{
 "cells": [
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   "execution_count": 2,
   "metadata": {
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    "hidePrompt": false,
    "slideshow": {
     "slide_type": "skip"
    }
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   "outputs": [],
   "source": [
    "%load_ext rpy2.ipython\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "hideCode": false,
    "hidePrompt": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import numpy.random as rnd\n",
    "from scipy import stats\n",
    "import sympy as sym\n",
    "from IPython.display import Image\n",
    "\n",
    "plt.rcParams['figure.figsize'] = (20, 7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "\n",
    "# Rare-event simulation\n",
    "## Lecture 3\n",
    "### Patrick Laub, Institut de Science Financière et d’Assurances"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Agenda\n",
    "\n",
    "- Show you Markov chain Monte Carlo (MCMC)\n",
    "- Go back to finish Markov chain example\n",
    "- Explain MCMC"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## MCMC: inputs\n",
    "\n",
    "_Inputs_:\n",
    "- $f_X(x)$, the _target density_ (known up to a normalising constant),\n",
    "- $q(y \\mid x)$, a _transition kernel_, gives the density of proposing a jump to $y$ given we're currently at $x$,\n",
    "- $X_0$, our _starting position_, and $R$ the number of _replicates_ we want.\n",
    "\n",
    "_Outputs_: $X_1, \\dots, X_R \\sim f_X(x)$, dependent but i.d.\n",
    "\n",
    "_An example_:\n",
    "- target is $f_X(x) \\propto 2 + \\sin(x)$ for $x \\in [0, 4\\pi]$,\n",
    "- we propose $(Y \\mid X) \\sim \\mathsf{Uniform}(X-1, X+1)$, so $q(y \\mid x) = \\frac12 1\\{ |y-x| \\le 1 \\}$,\n",
    "- start at $X_0 = 2\\pi$, and ask for $R = 10^6$ samples."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## MCMC: Metropolis–Hastings algorithm\n",
    "\n",
    "_Inputs_: target density $f_X(x)$, transition kernel $q(y \\mid x)$, starting position $X_0$, and desired number of replicates $R$.\n",
    "\n",
    "_Definition_: $$\\alpha(X,Y) := \\min\\Bigl\\{ \\frac{ f_X(Y) \\, q(X \\mid Y)\n",
    "}{ f_X(X) \\, q(Y \\mid X) } , 1 \\Bigr\\} .$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "To generate the $r$-th random variable:   \n",
    "$\\quad$ Make a proposal $Y$ from $q(\\,\\cdot\\, \\mid X_{r-1})$  \n",
    "$\\quad$ With probability $\\alpha(X_{r-1}, Y)$:  \n",
    "$\\quad$ $\\quad$ We accept the proposal  \n",
    "$\\quad$ $\\quad$ $X_r \\gets Y$  \n",
    "$\\quad$ Otherwise:  \n",
    "$\\quad$ $\\quad$ We reject and stay where we are  \n",
    "$\\quad$ $\\quad$ $X_r \\gets X_{r-1}$  \n",
    "Return $(X_1, \\dots, X_R)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "For $r = 1$ to $R$  \n",
    "$\\quad$ $Y \\sim q(\\,\\cdot\\, \\mid X_{r-1})$  \n",
    "$\\quad$ $U \\sim \\mathsf{Unif}(0,1)$  \n",
    "$\\quad$ If \n",
    "$U \\le \\alpha(X_{r-1}, Y) = \\min\\bigl\\{  \\frac{ f_X(Y) \\, q(X_{r-1} \\mid Y) \n",
    "}{ f_X(X_{r-1}) \\, q(Y \\mid X_{r-1}) } , 1 \\bigr\\} $  \n",
    "$\\quad$ $\\quad$ $X_r \\gets Y$  \n",
    "$\\quad$ Else  \n",
    "$\\quad$ $\\quad$ $X_r \\gets X_{r-1}$  \n",
    "$\\quad$ End If  \n",
    "End For  \n",
