2 = fig.add_subplot(grid[1, 0])\n",
    "ax2.loglog(x_left, y_left, 'r', label='fitted power law')\n",
    "ax2.scatter(a_left, b_left, label='empirical distribution')\n",
    "ax2.set_yscale('log')\n",
    "ax2.set_xscale('log')\n",
    "ax2.set_ylim(0.1, 100)\n",
    "ax2.set_xlabel('r')\n",
    "ax2.set_ylabel('f(r)')\n",
    "ax2.set_title('Left tail')\n",
    "ax2.legend(loc='best')\n",
    "\n",
    "# Format axis in scalar form\n",
    "formatter = FuncFormatter(lambda y, _: '{:.16g}'.format(y))\n",
    "ax2.xaxis.set_major_formatter(formatter)\n",
    "for tick in ax2.xaxis.get_minor_ticks():\n",
    "    tick.label.set_fontsize(8) \n",
    "ax2.xaxis.set_minor_formatter(ScalarFormatter())\n",
    "\n",
    "print('tail exponent (right tail): ' + str(round(alpha_right, 2)))\n",
    "print('tail exponent (left tail):  ' + str(round(alpha_left, 2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "recognized-worst",
   "metadata": {},
   "source": [
    "## Bootstrap\n",
    "\n",
    "Maximum likelihood only provides one estimate for the parameters. We can complement this with a confidence interval via bootstrapping. It involves randomly resampling portions of our data. For example, we could randomly select 70% of our dataset and compute a new estimate of the parameters. Repeating the process many times allows to build a confidence interval.\n",
    "\n",
    "Let's create a 90% confidence interval by estimating our parameter, $\\alpha$, 500 times using a randomly selected portion of 80% of the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "fifty-inclusion",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Confidence interval (right tail exponent): [2.09 2.48]\n",
      "Confidence interval (left tail exponent):  [1.5  1.74]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "r = np.array(btc['log_return'].dropna()) # Vector of log returns\n",
    "\n",
    "# Bootstrap parameters\n",
    "bts = 0.8 # Fraction of data to be retained in each bootstrap sample\n",
    "nbts = 500 # Number of bootstrap samples\n",
    "alpha = 0.9 # Significance level\n",
    "\n",
    "# Initialize arrays\n",
    "alpha_right_bts = []\n",
    "alpha_left_bts = []\n",
    "\n",
    "# Calculate alpha for each sample\n",
    "for i in range(0, nbts):\n",
    "    r_bts = np.random.permutation(r)\n",
    "    r_bts = np.sort(r_bts[:round(bts*len(r))])\n",
    "    r_right_bts = r_bts[round((1-p)*len(r_bts))-1:]\n",
    "    r_left_bts = np.abs(r_bts[:round(p*len(r_bts))])\n",
    "    alpha_right_bts.append(len(r_right_bts)/np.sum(np.log(r_right_bts/min(r_right_bts))))\n",
    "    alpha_left_bts.append(len(r_left_bts)/np.sum(np.log(r_left_bts/min(r_left_bts))))\n",
    "    \n",
    "alpha_right_bts = np.sort(alpha_right_bts)\n",
    "alpha_left_bts = np.sort(alpha_left_bts) \n",
    "\n",
    "# Confidence intervals\n",
    "ci_right = [alpha_right_bts[round(0.5*(1-alpha)*nbts)],alpha_right_bts[round(0.5*(1+alpha)*nbts)]]\n",
    "ci_left = [alpha_left_bts[round(0.5*(1-alpha)*nbts)],alpha_left_bts[round(0.5*(1+alpha)*nbts)]]\n",
    "\n",
    "# Plot alphas\n",
    "fig = plt.figure(figsize=(10, 7))\n",
    "ax = fig.add_subplot(111)\n",
    "ax.boxplot([alpha_left_bts, alpha_right_bts], sym='r+', labels=[r'$\\alpha_{left} $', r'$\\alpha_{right} $'])\n",
    "ax.set_title('Bootstrap analysis')\n",
    "for tick in ax.xaxis.get_major_ticks():\n",
    "    tick.label.set_fontsize(14) \n",
    "    \n",
    "print('Confidence interval (right tail exponent): ' + str(np.round(ci_right, 2)))\n",
    "print('Confidence interval (left tail exponent):  ' + str(np.round(ci_left, 2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "increasing-latvia",
   "metadata": {},
   "source": [
    "We can see that the estimated value of $\\alpha$ falls in the confidence interval.\n",
    "\n",
    "## Body-tail fitting\n",
    "\n",
    "In order to capture different regimes in the body and the tails of the empirical distribution, it is possible to fit two distributions. In order to do so, we can use maximum likelihood to find an optimal point $r^*$ such that\n",
    "\n",
    "\\begin{equation}\n",
    "    f(r) = \\begin{cases} f_{body}(r), & |r| < r^* \\\\\n",
    "    f_{tail}(r), & |r| \\geq r^* \\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "with $f_{body}(r^*) = f_{tail}(r^*)$.\n",
    "\n",
    "Considering only the (absolute value of the) negative returns, let assume that the body of the distribution follows an exponential distribution and the tail follows a power law distribution.\n",
    "\n",
    "The exponential distribution PDF is given by\n",
    "\n",
    "\\begin{equation}\n",
    "    f(x; \\lambda) = \\begin{cases} \\lambda e^{-\\lambda x}, & x \\geq 0 \\\\\n",
    "    0, & x < 0 \\end{cases}.\n",
    "\\end{equation}\n",
    "\n",
    "Therefore, we have\n",
    "\n",
    "\\begin{equation}\n",
    "    f_{body}(r) = C^{-1} \\lambda e^{- \\lambda r}\n",
    "\\end{equation}\n",
    "\n",
    "and \n",
    "\n",
    "\\begin{equation}\n",
    "    f_{tail}(r) = C^{-1} \\frac{\\alpha}{r_{min}} \\left( \\frac{r}{r_{min}} \\right)^{-(\\alpha + 1)}\n",
    "\\end{equation}\n",
    "\n",
    "where $C$ is a normalization constant given by\n",
