notebooks/NB34 - Tail Alpha Distribution.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# About.\n",
    "This notebook is about Power Law Data. It explores the distribution of **finite sample** estimates for the mean and tail exponent.\n",
    "\n",
    "### Permalink\n",
    "[Click here](https://github.com/FergM/fattails/blob/main/notebooks/NB34%20-%20Tail%20Alpha%20Distribution.ipynb) for latest version of this notebook.\n",
    "\n",
    "### Mean Estimation Methods\n",
    "1. Naive Sample Mean\n",
    "    * i.e. just calculate the sample mean. This can be very biased in the presence of skewness.\n",
    "2. Plug-in Estimation\n",
    "    * i.e. estimate the tail alpha first and use that to estimate the sample mean.\n",
    "\n",
    "### Context\n",
    "For background info see the Technical Incerto chapter called *Calibrating Under Paretianity*.\n",
    "\n",
    "I use the Numpy package's Pareto Distribution:\n",
    "* [Numpy Documentation](https://numpy.org/doc/stable/reference/random/generated/numpy.random.Generator.pareto.html#numpy.random.Generator.pareto)\n",
    "* [Wikipedia](https://en.wikipedia.org/wiki/Pareto_distribution)\n",
    "* Properties\n",
    "    * pdf = alpha*x^(-alpha-1)\n",
    "    * CDF = 1 - x^-alpha\n",
    "    * Survival Function = x^-alpha\n",
    "    * Mean = alpha/(alpha-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generate Samples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Settings\n",
    "alpha = 1.2  # The tail exponent\n",
    "\n",
    "n = summand_count = 100  # The number of values in your sample average\n",
    "\n",
    "sample_size = 100_000  # The number of monte carlos you want to run"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Random number generator\n",
    "rng = np.random.default_rng()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([7.71874151, 1.328448  , 1.46391072])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Samples\n",
    "# ToDo: Explain the +1 term here...\n",
    "samples = [rng.pareto(alpha, summand_count)+1 for idx in range(sample_size)]\n",
    "samples[0][0:3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mean Estimates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.000000000000001"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# True Mean\n",
    "true_mean = alpha/(alpha-1)\n",
    "true_mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3.060973357639481, 3.140643943814928, 2.9647158997498657]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Sample Mean Values\n",
    "sample_means = [sample.mean() for sample in samples]\n",
    "sample_means[:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1.2665939829546191, 1.24482971091516, 1.2875449608466047]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# MLE Alpha Values\n",
    "mle_alphas = [(summand_count-1)/(np.log(sample).sum()) for sample in samples]\n",
    "mle_alphas[:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[4.751022393368213, 5.08447159563296, 4.477717004866816]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# MLE Mean Estimates\n",
    "alpha_implied_mean = lambda alpha: (alpha/(alpha-1))\n",