    "Return $(X_1, \\dots, X_R)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Prepare yourself to see the coolest animation ever..\n",
    "\n",
    "[Animation](https://chi-feng.github.io/mcmc-demo/app.html#HamiltonianMC,banana)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## How does MCMC help with rare event estimation?\n",
    "\n",
    "Multiple ways. One method is the _Improved Cross-Entropy method_.\n",
    "\n",
    "To estimate $\\ell = \\mathbb{P}(X > \\gamma)$, with optimal IS density $g^*(x) \\propto 1\\{x > \\gamma\\} f_X(x)$, then:\n",
    "  \n",
    " 1. Choose a family $f( \\,\\cdot\\, ; \\mathbf{v})$, $R$ (e.g. $R=10^6$).\n",
    " 2. Simulate $X_r \\overset{\\mathrm{i.i.d.}}{\\sim} g^*( \\,\\cdot\\, )$  for $r=1,\\dots,R$ using MCMC.\n",
    " 3. Set $\\mathbf{v}_*$ to be the MLE estimate of fitting $\\{X_1,\\dots, X_R\\}$ to $f( \\,\\cdot\\, ; \\mathbf{v})$. That is,\n",
    "     $$\n",
    "     \\DeclareMathOperator*{\\argmax}{arg\\,max}\n",
    "     \\mathbf{v}_* = \\argmax_{\\mathbf{v}} \\frac{1}{R} \\sum_{r=1}^R \\log \\bigl[ f(X_r; \\mathbf{v}) \\bigr] .\n",
    "     $$\n",
    " 4. Return the result of IS with $f( \\,\\cdot\\, ; \\mathbf{v}_*)$ proposal.\n",
    " \n",
    " \n",
    "This is _so much simpler_..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## A very strange Markov chain example\n",
    "\n",
    "Given $X_{i-1} = x_{i-1}$, how to get the next $X_i$?\n",
    "\n",
    "Sample $E_i \\sim \\mathsf{Exponential}(\\lambda)$ and either _jump left_ taking $X_i = x_{i-1} - E_i$ or _jump right_ taking $X_i = x_{i-1} + E_i$.\n",
    "\n",
    "What are the rules for jumping left or right? \n",
    "\n",
    "- If $x_{i-1} < -1$ we jump right\n",
    "- If $x_{i-1} > 1$ we jump left. \n",
    "- If $x_{i-1} \\in (-1, 1)$ we jump left with probability \n",
    "$$ \\frac{ \\frac{1}{(x+1)^2} }{ \\frac{1}{(x+1)^2} + \\frac{1}{(x-1)^2} } .$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## R to generate a transition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Given $X_{i-1} = x_{i-1}$, how to get the next $X_i$?\n",
    "\n",
    "Sample $E_i \\sim \\mathsf{Exponential}(\\lambda)$ and either _jump left_ taking $X_i = x_{i-1} - E_i$ or _jump right_ taking $X_i = x_{i-1} + E_i$.\n",
    "\n",
    "What are the rules for jumping left or right? \n",
    "\n",
    "- If $x_{i-1} < -1$ we jump right\n",
    "- If $x_{i-1} > 1$ we jump left. \n",
    "- If $x_{i-1} \\in (-1, 1)$ we jump left with probability \n",
    "\n",
    "$$ \\frac{ \\frac{1}{(x+1)^2} }{ \\frac{1}{(x+1)^2} + \\frac{1}{(x-1)^2} } .$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1] 0.05895065\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "\n",
    "lambda <- 5\n",
    "\n",
    "rtransition <- function(x) { \n",
    "\n",
    "    E <- rexp(1, lambda)\n",
    "    \n",
    "    probJumpLeft <- (1 / (x+1)^2) / \n",
    "        ((1 / (x+1)^2) + (1 / (x-1)^2))\n",
    "\n",
    "    if (x > 1) {\n",
    "        return( x - E )\n",
    "    }\n",
    "    if (x < -1) {\n",
    "        return( x + E )\n",
    "    }\n",
    "    \n",
    "    if (runif(1) < probJumpLeft) {\n",
    "        return( x - rexp(1, lambda) )\n",
    "    } else {\n",
    "        return( x + rexp(1, lambda) )\n",
    "    }\n",