    "\n",
    "\\begin{equation}\n",
    "    C = \\int_{0}^{r_{min}} f_{body}(r) \\, dr + \\int^{\\infty}_{r_{min}} f_{tail}(r) \\, dr = 2 - e^{- \\lambda r_{min}}.\n",
    "\\end{equation}\n",
    "\n",
    "The likelihood function is given by\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mathcal{L}(\\{r_i\\} \\mid \\alpha, \\lambda, r_{\\min}) = \\prod_{r_i < r_{min}} f_{body}(r_i) \\prod_{r_i \\geq r_{min}} f_{tail}(r_i)\n",
    "\\end{equation}\n",
    "\n",
    "and the log-likelihood function is given by\n",
    "\n",
    "\\begin{align}\n",
    "    \\ell (\\{r_i\\} \\mid \\alpha, \\lambda, r_{\\min}) &= \\sum_{r_i < r_{min}} \\log f_{body}(r_i) + \\sum_{r_i \\geq r_{min}} \\log f_{tail}(r_i) \\notag \\\\\n",
    "    &= \\sum_{r_i < r_{min}} (\\log \\lambda - \\lambda r_i - \\log C) + \\sum_{r_i \\geq r_{min}} \\left( \\log \\left( \\frac{\\alpha}{r_{min}} \\right) - (\\alpha + 1) \\log \\left( \\frac{r_i}{r_{min}} \\right) - \\log C \\right).\n",
    "\\end{align}\n",
    "\n",
    "Nothing guarantees that the PDF resulting from the tail-body likelihood will be continuous in $r_{min}$. To enforce continuity, we add a penalty term, $\\phi$ > 0, which penalizes discontinuities in $r_{min}$. The resulting function is\n",
    "\n",
    "\\begin{equation}\n",
    "    L = \\ell(\\{r_i\\} \\mid \\alpha, \\lambda, r_{min}) - \\phi \\big(f_{body}(r_{min}) - f_{tail}(r_{min})\\big)^2.\n",
    "\\end{equation}\n",
    "\n",
    "The optimization algorithm needs an initial guess to work. We select $\\lambda = 10$, $r_{min} = 0.05$ and set $\\alpha$ equal to the value we previously estimated. Also, we set the penalty $\\phi = 10$. Feel free to try other parameters and see how it affects the result. Note that in the function below, we use $-L$ because ```fmin``` minimizes the function. Minimizing the negative of a function is equivalent to maximizing the function. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "separate-honey",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: -949.979238\n",
      "         Iterations: 246\n",
      "         Function evaluations: 510\n",
      "lambda: 118.66\n",
      "alpha:  0.37\n",
      "r_min:  0.0084\n"
     ]
    },
    {
     "data": {
      "image/png": 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EpApo1gz+8Q/IyICxY0nyOx58dwyfPzeSy+Z/QPM4b8J09R+r/Iq73Fzc+ooQVdAH0efzHfz+wOPcgmFX7IhBi498DBAXF1fk+Z1zRe5/ww03MGrUKBYuXMgzzzzD/gNzwgZZlQxkGvZCRKQKiY2F66+HhQvh3XdJbd2Ev3/0JNP/fS1Dp73uzbsplVqjpJgyrQ+VWbNmsXLlSvLz83njjTfo06dPqY8988wzefPNN9m6dSsA27ZtA2Dnzp2kpnqtfi+99FLFF12MKhnIRESkCvL5vOmYvvkGPv0UOnSAP/wBZs6E774DXfWotG4d2JaYgP+wdTEBP7cObFuu8x7Zh+z2228v0/G9evXi9ttvp1OnTjRv3pwLLrig1Md27NiRP/3pT5x++ul07dqVm2++GYB77rmHiy++mFNPPfXg5cxQMOdcyJ6soqWnp7s5c+aEuwwRETleX33FsBdmQ+Z+3vj6afjPf7yxzSTolixZQvv27Uu9/8R5axnz0TLW7cikUVIMtw5sW67+Y+U1Y8YMHnroId57772w1VCSon6+ZjbXOZde1P7q1C8iIuHTuzcs8nmtY9/6vQFnb7vNu2MzMjLc1UkhQ7unhjWAVXe6ZCkiIuGXmAjz53uDyo4eDT17wpIl4a5KKrG+fftW2tax46FAJiIilUNCAjz3nDd/5s8/exObP/kkVOGuNZVdVe62VJkdz89VgUxERCqXoUO9OzL79vUmMD/7bFi/PtxVVTvR0dFs3bpVoayCOefYunUr0dFlG19PfchERKTyadgQpkzxBpb9wx+gc2dvgNky3EUnJWvcuDEZGRls3rw53KVUO9HR0TRu3LhMxyiQiYhI5WQGv/0tnHEGXHGFN6H5ddfBo496lzelXAKBQMhGoZdj0yVLERGp3Nq398Yqu+MOeOEF6NbNeyxSjVTJQKapk0REapjISPj73+GzzyAvD/r0gbvv1ij/Um1UyUCmqZNERGqoU0+F77+Hyy+Hv/7VC2b/+1+4qxIptyoZyEREpAZLTISXX4Y33vDCWLduMHbsYcNjTJy3lt6jp9H89vfpPXoaE+etDV+9IqWgQCYiIlXTL38JCxZAr17w61/D+efDpk1MnLeWO95ZyNodmThg7Y5M7nhnoUKZVGoKZCIiUnU1bgwffwyPPAIffQSdO/PlP18iMyfvsN0yc/IY89GyMBUpcmwKZCIiUrX5fPD738OcOVC/Pg+99Cf+9vFTROfsP2y3dTsyw1SgyLEpkImISPXQuTPMmsWrp17MlfOm8P6439F5/aEO/42SYsp0OvVDk1BSIBMRkeojOprYfz7KNZf/ndjs/bzz3z/w26/fIM4Ptw5sW+rTqB+ahJpG6hcRkWplaPdUuOUqrmnZgVFvPsytX/yHa3cvJflXb5T6HGM+WlZsP7Sh3VOLPGbivLWM+WgZ63Zk0igphlsHti12X5EjqYVMRESqnaHdU/nw3vM5d/Fn8J//kLxiGXTtCi+9dNjwGMUprr9ZcevVoiblpUAmIiLVl5k3D+aCBdC9OwwfDhdfDFu3lnhYcf3NiltfUouaSGkokImISPXXrBlMmwajR8OkSd4NAB9/XOzutw5sS0zAf9i6mIC/2H5oZW1REzmSApmIiNQMfj/cdht8+y0kJcHAgfC730Hm0aFpaPdUHriwM6lJMRiQmhTDAxd2LrZPWFlb1ESOpE79IiJSs3TvDnPneuHs8cdh6lR45RVvCqZChnZPLXWn/FsHtuWOdxYedtmypBY1kSNVmhYyM2thZs+b2fhw1yIiItVcTIwXxj78ELZtg5NOgjFjIC/v2McWoawtaiJHCmoLmZm9AJwLbHLOdSq0fhDwT8APPOecG+2cWwFcp0AmIiIhM3AgLFwII0bAH/8IU6Z4d2I2bVrmU5WlRU3kSMFuIRsHDCq8wsz8wJPAYKADcKmZdQhyHSIiIkVLSYG334YXXvCmX+rSBV59NdxVSQ0T1EDmnPsc2HbE6pOA5c65Fc65bOB14PzSntPMRpjZHDObs3nz5gqsVkREaiwzuOYamD8fOnSAyy+Hyy6DHTvCXZnUEOHoQ5YKrCn0OANINbNkM3sa6G5mdxR3sHNurHMu3TmXXrdu3WDXKiIiNUnLlvD55/DXv8Kbb3qtZdOnh7sqqQHCEcisiHXOObfVOTfSOdfSOfdAyKsSEREBiIiAv/wFvv4aoqPhzDO9/mVZWeGuTKqxcASyDKBJoceNgXVlOYGZnWdmY3fu3FmhhYmIiBx00kkwb57X4X/MGDj5ZFi0qNynnThvLb1HT6P57e/Te/Q0Ta8kQHgC2WygtZk1N7NI4BJgUllO4Jyb7JwbkZiYGJQCRUREAIiLg6ef9kb3X7cO0tPhn/+E/PzjOp3mvJTiBDWQmdlrwEygrZllmNl1zrlcYBTwEbAEeNM590Mw6xARESmX887zhscYMABuugkGDYK1ZQ9RmvNSihPsuywvdc41dM4FnHONnXPPF6yf4pxrU9Bf7P6ynleXLEVEJOTq1/dayp5+Gr780psPc3zZhs7UnJdSnEozUn9Z6JKliIiEhRn8+tde37KWLeHii2H4cNi1q1SHa85LKU6VDGQiIiJh1batdxfmX/4C//kPdO3qtZodw60D2xIT8B+2rqxzXuqmgOpJgUxEROR4BALeeGVffOG1nJ1+OvzpT5CdXewh5Z3zUjcFVF9BncsyWMzsPOC8Vq1ahbsUERGp6U45Bb7/Hn73O/j73+Gjj+C//4V27YrcvTxzXpZ0U4Dm0azaqmQLmfqQiYhIpZKQ4M2F+fbbsHIlnHAC/Pvf4Nxxna64y5K6KaD6KnMgM7O4ggnCRUREpLALL/SGxzjtNPi//4Nzz4UNG8p0ipIuS+qmgOrrmIHMzHxmdpmZvW9mm4ClwHoz+8HMxphZ6+CXKSIiUkU0agQffACPPw7TpnnDY7z7bqkPL+myZEXcFCCVU2layKYDLYE7gAbOuSbOuXrAqcA3wGgzuyKINR5F45CJiEilZgY33ABz50LjxjB0qDcF0549xzy0pMuS5b0pQCqv0nTq7++cyzlypXNuG/A28LaZBSq8shI45yYDk9PT068P5fOKiIiUSYcO8O23cNdd8OCDMH261+H/5JOLPaRRUgxriwhlBy5LluemAKm8jtlC5pzLKbhsWeyMqkUFNhEREQEiI2H0aC+MZWdD795w772Qm1vk7rosWTOVqlO/cy4f+N7Mmga5HhERkerp9NNhwQK49FK45x7o0weWLz9qN12WrJnKMg5ZQ+AHM5sF7D2w0jk3pMKrOgaNQyYiIlVSYqI3sv8558BvfgPdusFjj8F113n9zgqE+rLkxHlrGfPRMtbtyKRRUgy3DmyrABhiZQlk9watijJSHzIREanSLrnEu3R59dVw/fXw3nvw7LNQt27ISzkwzMaBOzsPDLMBKJSFUGmGvTAA59xnRS2F9xEREZFSatIEpk6Fhx7yhsno3Nn7GmIlDbMhoVOqYS/M7IYj+4+ZWaSZnWFmLwFXB6c8ERGRaszng1tugdmzvdaxs8+GUaNg376QlVDcMBtrd2RqAvMQKk0gGwTkAa+Z2XozW2xmK4D/AZcCjzrnxgWxRhERkeqtSxcvlP3+9/Dkk9CjB3z3XUieuqRR/jWBeeiUZtiL/c65p5xzvYGmwJnACc65Zs65651z84NdpIiISLUXHQ2PPAKffAK7dnljlT3wAOTlHfvYcihqmI0j6RJm8B2zU7+ZRQMjgVbAAuAF51zRg6eEiO6yFBGRaqt/f28+zJEj4c47YcoU787MtLQSDyvuTsmi1gOHrftFj1SmL93MuoL5M4uiCcyDy9wxZqI3szeAHOALYDCw2jn3uxDUdkzp6eluzpw54S5DRETKYdgzMwF449e9wlxJJeOcN6r/b3/rPX7ySbjiisOGxzjgyDslwRtM9hc9Unl77trD1gd8BgY5ee6wfQ+MddZ79LQiZwpITYrhq9vPqMAXWPOY2VznXHpR20rTh6yDc+4K59wzwEV4c1iKiIhIMJnBlVd6g8l27QpXXQXDhsG2bUftWtydkq99u+ao9Tn57rAwdmDfA5ckNVNAeJQmkB2cFinclypFRERqnLQ0mDHD6082YYI3PMbUqYftUtzlxLxjXAUr6hyaKSA8SjMwbFcz21XwvQExBY8NcM65WkGrTkRERMDvh9tvh7POgssvhwEDvDsy//53iI4udkJyv1mpQ1nhuy2PNVNA4X5piTEBzGDHvhyN8l8OpbnL0u+cq1WwJDjnIgp9rzAmIiISKiecAHPnev3KHn0UTjwRFiwo9jLjpSc3OWp9wGcE/HbUvqW9JHmgv9raghsAdmTmsH1fjobIKKdSTS4uIiIilURsLDzxhHf35ebNcOKJDJ32Og8M7XjUZcb7hnY+6vLjmIu7Muairsd9SbKo/mqFaYiM41OWuSwrDQ17ISIiNd7gwd7wGCNGwB/+wNB+7zP0pZe8KZkKKe7y4/FeVizN8BcaIqPsqmQLmXNusnNuRGJiYrhLERERCZ+6deGdd+C552DWLG/E/9dfD+pTljSyf1n2kcNVyUAmIiIiBczguutg/nxo1w4uvdQbr2zHjgp7ionz1tJ79DSa3/4+e7Nyj+qDVpiGyDg+CmQiIiLVQatW8MUXcO+9XitZ167w2WflPm1RnfhxUDs2gAFJMYGD32uIjONXJfuQiYiISBEiIuCuu2DgQK+VrF8/uPVW+OtfISrquE5ZVCf+nHxHbGQE8+46qyKqFtRCJiIiUv2cfDLMmwe/+hU8+CD07AmLFx/XqYrroK+O+xVLgUxERKQ6io+HsWPh3XchIwN69IB//Qvy88t0muI66KvjfsVSIBMREanOhgzxhsc44wy48UY4+2xYt67Uh2tuy9BQHzIREZHqrkEDeO89ePppuOUWbz7MZ5+FCy885qEHOugfmCqpvNMjFZ526cC5KvL8VZW5Mkw8Wtmkp6e7OXPmhLsMEREph2HPzATgjV/3CnMlNcSyZd58mHPnwjXXwD//CQkJIXnqA3dsFr5JIOAzMMjJO5RHYgL+anm3ppnNdc6lF7VNlyxFRERqkrZt4euv4U9/gpdegm7dvMchUNwdm4XDGNTM6ZeqZCAzs/PMbOzOnTvDXYqIiEjVExkJ993njVOWnw+nngp/+Qvk5AT1actyZ2ZNu4uzSgYyTZ0kIiJSAfr0ge+/hyuv9AJa797w449Be7qy3JlZ0+7irJKBTERERCpIrVowbhy89Rb89BN07w7PPANB6GNe1B2bAZ8dNRVTTbyLU4FMRERE4KKLYMECr5Vs5EhvuIyNGyv0KYZ2T+WBCzuTmhRzcKqlMRd3ZcxFXQ9bVx079B+Lhr0QERERT2oqfPihN4Dsbbd5w2M8/zycd16pDi9qSIsjg9XQ7qkM7Z56cN/fvzH/qH0PTGZek4bBUCATERGRQ3w++N3voH9/b3iMIUPg17+Ghx+GuLhiDztySIu1OzK5452FAEeFqZL2BUp9nupElyxFRETkaB07wrffepOTjx3r9S2bPbvY3Ysa0qK44StK2rcs56lOFMhERESkaFFR3uTk06bB/v3Qqxf87W+Qm3vUrmWZhLykfWvqZOYKZCIiIlKyvn29Dv/DhsFdd8Fpp3l3ZBZSlknIS9q3pk5mrkAmIiIix5aUBK+84i2LF3sj/L/wwsHhMcoyCXlJ+9bUyczVqV9ERERK77LLvAFlr74arrsO3n8fnnmmTJOQl2bfmjbZuCYXFxGRsNLk4lVUfj488gjceSckJ3uDyw4cGO6qKjVNLi4iIiIVy+eDP/wBZs3yAtmgQXDjjZBZvTvfB4sCmYiIiBy/bt284TB+9ztvQNkePWDevHBXVeVUmkBmZnFm9pKZPWtml4e7HhERESmlmBh47DH4+GPYuRNOPhn+8Q/IyzvmoeIJaiAzsxfMbJOZLTpi/SAzW2Zmy83s9oLVFwLjnXPXA0OCWZeIiIgEwYAB3vAYQ4bA7bfDGWfA6tXlPu2BqZSa3/4+vUdPY+K8tRVQbOUS7BayccCgwivMzA88CQwGOgCXmlkHoDGwpmA3RWoREZGqKDkZ3nrL6+Q/bx506eINlXGcNxEemGZp7Y5MHIemUqpuoSyogcw59zmw7YjVJwHLnXMrnHPZwOvA+UAGXigLel0iIiISRGbesBjff+9NUH7FFXDppbB9e5lPVVOmUgpH8EnlUEsYeEEsFXgH+IWZ/RuYXNzBZjbCzOaY2ZzNmzcHt1IRERE5fs2bw2efwf33w9tve61l06aV6RQ1ZSqlcAQyK2Kdc87tdc5d45z7jXPuleIOds6Ndc6lO+fS69atG8QyRUREpNz8fm+sspkzIS4OzjzTGy4jK6tUh9eUqZTCEcgygCaFHjcG1pXlBGZ2npmN3blzZ4UWJiIiIkGSng7ffQf/93/w8MNw4omwcOExD6spUymFI5DNBlqbWXMziwQuASaV5QTOucnOuRGJiYlBKVBERESCIDYWnnwS3nsPNm70Qtmjj3qj/hdjaPdUHriwM6lJMRiQmhTDAxd2rnZTKQV1Lkszew3oC6SYWQZwt3PueTMbBXwE+IEXnHM/BLMOERERqUTOOcdrHbv+erj5Zm8+zHHjoHHjIncf2j212gWwIwX7LstLnXMNnXMB51xj59zzBeunOOfaOOdaOufuL+t5dclSRESkiqtXDyZOhLFjvf5lnTvDm2+Gu6qwqZLDS+iSpYiISDVg5rWSzZ8PbdrAsGFw1VXeaP81TJUMZCIiIlKNtG4NX34Jd98Nr74KXbvCF1+Eu6qQUiATERGR8AsE4J57vCAWEQGnnw533AHZ2eGuLCSqZCBTHzIREZFqqlcv7xLmddfB6NHQsycsWRLuqoKuSgYy9SETERGpxuLj4dlnYcIE+PlnOOEEb7iM45wPsyqokoFMREREaoChQ73hMfr2hVGj4OyzYcOGcFcVFApkIiIiUnk1bAhTpngtZDNmQKdOXstZNVMlA5n6kImIiNQgZt6US999B82awYUXwq9+BXv2hLuyChPUkfqDxTk3GZicnp5+fbhrERERkYoxcd5axny0jHU7MmmUFMOtA9sePkJ/+/beILL33ON1+J8xg8/+/Ah3bog/7BiAMR8tY+2OTPxm5DlHUkwAM9ixL4dGSTH0a1eX975fz47MHABqxwa4+7yOYZsRwFwV7iCXnp7u5syZE+4yRESkHIY9MxOAN37dK8yVSDhNnLeWO95ZSGZO3sF1MQF/8fNWfvEF+4ZdRtSGdTzRaxj/OmUYuf4IAj4Dg5y8suebgN8Yc1HXoIUyM5vrnEsvaluVvGQpIiIi1cuYj5YdFsYAMnPyGPPRsqIPOPVUho54kokd+/K7r19j/Ct/JG3bWnLy3XGFMfBCXLHPF2RVMpCpD5mIiEj1sm5HZpnWA/xvv59bzrmZ3w65jbTt65gy7kYumf9huYbHKOn5gqlKBjKNQyYiIlK9NEqKKdP6wtveb38qA699grmN2jP6oyd49p37SN67o0LrCLYqGchERESkerl1YFtiAv7D1sUE/Ac76R/rmI0JKVw17K/cf+b1nLbyOz58YRT9fppdphoCfivx+YJJgUxERETCbmj3VB64sDOpSTEYkJoUU3yH/mKOaVQ7jo5j7uarV6ews1ZtXhx/L/d9/BTROftJiglQOzZw8NxX9GxKUkzg4LlqxwaC2qH/WHSXpYiIhJXuspSgyMqCP/8ZHnoI2raF//4X0ou8wTFkdJeliIiI1CxRUTBmDHz6Kezd601afv/9kJd37GPDoEoGMt1lKSIiIqVyxhmwYAH84hdei9npp8PKleGu6ihVMpDpLksREREptdq14bXXvMuWCxdC167w0kvlGh6jolXJQCYiIiJSJmZw+eVea1n37jB8OPzyl7B1a7grAxTIREREpCZp1gymTYN//APefRc6d4aPPw53VQpkIiIiUsP4/fDHP8K330JSEgwcCE89FdaSFMhERESkZureHebOhVtugbPPDmspEWF9dhEREZFwionxxioLM7WQiYiIiIRZlQxkGodMREREqpMqGcg0DpmIiIhUJ1UykImIiIhUJwpkIiIiImFmrhJNG1BWZrYZWF3CLolASR3NStpe3Lai1qcAW0p4nlA61msO