    "mle_means = [alpha_implied_mean(alpha) for alpha in mle_alphas]\n",
    "mle_means[:3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mean Distribution\n",
    "Plot the distribution of sample mean estimates."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Put all the data in one table:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "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>mle_alpha</th>\n",
       "      <th>mle_mean</th>\n",
       "      <th>sample_mean</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>99995</th>\n",
       "      <td>1.056078</td>\n",
       "      <td>18.832402</td>\n",
       "      <td>4.203487</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99996</th>\n",
       "      <td>1.160875</td>\n",
       "      <td>7.216024</td>\n",
       "      <td>3.934823</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99997</th>\n",
       "      <td>0.996467</td>\n",
       "      <td>-282.009705</td>\n",
       "      <td>5.454379</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99998</th>\n",
       "      <td>1.018729</td>\n",
       "      <td>54.392722</td>\n",
       "      <td>5.366482</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99999</th>\n",
       "      <td>1.234867</td>\n",
       "      <td>5.257724</td>\n",
       "      <td>3.984053</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       mle_alpha    mle_mean  sample_mean\n",
       "99995   1.056078   18.832402     4.203487\n",
       "99996   1.160875    7.216024     3.934823\n",
       "99997   0.996467 -282.009705     5.454379\n",
       "99998   1.018729   54.392722     5.366482\n",
       "99999   1.234867    5.257724     3.984053"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Put data in a DataFrame\n",
    "sample_means = pd.Series(sample_means)\n",
    "mle_alphas = pd.Series(mle_alphas)\n",
    "mle_means = pd.Series(mle_means)\n",
    "df = pd.concat([mle_alphas, mle_means, sample_means], keys=['mle_alpha', 'mle_mean', 'sample_mean'], axis=1)\n",
    "df.tail()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the truncation conditions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Filter Conditions\n",
    "\n",
    "# Only count alpha>1 cases for mle method\n",
    "alpha_condition = df.mle_alpha > 1 # Truncate to remove alpha estimates less than one.\n",
    "\n",
    "# Truncate the range before plotting for faster plots\n",
    "mle_xlim_condition = df.mle_mean < 20  \n",
    "sample_xlim_condition = df.sample_mean < 20  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the distribution of mean estimates:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Mean Estimation Method Comparison\n",
    "\n",
    "# Mean extimated from MLE Alpha estimate\n",
    "ax = df.mle_mean.loc[alpha_condition & mle_xlim_condition] \\\n",
    "       .plot(kind='hist', bins=200, xlim=[0, 20], alpha=0.5, \\\n",
    "             density=True, \\\n",
    "             title='Distribution of Mean Estimates');  # The alpha=0.5 parameter is just a matplotlib thing for plot shading\n",
    "\n",
    "# Naive Sample Mean\n",