    "}\n",
    "\n",
    "rtransition(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Plot transition densities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "%%R\n",
    "\n",
    "dtransition <- function(y, x) { \n",
    "    \n",
    "    leftJump <- dexp( -(y-x), lambda )\n",
    "    rightJump <- dexp( (y-x), lambda )\n",
    "    \n",
    "    if (x < -1) {\n",
    "        return(rightJump)\n",
    "    } \n",
    "    if (x > 1) {\n",
    "        return(leftJump)\n",
    "    }\n",
    "    \n",
    "    probJumpLeft <- (1 / (x+1)^2) / \n",
    "        ((1 / (x+1)^2) + (1 / (x-1)^2))\n",
    "    \n",
    "    return(probJumpLeft*leftJump + (1-probJumpLeft)*rightJump)\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
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   },
   "outputs": [
    {
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/wecFV7nrssxlPAbn+wa9ezeuXMH27bzGBwoL4eEBAH37wtub3zyknkldxSGXy999991163ivEya8hAJhAAAPzv2Senn2UsgUNtlulH8Vh4sLAgMBYPZs3L7Nbx5Sz2ihCmHzDNAHAODGOUAPvjHYSeZkw1UcPAP0e+89+JGwY0d4evKbh9QzCtCETSJQvcrIAyjkOdGb/m9Waio5pTi4i4vDtGn8hk9IeNCLqU8f+Pjwm4fUMwrQhM1l4BIAIBhw4jlRP69+aqhttYpDoQDPetDffkNlJQDMmYPkZH7zkHpGAZqwqWmW1AGYwHOi6hTHRyEf8ZyEj+pOSTt38pth2bIHPxJmZUmwgTipNxSgCRvZww8FAMcIBIwLGFepqVyauZTH4NyrOAD861+8xgc2bHjwgeqg7RsFaMKmJhrkApt4TtTXs68GGlvtxcHZV189+NCoEZXZ2TMK0MRMXH8kzFHljL492lHm+HHIxzzGl6Kji0xm+hqLvfMOwsMlmIfUDwrQhE1foB8AzvvGuincXvF9RaVRcUpxQIIY/e23/MauadL/44/IyOA3D6lntBkDYVNTdOsIVBi70CJOMqduHt0qNZX8qjh4DPvA6tUAEBXFb4affnrwISsL5eX85iH1jN6gCZtNnFPP1Y6XHJ+bNlcpU3JKcYBrjK6u4vj1V1RV8Rheo0Fq6oPPEyciIoLHJMQqUIAmbGrajQLgFTuBSMfI/l79uaY4OKqu4lizBjk5PIbPy8NHD4sPf/wR9+/zmIRYBQrQhFnNj18duU0R6BDYxrWNBpqMSi4ZVr5ldtVVHNxWe/v4YNWqB5+PHqUctD2jAE3Y1I5qA7nN8n3O9xtzNyplyrmhc7lNwhm3AF1aiqxaLaSoDtqOUYAm5qvi9jthW9e27d3aqzSqJRlLeIzPvVkSgBkzEB/PY+DTp7FmzYPPSiXtSWjPqIqDsAmr9ae6J1AA8Ng7JN45HgC/Kg6+qqs4TO0ZZLaaZtAA5s+nAG3P6A2asOkF9Hj4mV8p9Ky0WWdKz/BLcfB9g66u4rh0Cdev8xi+doA+e5ZrW1NSzyhAEzZHgcSHn6vfoHno69k3xjGmSlPFKcXBV3UVx/Hj+PNPHsN36ID+/R98/vFHpKTwmIRYBQrQhM0l4PLDz7GAgs8sHd06eio8+fXikKiKo5DLYvjg4EeLYD77DE2b8piEWAUK0ISNplaZ3SSgGZ9ZRtwaUaQucpA52HAVh6cnp+zDwoU4ePDB53PnKMVhzyhAEza1OwCdAM7zmWWM3xg3uRu/FIcUVRxPPolx43gM3L8/WrZ88HntWty4wWMSYhWoioOwqR3VLgIqoDmHWXp69gSggSZDZYPLMKqrOJyc4MRlz5m4ODg68hiYWB16gybm8+TWcfTZG88CcJA5fBL6CY/xpajiqKh48Guh2MaNQ1rag8/UsN++UYAmbAY83NUbgAe3Ko6JARMBVGmqvsz4ks8MPFXH5dJSrFzJY/jcXLi4PPj81lto0oTHJMQqUIAmbDyBhzW4HHv29/LsVf3Bhqs43N05/X5Xuw46MJAWqtgzCtCEzQZgy8PPnOqgNdAMvDEQgEKm4JTikIJCwSn7MHgwFA/LGz/+GOfO8ZiEWAUK0ISNutbvhNHA63xmse0UR43163mMOnHio8/jx6NxYx6TEKtAAZqwqf1O6AS04DCFDLLqKg4AnKo4+KY4qqs4AISG8hh+6tRHnwMDqaLDnlGAJhbh0XH0evn1SXcnAVDIFJ+GfsphBs6qqzgAbNli9DozNWr06DOlOOwbBWjCRlN3rUoZhymCHIJG+Y4CoIaaU4qD7xt0TXXdypVQqUQffvz4R581GqjVhi8lNo4CNGETDnD5e3st7gr31i6tAWg0trlQpbqKA1x69hcXY+bMR1+pDtq+UYAmbPoC3ThPsSN/x6rsVeCZ4pBiqTcAT0/R+yVlZ+P27Udf+/RBgwbizkCsCAVowuYwcILzFK1cWvX06AlADfUXGV9wno2nBQsQFibukAUF8PR89LVnT/jz2DGBWAcK0ITNReBKra/Nam3yLZYQZUis04Pf2WxyoUpNFUdwMOQi/yemUNT5kXDePFy9Ku4MxIpQgCZstH4kXM7h36GFGQv/LPoTgBxy267iOHECycnijt2kCSZNevS1XTv4+oo7A7EiFKAJG1ndr1uBXLGn6OvZt7lzcwAaaDilOCSq4khMRGKi0UuZnT2LE7VyTJTisG8UoAkbrah2CEgWe4qmzk19HXxhu+1Ga1dxFIi8GP72bWTU+kfy2We4csXw1cTGUYAmFuHRjmPs7bFZqiwAcsjnhc4Te3hAsioOb2/RqziefRb9+j36mp1NO6rYMwrQhM1AoFetr55AvthTvOH/ho/CBzxTHBLp2ROvvSbukHv2IDX10Veqg7ZvFKAJG69a7Uarv4r+Bt3erb1C9qBdm2334nB2ho+PuGPv2YP79x99jY2lHwntGQVowmY9kFDrK4836EE3BlV/kEHGKcXBV00VR2kpduwQd+z8fHh5Pfo6fjxiYsSdgVgRCtCEjbpu4XN3oL/YU0wLnlb9gWuKQ4oqjuJi/Pe/4o5dXAxv70dff/qpTsaD2BkK0IRZ7Uo7fyBA7PE7uXWq+Xxfdd/IlWbj+wthTRWHlxfyRf4LxurV8PN79DU7G+Xl4s5ArAgFaMJGK7BdAaaLPL5m8M3B1Z9lkH0W+pmow9eaSIIf15RK0XvNaRXVUYrDvlGAJhZxE3uhigyyaUGPUhyLMxaLOvwDUkTnaj/8IO540+v+ebh5M6U47BkFaMImHAip9dWLw4+EHd061ny2yYUqNVUcAIKCxB1727Y6XxMTce+euDMQK0IBmrAZAHSt9dUdEHedxNXyq/+X+n/Vn/mlOPiW2dVUcQDYtEncsXXXDVIdtB2jAE3YHAJO1/oq08lKWyhUGfqq76s1XzmlOPiqqeIAsHo1KirEGriyUjvFoVRCqRRreGJ1KEATNucBrfaWzUUd31Xu+pjLYzVfbXKhSk0VBwBPTxHbcWg1gwYwZw5atxZreGJ1KEATNhqdhnbLRB1/W962b7O/rfnKr4pDIt7eIgbovLw6RdAALlwQvR0TsSIUoAkbmc6RVaJmOdq5tuvl+ajbB78qDokKOT77DFFRYg0WHo53361z5KefcOOGWMMTq+NQ3w9AbIxuVPsFGF63QYclQpQhDrJH/1rer+SyUIWv2lUcAWKu41EotKue58yhHLQ9ozdoYikvIE+80T5J/ySx+FGT+/lh88Ub+xHpqjhOnxaxYfOuXXWCP4ALF0Rfq0isCAVowmYw8EzdI+KWQg/xHtLUuWnNV5uv4khKwpEjYg3coQOee67OkQ0bcPmyWMMTq0MBmrDxBtx1joj4Bt3EuYmH4lG+hFOKQ7oqDm9vEV9x3d0REmL6MmI3KEATNt8BWg00xX2DHn5reIm6pOYrpxSHdETdVGXhQhw+XOcINey3bxSgCRutdqMAhgPtxRt/auBUV7lrzVebTHHU9uSTmDxZrMFyc+Fe9+8vr7+Oli3FGp5YnfoJ0CqVql7mJaLQqrQLAVz1X2iOdm7tan/ll+IAv5ZJtX/Ic3CAh1gVLsjP166DDgiAo6NYwxOrI0WAzsjIGD9+fJcuXT744IO0tLTWrVs7Ozt37tz51q1bEsxOxKUb0g4CX4o3/rM3nq391SZTHLWrOKqqRGzH0bOndg56/nycPCnW8MTqSBGgx4wZk5qaOnHixLt37z7++ONvvfVWVlbWgAED3n77bQlmJ7x5iNpxdFbwrNpfubYb5fUGXbuKA8B//iPWwMOHw8WlzpGxY/HYYwauJrZPioUqhw8fvnHjhq+vb9OmTbds2fLaa68plcpp06Z98YWx3Yy2bNny1VdfaR28evVqw4YNeT4sMSEC8Kt7RNwqDmlSHHwtWIA+fR58VihQVSXWwOPGaUd7SnHYNykCtIeHR05Ojq+vb3x8/Lp165RKJYDs7GwHB2OzDx06dOjQoVoHp0yZkpaWxvFZiSmDdI6IGKDzqvLGpYzbGL2x5gi/hSqQrG2/QiHWSB06aB9ZsAADB6K9iL/SEmsiRYrjgw8+aNmyZa9evRQKxXPPPQdg7dq13bt3HzNmjASzE3EdBJLqHhExQLvL3acGTq19xOarOABs2SLWSKNHax/RaETfVItYESkC9Lhx406dOjVq1KiaI6WlpXPnzp07d64EsxNxnQOu1T3iCIiVBVXIFK1cW9U+YvNVHIB24YW5NBqMH699kOqg7ZtEzZIaN27cuHHjmq/jxo2TZl4iOt12owAWijT4gcIDx4qPvR/8fs0Rm6/iALB5M559Fk5OFo5aWKhnd6sePWjTWHtGC1WICJaKNM5jLo8N8xlW+4hNbhqrVcXxyy9IT7d81Nxc+PpqH+zWDcHBlo9NrBQFaMJG7xv0FpFaQnspvGKd6rx+cq3i4BWma/fiAODjg1wRChEVCrRqpX1w4UKcO2f52MRKUYAmInAHisUYZ2nm0r0Fe2sf4VrFIRFfX1F2PQkP17NovE0bcTtOE+tCAZqwGQo8rXPQR6S1KoO8BrV3q1MyxrWKQ6IwPXUqOnWyfJikJBw8qH2QUhz2jQI0YeOj024U4gXoIGWQj8Kn9hGuVRy8aFVxuLnBaMm/QJmZKC3VPrhoEc6etXxsYqUoQBM2a4BdOgd9RCqFHp8yXisic63i4BWmtao4bt3S8+rLrkePR+sTa+TkiNjNlFgd/QH66tWrACorK7du3frtt99WVlZK+1TEeum2GwUwBtD5+cocUwOnBijrpFRtcqGKVhVHair27bN81K1bceeO9kGqg7ZvegL0rFmzWrduXVVVtWjRopkzZ/7nP/+hrkakNt0qjmCRKuobOTeS1/130iYXqvCp4vjjD+Tp/D0lOhr+/paPTayUngC9YsWK8+fPKxSKZcuWrVmzZvfu3QkJCdI/GbFOekPaXuBrMQZ/7sZzWkdscqGKFl9fUdIQOTl66qDHjEF8vOVjEyulJ0BrNBpvb++TJ09WVVW1a9fOwcGhoqJC+icjNsQLyBFjnAVhC7SO2GS7US0hIVi2zPJhysrg46N9MCEBycmWj02slJ6/mA4bNqxbt25FRUVTpkzJzs4eOHDgk08+Kf2TEevUANAt6xKriqORUyOtIzbZblSrigOAp6flo27YAKVS+2BODsrKLB+bWCk9AXrFihXbtm1TqVTDhg3LzMwcPHjwhAkTpH8yYp20cxAAAF8x3qCL1EWv3H5lW8y22gdtst2oVhUHgDVrYHHvxsOH8fTT2gepI6R9q5Pi8Pf3B6BQKIYNG6ZWqx0cHEJCQqZNm+Ypxp//xD7sB3SXFovyBu0qd50TMkfroD1UcQBYu9byUT//XM/Bn3+mFIc9qxOg8/Pzaz6/+uqrkj8MsQFngRs6B12BFmIM3tBJe7sce6jigDjVcNu36zl48qSe2jtiN2ihCmGjt1kSAMszEYeKDum+L9tDFQcALy+UlFgygEqFY8f0n6I6aDtGAZqw0RudASyyeOQWLi1e9n1Z66A9VHEAWL8ebm6WDJCailWr9BxXKvX8ckjsRp0fCTUaTVJSku5nAC1btpT0uYi1MhTSfgGmGA7fQjjLnN2V2n0+7KSKw123fwkbvUXQAKZPpwBtz+oEaB8fnx49euh+BpCZmSnpcxFrZSgEewL5gCWbOy3LXNbCpUVvz961D9pJFceBAwgLQyPtIkLhDAXoy5cRFqb/FLEDdQI0RWFi0jBA79/V/YAcywL0UO+h/g7ay5YXZyyeEjjFglHrw+7d2m2NLlxARoYlAfqJJ9C0qZ7jCQno0QMdO5o9MLFqlIMmbHwMBGjLS6H9HPw8FdoFnTbZblS3isPXFzkW/eOpqtKzjBDA9OkUne2ZdoA+ffr08OHD4+LiPDw84uLiRowYcebMmXp5MmKdVgF79R23PEC/fvv13CrtcmqbbDeqKyDAwk1VvvoKhw7pORIvtCoAACAASURBVH7lioWRn1i1OgH6wIED3bp1a9iw4TfffHPixIlvvvmmYcOG3bp1OyhGN1tiH/S2GwUwFrDwTW5myExvhXaOxCYXqujq3l3PdlUsRo1C5856jm/eTHsS2rM6Oejp06evWLFi5MiR1V8bN2781FNPNWvW7P333z9y5Eh9PB6xRnp/J/QCdLb7YNPAsYFMZ2ybXKiiW8Uhl8PR0ZIhFQq4uloyALFJdd6gz549O2jQIK0rBg0adJY21SEPGQppRwELO7YNuzlM96BNLlTRreKoqtITtVmMH68/R0IN++1bnQBdWVnpplNO7+LiUlVVJeEjEZvkA2RbNsKicD2LXWxyoYpuLw65HN9/b8mQ+fnw8NBz/JVX8PjjlgxMrJr2QpULFy7oXqRW6806kv9FkUCQvuN+FgfoBo4NdA/a5EKVBQu0y+wsftHVaCDTl1ry94ezsyUDE6tWJ0B7eHjobf3sStkv8pCeNAQAiwP0nYo7M9Nm/jfyv1rHbXKhil5OTpbcPXGi/uOLF+Ppp9GliyVjE+tVJ0DnirFzGrFvvwEhQDOd466AJd2AgpXB04Om6x63yYUqev3yiyV3Dx2q//ioUQgI0H+K2AHtOuiLFy9u2LDh/Pnz9fI0xPolATcNnHragmHVUEuZ4pC6igOwpIqjsBBvvqn/VEAApTjsWZ0A/fXXX7dt23b+/Pnt2rVbsWJFfT0TsWaG2o0CmGPBsD/n/bwuZ53ucTup4gDw44/IyjJvPKUSgwfrP7V4MQ4fNm9UYgPqBOjPPvts/fr1Z86c2bBhw6efflpfz0Rs1AJAZe693T26D/XW89d4O6niAJCUhOvXzRvP2Rn9+uk/pVaDfsK3Y3UC9J07dwYMGACgX79+GRkZ9fRIxKoZeYM+ZcHvhDLIfBR6mk1wreLgVZ6k24sDgJ+f2W/Q+/fD0F9oqQ7avtUJ0DKZzMnJCYBSqXRw0LOfLCFGOj77AWZGIODj9I8vlV3SPc4pxZGfny+TyQos64/BJijI7HYcd+8arAHp3h1xceY/FLFy2nXQJ0+erP6sVqtrPgNo27atpM9FrNWLgKEthP0teIOeFDApwjFC9zinKo7MzMzIyMisrKxYvfliHl56yexbMzMNdirVVxZL7Id2w/4+DwvsPT09+9QqtqdW0aSaL2CoHMEfMPvfEje5m1KmZ2sQTimOrKys+Pj4LHNzDiboreJQKMweLzwcjRvrP7VsGTp0oMWEdosa9hM2XwFNgb76TgVY8AY9Mnnk741+1z3OKcWRk5MTFxeXnW3h4kcD9L6VFxRgyxaMHm3GeC++aPBUq1YIDjZjSGIbqGE/YaM23C9pIPC8ucOubLBS73FOVRyZmZlNmjTh9Uait4pDpcL27eaNN38+Kir0n3rySYSHmzcqsQF13qDbtWtn6Lrjx4/zfxhiA4yUDDgDWYC+fT9M85Trz2xzSnFkZmY2btx47169ew9YRqPB559r9+IA4OODvDzzhqyqgqHf7JcvxxNPwPB/uMS21fm/fbVlHRHJ/7g04ANgPfuNJeqSN1Le+CVWz2JoTimOioqK8PBwLjloQ1VvMhlcXMwb8oMPDJ7KybFwqxZi1eoE6JYtW9bXcxBbYaQO2t/cMjsXucsX4V/oPcWvF4e/vz+vFIfevnMAfv7ZvPE++gizZxuciuqg7RilOAibKCDQwClnczdVqdRU6m52VY1TikMmk3l7e+eZm3MwRi7H11/rP6XUU6ZiUlUVjGw5Fx1NzZLsGaU4CBvDBQWA0WUsRuwv3H+29Ox7Qe/pnuKR4igoKPDw8JDJZLyWehuqrV61CsOH62+8b1h2Nvz9DZ61oLqa2ABBKY5t27ZR9oNU2wNEAvGijtnJrVMb1zZ6T/FIcWRlZQUEBACQGcpFWGj3bj0/EgI4fRqdOqGZbq9WY0pLERNj8OyuXWjUyOAyFmLr9Pw2fOXKlcWLF+fn51d/LSoqSk1Nfe6556R9MGKlzgBqwwG6L6BmL96s0FRImeLIzMz09/cHoFQqKysrlWZlHgyqqsK//60/QAcGgj3rHRmJ+Yb/FpGbixJL+nAT66bnP6VRo0YBCAkJSUtL69u3b2Fh4Ycffij5gxErZeRHQgDTzCqtX5ix8EzJGb2neKQ4MjIyAgMDAfj7+/NaTKhXaKgZJReXL8NIe/aXXgL95daO6XmDvnjx4qFDh9RqdZcuXV555ZVu3bo9//zzQ4YMkf7hiBUynhT4HmgHsHbvecv/rWCl/vVwPFIcNQE6ICAgIyMjJCRE3PENMtR136hTp+DpiebN9Z/dtQsNG1K/JLul53XH09Pz2rVrLi4uarU6NTU1LCwsOTlZ8gcjVsr4z2ppwBX2MeUyud5GHOCT4sjKyqpOcQQEBIhfaadQ4D//EXG8kSMxYIDBs6dP4/ZtEWcj1kVPgJ42bVq7du3S0tIGDRrUt2/fvn37NmnSRPonI9bJ+Bt0IGBGH/FRyaMMneKR4rh//35QUBCAwMBALn3PDVVx5ORgzRrWwXbtQnGxwbNUB23f9AToSZMm3bhxw9fX98MPP5w5c2b//v23bt0q/ZMR6zQC6GT4rHkB+pvIbwyd4tGLoybFwSVAazT6e3EAUKuxcyfreGvWGPsZUKk0r7qa2Ab9K/xDQ0OrPzz/vNndb4h98jPcbhTmBmiZ4fdyHimOjIyM6jK7wMDAEydOiDy6SoVFi/RXcfj5ITeXdbycHPj5GTw7ZYolfUyJtaNudoTNMuCA4bNB7Ku9i9XF79x5x9BZHimO8vJyZ2dnAIGBgffvc1ipaKQdB+MqlWpyw/+ZXr1q9kZaxAZQgCZs1ICRjfzCgIWMA7rJ3Qz1GgWfFEfN+pSgoCAuAdrI+hf2dhw//mjs7M6dxorwiK2jjQeJyHINN+vQq0RdooDBv6WLnuJQq9Xyh2+knp6ehYWF4o4PpRIrDf55w0qjQX4+goIMXkApDvtGb9CEjfGFKgDGMQ64r3DfxtyNhs6KnuLIycnx8XnUs5pLOw4j+xyuXAmWbVzS0/Hxx8YuuHaNUhz2jAI0EZkcqGK5/hmPZ17ze83QWdFTHPfv3w/mukmURoM9ewyevX4dt24JHyw4GF/o78P6QEICTp0SPh6xMRSgCZsoUxkM1q7QuapcF7nBTvaipzjS09ODaqUMvLy8atrOiKOyEosWGTwbFASWrHdOjsHNrsj/AukCtG7vXZH/wyCSGAm0MnpBEMAUUz+///m18muGzoqe4qgpgq4m/u+ExnMmkZEoKhI+2Pbt2LfP2AWc+vERKyFFgL506VKzZs38/PxiY2O3bdtWfbC8vNzbW38DM2LNfgWuGr2ANUB/EPxBE2eDS1VFT3Gkp6fXTnEEBweLX8hhJGq++KKxPbp1pKcb+4UQwMiR6NhR+HjExkgRoCdMmPDaa6+VlZWtXr36rbfeOnz4sASTEk5OAcZ7P3QBmLb4KFQXyg3/eyh6ikMrBx0UFJSeni7mBE5OWL5crMHu34fxhLmvr9k7HRIbIEWZXVJS0v79++Vyebdu3VauXPnGG28kJSWZvOvs2bPHjh3TOnju3DkX+vexXpms4ujCOOD4lPG/N/rd0FnRUxxaOWjxAzSMVnGUlmLpUrynZ+8YvV56CfFGN0dYsQKtWqFXL5bHI7ZDijdof3//v//+u/rz4MGDmzZtOmWK6QaSzs7OPjqcnZ3lRpZVEf5M5jyTgSUsA34f9b2RszyqOGoHaPFTHMarOBwdDXbq0KdlSzg7G7tg+HB06CB8PGJjpHiD/uyzz/r06fP444//9NNPvr6+q1at6t27d+fOnY3fFRcXF6fT5vbIkSNpaWncnpSYZrJs2AH4W/BoaqiL1MZ+NBM9xVFSUuLq6lrzNTg4WOQ36IoKLF6M3r31n1UoUMVQhfjss8aiPQA/Pzga6Y1CbJwUAXrYsGGdO3f++++/nZycAPj4+Bw+fPjnn38Wv08N4c/kGzTTj4TJ5ckL7i9Y1WCVoQt49OKoLSQkROQAbXLly8NOZELMmmXighUr0KKFwT8OiK2TKF0QEhIyePBgNze36q9KpXLYsGHzjWy1RqzVS0B7oxcogUrBo0U5RX0W+pmRC8RNcahUKgeHOi8l7u7u4q/2Nl77Zry5Rl2PP27iArUaaiO9UYiNo3wuYeMPiPgrbbYq20ivUYid4tBKQHPh7Ixly0QZKTkZEyeKMhKxVRSgCZslwJ+mrnET/BK9IXfDoaJDRi4QN8WRlpamuwOhXC6vYskLm2akigPA+vW4eFHIMKmp8Pc3cU2XLqD9juwYBWjCRi3gd8JNgMBdPsb4jenv1d/IBeKmOLRWqVQTeV8V41UcAPLycOGCkJHu3TOdr+7YEVFRQh+N2BwK0ISNkOZvGUCOsNFult90kBn7pVrcFEdqamqoTswLDQ0VszSorAxffmnsgpAQ3LsnZKSoKDz5pIlrVq3CX38JfjZiayhAE/HtAPYLu3Ja6rQKjbFuQKKnOHQDdEhIyD1hEVMQk1UcDRuiUlAGqF07tDLe9wR47DGEhQl7MGKDKEATNiZXEgIIBQQGvB+if3CSORm5QIIUR2hoqJgBGqaqOFq2xD//KWSYDz5Ajqm/iXTogOhowQ9GbA0FaMImWkCrjTBAYMrgdoXxxh7ipzjCdF44RQ7Qrq5YwrSU0qDGjU1vYbh6NQ4Z+5GV2DYK0ITNKOAxU9eECXuDrtBUzLg3w/g14qY48vPzPT09tQ6KnIOGqSoOANOmCRnmlVegNPVja24uCgqEPRWxQRSgCZudwHVT1wQLe4N2lDn+GGVi1Ya4KQ69jVzCwsLEfINWq7F3r4lrEhOFjDR+vAiPQ2waBWjC5iSQYuoaR2CkgKFyVDm5VbnGrxExxaFSqRz19a3w8PAoYmmib0JpqekUh6sriouNX1JSgjt3TM8WGWmiYTSxaRSgCRshPxICeE3ANb8X/r6nwGjJsKgpjrS0NL67EQrXti1MrS+/exfh4aZHeuEF08vBie2iAE24mAKYeEUEBnkPMrJdbDURUxz37t3TrbGrplQqK6Tc++/jj0304QfUajzxhOmR9u0TuCyR2CQK0ISZkDfoUsDkX9Avl102OY6IKY67d++GG3gpFfN3Qjc3EwtVhImPx9ixpi/LyzOZLCE2jAI04SIcSDV1zfz0+QVVJkoQRExx3L17V7fGrlp4ePjdu3fFmsh0FceZM1i3zvglO3bgxg3TUz3/PNq1E/xgxNZQgCZsXgGE5DzDAZMB79uob/0dTHQDEjHFobcIulpYWJhoAVpIFYdCYbKQ4+5dCNk76LffBDb2IDaJAjRhEwAY3YPpgXABKY6TJSdNjiNuiiMiIkLvKTHfoEtKsHSpiWvCw2FquvHjBS0RTErCzZuCn43YGgrQhM0i4IiAyyKBbKMXaKD5OO1jk+OImOK4f/9+YGCg3lNiBmiTvTgA+PiY7FO3WOS9GIlNogBN2AhpNwqgEWA8wsgg2x673eQ4IqY4qqqqFAqF3lMRERF3hFQdC2S8F0e15cuNn//5Z0FTKZWmVxsS2yXFnoTEzggJ0AD2AEa2ystUZWarsuOd440PIlaKQ61WG4rOAPz9/bOzjb/xC+bhYfnbb3k5nIy1kHrkrbdg+H8WsXn0Bk2YCSmzA2Bsq0HgcNHhg0UHTQ4iVoojPT2d+2ZXNUxWcQD49FMjaejsbKE96m7exH2R9z0nVoQCNOHFAzCyWm6g98DX/V43OYhYKY7bt29HRkYauUC0tSpVVaarOACoVLh2zdDJ0FB89ZWg2fbswenTgp+N2BoK0IRNNGBqn7wHGhgt5EgsTlRBZXIQsVIcd+7cMVTCUU203wmLirBihenLIiKM9Nq4dAnJyYJmmzABffoIfTRicyhAEzavAs2EXRlhtK3S0sylpepSk4OIleIwGaAbNGiQkmKyDZQwQgo5WrY08uvenj24baJR9gOU4rBvFKAJm+3ALWFXRgJGgsx3kd95K7xNDiJZikPMAC2kiqNtW4wYYejk5Mno2lXQVDt34u+/BT8YsTUUoAmbkwJWoFTrChjp9rO/UNC2hWKlOFJSUho0aGDkgsjIyNsC31qN8/LCwoUWjrFjhwgPQuwABWjCRmC7UQChQGsDpyo1lf/J+o+QQcRKceTm5np7G3thF/MNWkgVB4BXXzV05t//FjqVkJd1YrsoQBOOBhs4rpQpt8RsETKCWCkOmalIJlqArqrCvn2CrkxJ0Zut1mgYwu6wYUKTIcQWUYAmHOUZOH6r4paQXqMQKcVRUFCguxWhFicnp/