5fnKcmxp9j3e91PvZfnPWVney5K21bT3s6b9bkLleT/1u1m2bfrdPKSZc65ukVucc9V2AcYe7/bithW1HpgT7tda2tccyvOV5djS7Hu876fey/Kfs7K8l3o/K+Z8leX9PI71leL91O9m+d/nyvJeBuP9PN7zVfdLlpPLsb24bcc6Z7hVdH3lOV9Zji3Nvsf7fuq9LP85K8t7WdK2mvZ+6nczfPS7WbZtNe39PK7zVelLlpWFmc1xzqWHuw4pP72X1Yvez+pF72f1offyaNW9hSxUxoa7AKkwei+rF72f1Yvez+pD7+UR1EImIiIiEmZqIRMREREJMwUyERERkTBTIBMREREJMwWyIDOzvmb2hZk9bWZ9w12PlI+ZxZnZXDM7N9y1SPmYWfuC38vxZvabcNcjx8/MhprZs2b2rpmdFe56pHzMrIWZPW9m48NdSygpkJXAzF4ws01mtuiI9YPMbJmZLTez249xGgfsAaKBjGDVKiWroPcS4DbgzeBUKaVVEe+nc26Jc24k8EtAt9+HSQW9lxOdc9cDw4FhQSxXjqGC3s8Vzrnrgltp5aO7LEtgZqfhhamXnXOdCtb5gR+BAXgBazZwKeAHHjjiFNcCW5xz+WZWH3jEOXd5qOqXQyroveyCN91HNN77+l5oqpcjVcT76ZzbZGZDgNuBJ5xzr4aqfjmkot7LguMeBl5xzn0XovLlCBX8fo53zl0UqtrDLSLcBVRmzrnPzSztiNUnAcudcysAzOx14Hzn3ANASZextgNRQSlUjqki3ksz6wfEAR2ATDOb4pzLD27lUpSK+t10zk0CJpnZ+4ACWRhU0O+mAaOBDxTGwquC/92sURTIyi4VWFPocQZwcnE7m9mFwEAgCXgiqJVJWZXpvXTO/QnAzIZT0PIZ1OqkrMr6u9kXuBDvP0pTglmYlFmZ3kvgBqA/kGhmrZxzTwezOCmzsv5uJgP3A93N7I6C4FbtKZCVnRWxrtjrvs65d4B3gleOlEOZ3suDOzg3ruJLkQpQ1t/NGcCMYBUj5VLW9/Jx4PHglSPlVNb3cyswMnjlVE7q1F92GUCTQo8bA+vCVIuUj97L6kXvZ/Wh97J60ftZCgpkZTcbaG1mzc0sErgEmBTmmuT46L2sXvR+Vh96L6sXvZ+loEBWAjN7DZgJtDWzDDO7zjmXC4wCPgKWAG86534IZ51ybHovqxe9n9WH3svqRe/n8dOwFyIiIiJhphYyERERkTBTIBMREREJMwUyERERkTBTIBMREREJMwUyERERkTBTIBMREREJMwUyERERkTBTIBMRKYZ59HdSRIJOf2hERAoxszQzW2JmTwHfcfgcfCIiQaGR+kVECjGzNGAFcIpz7pswlyMiNYRayEREjrZaYUxEQkmBTETkaHvDXYCI1CwKZCIiIiJhpkAmIiIiEmbq1C8iIiISZmohExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgE5Fqy8wuM7M5ZrbHzNab2Qdm1qdgWxsze8vMtpjZTjNbYGY3m5nfzNLMzBUct8fMNprZe2Y24IjzrzKzzEL77TGzRuF5tSJSlSmQiUi1ZGY3A48BfwfqA02Bp4Dzzawl8C2wBujsnEsELgbSgYRCp0lyzsUDXYFPgAlmNvyIpzrPORdfaFkXxJclItWUOefCXYOISIUys0RgLXCNc+6tIrb/F6jtnDunmOPTgJVAwDmXW2j9H4BbgYbOuXwzWwX8yjk3teJfhYjUJGohE5HqqBcQDUwoZnt/YPxxnPcdoB7Q9jjrEhEpkgKZiFRHycCWwq1bRWxffxznPXA5sk6hdRPNbEfBMvE4zikiQkS4CxARCYKtQIqZRRQTyrYCDY/jvKkFX7cVWjdUlyxFpLzUQiYi1dFMYD8wtJjtU4FfHMd5LwA2AcuOrywRkaIpkIlIteOc2wncBTxpZkPNLNbMAmY22MweBO4GTjGzMWbWAMDMWpnZf80s6cjzmVl9MxtVcNwdzrn8EL4cEakBdMlSRKol59wjZrYR+DPwCrAbmAvc75z7ycx6AfcBP5hZBLAKeLFgv6SC0+wwMwP2AnOAi51zH4b0hYhIjaBhL0RERETCTJcsRURERMJMgUxEREQkzBTIRERERMJMgUxEREQkzKr0XZYpKSkuLS0t3GWIiFQbKzbvBaBF3bgwVyJS/cydO3eLc65uUduqdCBLS0tjzpw54S5DRKTaGPbMTADe+HWvMFciUv2Y2eritumSpYiIiEiYhSSQmdkLZrbJzBYVs93M7HEzW25mC8zshFDUJSIiIlIZhKqFbBwwqITtg4HWBcsI4N8hqElERESkUghJHzLn3OdmllbCLucDLztv2oBvzCzJzBo659aXfF7Iys3DZ1awgDfLiYiIiJQkJyeHjIwM9u/fH+5Sqp3o6GgaN25MIBAo9TGVpVN/KrCm0OOMgnVHBTIzG4HXikYXn58tdRqS7/ORb0ae+Qq+L1h8Plzhdb4D3/txPiPffDhfwfc+/8H9nc9Hvs/vfe8vtM7vhwPn9fvBjHy/H+fzg8/bxxXs4/z+gscRuAg/BCIhIgKLDGCBAEQEsMgIfBEBLDISX2QEFojEH4jAIiPxRwbwRQXwFXzvIgLe8YEABCIgwvvqncv7aj7D8IKpz2eY9/M6GFR9Br6CwOozw+fj4P7F7ucrWF9oPyvYfmC9+bzHVnBeMwrto6AsIlIZZWRkkJCQQFpamv5GVyDnHFu3biUjI4PmzZuX+rjKEsiK+iQUOcmmc24sMBagU2Idt6vXqVh+PuTnQ14elp+P5efhy8vD75y3Li8Pc/lYnreN/HwsPx9ffi7kZ2M53npvXcH3zmF5ed5jV7C/Kzh3fj6W7/Dl5xWs87Yd2NcXpvlBc81Hrj+CHJ+fPJ+fHF8EeT4fOf4A2f4A2f4IsiIOfF+wRATI8UWQHREg64j1R+6XXbBfdqHzHXicGYgiMyKK/RFRZAai2B8RSVZEpJfMChwIaYWDW62YAI2SYkhNiqZRYgyNkrylbkIkMYEIYiP9xEb5iY2MIDbgx+fTHw0RkYqwf/9+hbEgMDOSk5PZvHlzmY6rLIEsA2hS6HFjYN2xDopu3YL2H74dtKKOW0EQPLjk5HhLbm6RX/Oyc8jNyiIvK4e8rGzysrLJzckmLyuH/Owc8rKzCx2TBzmFHhecx3JyvXW5OVih81tuLoGcbCKzsvBlZ2PZWVh2Nr7sgsc5e/Flet/7cg6t9+Vk48vLK9+PwYzcqGhyI6PIiYohNyqanIIlNzKKnMgY9kVEspMIthHB1vwIdvsCfF8Q6PYFotkbGXPYkhMbR358ApGJCdStX4emyXE0SIwmOS6SOnGRJMdH0aR2DPVqRVfQmykiUn0pjAXH8fxcK0sgmwSMMrPXgZOBncfqP1apmUFEhLeUgr9gqXTy8iA7G7Kyjl4Kr9+/HzIzYd++Q1/37cMyMwns20dg3z5iCq0/tM922HlovcvMxPbtK3155mNfZDR7AoeHtu8jY3AJCeTHxZMbG0tOfAI5CYnk1Eokv1YS+Ym1cEm1oXYSLqk2Fh2NP8JPhM/w+wy/GRF+73tvne/gtgif4Tu43ojw+byvhfb3FTo+OS4Kv1r1RETkGEISyMzsNaAvkGJmGcDdQADAOfc0MAU4G1gO7AOuCUVdcgx+P8TEeEsIGHitiwcC3p493rJ79+FLwTr/7t0k7N5NzM5dJO7cRd6OXeTv2kXuzp24jAwiM/cSlbWPqJzsEp83yx/Brqh4dkXHsysqjl3RceyMjmdHoe+3xdRie0wi22JrsS2mFttiE9kdFYuzkm9UTowJ0K1JEg0To6lfK5p6taKIj4ogKsJHZISPqAh/wVfvcaTfR1TAX/C14HGET/+LFZFqye/307lz54OPL7nkEm6//fZyn/fss8/m1VdfJSkpqUzbSjJu3DjmzJnDE088Ue76ihKquywvPcZ2B/w2FLVIJWd2KATWqVOqQyI4xgc5Kwt27oQdO2D7dtixA7d9O7lbt5G3zXscv3078Tt2kLpjB7ZjB7ZzLb5NO/Hv3OFdAi5Cvt9PTlJtspPqkF07mazE2uxPqsP+xNrsr1WbPbVqszQ/lnn/i2aWxbMyPxJXZHfJkkVF+EhLjqNVvXgGdWpAg8RoYiP9xEVGqH+diFRpMTExzJ8/v8LPO2XKlKPWOedwzhW5rTKoLJcsRYInKgrq1fOWAobXRHvMG5Kdg717YcuWoxbf5s1EbdlC1IF161bCgtmwdat3kwlwWuFTBQLk169PTt0G5NStR1ZKXfan1CMzuS77atdlT5267E5KZledumThJys3n6zcfLbvzWbV1r18u3Ib7y8s/kp+TMBP85Q4+rROoWmdWJrUiaVJbe8miehApbwoLiKVxL2Tf2Dxul0Ves4OjWpx93kdj+vYtLQ0LrvsMqZPn05OTg5jx47ljjvuYPny5dx6662MHDmSGTNmcNddd5GcnMyyZcs47bTTeOqpp/D5fAenVtyzZw+DBw+mX79+zJw5k4kTJ3L66aczZ84cUlJSePnll3nooYcwM7p06cJ//vMfJk+ezH333Ud2djbJycm88sor1K9fv0J/NkVRIBMpiRnEx3tLaSeyz8/3WuM2bYKNG2H9etiwAduwAf/69fg3bCB6QwYJ82Z7Qe7Iu3LNoEEDaNLksCW7ZSoromuzM6UB22slsy/PsTc7j8zsXPZm5bE3K5f5a3Yw7qtVZOflH3bK+rWiOLV1XU5tnUL7hrVoWidWIU1Ewi4zM5Nu3bodfHzHHXcwbNgwAJo0acLMmTP5/e9/z/Dhw/nqq6/Yv38/HTt2ZOTIkQDMmjWLxYsX06xZMwYNGsQ777zDRRdddNhzLFu2jBdffJGnnnrqsPU//PAD999/P1999RUpKSls27YNgD59+vDNN99gZjz33HM8+OCDPPzww0H8KXgUyEQqms/nXW6tUwfatSt535wc2Lz5YGhj3TrIyIA1a7xl0SL44APYt49I4ODZIiKgUSMvrDVvDi1aeF87tiD/nE5sSkhmzc79rNm2j4ztmazYvIcPFq5n/NwMoCDz1Yqmce0Yr29bQjT1a0VRr1YU9ROiqVcrmoaJ0cRF6U+ESE1wvC1Z5VXSJcshQ4YA0LlzZ/bs2UNCQgIJCQlER0ezY8cOAE466SRatGgBwKWXXsqXX355VCBr1qwZPXv2POr806ZN46KLLiIlJQWAOgXdZDIyMhg2bBjr168nOzu7TGOJlYf+2oqEUyDgBatGjYrfxzmv79uBkHbk8sUX8OqrBy+T+oAGkZE0aN6cE1u0OBjWcrumsaZ2QxbG1GPF3nx+LghrP6zbxbRdm9iXffgwJxE+o2/butRNiKJWdIBaMd5SOzbAqa3rkhhT+hGoRUTKKioqCgCfz3fw+wOPcwv69h55w1NRN0DFxcUVeX7nXJH733DDDdx8880MGTKEGTNmcM899xzvSygTBTKRys7sUItb165F75OdDT//DCtWeMvKlYe+//pr2LmTCKA50NwMmjaFtm0PLd3asjetLRtqpbBpdzabdu/ns2Wb+T5jB/PX7GRXZs5hl0FjI/2c360R3ZvW5tTWKTRMDM2duCIihc2aNYuVK1fSrFkz3njjDUaMGFHqY88880wuuOACfv/735OcnMy2bduoU6cOO3fuJDU1FYCXXnopWKUfRYFMpDqIjIRWrbylKNu3e+Fs+XJYtuzQ8uKL3jAiQBzQMjaWlgUh7fwOHaBzZ+jUCVq0YH+eY1dmDmu2Z/LKN6uZNH8dr81aQ0zAzwvDT6RXy+TQvV4RqRaO7EM2aNAgRo8eXerje/Xqxe23387ChQs57bTTuOCCC0p9bMeOHfnTn/7E6aefjt/vp3v37owbN4577rmHiy++mNTUVHr27MnKlSvL8pKOm7kwTfNTEdLT092cOXPCXYZI1eWc12+tcEhbtgyWLoVVqw7tFxsLHTt64axzZ+jcmfwOHfkuK5Jrxs0hOy+fWwe2pUOjWrSqF0/d+CiNnVZFDXtmJgBv/LpXmCuRYFuyZAnt27cPdxnHbcaMGTz00EO899574S6lSEX9fM1srnMuvaj91UImUpOZQWqqt5xxxuHb9uyBxYth4UJvWbQI3n/fa1XD66uWnpLCt+068kFkQ2YsaMJ/G7Ride2GRAYiSE3ybhiIOzBWWqSfmEi/Nz9pweO4qAgaJkbTuHYsjZKiiYrQnZ8iUjMpkIlI0eLj4aSTvKWwTZu8cFYQ1GLnz+fCWe/wi2xvRoTsuHjWt2jP8sZtWNigFYsbtGJhYkP25joys/PYl5NHXv7RLfN+n9GhYS06pdaiSZ1YWqTE0a9dPYU0ESlS37596du3b7jLqDAKZCJSNvXqea1phVrULDsbfvgBvvuOyLlzaTZ3Ls2mjefMrCxvh/h46NEDevbE9exJ1oknk5mUzJ6sXNbtyGTtjkxWbN7LnNXb+PiHjWzde2i6q1Nbp5CaFEPj2jGk1o6hWXIc3RonaWYCEalWFMhEpPwiI6F7d2+57jpvXU4OLFkC330Hc+fCrFnwyCNYTg7RQHTz5tTu1YsmvXpBr15wRhcItAVgT1Yu/56xnFVbvaE5lizZxJY9WQefrkXdOG4e0IZzu5QwXIiISBWiQCYiwREIQJcu3jJ8uLcuM9MLaDNnesv06d4YauDNX5qeDqedRvzpp3Nrn1Og0PhB+3PyWLsjkwUZOxj7+UpGvTqPifPW0rNFMhenN9G4aCJSpSmQiUjoxMRA797eAt5dnmvWHApoX38No0fD/fd7sxGkp0PfvnD66UT37k3Lugm0rBvPWR0acOeEhcxauY2pSzbxjw+X0rZBAoM7NeSEprVpkBhNg1rRxESq/5mIVA2+cBcgIjXYgUFqhw2Dxx7zLmtu3w4ffgi33uptf+ghGDwYatf2bjD44x+Jm/Ep/xzSlpl3nMm7v+3NtX2aE+n3MeajZVz67Df0e2gG7e/6kNvfXkDmETMQiEjlsmHDBi655BJatmxJhw4dOPvss/nxxx+L3T8+Ph6AdevWHZwmady4cYwaNapcdTz22GPs27evTMfMmDGDc889t1zPe4BayESkcklIgIEDvQVg716v9eyzz7zln/+EMWMgKgr69KHrWWfR9ayzcGf1ZPX2/azdkcmGnfuZtnQTr89ew7a92dw2uB0tUuI0NppIJeOc44ILLuDqq6/m9ddfB2D+/Pls3LiRNm3alHhso0aNGD9+fJmeyzmHz1d0W9Rjjz3GFVdcQWxsbOlfQAVSIBORyi0uDvr39xaAffu8+Ts/+QQ+/hhuuw1uuw2rV4+0/v1JO+ssGDCAX/Q4geYfLePJGcv5ePFGmtSJ4aGLunJyC80oIHKUm26CYib5Pm7dunkt3yWYPn06gUCAkSNHFjqsG3v27OHMM89k+/bt5OTkcN9993H++ecfduyqVas499xzWbRoEQBr1qxh0KBBrFy5kssuu4y7776bVatWMXjwYPr168fMmTOZOHEio0ePZvbs2WRmZnLRRRdx77338vjjj7Nu3Tr69etHSkoK06dP5+OPP+buu+8mKyuLli1b8uKLLxIfH8+HH37ITTfdREpKCieccEKF/bh0yVJEqpbYWK/17KGHYMECWLsWXnoJBgyAqVO9GwhSU6F7d/7w1SvM6p/A385rz67MXK58fhYf/7CB/Tm6jClSGSxatIgePXoctT46OpoJEybw3XffMX36dG655RaONbPQrFmzeOWVV5g/fz5vvfUWB2byWbZsGVdddRXz5s2jWbNm3H///cyZM4cFCxbw2WefsWDBAm688UYaNWrE9OnTmT59Olu2bOG+++5j6tSpfPfdd6Snp/PII4+wf/9+rr/+eiZPnswXX3zBhg0bKuxnoRYyEanaGjWCq67ylvx8b8DaDz/0ZhW4/37q/u1vXFm/PuedcRZ35TXnpuf3khsTR6fUWow4rQUDOzbQpUyRY7RkhZpzjjvvvJPPP/8cn8/H2rVr2bhxIw0aNCj2mAEDBpCc7LWAX3jhhXz55ZcMHTqUZs2a0bNnz4P7vfnmm4wdO5bc3FzWr1/P4sWL6dKly2Hn+uabb1i8eDG9C25Ays7OplevXixdupTmzZvTunVrAK644grGjh1bIa9ZgUxEqg+fD7p29ZbbboOtW71w9t57JE2ZxOM7d5IfiGRV55OY1PQE7v7hBF7u1paL0xvTsVEiLVLiiPDrwoFIqHTs2LHIfmCvvPIKmzdvZu7cuQQCAdLS0ti/f3+J5zryP1YHHscVGj5n5cqVPPTQQ8yePZvatWszfPjwIs/rnGPAgAG89tprh62fP39+0P4DF7K/PGY2yMyWmdlyM7u9iO2JZjbZzL43sx/M7JpQ1SYi1VRyMlx+Obz2GmzeDNOn47thFC32bOKmiY/z7VPD+cO917Lolnu49p636Hj3R5z/5FfcOWEhr3y7mgUZO455mUREjt8ZZ5xBVlYWzz777MF1s2fPZvXq1dSrV49AIMD06dNZvXr1Mc/1ySefsG3bNjIzM5k4ceLB1q3Cdu3aRVxcHImJiWzcuJEPPvjg4LaEhAR2794NQM+ePfnqq69Yvnw5APv27ePHH3+kXbt2rFy5kp9++gngqMBWHiFpITMzP/AkMADIAGab2STn3OJCu/0WWOycO8/M6gLLzOwV51x2EacUESmbQMAb06xvX3j4YVi6FN5+m+5vv80J05/nL9OfZ12rjnzW6VRebZzOq3HepZGeLerw6LBuNEyMCWv5ItWRmTFhwgRuuukmRo8eTXR0NGlpadxzzz3ceOONpKen061bN9q1a3fMc/Xp04crr7yS5cuXc9lll5Gens6qVasO26dr1650796djh070qJFi8NC24gRIxg8eDANGzZk+vTpjBs3jksvvZSsging7rvvPtq0acPYsWM555xzSElJoU+fPgdvKij3zyIU//szs17APc65gQWP7wBwzj1QaJ87gCZ4wSwN+ARo45zLL+686enp7kCnPRGR47ZiBbz9trd8+y0A2R06MqnVKfy74Yn8lNSIWwe25dwuDWmWHHeMk1Vtw56ZCcAbv+4V5kok2JYsWUL79u3DXUa1VdTP18zmOufSi9o/VH3IUoE1hR5nACcfsc8TwCRgHZAADCsqjJnZCGAEQNOmTYNSrIjUMC1aeAPR3nqrN3PAO+8Q+fbbXDT5OS5yz7K0STvenH0qF7U/FdegAfUSoqlfK4qBHRtwyUn6OyQi5ReqPmRF9YA7smluIDAfaAR0A54ws1pHHeTcWOdcunMuvW7duhVdp4jUdE2awO9+B59/Dj//DA89RLu6sdw17Vm+/fdwXnvrLi5a8AlzFqzi9ncW8o8Pl4a7YhGpBkIVyDLwLkce0BivJaywa4B3nGc5sBI49kVjEZFgadwYbrkF5s6FJUvw/elPtN67mWtf+BsL/n0VE6Y/yoqx/+HPb3zHsg27w12tSJnpppXgOJ6fa6gC2WygtZk1N7NI4BK8y5OF/QycCWBm9YG2wIoQ1SciUrJ27eCvf4X//Q++/RYbOZJuqxfxzIS/8/tr+vHleVdyy59f5rGpPzJr5Tb9QyeVXnR0NFu3btVntYI559i6dSvR0dFlOi4kfcicc7lmNgr4CPADLzjnfjCzkQXbnwb+Bowzs4V4lzhvc85tCUV9IiKlZuZNcn7SSdhDD+E++ojosc9x9ZT3iZjzLgufa8lbnfvzz/5D6HVia3q3SqF709rhrlrkKI0bNyYjI4PNmzeHu5RqJzo6msaNG5fpmJDcZRksustSRCqNrVvh1VfJe+EF/PPnk+0P8HGrkxnfuT+Nfnk+t53TkcTYQLirPCbdZSkSPCXdZakhqUVEKkJyMtxwA/5582D+fCJH/R9nb13KuPH3cMO1/Xl58LX84/mp/LR5T7grFZFKSIFMRKSide0Kjz2Gb+1aePttEk7szm+/eIVbrh/Iit4DePv+59iyKzPcVYpIJaJAJiISLJGRcOGFxE/7BN/y5WT//mZ6bvyRX/z5evY3a87iG+7AbdgQ7ipFpBJQIBMRCYUWLYh9eAwJm9ez7plxbK3XmA5PjCavcWN2nHsB+VM/hSrcp1dEykeBTEQklCIjaTTiajotmc3E1z/lvycNxX36Kb4B/dnbpj08/TTs3RvuKkUkxBTIRETCwO8zhg47g/M/eZUZn37H7efdzE978uA3vyGzQSP23XATrFwZ7jJFJEQUyEREwqh2XCQXnNKKfg/cyqP3/5cbRv2LT5p0I/DUE+S3bMmK3v1ZO/49Xc4UqeZCNbm4iIiUYGDHBgzs2AA4mUc/OYvrv/2BPlPHM3T2+6RcfB507Ag33ABXXAFxceEuV0QqmFrIREQqmd8PaMO4P1/Ar2a8wqtvfsEt5/yejH35MHKkN7/mHXfAuiOnAxaRqkyBTESkEht+Zjsir72GMy55iHtuewbXvz88+CCkpcHVV8