    "df.sample_mean.loc[sample_xlim_condition] \\\n",
    "  .plot(kind='hist', bins=200, xlim=[0, 20], alpha=0.5, \\\n",
    "        density=True, \\\n",
    "        title='Distribution of Mean Estimates');\n",
    "\n",
    "# True Mean\n",
    "ax.axvline(true_mean, linewidth=1, color='k', label='True Mean');\n",
    "plt.xlabel('Mean Estimate')\n",
    "plt.ylabel('Relative Frequency')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Extra Stuff\n",
    "Further exploration beyond what's in Incerto appendix C."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Min Max Summary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "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>mle_alpha</th>\n",
       "      <th>mle_mean</th>\n",
       "      <th>sample_mean</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>0.81</td>\n",
       "      <td>-47325.17</td>\n",
       "      <td>1.9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>1.98</td>\n",
       "      <td>197493.97</td>\n",
       "      <td>6265.9</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     mle_alpha   mle_mean  sample_mean\n",
       "min       0.81  -47325.17          1.9\n",
       "max       1.98  197493.97       6265.9"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Without Truncation\n",
    "df.agg(['min', 'max']).round(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "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>mle_mean</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>2.02</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>20.00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     mle_mean\n",
       "min      2.02\n",
       "max     20.00"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# MLE Truncated\n",
    "df.mle_mean.loc[alpha_condition & mle_xlim_condition] \\\n",
    "  .agg(['min', 'max']).round(2).to_frame()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "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>sample_mean</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>1.9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>20.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     sample_mean\n",
       "min          1.9\n",
       "max         20.0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Sample Mean Truncated\n",
    "df.sample_mean.loc[sample_xlim_condition] \\\n",
    "  .agg(['min', 'max']).round(2).to_frame()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Alpha Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Look at things in Alpha space:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Convert sample means to implied alphas\n",
    "mean_implied_alpha = lambda mean: (mean/(mean-1))\n",
    "\n",
    "df['sample_mean_alpha'] = df.sample_mean.apply(mean_implied_alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = df.filter(like='alpha') \\\n",
    "  .plot(kind='hist', bins=200, xlim=[0, 2], alpha=0.5, \\\n",
    "        title='Distribution of Alpha Estimates');\n",
    "\n",