LycsvnQmGhoCoOAIGByMjQPZyVBQMr0vXw9YWrq+BnI7aGAjThyBHQW1p8sPCgwAAtSorj1q1b0QIWfshkMrVabfl0gqo4ADz9NPT9keDnx9BQevVq7N4t+MGIraEATdi8BrQRfHEDA5V2r/m9NtjbUP6jDlFSHCZLOKqJ1rZfYIZiwgTo+90yOxvOQhoGAgCGDsVTTwl+MGJrKEATNgLbjVaLApL1Hd+Zv1MjrKWHKCkOgW/QUVFRyQLXhxjh7c3wG58+//0vjh0TerGPD9zcLJmNWDUK0ITNvwDhdbfNAb07pnyb/a1MWDGIlCmO6OjoW7cEFnkbJbCK4+5d/N//6R4eOxbPPCN0qm++wa+/Cn4wYmsoQBM2AtuNVhsMDNF3fHPMZoEjiJLiuHnzpsA36JuWd78XXsXh749z53QPZ2QwdBBVqyFK2pxYJwrQhJnwAF0FbNQ5WKou3ZW/S+AIYlVxeHl5mbwsJiZGhDfoggKsXCnoSmdnlOpZ7D5+vKWPQOwGBWjCTPjaCDmgG6vuVd47ViI0yWp5ikMjsKZCrFJowdMBQM+eWgeqqgRtRVijc2c89hjD9cS2UIAmHMn0vW7HOsV+HGJ6s6tqlqc47t+/HxwcLORKBwcHlcp0gz3ThK8z+fBDrQN37uit7DDo8cfRsCHD9cS2UIAmbKIBP5brvXWWqxwsOphemS7wdstTHDdv3oyJiRF4sYuLS6m+tAMDHx98/rnZdwcEYPJkhuvXrcPvv5s9G7F2FKAJm7GAiV2q6ooFtH5321+4v1QjNAhanuK4fv16rMCyClHS0DKZ0CoOAAkJWLOm9gGlEi1bMszWpAnbGzexLRSgCZsEo01EdcUC1+se+Sjko2hHYRs6iZHiuHHjhvAAHRsbe/36ddPXGaFS4bffhF4cEoLLdVZUzprFtkPK44+jUSOG64ltoQBN2JwAmPYdGQp0rnvk+5zvhd9ueYrj+vXrDQWnaUUI0Pn5QrerAtCwIepO9/LLaCZwQwQAwPr1DH8cEJtDAZqwEd5utFow4FHndo2QvWJrWJ7iSE1NDQsLE3hxo0aNLA3QTPz8tFZqR0YK3c+7Wm4u8vNFfihiPShAE740ddeqyCD7usHXwm8XZaGKyV6jNWJjY2/cuGH5jAymTKn5qNFgiN6FPeR/FQVowob1DVoGVNb6mlSadLLkpPDbLUxxZGVl+fkxVJ04OzuXlZVZMiN8fbFgAcP1lY/+8dy9y9BotFqDBhBWQ0hsEgVowp1XrUq7i2UXS9Qlwu+1MMVx9erVuLg4plucnJwsqrRjquIAsHAh/vij+uO1a2B8WAwahM6dTV9GbBQFaMJmLMBSBgYAccDVh59H+Ix4yp2hP6aFKQ4zArSlaWimKg4AUVE1hRwhIRg0iG22Awdw5gzbLcSGUIAmbAIBlh+xgLoB+qsswRUOACxOcVy+fLlx48ZMt8TFxV25csX8KfPy8B9B2y0+0LgxHk7XpAlatWKbraAAxcVstxAbQgGasJkPnGC8pT3g8PDz1bKrxi7VnU7yFEd8fLxFAZqpFweA+Hg8/nj1x1deQVUV290DB1KKw55RgCZsmNqNVmsJDH/4eVH4IqZ7LUxxZGdn+/v7M91iaYAG41bbLi54+eXqj88+C4WCbaqDBynFYc8oQBNmrAEawKcAgCtlV4Q3Gq1mSYpDpVIpWAMeEBERYVFPO39/zGd86y8oqH7vfv555tnOn8e1a8x3EVtBAZowYyqzq/YboALSKtOElyRXsyTFwbSGsIZcLler1cKblGqTySC4N9MDc+YgKenCBcycaeacxF5RgCZSqG6Z9LTH0309+zLdaEmK49KlS/HxTJ2dHmjQoMGdO3fMnLWykrm/XFwcLl48exahocyzKZUM268Qm0MBmrCJYWw3Wq0pcBGYlz6P9UZLUhwXL15s2rSpGTfGx8dfunTJzFlzc/E1w1JJAGjSBBcvXrwIMx527FgMHMh8F7EVFKAJmzcAM7qnNQUuABWaCtYbLUlxmB2gmzVrduHCBTNnNSM30qYNOnUKDWVrNFotORlpacx3EVtBAZqw2crYza5aM+AKMCdkDuuNlqQ47ty5ExERYcaNzZo1u3jxotnzslVxAPDwQL9+EybA25t5qv37kZjIfBexFRSgCZtEIJX9rgjgzfJrm3I3sd5odoqjuoSD9TfJahY1HQ0MxDzmTE5ZcvrsWeb8LDlmDPPiQ2JDKEATiaSoS30d2EqSYUGK49q1a43MbWWvUCg0Go2ZhRxmVHEApZ8vjXEwp7YvORn37plxH7ENFKAJG9ZudjX+dGkR4tGd9S6zUxxnz55t0aKFefcCiI6ONnPvKzOqOACfNtGvRjG0ya6xdy8OHzbjPmIbKEATidzISzjHfpfZKY7z5883b97cvHsBtGjR4tw5M54XyM7G6tWsN/37XO/kv8zI7RM7RwGaSKSpptKMgGd2iuPs2bMtzaiKeKhFixZJSUlm3syeG9l/JSxgaBczpjIrx05shoPpSyxmpLOBkU5jKpWqsLBQ62B5ebloj0XM8gYgdP+ouub6PP8S+12LMxZPCZxi+jodubm5Pj4+ZtxYrUWLFl8J31pQC3vULCqRu/V+0oypBg2Co6MZ9xHbIEWAnjRp0p49e1xcXLy8vLROpRmu4dy+ffvKlSu1Dl69etXsX36IKMxoNwrgYNHBjMqMfB/mZhPmpTjy8vK8zahZqyUwMDAjI8OcO4OC8OmnTHeUleHJJ4GrVxERARcXpnu9vSGnvwbbLykC9O7du8ePHy+TyXQDrhFDhgwZorNB25QpU4zEdCKBT4EXgLaMdznKHJs4NxnPPp15KY7Tp0+3bt3ajBtr8/T0zM/P132rMIG9isPZGfPmAfO3omVL9GVbCr9uHby98cILTDcRmyHRH74jRoyIjo6WZi7ClRntRgF0dOvY3KX5SPYbzaviOHPmTCvW1vc6WrVqZU4auqIC+/cz3bFlC27fBlq1wunTrLM9+yyeeYb1JmIzJArQXbt2fe+996SZi/BmRoB+I+UNAMeAJYw3mpfiOHnyZJs2bcy4sbY2bdqcOnWK+Tb2Ko7kZLi5AW3agH06b2+4u7PeRGwGpa8IMzMKB7q5dwMQDRxivNG8FMft27cjIyPNuLG2Nm3anDzJsAH5A+wlHFOnwt8fCAzEBx+w3rt+PRISWG8iNoMCNJHCSN+RAIKAdMYbzUhxFBYWenh4sN6lKyoqKjk52Zw7Wao4cnOxdOnDL+x5c7UaajXrTcRmUIAmbGIBX8ZbFmUsSix+0NHHH