OCBeEuUUQqgAKZiEglVis6wAMXduaOc9ozjlT+c/MYb4Lz3/wG3n7bG4T2rLPgo4/Uz0ykClMgExGpAq7ulcYpLZO5690feG4t7H3wYVizBh54ABYtgkGDvHA2bhxkZYW7XBEpIwUyEZEqwOcz7jy7Pa3qxXPf+0vocu/HnPHC91xT/wxmTZvrBTGAa66B5s29oLZtW1hrFpHSUyATEakiOqUmMvXm03n1+pP5VZ/mtGuQwHc/7+CX475jpK8T89/7jLwPPoTOneHOO6FJE7jxRli1Ktyli8gxmKvCfQ7S09PdnDlzwl2GiEjYLNuwm398uJRpSzcBEOn30aJuHOf7tnDtrIlEvfk65OfDsGFw223QpUuJ5xv2zEwA3vh1r6DXLlLTmNlc51x6UdvUQiYiUoW1bZDAC8NPZNKo3oy+sDPX9EkjJT6Kf6yLokfLy7n3scmsvuJXMGmS18fs7LPhs890A4BIJaOBYUVEqoEujZPo0jjp4OO5q7fx8szVvPu/LbzY4Dy6/PEcfv/jVE794FUi+vaFk0/2WszOPx98+r+5SLgpkImIVEM9mtWhR7M65OTl89qsn5m6ZBPX7D2Ldn84mykJP+F75GG48EJo2xZuvdWbASAqKtxli9RY+m+RiEg1FvD7uKpXGi9fexJ3n9eBpTtyuafxaWT/sARefx1iY+FXv/LuzBwzBvLywl2ySI0UskBmZoPMbJmZLTez24vZp6+ZzTezH8zss1DVJiJSEww/JY0Rp7Xg5Zmr6fHAdO6J6cTaT7+Ejz+GDh3gj3+EmV/DvHmwbFm4yxWpUUISyMzMDzwJDAY6AJeaWYcj9kkCngKGOOc6AheHojYRkZrCzLhjcDteGJ5OZISPcV+vou9DM3gq0Jz8jz+B2bMhJgZ27fKGzrj2Wvjhh3CXLVIjhGTYCzPrBdzjnBtY8PgOAOfcA4X2+T+gkXPuz6U9r4a9EBE5Pnn5jvlrdvC39xYzf80OakVH0KVxEj9t3kNiXhYfbJiCvfgi7NsHgwfDH/4A/fqBWbhLF6myKsOwF6nAmkKPMwrWFdYGqG1mM8xsrpldVdSJzGyEmc0xszmbN28OUrkiItWb32f0aFabCf93Co8O68rgTg3ZmZnD+p37WbrH8eC5o8hftRruuw+++w7OPBN69IBXX4WcnHCXL1LthCqQFfVfqiOb5iKAHsA5wEDgL2bW5qiDnBvrnEt3zqXXrVu34isVEalBzIwLujfmHxd1YfINfTipeR0SoiP494yf+MO0DHbe/EdvpP/nnoPMTLj8cmjZEh55xLu0KSIVIlSBLANoUuhxY2BdEft86Jzb65zbAnwOdA1RfSIigve/5w4Na3FT/9a8M28tl4z9hplr97L/quFef7L33vMC2S23eFMz3XqrN8m5iJRLqALZbKC1mTU3s0jgEmDSEfu8C5xqZhFmFgucDCwJUX0iIlLITf3b8LehnVi+aTeXPvsNvR74lEkLN8A558D06d4NAGefDY8+Ci1awJVXwvz54S5bpMoKSSBzzuUCo4CP8ELWm865H8xspJmNLNhnCfAhsACYBTznnFsUivpERORoV/Zsxpw/D+C5q9JpkBjDn95ZyLa92d7G9HR47TVYvhxGjYIJE6B7d6+v2YQJkJsb3uJFqhhNLi4iIgcVN7n4orU7ufDfX9O2fgLPXNmDhonRWOE7Lrdvh7Fj4cknvUuYTZrAb37jDTqr/r4iQMl3WSqQiYjIQcUFMoBPl2zk1/+ZS26+IyU+iuYpsTSuHUuTOrH0bFGHns2T8eXnef3MnngCPv0UIiPhkkvgt7+Fk04K9csRqVQUyEREpFRKCmQAyzbs5uuftrB43S5+3raPjO2ZrN+ZSb7zbgY4v1sjBndqSNPkWFi8GJ56Cl56CfbsgRNP9C5v/vKXEB0dypclUilUWCArGOD1CuBUoCGQCSwC3gf+65zbWf5yS0+BTESkYh0rkBVlT1YuT8/4ifcWrGPV1n34fcaofq24+pQ06sRFesNj/Oc/XqvZ0qWQkuJdyhw5Epo1C9ZLEal0KiSQmdkHeENVvAvMATYB0XgDuvYDzgMecc4defdk0CiQiYhUrOMJZIWt2baP295ewNc/bcXvM5rWiaVZciydGiUyqGN9Oi2b6/Uze/dd74Bzz4URI2DQIPD7K+pliFRKFRXIUgrGByvXPhVJgUxEpGKVN5AdsGzDbt5fsI6fNu9lxZa9LFm/C7/PeO+GPrRvWAt+/hmefhpeeAE2boTGjb25M6+7Dpo2rYiXIlLpVMjUSc65LWbmN7OpJe1zPAWKiEj10rZBAjef1ZYnLz+BD353KuNH9iLS7+MX//6a5Zt2e6Hr73/37sh8+23o1An+9jdIS/PGN5swQVM0SY1SpnHInHN5wD4zSwxSPSIiUg2lp9Xh9RE9yc13nP/EV2zatd/bEAjAhRfCBx/AihXw5z/DggXeuqZN4Y474Kefwlu8SAgcz8Cw+4GFZva8mT1+YKnowkREpHrp2iSJl689ib3ZeZw2ZjrDX5zF23MzONh1Ji0N/vpXb+7MSZO8uzIffBBatYL+/eGVV2DfvnC+BJGgiTiOY94vWERERMqkZ4tkHr+0O5//uJm5q7dzy1vfsy87l8tObobfVzDQbEQEnHeet6xdCy++6PU1u+IKSEiAYcNg+HA45RQoPDitSBWmcchEROSgiurUXxrOOYY++RXfZ+wkLtLPjWe2ZnjvNKIiirjbMj8fvvgCxo2Dt96CvXu9lrPhw715NHUjgFQBFXWX5WRgLPChcy7niG0tgOHAKufcC+Urt/QUyEREKlYoAxnAzn05fLJkI09NX86KLXuJjPDRrXES6Wm1SUuJIyU+kuS4KJLjI0lNivGma9qzx7sRYNw4mDHDayU780y46ioYOtRrRROphCoqkDUAbgZ+AWwDNuONQ5YG/AQ84Zx7tyIKLi0FMhGRihXqQHZAXr7j8x838/VPW5i1ajuL1u4kL//wf5/ObFePR37ZjcTYwKGVK1fCyy97swGsXAkxMTBkCFx2mTe2WWRkSF+HSEkqKpC1Aho45740szQOjdRfC1jrnAv5bTAKZCIiFStcgexI+3Py2Lw7iy17sti6J5uPftjAO/PW0jwljocv7krXJkmHH5CfDzNnwquvwptvwpYtULs2XHSRF85OOw18x3Mfm0jFqZBxyIDHgF0AzrlVzrmZzrn5wL6CbSIiIhUiOuCnSZ1YujetTf8O9RlzcVfuH9qJn7ft4/wnv+LK57/l658KDX3p80Hv3t4sAOvWwZQpcM45XkDr18/rY/aHP8B330EV7jst1VdZAlmac27BkSudc3PwLluKiIgEzSUnNWXOn/vzm74t+eJ/W7js2W/5x4dLyT/i0iaBAAwe7M2fuXEjvP469OgBjz/ufW3f3hte43//C88LESlCWQJZdAnbYspbiIiIyLHUig5w26B2fHlbPy48IZV/z/iJng98yo2vzeO5L1awNyv38APi4rxhMt59FzZsgLFjoWFDuOceaNMGTjoJHn3Ua1UTCaOyBLLZZnb9kSvN7DpgbsWVJCIiUrLGtWN5+OKuPHnZCZyYVoc5q7Zx3/tLuPCpr5n509aiD6pTB66/HqZP9+bSfOghyMuDm2+G1FTvkuejj8Lq1aF9MSKUrVN/fWACkM2hAJYORAIXOOc2BKXCEqhTv4hIxaosnfqPx4eLNvC39xazfmcm1/VpTp/WdTm1VQo+3zEGj126FMaP95bvv/fWnXgi/OIX3tKqVfCLlxqhQu6yLHSyfkCngoc/OOemlbO+46ZAJiJSsapyIAPYuieLOycsZNrSTeTkOVrXi+eSk5pybe80bwyzY1m+3Bvj7O23YfZsb13Xrt7dmr/4hdf/TOQ4VdRdlgA456Y75/5VsJQ6jJnZIDNbZmbLzez2EvY70czyzOyistYmIiI1W3J8FM9cmc73d5/Fwxd3xWfG395bzPsL15fuBK1awW23waxZ3pyajzzi9UP7y1+gQwfo2BHuustrSdPdmlKBQjJ1kpn5gR+BAUAGMBu41Dm3uIj9PsGbwPwF59z4ks6rFjIRkYpV1VvIjpSTl895//qSpRt207tVMqe1rssJzWrTrkECCdGBY5/ggLVrYcIEr+Xs88+9cc+aNfMGoT3/fG+cs0AZzic1UoW2kB2nk4DlzrkVzrls4HXg/CL2uwF4G9gUorpERKQaC/h9PHtVOtef2pzNu7N44IOlXPz0TLrc+zFjPlpKdm5+6U6UmgqjRnk3BKxfD889B126wLPPQv/+ULeuNwDt66/Dzp3BfVFSLYUqkKUCawo9zihYd5CZpQIXAE+XdCIzG2Fmc8xszubNmyu8UBERqV6a1InlT+d04OPfn86sO8/k0WFdqRUd4MnpP9H/kc+Y9H0Zh7yoVw+uuw4mTYKtW2HiRK9/2dSpcOmlkJICAwbAv/6lOzal1EIVyIrqSXnktdLHgNucc3klncg5N9Y5l+6cS69bt25F1SciIjVAvVrRXNC9MfPvGsDzV6cTFxXBja/N47GpPx49wGxpxMZ6lyyff95rOfvqK7jlFsjIgBtvhLQ06NQJ/vhHr3UtO7vCX5NUD6EKZBlAk0KPGwNH/pckHXjdzFYBFwFPmdnQkFQnIiI1iplxZvv6TPi/Uzi3S0Mem/o/Ji8o5+Cwfj+ccgqMHg1LlsCyZd5YZw0awGOPwRlneK1nF17oXfJcu7ZCXotUD6EKZLOB1mbW3MwigUuASYV3cM41d86lOefSgPHA/znnJoaoPhERqYGiA34eG9aNBrWi+cvERazYvIe842kpK0qbNl5r2dSphy5tXnopzJnjDVDbuLE3pMYdd3g3CuTkVMzzSpUUEYoncc7lmtko4CPAj3cH5Q9mNrJge4n9xkRERIIlwu/jN31bcvekHzjj4c+IDvhoXS+BDg1r8dehHYmK8Jf/SRISvEub55/vDZfxww/wwQfeJOgPPeS1qiUmen3PBg/2vjZpcuzzSrURkmEvgkXDXoiIVKzqNuxFaTnnWLR2F0s27GLZht1MnLeWrXuzqR0b4InLTuCUlsmlG1j2eOza5bWiTZnihbQD82q2bevdwdm/P/Tr5wU2qdIqdKT+ykSBTESkYtXUQHak/HzHhHlruXfyD+zan0urevE8eFEXTmhaO7hP7BwsWuQFtKlT4bPPYO9e8Pm8idD79/daz3r2hMjI4NYiFU6BTERESkWB7HB7s3J5f