    "ax.axvline(alpha, linewidth=1, color='k', label='True Alpha' );\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Alpha-Mean Relationship"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the Relationship\n",
    "get_mean = lambda alpha: (alpha/(alpha-1))\n",
    "get_alpha = lambda mean: (mean/(mean-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Tail Alpha vs. Mean \n",
    "\n",
    "# Generate Points\n",
    "alpha_range = np.linspace(1.01, 2)\n",
    "\n",
    "# Plot\n",
    "plt.plot(alpha_range, get_mean(alpha_range))\n",
    "\n",
    "# Labels\n",
    "plt.xlabel('Tail Alpha')\n",
    "plt.ylabel('Mean')\n",
    "plt.title('Mean = f(Tail Alpha)')\n",
    "\n",
    "# Asymptote\n",
    "plt.axhline(y=1, color='g', linewidth=1, label='Converges to 1')\n",
    "plt.axvline(x=1, color='g', linewidth=1);\n",
    "\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAEWCAYAAABrDZDcAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuNCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8QVMy6AAAACXBIWXMAAAsTAAALEwEAmpwYAAAnLElEQVR4nO3dfZyUdb3/8ddn5253ZxeQexQUEAQFEYRAUtQwjzdpHs1OkJV2c9ROZXo6p7SOv7RzuvN0urU6mWV5k2mapmlm5R15SgVUVERERFkRWEFg728/vz/m2mVFWHaXnfnO7Lyfj8c8Zuaaa67rM4POe6/v9b2+X3N3RESkeJWELkBERMJSEIiIFDkFgYhIkVMQiIgUOQWBiEiRUxCIiBQ5BYEEZ2a1ZjYxevwLM/uvPm5nvJm5mcX7c91sMrNPmtmm6DsYFi272cz+MWRduzKzUWb2vJmlQtci/U9BIP0m+jHruLWbWUOX5+fs6X3uXuHua3uxn+OjH/Ev9E/lYZhZAvg28A/Rd7DFzGYARwC/C1vdW7n7JuBB4PzQtUj/UxBIv4l+zCrcvQJ4FTi9y7Kb+nFX5wJbgY/04zZDGAWUAs91WXYBcJPn55WeN5GpTwYYBYFknZnNNbO/mdk2M3vdzK42s2SX193MJvVwW2ngbOBTwGQzm9PNug+Z2dfN7HEz22FmvzOzobusdo6ZvWpmb5jZl3pa874ys0OAF6Kn28zsgejxKcDDXdY7z8weNbPvRLWsNbN3RsvXm9lmMzu3y/opM/tW9Jk2mdn/mllZ9Np+ZvZ7M6s2szejx2N3+b7+M9pfjZndb2bDu5T9GDDRzA7qr+9B8oOCQHKhDbgEGA7MB04A/qWP2zoLqAV+A/yRzNFBdz4CfAwYA7QC39/l9WOAKVFN/8/MDu1LzdGP9J5ul+66vruvBqZFT4e4+8Io5CawMyA6zANWAMOAXwG/Bt4BTAI+BFxtZhXRut8ADgFmRq8fAPy/6LUS4DrgIOBAoAG4epd9fRD4KDASSAL/1qXmVmANmaYrGUAUBJJ17r7M3f/u7q3uvg74CXBcHzd3LnCLu7eR+VFcFLW178kN7v6su9cBlwP/ZGaxLq9f6e4N7v408DTRj1xva3b3Id3cvtHDzzYkuq/ZZfnL7n5d9JlvAcYBX3H3Jne/H2gGJpmZkWnDv8Tdt7p7DfA1YFFU4xZ3v93d66PXvrqbz3Sdu6929wbgVjKB0lVNlzplgFAQSNaZ2SFRM8RGM9tB5sdp+N7et5vtjAPeRaatGjInVEuB93TztvVdHr8CJHbZ98Yuj+uBiv6suZe2RfeVuyzf1OVxA3SevO26rAIYAZQDyzqORoD7ouWYWbmZ/cTMXok+0yPAkF2CcbffRxeVXeqUAUJBILnwY2AVMNndBwFfBKwP2/kwmf9m7zazjcBaMkHQXfPQuC6PDwRagDf6u+ZdekztevtiD/ZHdNTyEpmmnb54g0woTOtyNDI4OnkP8DkyzWDzos90bEf5Pdl41NV2EpkjJxlAFASSC5XADqDWzKYCn+zjds4FriTTXNFxex9wakcf/N34kJkdZmblwFeA26Imln6tuWuPqd3cvtaTDxe5lz42m7l7O/BT4DtmNhLAzA4ws5OiVSrJBMW26KT5l3u5i7nAOnd/pS/1Sf5SEEgu/BuZk5A1ZH6obuntBszsKDInOX/o7hu73O4icwJz8R7eegPwCzJNHqXARbmquY+uIdOTqS9HTABfIPN9/D1q/vkzmaMAgO8CZWSOHP5OptmoN84B/rePdUkes/zsriyy78zsIeBGd782dC29YWa/Am519ztD19IhOsJ4GJjl7o2h65H+FfTyehF5O3f/YOgaduXum4FD97qiFKSsNQ2Z2c+ji12e7bJsqJn9ycxejO73y9b+RUSkZ7LWNGRmx5K58Od6d58eLbsK2Oru34gustnP3Qt6vBgRkUKX1XMEZjYe+H2XIHgBON7dXzezMcBD7j6lu22IiEh25focwSh3fz16vJHMoFu7ZWbnE410mE6nZ0+dOjUH5WXHsg3LmL3/7M7ntU2tvPxGHROHp0mndJpGRLJj2bJlb7j7iL2tl+sjgm3uPqTL62+6+17PE8yZM8eXLl2atTqzza40/Ms7v+en12/jjB8+ys/Pm8PCqXvMQhGRfWJmy9x9jwMzdsj1dQSboiYhovvNOd5/XkinMlf01zb15LomEZHsynUQ3MXO4QDOJc8m38iVjuag+qbWwJWIiGS3++jNwN+AKWZWZWYfJzNE7olm9iLw7uh50SlPZoKgVkEgInkga2cq3X1Pl/yfkK19Fop0MtM0VN+spiHJHy0tLVRVVdHYqAuHC01paSljx44lkehuRPY9U5eVAOKxElLxEuqadUQg+aOqqorKykrGjx9P34c6klxzd7Zs2UJVVRUTJkzo0zY06Fwg6VScOjUNSR5pbGxk2LBhCoECY2YMGzZsn47kFASBpFMx6tVrSPKMQqAw7eu/m4IgkHQyrpPFIpIXFASBpFNxnSwW2cXGjRtZtGgRBx98MLNnz+bUU09l9erVocvqN1/7Wm/mKMr40pe+xLhx46io2HXW0P6jIAikPBnTyWKRLtydM888k+OPP56XXnqJZcuW8fWvf51Nmzbt/c39rK0tO3+k9SUITj/9dB5//PEsVLOTgiCQdFIni0W6evDBB0kkElx44YWdy4444ggWLFiAu/Pv//7vTJ8+ncMPP5xbbslMGPfQQw9x/PHHc/bZZzN16lTOOecc3J377ruP97///Z3beeihhzjttNMAuP/++5k/fz5HHnkk73//+6mtrQVg/PjxfOELX+DII4/kN7/5Dffeey9Tp05l9uzZXHTRRZ3vr6ur42Mf+xhz585l1qxZ/O53metin3vuOebOncvMmTOZMWMGL7744ls+36WXXkpDQwMzZ87knHPOAeDb3/4206dPZ/r06Xz3u9/d7fdy1FFHMWbMmH74hrvh7nl/mz17thcyruBty/71lqf8nV//S4BqRHZv5cqVQff/ve99zy+++OLdvnbbbbf5u9/9bm9tbfWNGzf6uHHjfMOGDf7ggw/6oEGDfP369d7W1uZHHXWUL1myxFtaWnzcuHFeW1vr7u4XXnih33DDDV5dXe0LFizoXP6Nb3zDr7zySnd3P+igg/yb3/ymu7s3NDT42LFjfe3ate7uvmjRIn/Pe97j7u6XXXaZ33DDDe7u/uabb/rkyZO9trbWP/3pT/uNN97o7u5NTU1eX1//ts+RTqc7Hy9dutSnT5/utbW1XlNT44cddpgvX758j99P1/fuzu7+/YCl3oPfWF1HEEg6paYhyW92Zf/3IOo6+GJv/PWvf2Xx4sXEYjFGjRrFcccdxxNPPMGgQYOYO3cuY8eOBWDmzJmsW7eOY445hpNPPpm7776bs88+m3vuuYerrrqKhx9+mJUrV3L00UcD0NzczPz58zv384EPfACAVatWMXHixM5++YsXL+aaa64BMkcUd911F9/61reATLfbV199lfnz5/PVr36VqqoqzjrrLCZPnrzXz3TmmWeSTqcBOOuss1iyZAmzZs3q03e0LxQEgaRTcXUflbzW1x/tvpo2bRq33XZbr9+XSqU6H8diMVpbM39gLVq0iKuvvpqhQ4cyZ84cKisrcXdOPPFEbr755t1uq+NHuTvuzu23386UKW+dSuXQQw9l3rx53HPPPZx66qn85Cc/YeHChb3+PCHoHEEg6WSM5rZ2mlvbQ5cikhcWLlxIU1NT51/eACtWrGDJkiUsWLCAW265hba2Nqqrq3nkkUeYO3dut9s77rjjWL58OT/96U9ZtGgRkGlvf/TRR1mzZg2Qae/fXa+kKVOmsHbtWtatWwfQeU4C4KSTTuIHP/gBHg3h/+STTwKwdu1aJk6cyEUXXcQZZ5zBihUr3rbdRCJBS0sLAAsWLODOO++kvr6euro67rjjDhYsWNDTr6tfKQgC6Rh4rl7NQyJA5qKoO+64gz//+c8cfPDBTJs2jcsuu4zRo0dz5plnMmPGDI444ggWLlzIVVddxejRo7vdXiwW47TTTuMPf/hD54neESNG8Itf/ILFixczY8YM5s+fz6pVq9723rKyMn70ox9x8sknM3v2bCorKxk8eDAAl19+OS0tLcyYMYNp06Zx+eWXA3Drrbcyffp0Zs6cybPPPstHPvKRt233/PPPZ8aMGZxzzjkceeSRnHfeecydO5d58+bxiU98YrfNQp///OcZO3Ys9fX1jB07liuuuKK3X+1eZXVimv4y0CamAbj1ifV8/vYVPHrpQg4YUhaoMpGdnn/+eQ499NDQZeSN2tpaKioqcHc+9alPMXnyZC655JLQZe3R7v798nViGomUR5PTqAupSH766U9/ysyZM5k2bRrbt2/nggsuCF1S1uhkcSAdk9MoCETy0yWXXJLXRwD9SUcEgaSTHUGgnkOSPwqhqVjebl//3RQEgZRHk9PoWgLJF6WlpWzZskVhUGA8mo+gtLS0z9tQ01AgFSn1GpL8MnbsWKqqqqiurg5divRSxwxlfaUgCKTjZHGtmoYkTyQSiT7PcCWFTU1DgXQeEehksYgEpiAIpCwRw0y9hkQkPAVBIGaWGYpak9OISGAKgoDKkzGdLBaR4BQEAaVTcZ0sFpHgFAQBpVMxnSwWkeAUBAGVJ+PUKghEJDAFQUAVqTj1OlksIoEpCAIqT8bUfVREglMQBJTpPqogEJGwFAQBad5iEckHCoKA0qkYdc2tGu1RRIJSEASUTsVpd2hs0QT2IhKOgiCgdLJjBFKdJxCRcBQEAZUnNSeBiISnIAho57zFOmEsIuEECQIzu8TMnjOzZ83sZjPr+xxrBSyd0nSVIhJezoPAzA4ALgLmuPt0IAYsynUd+WDnEYGCQETCCdU0FAfKzCwOlAMbAtURVDqppiERCS/nQeDurwHfAl4FXge2u/v9u65nZueb2VIzWzpQJ9MuT6ppSETCC9E0tB9wBjAB2B9Im9mHdl3P3a9x9znuPmfEiBG5LjMnNG+xiOSDEE1D7wZedvdqd28Bfgu8M0AdwZV3nixW05CIhBMiCF4FjjKzcjMz4ATg+QB1BJeKx0jETCeLRSSoEOcIHgNuA5YDz0Q1XJPrOvJFeTKuIBCRoOIhduruXwa+HGLf+SadjKlpSESC0pXFgaVTOiIQkbAUBIGVp+I6IhCRoBQEgVWkYuo+KiJBKQgCK0/GNQy1iASlIAisIhWnXk1DIhKQgiCw8mRMJ4tFJCgFQWDpVFxjDYlIUAqCwNLJOI0t7bS1awJ7EQlDQRCYJqcRkdAUBIGlO0cg1QljEQlDQRBYx5wE6kIqIqEoCALrmKWsXk1DIhKIgiCwnfMWq2lIRMJQEATWebJYTUMiEoiCILDOIwI1DYlIIAqCwDrOEahpSERCURAE1jFvsU4Wi0goCoLAOo4I1H1UREJREAQWKzFKEyUagVREglEQ5IEKTVcpIgEpCPJAeVJBICLhKAjyQHkypnmLRSQYBUEeUNOQiISkIMgD5am4jghEJBgFQR6oSMWo1xGBiASiIMgDOlksIiEpCPJAhZqGRCQgBUEeKE/GqGtqxV3zFotI7ikI8kA6Fae13Wluaw9diogUIQVBHkhH01Vq3mIRCUFBkAfKUxp4TkTCURDkgYpUx7zFOiIQkdxTEOSB8qhpSEcEIhKCgiAPpDuPCBQEIpJ7QYLAzIaY2W1mtsrMnjez+SHqyBc7p6tUEIhI7sUD7fd7wH3ufraZJYHyQHXkhXQ0XaXmLRaREHIeBGY2GDgWOA/A3ZuB5lzXkU/UNCQiIYVoGpoAVAPXmdmTZnatmaV3XcnMzjezpWa2tLq6OvdV5tDOeYt1RCAiubfXIDCzo8zsCTOrNbNmM2szsx37sM84cCTwY3efBdQBl+66krtf4+5z3H3OiBEj9mF3+a80UUKJ6YhARMLoyRHB1cBi4EWgDPgE8MN92GcVUOXuj0XPbyMTDEXLzEgn4+o+KiJB9KhpyN3XADF3b3P364CT+7pDd98IrDezKdGiE4CVfd3eQFGeimmICREJoicni+ujnj1PmdlVwOvs+7mFzwA3RdtdC3x0H7dX8NKpOHVqGhKRAHoSBB8m88P/aeASYBzwvn3Zqbs/BczZl20MNGlNTiMigew1CNz9lehhI3BldsspXuXJmCanEZEgetJr6Ggz+5OZrTaztR23XBRXTCpSOiIQkTB60jT0MzJNQssA/cmaJeWpuEYfFZEgehIE2939D1mvpMhVpGI6IhCRIPYYBGbW0bf/QTP7b+C3QFPH6+6+PMu1FZVBZQm2NbTQ1u7ESix0OSJSRLo7IvifXZ537eXjwML+L6d4HTy8gubWdl57s4EDhxX1GHwikmN7DAJ3f1cuCyl2k0ZVALB6U42CQERyqie9hoaZ2ffNbLmZLTOz75nZsFwUV0wmjcwEwYubawNXIiLFpidXCP+azGih7wPOjh7fks2iitGg0gRjBpfy4uaa0KWISJHpSa+hMe7+n12e/5eZfSBbBRWzSSMreHGTjghEJLd6ckRwv5ktMrOS6PZPwB+zXVgxmjyykjWba2lv99CliEgR6UkQ/DPwKzJdR5vJNBVdYGY1+zgvgexi8qgKGlraeG1bQ+hSRKSI9GSsocpcFCJwyKiOE8Y1jBuqnkMikhs9uaBst3RBWf+bNCKTuS9uqmXh1FGBqxGRYtGbC8q60gVlWTC4PMHIypS6kIpITvXpgjIzS2SnHDlkVKWCQERyqsczjVnGCWb2MzLzDksWTBpZwZpNNbir55CI5EZPriw+ysy+D7wC/A54BJia7cKK1eRRFdQ1t7Fhe2PoUkSkSOwxCMzsa2b2IvBVYAUwC6h291+6+5u5KrDYTB7ZccJYVxiLSG50d0TwCWAT8GPgBnffQuYksWTR5I4xh3SFsYjkSHdBMAb4L+B04CUzuwEoM7OeDEshfbRfOsnwipTGHBKRnOmu11AbcB9wn5mlgNOAMuA1M/uLu38wRzUWnckjK9RzSERypke9hty9yd1vd/ezgclkAkKy5JBRFazZVKueQyKSEz3uPtrB3Xe4+/XZKEYyJo2qpKaplY071HNIRLKv10Eg2acTxiKSSwqCPNQRBKvVhVREcqC7QefO6u6N7v7b/i9HAIZVpBiWTrJGJ4xFJAe66wp6ejevOaAgyKJJ6jkkIjnSXffRj+ayEHmryaMquOupDbg7Zha6HBEZwLprGvqQu99oZv+6u9fd/dvZK0sOGVXJjsZWNtc0MWpQaehyRGQA665pKB3da4ayACZ16TmkIBCRbOquaegn0f2VuStHOnQOPre5hmMmDw9cjYgMZHsdN8jMSoGPA9OAzj9N3f1jWayr6A2vSDKkPMFqXUsgIlnWk+sIbgBGAycBDwNjAXVwzzIz45CRlazR4HMikmXdzUfQcbQwyd0vB+rc/ZfAe4B5uSiu2E0aVcFqjTkkIlnW3RHB49F9S3S/zcymA4OBkfu6YzOLmdmTZvb7fd3WQDV5ZAXbG1qorm0KXYqIDGA9aRq6xsz2A/4DuAtYCXyzH/b9WeD5ftjOgHXIqMwJ4zU6TyAiWdRdEIyMriEYBHwUmAP8kEwIpLt5316Z2VgyTUzX7st2BrrOwed0hbGIZFF3vYZiQAWwu8ta97XR+rvA5+nmGgUzOx84H+DAAw/cx90VphGVKQaVxjX4nIhkVXdB8Lq7f6W/d2hmpwGb3X2ZmR2/p/Xc/RrgGoA5c+YU5dlSM2P6AYN57OWtGmpCRLKmu6ahbP3qHA2818zWAb8GFprZjVnaV8E75fAxrNlcyws6KhCRLOkuCE7Ixg7d/TJ3H+vu44FFwAPu/qFs7GsgOHX6aGIlxl1PbQhdiogMUHsMAnffmstCZPeGVaQ4etJw7l6xQdcTiEhWBJ2hzN0fcvfTQtZQCE6fMYb1Wxt4av220KWIyACkqSoLwEnTR5OMlXDX02oeEpH+pyAoAINKExw/ZQT3rHidtnY1D4lI/1IQFIj3ztyfzTVNPPbyltCliMgAoyAoECdMHUV5MsbdT78euhQRGWAUBAWiLBnjxMNG8YdnX6e5tT10OSIygCgICsjpM/ZnW30Lj655I3QpIjKAKAgKyLGHjGBQaVy9h0SkXykICkgyXsIp08dw/3MbaWxpC12OiAwQCoIC896Z+1PX3MYDqzaHLkVEBggFQYE5auIwhlekuFvNQyLSTxQEBSZWYpw2Ywx/WbWZmsaWvb9BRGQvFAQF6PQjxtDc2s69z+iaAhHZdwqCAnTkgfsxbf9BfOdPL1LX1Bq6HBEpcAqCAmRmXPneaWzc0cgPH1wTuhwRKXAKggI1Z/xQzpp1ANcueZl1b9SFLkdECpiCoIBdespUkvESvvL7laFLEZECpiAoYCMHlXLRCZN4YNVmHli1KXQ5IlKgFAQF7rx3TmDiiDRfuXslTa262lhEek9BUOCS8RKuOH0a67bUc+2Sl0OXIyIFSEEwABx7yAhOmjaKqx9Yw+vbG0KXIyIFRkEwQPzHew6j3Z2v3bsqdCkiUmAUBAPEuKHlXHjcwdz99AaNQyQivaIgGEA+efzBvGP8fnzu1qf5v5c0eY2I9IyCYAApTcS49iPvYPzwci64fhkrN+wIXZKIFAAFwQAzuDzBLz46l3QqznnXPc76rfWhSxKRPKcgGID2H1LG9R+fS2NLG+de9zhb65pDlyQieUxBMEAdMqqSa899B1VvNvDxXz5BQ7MuNhOR3VMQDGBzJwzl+4tm8fT6bXzypmUaslpEdktBMMCdPH00Xz3zcB5ZXc2ZP3qUtdW1oUsSkTyjICgCi+ceyPUfm0d1TRNnXP0of1qpAepEZCcFQZE4ZvJw7v7MMYwfnuafr1/K/9z/Am3tHrosEckDCoIiMna/cn5z4Xz+ac5YfvDAGj76iyfYVq8eRSLFTkFQZEoTMb75vhl87czD+dtLb3Didx7h9mVVtOvoQKRoKQiKkJnxwXkHcse/HM3+Q8r43G+e5n3/+3+sqNoWujQRCSDnQWBm48zsQTNbaWbPmdlnc12DZEw/YDB3fPKd/PfZM1i/tYEzfvgoX7htBW/UNoUuTURyKB5gn63A59x9uZlVAsvM7E/urol3AygpMd4/ZxwnTx/NDx5Yw8//+jL3Pvs6nzhmIh+efxBD08nQJYpIluX8iMDdX3f35dHjGuB54IBc1yFvVVma4IunHsp9Fx/LvAlD+c6fV/POb/yFy+98lnVv1IUuT0SyKMQRQSczGw/MAh7bzWvnA+cDHHjggbktrIhNGlnBtee+gxc31XDtkpe55Yn13PjYK5x02Gj++dgJHHngfphZ6DJFpB+Ze5jeImZWATwMfNXdf9vdunPmzPGlS5fmprAssCsN/3Jh9srZXNPI9f/3Cjf8/RW2N7QwaWQFZ846gDNm7s/Y/cpDlyci3TCzZe4+Z6/rhQgCM0sAvwf+6O7f3tv6CoLw6ptbufPJDdzxZBVPrHsTgHkThnLmrAM45fAxDC5LBK5QRHaVt0FgmXaFXwJb3f3inrxHQZBf1m+t584nX+OOJ19j7Rt1JGLG3AlDedeUkSycOpKJIypClygi5HcQHAMsAZ4B2qPFX3T3e/f0HgVBfnJ3nnltO/c88zoPrtrM6k2ZAe0mDE/zrikjOfaQ4cwZP5SKVNBTUSJFK2+DoC8UBIVh/dZ6HnxhM395fjN/W7uF5tZ2YiXG9P0HMW/iMOaOH8o7JgxVM5JIjigI8kixBEFX9c2tLHvlTR5bu5XHX97KU+u30dzWjhlMGlHBjLFDOGLcYGaMHcKhYypJxWOhSxYZcHoaBDpml6woT8ZZMHkECyaPAKCxpY0nX93G4y9v5emqbTy8ejO3L68CIBEzpo4exKFjKpkyehBTR1cyZXQlwytSIT+CSNFQEEhOlCZizD94GPMPHgZkzi9s2N7IivXbeLpqO8+8to2/PL+ZW5dWdb5neEWSKaMrmTi8gokj0hw8InO//+AySkp0LYNIf1EQSBBmxgFDyjhgSBmnHD6mc3l1TRMvbKxh1cYdvLCxhtWbarjzydeo6TLNZmmihPHD0owbWs6BQ8s5aFh55+MDhpRRmlAzk0hvKAgkr4yoTDGiMsUxk4d3LnN3qmubWFtdF91qWbelnle21LHkxWoaW9rfso3hFUn2H1LG/oPLMvdDShkzuIxRg1KMGlTKqEGlJOMaeFekg4JA8p6ZMbKylJGVpRw1cdhbXusIiVe31PPKlno2bGtgw/YGXtvWyJrqWh55sZr65ra3bXNYOsnIQaWZ4KlIdQbQiMoUwyuSDEunGFaRZL/yJDE1Q8kApyCQgtY1JOaMH/q2192d7Q0tbNzRyMbtjWza0cimHU1s3NHIpu2NvFHbxJpNNVTXNtHS9vaeXWawX3mSoekkQ8uT7JdOsF95kiHlSYamEwwpTzKkLLovTzC4LHNT85QUEgWBDGhmFv1IJ5k6etAe1+sIjOqaJqprm9ha18yW2ma21DWzta6JLbXNvFnfzLo36nmyfhtv1jfvNjg6pOIlDC5LMKgswaDSeHSfYFBZnMrSBBWpOINKM48rS+NUpOJUdNyn4qRTcVLxEg3wJzmhIBDhrYExeVTlXtd3d+qa23izrpntDS1sq29hW8POx9sbWtjR0MKOxhZ2NLSyta6ZdW/UsaOxlZrGlm5DpEMiZpQnM8FQnoyRTsVJp2Kkk5nn5ak45YnoPhmjPBmjLBGjPHq9NBHrXF6aiFEWvV6aiKm5S95CQSDSB2bW+df7uF6+191pam2nJgqFmsZWapuiW2Mrdc2tncvqm1qpa26jrsv9ltp66pvbolvrbs+B7E0yXhKFQkkmJBIxUokYpfHM89JECan4ztdT8Z33qXiMVKJk5+N4CalECclYLLov6bxPRutk7jPL1PU3/ygIRHLMzKIf2xgjKvf9orn2dqextY26pjYamttoaMkEREMUFg0tmVtjy87XG5ozzxtb2mls3fm4oaWNbfXNNLW2v+31ptZ2+mMggniJkYhCIhnfGRjJWAmJuGXuo2WJWMfyEhKxna/Fd3ncsV7H40R0H4+VkCixt6wXL7HM8pgRL4nuO5fvXBYr2bl+rMQGdDOdgkCkwJWUWNQclN3/nd2dljanqbWtMyiaW9tpam3vvG9qbaOppZ3mtsyy5o5lrW9d1tLxuC3zvpY2pyV63hItq21qpaWtnZZWz6wfvae1PbNuS3vmfW3tuRm+pSMQOkKl43m8JPM8VmIkSkoyy2Idr+1c5y3PY0asZGfIxMyIRdsssZ3rdbynpPO9JcRKyNwb0TqZZSWW2W6J7Xxfjz9bFr83ERlAzIxk3EjGS9j7WZTcaWvPBMWuAdEahUpLm3fet3as19ZOa5vT2rFul1Bpbdv5uKW9Y73M8sy+nLb2dlranbaO19oz222LttXWnlnecV/f3EqbQ1u0vbb2ndtvbyd6T/R6+87XO96fbQoCESlomb+Ko+66A3R4qvYuwdLmbw2KjmXtXYKjPVrnsG/2bPsKAhGRPFdSYiSzeJJd19mLiBS5gpiPwPY354LQVYiIFJgr0MQ0+aIYJ6YRkfB6OjGNmoZERIqcgkBEpMgpCEREipyCQESkyCkIRESKnIJARKTIKQhERIqcgkBEpMgpCEREipyCQESkyCkIRESKnIJARKTIKQhERIqcgkBEpMgpCEREipyCQESkyCkIRESKXJAgMLOTzewFM1tjZpeGqEFERDJyHgRmFgN+CJwCHAYsNrPDcl2HiIhkhDgimAuscfe17t4M/Bo4I0AdIiICxAPs8wBgfZfnVcC8XVcys/OB86OnTWb2bA5qy5bhdoW9EbqIPhoOFGrtoPpDU/1hTenJSiGCoEfc/RrgGgAzW+rucwKX1GeFXH8h1w6qPzTVH5aZLe3JeiGahl4DxnV5PjZaJiIiAYQIgieAyWY2wcySwCLgrgB1iIgIAZqG3L3VzD4N/BGIAT939+f28rZrsl9ZVhVy/YVcO6j+0FR/WD2q39w924WIiEge05XFIiJFTkEgIlLk8joICn0oCjP7uZltLsRrIMxsnJk9aGYrzew5M/ts6Jp6w8xKzexxM3s6qv/K0DX1lpnFzOxJM/t96Fr6wszWmdkzZvZUT7sx5gszG2Jmt5nZKjN73szmh66pp8xsSvSdd9x2mNnF3b4nX88RRENRrAZOJHPR2RPAYndfGbSwXjCzY4Fa4Hp3nx66nt4wszHAGHdfbmaVwDLgHwvl+zczA9LuXmtmCeCvwGfd/e+BS+sxM/tXYA4wyN1PC11Pb5nZOmCOuxfcBVlm9ktgibtfG/VuLHf3bYHL6rXod/Q1YJ67v7Kn9fL5iKDgh6Jw90eAraHr6At3f93dl0ePa4DnyVwVXhA8ozZ6mohu+flXz26Y2VjgPcC1oWspNmY2GDgW+BmAuzcXYghETgBe6i4EIL+DYHdDURTMD9FAYmbjgVnAY4FL6ZWoaeUpYDPwJ3cvpPq/C3weaA9cx75w4H4zWxYNGVMoJgDVwHVR09y1ZpYOXVQfLQJu3ttK+RwEkgfMrAK4HbjY3XeErqc33L3N3WeSuXp9rpkVRPOcmZ0GbHb3ZaFr2UfHuPuRZEYa/lTUVFoI4sCRwI/dfRZQBxTiOcok8F7gN3tbN5+DQENRBBa1rd8O3OTuvw1dT19Fh/UPAicHLqWnjgbeG7Wx/xpYaGY3hi2p99z9teh+M3AHmebeQlAFVHU5gryNTDAUmlOA5e6+aW8r5nMQaCiKgKKTrT8Dnnf3b4eup7fMbISZDYkel5HpdLAqaFE95O6XuftYdx9P5r/7B9z9Q4HL6hUzS0edDIiaVf4BKIjec+6+EVhvZh0jd54AFEQniV0spgfNQpDfo4/2ZSiKvGJmNwPHA8PNrAr4srv/LGxVPXY08GHgmaidHeCL7n5vuJJ6ZQzwy6jXRAlwq7sXZDfMAjUKuCPz9wRx4Ffufl/YknrlM8BN0R+ha4GPBq6nV6LwPRG4oEfr52v3URERyY18bhoSEZEcUBCIiBQ5BYGISJFTEIiIFDkFgYhIkVMQiHRhZt714i0zi5tZdaGOACrSEwoCkbeqA6ZHF6FBpi+2rmiXAU1BIPJ295IZ+RN2uTozumL259FcB0+a2RnR8vFmtsTMlke3d0bLjzezh7qMbX9TdNW2SN5QEIi83a+BRWZWCszgraOufonMkA9zgXcB/x1dxbkZODEaZO0DwPe7vGcWcDFwGDCRzFXbInkjb4eYEAnF3VdEQ28vJnN00NU/kBkQ7t+i56XAgcAG4Gozmwm0AYd0ec/j7l4FEA3XMZ7MRDkieUFBILJ7dwHfIjNW1LAuyw14n7u/0HVlM7sC2AQcQeZIu7HLy01dHreh/+8kz6hpSGT3fg5c6e7P7LL8j8BnOtr5zWxWtHww8Lq7t5MZrC+Ws0pF9pGCQGQ33L3K3b+/m5f+k8y0lyvM7LnoOcCPgHPN7GlgKpneRyIFQaOPiogUOR0RiIgUOQWBiEiRUxCIiBQ5BYGISJFTEIiIFDkFgYhIkVMQiIgUuf8P+uMxxoX2+48AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Mean vs. Tail Alpha \n",
    "\n",
    "# Generate Points\n",
    "mean_range = np.linspace(1.1, 7)\n",
    "\n",
    "# Plot\n",
    "plt.plot(mean_range, get_alpha(mean_range))\n",
    "\n",
    "# Labels\n",
    "plt.xlabel('Mean')\n",
    "plt.ylabel('Tail Alpha')\n",
    "plt.title('Tail Alpha = f(mean)')\n",
    "\n",
    "# Range\n",
    "plt.ylim([0,10])\n",
    "plt.xlim([0, 7])\n",
    "\n",
    "# Asymptote\n",
    "plt.axhline(y=1, color='g', linewidth=1, label='Converges to 1')\n",
    "plt.axvline(x=1, color='g', linewidth=1);\n",
    "\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### More Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Plot the Sample Mean Distribution on its own\n",
    "#df.loc[sample_xlim_condition].sample_mean \\\n",
    "#  .plot(kind='hist', bins=200, xlim=[0, 20], \\\n",
    "#        title='Distribution of Sample Mean Estimates');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Plot the MLE Distribution on its own\n",
    "#df.loc[alpha_condition & mle_mean_condition].mle_mean \\\n",
    "#  .plot(kind='hist', bins=200, xlim=[0, 20], \\\n",
    "#        title='Distribution of MLE Mean Estimates');"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "fattails",
   "language": "python",
   "name": "fattails"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}