+F6Hv54GRt3ZTGwY316tkime9PadGhYi8iIIHfFzs6Gb77xwtknn3gzCOTne3d3nn76oYDWqRMEqwVPKowCmYiIlIoCWdH2ZOXy6Cc/MmXhetbv3A9Ai5Q4Jt3Qh/iokHTH9uzcCTNmHApoy5Z56+vX9+7i7NfPC2qtWyugVUIKZCIiUioKZMe2fmcmk79fx9+nLOX2we0YeXrL8BWzZg18+umhS5wbN3rrGzTwgtmBpX17BbRKoKRAFsJYLyIiUvU1TIxhxGktmfT9Op757CfO6lCfFnXjw1NMkyYwfLi3OAc//uj1OzuwvPGGt1/duocHtI4dvX5pUmno3RARETkO9w7pSG6e4/wnvmLrnqxwl+O1gLVtCyNGwCuveK1ny5d7g9AOGuT1P7vhBm8Ozrp14YILvAFr582DvBInyZEQUAuZiIjIcejRrA5v/LoX5/zrC64ZN5vXR/QkNrIS/bNqBi1best113nrVq06vAVt4kRvfUKCd+dm797ebAMnnwy1aoWr8hqpEn1yREREqpYOjWrxp7Pbc9/7S7j46ZncO6QjPZrVDt6YZeWVluYtV1/tPc7I8ILZ119783D+9a/eXZw+H3Tu7IWzU07xglpamvqhBZECmYiISDlc16c5Wbn5vPjVSi56eiYNakVzZvt6nNWxAae1Tqm84Qy86Zsuv9xbwBuk9ttvvYD29dfw3//Cv//tbWvQ4FA4O+UU6N4doqLCV3s1o7ssRUTkIN1lefx27c/ho0UbmLpkIzOWbSYrN5+Tmtfh6l5p9GtXt3JdziytvDxvmqevvjoU0las8LZFRXmh7KSTDi2tWqkVrQQa9kJEREpFgaxi7M/JY/QHS5n0/Tq27c0mMSbAwI71aVw7ll4tkzkxrU64Szx+69fDzJleOJs1C+bOhX37vG21a8OJJx4e0urXD2+9lYgCmYiIlIoCWcXKy3e8v3A9L329ijXb9rFpt3c3Zlykn6TYSBJjAiTFBujWJIkRp7UgKbYKToeUmwuLF3vhbPZs7+vChYfu3Gza9PCA1qMHxIdpmJAwUyATEZFSUSALrqUbdvHJDxvZkZnDjn057MzMZs22TJZt3I3PIL1ZHW4/u13w58wMtn37vOE0Zs06tBy41OnzeQPV9ujhXfI84QTo1q1G3NWpgWFFREQqgXYNatGuwdHB4/s1O5i6ZCP/mracm9+Yz9u/OYXk+CrcYT421uv837v3oXVbthxqQZs925v66eWXD21v1coLZwdCWvfu3nhpNYQCmYiISJh1bZJE1yZJdGxUixtfn885j3/Jb89oRfsGCbSun0BiTCDcJZZfSgoMHuwtB2zY4LWkffed93X2bHjzzUPbGzc+PKSdcAKkplbLGwcUyERERCqJQZ0aMuH/YrnxtXn8ZeIiACIjfPzjF525oHvjMFcXBA0aHB3Stm+H+fO9kHZgmTzZmxoKvGDXpcuhpXNnbyqomJiwvISKokAmIiJSiXRslMjHvz+djO37WLF5L499+j9+/8b3TFm4gd+d2ZpOqYnhLjG4ateGfv285YA9e2DBgkOtaQsXwjPPQGamt93ng9atvXBWOKilpVWZOTsVyERERCoZv89olhxHs+Q4erZI5vkvV/DMZyv4ZPFGbh7QhhvPbB3uEkMrPv7QrAEH5OV5NwosWOAFtAOB7e23D7Wmxcd7waxwUOvUyQt9lUzIApmZDQL+CfiB55xzo4/YfjlwW8HDPcBvnHPfh6o+ERGRyigm0s+oM1pz1Slp3DZ+AY988iPRAR8jTmsZ7tLCy+/3WsVat4Zf/OLQ+j17vMFsCwe1t96CsWMP7dOggXenZ4cOhy9164atf1pIApmZ+YEngQFABjDbzCY55xYX2m0lcLpzbruZDQbGAieHoj4REZHKrlZ0gId/2ZVte7P5+5SlvPDlKk5vU5fLezalS+OkcJdXecTHe5Ojn1woQjgH69Z54Wzx4kPLf/7jTRd1QJ06R4e09u1DciNBqFrITgKWO+dWAJjZ68D5wMFA5pz7utD+3wDVsPeiiIjI8YuNjGDsVemMn5vBvJ+3M+n7dbwxZw1t6sdz39DOnNS8Cs8AEExmXqhKTT38BoIDQa1wSFuyBMaPh23bDu2XkHAooLVrB23aeEvLlhU2n2eoAlkqsKbQ4wxKbv26DvggqBWJiIhUQYkxAa7r0xxozq79Obw7fx1PTV/Or16azRU9m9GqXjyt6sXTsm48cVHqKl6iwkFtwIBD652DzZsPD2qLF8OUKfDii4f28/m8GwcOBLS2bQ9937hxmW4oCNU7VVQ7X5FTBJhZP7xA1qeY7SOAEQBNmzatqPpERESqnFrRAa7s2YweTWtz54SFjP18Bbn5h/557ZyayF/P70iTOrHUjo3E76t+43cFhRnUq+ctffsevm3HDvjf/2DZMvjxx0PLF1/A3r2H9ouJ8fq3HQhobdqU+JShCmQZQJNCjxsD647cycy6AM8Bg51zW4s6kXNuLF7/MtLT06vuvE8iIiIVpEOjWkz8bW+yc/P5edtelm/aw/w1Oxn39UoueMrrEWQGSTEBGiTG8K9Lu9GqXkKYq66ikpK8CdRPPPHw9Qcufx4IaAcC2/ffw4QJh+b2LEaoAtlsoLWZNQfWApcAlxXewcyaAu8AVzrnfgxRXSIiItVGZISPVvUSaFUvgUGdGnJdn+bMXLGVbXuy2LY3m5Vb9zH5+3UMeeIrTm2dQoeGiZzSKpn0ZrWxajj6fUgVvvxZeAw1gJwcb4iOdu2KPTwkgcw5l2tmo4CP8Ia9eME594OZjSzY/jRwF5AMPFXwocgtbgJOEREROba6CVEM6drosHVXnNyUN+dk8N3P2/l48UYenQrndGnImIu6EBupPmdBEQh4/ctKELKfvHNuCjDliHVPF/r+V8CvQlWPiIhITXRyi2RObpEMwO79OYz9fAX/mracb1dsZVS/Vgzu3JB6CVFqMQsxRWEREZEaKiE6wC1ntaVv23o8+OFS7pm8mHsmLyY64OP0NnW5Z0hHGiZW7TkiqwoFMhERkRquR7PavHZ9T2au2MpPm/ewZP1uxs9dw9fLtzLm4q50aFiLJnVi1GoWRApkIiIigs9n9G6VQu9WKQAM6dqI4S/OYuR/5wIw8vSW3D64+E7pUj4KZCIiInKUXi2TmXVnf5Zv3s197y/h6c9+Ys22fbSqF0/TOrGc1LwOTerEhrvMakOBTERERIqUGBugR7M6PHtVOo988iMzlm5iyqL1OAfxURF88cd+1I6LDHeZ1YICmYiIiJQoJT6Kv1/QGYCs3Dw+XLSB370+n7+8u4gnLjshzNVVDwpkIiIiUmpREX6GdG3EtKWbeHf+Ouas+pRTW6fw236tSEuJC3d5VZYCmYiIiJSJmfHXIZ3onJrI9xk7eX/hej5evJFfpjemU2oigzo1ICrCH+4yqxQFMhERESmzxNgAvzq1BQBrtu3j1vHf89LM1WTn5lM7NsCoM1qTmhTNiWl1SI6PCnO1lZ8CmYiIiJRLkzqxvD6iF7l5+dw7eTFvzV3D395bDECDWtG8ev3JtKgbH+YqKzcFMhEREakQEX4ffxvaib+e35GdmTl88b8t3PDaPM54+DPO6dyQa3qnkZ5WJ9xlVkoKZCIiIlKhzIyk2EjO69qIRkkxTJq/lnfmreX9hev58zntuapXGpERvnCXWakokImIiEjQ9GhWmx7NavPHQe04919fct/7S3j0kx/p2SKZVvXiaVwnlu5NkuiUmhjuUsNKgUxERESCLi4qgik3nsqXy7fw2Y+bmPnTVr743xay8/IBSI6LJD46gpiAn5hIv/c14Cc60k9swE9CdIBBnRpwUvPqeclTgUxERERCIibSz4AO9RnQoT4A+fmO1dv28c53GWzenUVmTh6Z2Xlk5uSxPyePXftz2Jedx/7sPLbty+aFr1Yy8vSWnNY6ha5NkoiLqj4xpvq8EhEREalSfD6jeUoct5zV9pj77s3K5YbX5vH0Zz/x9Gc/kRwXyfWnteC01nVp3zABMwtBxcGjQCYiIiKVXlxUBC8MP5HNu7P4ePEGXv56NaM/WMroD5ZyZc9m/G1op3CXWC4KZCIiIlJl1E2I4vKTm3H5yc3YuGs/j039kf98s5pPl2ykWXIcTevE0jQ5liZ1YjkxrTYNE2PCXXKpKJCJiIhIlVS/VjT3DulE0zpxLN2wizXb9vHp0k1s2ZMFQFykn4vTm9C0Tiz1akVRLyGalnXjKuXMASELZGY2CPgn4Aeec86NPmK7FWw/G9gHDHfOfReq+kRERKTqiYzw8Zu+LQ9bty87l2UbdvPMZyt49dufD97JeUCLunEkRAeIjvARfeBuzoD3/aHFR1xkBImxAZJiAtSOiyQpJkByfBS1YwMV3mctJIHMzPzAk8AAIAOYbWaTnHOLC+02GGhdsJwM/Lvgq4iIiEipxUZG0L1pbZ6+sgf5+Y4dmTls3p3Fxl37Wbh2JwszdrKv4E7O7fuyWZ+Tx/6cfPbneHd4ZuXkHxXiCosO+GiUFENqUgwp8VGkxEeSEh9FnbhIEmMCJMYEqFXoa1yk/5gBLlQtZCcBy51zKwDM7HXgfKBwIDsfeNk554BvzCzJzBo659aHqEYRERGpZnw+o05cJHXiImnbIIHT2tQt1XF5+Y692bns3JfDjn057MjMZse+HLbsyWLt9kzW7shk/c79rNyyl827s8jKLT7AJcUGaHWMuTxDFchSgTWFHmdwdOtXUfukAocFMjMbAYwAaNq0aYUXKiIiIuL3GbWiA9SKDtDkGGPROufYk5XL9r057MzMYdf+gq+ZOezIzGH11n38tGlPiecIVSArqp3OHcc+OOfGAmMB0tPTj9ouIiIiEkpmRkJ0gIToQMn7/ab4baGa2TMDaFLocWNg3XHsIyIiIlLthCqQzQZam1lzM4sELgEmHbHPJOAq8/QEdqr/mIiIiNQEIblk6ZzLNbNRwEd4w1684Jz7wcxGFmx/GpiCN+TFcrxhL64JRW0iIiIi4Rayccicc1PwQlfhdU8X+t4Bvw1VPSIiIiKVRaguWYqIiIhIMcxrmKqazGwzsLoMhyQCOytov2PtU9L2FGBLKeqojEr7M6xsz1Wec5X12LLsX97P2rGOr6qfNX3OKvYY/U0rWlX9nJX3fMH6m6bPWfEOvK5mzrmiB0JzztWYBRhbUfsda5+StgNzwv2zCPbPsLI9V3nOVdZjy7J/eT9rpfgcVsnPmj5nFXuM/qYF/70P9XNVxr9p+pyV72dT0y5ZTq7A/Y61T2mfq6oJ5euqyOcqz7nKemxZ9i/vZ02fs8r1XKH8nJXlGP1NK1pV/ZyV93zB+pumz1nxjvm6qvQly6rKzOY459LDXYdUf/qsSSjocyahUN0/ZzWthayyGBvuAqTG0GdNQkGfMwmFav05UwuZiIiISJiphUxEREQkzBTIRERERMJMgUxEREQkzBTIRERERMJMgawSMrM4M5trZueGuxapnsysvZk9bWbjzew34a5Hqi8zG2pmz5rZu2Z2VrjrkerJzFqY2fNmNj7ctRwvBbIKZGYvmNkmM1t0xPpBZrbMzJab2e2lONVtwJvBqVKquor4nDnnljjnRgK/BKrtuD5SPhX0WZvonLseGA4MC2K5UkVV0OdshXPuuuBWGlwa9qICmdlpwB7gZedcp4J1fuBHYACQAcwGLgX8wANHnOJaoAvefF3RwBbn3HuhqV6qior4nDnnNpnZEOB24Ann3Kuhql+qjor6rBUc9zDwinPuuxCVL1VEBX/OxjvnLgpV7RUpItwFVCfOuc/NLO2I1ScBy51zKwDM7HXgfOfcA8BRlyTNrB8QB3QAMs1sinMuP7iVS1VSEZ+zgvNMAiaZ2fuAApkcpYL+phkwGvhAYUyKUlF/06o6BbLgSwXWFHqcAZxc3M7OuT8BmNlwvBYyhTEpjTJ9zsysL3AhEAVMCWZhUu2U6bMG3AD0BxLNrJVz7ulgFifVRln/piUD9wPdzeyOguBWpSiQBZ8Vse6Y14mdc+MqvhSpxsr0OXPOzQBmBKsYqdbK+ll7HHg8eOVINVXWz9lWYGTwygk+deoPvgygSaHHjYF1YapFqi99ziRU9FmTUKhxnzMFsuCbDbQ2s+ZmFglcAkwKc01S/ehzJqGiz5qEQo37nCmQVSAzew2YCbQ1swwzu845lwuMAj4ClgBvOud+CGedUrXpcyahos+ahII+Zx4NeyEiIiISZmohExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRERGRMFMgExEREQkzBTIRkWKYR38nRSTo9IdGRKQQM0szsyVm9hTwHYfPpyciEhQaqV9EpBAzSwNWAKc4574JczkiUkOohUxE5GirFcZEJJQUyEREjrY33AWISM2iQCYiIiISZgpkIiIiImGmTv0iIiIiYaYWMhEREZEwUyATERERCTMFMhEREZEwUyATERERCTMFMhEREZEwUyATERERCTMFMhEREZEw+39KDqr8zY6c9wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 720x504 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Get negative returns\n",
    "r_neg = np.sort(abs(r[r<0]))\n",
    "\n",
    "# Parameters for likelihood function\n",
    "penalty = 10\n",
    "lambdaa = 10\n",
    "xmin = 0.05\n",
    "alpha = alpha_left # left tail\n",
    "\n",
    "params = [lambdaa, alpha, xmin] # initial guess\n",
    "\n",
    "# Likelihood function\n",
    "def tail_body_likelihood(params, r, penalty):\n",
    "    l = params[0]\n",
    "    alpha = params[1]\n",
    "    xmin = params[2]\n",
    "    \n",
    "    C = 2 - np.exp(-l*xmin) # Normalization constant\n",
    "    L = 0 # Initialize likelihood\n",
    "    \n",
    "    # Find returns lower than xmin\n",
    "    r = np.sort(r)\n",
    "    idxmin = np.where(r>=xmin)[0][0]\n",
    "    \n",
    "    # Computing log-likelihood values for the body of the distribution \n",
    "    for i in range(0, idxmin):\n",
    "        L = L - np.log(C) + np.log(l) - l*r[i];\n",
    "    \n",
    "    # Computing log-likelihood values for the tail of the distribution\n",
    "    for i in range(idxmin, len(r)):\n",
    "        L = L - np.log(C) + np.log(alpha/xmin) - (alpha + 1)*np.log(r[i]/xmin)\n",
    "    \n",
    "    # Computing values of body and tail distributions at xmin\n",
    "    Fbody = l*np.exp(-l*xmin)/C\n",
    "    Ftail = alpha/(C*xmin)\n",
    "    \n",
    "    # Adding penalty term to enforce continuity\n",
    "    L = L - penalty*(Fbody-Ftail)**2\n",
    "    \n",
    "    # Changing sign to log-likelihood (fmin algorithm searches for minima)\n",
    "    L = -L\n",
    "    \n",
    "    return L\n",
    "\n",
    "\n",
    "# Minimize the likelihood function\n",
    "with np.errstate(divide='ignore', invalid='ignore'): # disable divide by 0 warning\n",
    "    par = fmin(tail_body_likelihood, x0=params, args=(r_neg,penalty), maxiter=1000, maxfun=5000, xtol=1e-12)\n",
    "\n",
    "lambdaa = par[0]\n",
    "alpha = par[1]\n",
    "xmin = par[2]\n",
    "\n",
    "\"\"\" PLOT CALIBRATED DISTRIBUTION \"\"\"\n",
    "\n",
    "C = 2 - np.exp(-lambdaa*xmin) # Normalization constant\n",
    "\n",
    "fig = plt.figure(figsize=(10, 7))\n",
    "r_neg_hist = plt.hist(r_neg, 40, density=True)\n",
    "b_neg = r_neg_hist[0] # Value (count)\n",
    "a_neg = r_neg_hist[1] # Bin Edges\n",
    "a_neg = np.mean(np.vstack([a_neg[0:-1],a_neg[1:]]), axis=0) # Bin Centers\n",
    "plt.clf()\n",
    "ax = fig.add_subplot(grid[0, 0])\n",
    "ax.scatter(a_neg, b_neg, label='Empirical')\n",
    "x = np.linspace(0, xmin, 1000)\n",
    "ax.plot(x,lambdaa*np.exp(-lambdaa*x)/C, 'r', label='Calibrated')\n",
    "x = np.linspace(xmin,max(r_neg),1000)\n",
    "ax.plot(x,alpha*(x/xmin)**(-alpha-1)/(C*xmin), 'r')\n",
    "ax.axvline(xmin)\n",
    "ax.set_xlabel('r')\n",
    "ax.set_ylabel('P(r)')\n",
    "ax.set_title('PDF')\n",
    "ax.set_yscale('log')\n",
    "ax.set_xscale('log')\n",
    "ax.legend(loc='best')\n",
    "\n",
    "y = np.linspace(1, len(r_neg), len(r_neg))\n",
    "y = 1-y/(len(r_neg)+1) # Calculating CCDF as rank-frequency plot\n",
    "\n",
    "ax2 = fig.add_subplot(grid[1, 0])\n",
    "ax2.plot(r_neg, y, label='Empirical')\n",
    "x = np.linspace(min(r_neg), xmin, 1000)\n",
    "ax2.plot(x,1-(1-np.exp(-lambdaa*x))/C, 'r', label='Calibrated')\n",
    "x = np.linspace(xmin, max(r_neg)*10, 1000)\n",
    "ax2.plot(x,1-(1-np.exp(-lambdaa*xmin)+1-(xmin/x)**alpha)/C, 'r')\n",
    "ax2.set_xlim(min(r_neg), max(r_neg))\n",
    "ax2.axvline(xmin)\n",
    "ax2.set_xlabel('r')\n",
    "ax2.set_ylabel('C(r)')\n",
    "ax2.set_title('CCDF')\n",
    "ax2.set_xscale('log')\n",
    "ax2.legend(loc='best')\n",
    "\n",
    "print('lambda: ' + str(round(lambdaa, 2)))\n",
    "print('alpha:  ' + str(round(alpha, 2)))\n",
    "print('r_min:  ' + str(round(xmin, 4)))"
   ]
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