8hkudeMFIco+Y1qnp6eBQUFbPcEB2PuXOGXHzuGRzPcuIGbN9mmI3aNAjRhMx5g7TTR2qV1E+cm1Z/bAcdZ7jUjxXH8+PF27dqx3qVX27ZtT5xg3CNXLmeq4jh+HI8e9sYNrF/PNFv79swbgRMbQgGasPkJYG3O082jm4fiQc6hK8DUE8iMFEdiYuITTzzBepdeTzzxRCJrN8/ycvzxh/DLs7JqBeh27XCc6c8vtGqFJk2Y7iC2hAI0YXOMMY+cUpEyIWVCzdfOAFM1tBkpjrS0tJCQENa79Grfvv3ff//Ndk9WFr75RvjlS5bg0YJHHx/k5zPNtmkTdu5kuoPYEgrQhI2MsczOXe4+PqBOTP4GEP6zFmuK4969e+Hh4Uy3GBEQEGDmekJh0tJ0wuuGDUwjxMWZ0dyU2AwK0IQNa7tRN4VbS5c6TYv+BoTv98ea4jh8+HCHDh2YbjEuJibmJrcf7q5dQ1FR3UOMG8dSisO+UYAmfPW/3l/rSCfgiODbWVMchw8f7ty5M9MtxnXu3PnIEeHPC4SE4OOPBV7bpQtefLHuofJy/PCD8NkoxWHfKEATvhaELdA68iQgvMU8a4ojKSnJ8i4ctT355JN//fUXww0sVRyLFukcUirB0kUvL481a01sCQVowmY8IHyj7IKqgmbOzbQONgSE7/fHlOIoKipydnZ2cBCzBVjTpk3ZNpAVXMWRk4MDB3SOyuXw8UF2tsDZZDJzthEntoICNGETBAjvP7wwY+HxEj11Y2GAwJ6ETCmOw4cPd+zYUfj1Qsjlcl9f38xMwctrMjOxZo2QCw8dwlNP6TvRpQsELzEPD2fNWhNbQgGasPkYOCv44tf8XuvopidiLgEChI3AlOI4ePBg165dhV8vUJcuXf7880/Rh7182UAjusmT0aOHwEH69kW3biI+FLEuFKAJG+HtRlUalRxyB5mehEOQ4EIOphTHsWPH2rdvL/x6gZ5++ukDepIRBmg0AntxTJsG/SvSFQrhTfiPHGFd2kJsCQVowsufRX+uzV5r6KzA5SrCUxwFBQWOjo7Ozs4CrxeuVatWp4V3ag4Lw0cfmbwqLw/bthk+vWSJwDR0QQF0NoYj9oMCNOGlg1uHyYGTDZ2NBm4IGER4iuPAgQM88hsA5HJ5SEhIamqqwKshYG+Kq1eRm2v4tEaD334TMlufPujeXdBzEVtEAZqwEb5Q5WLZxZoWHLp6AvsEDCI8xbFv376ePXsKvJhVjx499u0T8rxAWZm+4gxtTzyBMWOMzQdh01GKw75RgCZsGgJC9souqCr4LP0zIxf0AoS8IgpPcZw6dUqsLqO6evXqtXfvXkGXZmTg229NXvXvfxs93awZhBULXryIK1cEPRex9r9kTgAAIABJREFURVJsGkvsyT+EXeap8Pwx+kcjF4QAmUAloDQ6jsAUx61bt6KiouTcNriOjo5OSUmpqqpSKBQmLhXwI+HFi7h82dSUwparsOwNQGwPvUETNhuFdbPbmb/TUWaiYHqWgK5JAlMcO3fu7Mu4HzYrc1qPGrBrF+vm3QYplQJftYlNogBN2AhsN/pdzncmrxGyG7XAFMevv/7ap08fIVearV+/fjuFtL2IiMDs2cYvuXULprPlGg2ee87kbC+9hBdeMP1QxEZRgCZcbIreJOQykzFVSIqjoKCgoqLC399fyIxm69Kli6BqaAFVHMuXw9PT1DgyGSoqTBbb3bmDu3dNPxSxURSgCRshVRzrctaVqkuFjBYNnDd6gZAUhwSvzwAcHR0jIyOvXzfVR8RUFcfvv+O88f/NNQYMwI4dxi85dAiHhbeeIraGAjQR35mSM85yQQtGhgJGlmtAWIojISFhyJAhwh7NIkOGDElISDBx0f37+O9/jZw/dw5C3/WHDoW3t/FLRo7UaVhK7AgFaCK+heELZcKqpXuYqoY2meIoLS1NS0uLFrA2xHJ9+vQxnYY2VcUxeTKCg4XNFxiIQYOMX0IpDvtGAZqweQuIN3rBkowlGSqh20Q5AdFG+3KYTHFIUL9Rw83NLTAw8NatW2aPsGMHfv+d5YaKCuMBeP9+IctiiK2iAE3YBJmqXK5CVYCDwF51ADAJKDJ81mSKY+PGjcOHDxc+nYWGDx/+44/G6rvRoAFmzTJ08uuvGXeoys3FhAlGzlMdtH2jAE3YzAaMt6+fEjhFYH6jWhugneGzxlMcubm5+fn5kZGRwqezUL9+/Xbt2mXsCsNVHBkZUKkY2zcHBUGthuGNaylA2zcK0ISNxujqkgkpE7JUWaxjfgIcNXDKeIpj48aNzz//POt0lnBycmrevPlxI/0vSktx8KDeMzdv4p132KccMQKG39n79QP/AhZSbyhAEzbGm0H39+rv78Bcj/ws8LWBU8bT2Rs3bnxB8nUao0eP/tZIt43jxw3VxnXoYFYwffFFDB5s6KSnp4CSamKzKEATZob+Vl1QVTDAa4AZA7YEkgG9e59+Fmqw41JSUlJUVJSXl5cZM1qiffv2586dKy4u1n/622/x8su6h0+exNKlZs2nVMJwDmfTJmwStCSI2CQK0EQ0A24MqNRUmnfvy8A6A6feSHlD7/Gvvvpq3Lhx5k1nIYM/FZaX4+5dtGype0atRv/+5s5XWYklS/Se6dOHUhz2jAI0YdMIMLR24tvIb5Uy4yUeBo0AfjSQP+np0bNcU651MD8//9KlSx06dDBvOguNGjVq7Vp9m8U4OUFfobRKhcceQ0yMufMplfjlF+jbuJZSHPaNAjRhMxGI0Hf8r6K/Yp1izR7WFfiHgXq7F3z0ZJlXr149xljHe748PT1bt26tvzWHo54efgsWCOy/b9ibb+ptQEopDvtGAZqw+RHQ/dkuU5W5LHOZhSOPBJwMnBqfMv5q+dWar5WVlZs3b5ay/FnXpEmTFi+uW2GybRu2btW9sqwMO3ZY3F90yBDk5Oge1migNtmzldgsCtCEzRF9ATrAIWBD9AbLB38P0Ntx+ZPQT2ovflm/fv2wYcMc9b2rSiYmJsbNzS0pKenBd40GixejWzfdK7/7Di+/bHHXZoUCi/VXHGoE7rJObBAFaGKplIqUH3J+EGWot4BP9R0PU4ZlqbLSK9MBqFSqVatW1dfPg7XNmDFj3rx5D74kJOCpp+CjZzuwLl3whv6fOdllZKCqqvaBtm3x+OMiDU6sDwVowka33WhSaVJj58aiDB4H+ADH9J3KUeUsyVwC4Lvvvhs8eLC7u7soM1qiefPmcrn8zJkzAODqiqlTda9JTERMjN68tFm2btXKRDdvjsceE2lwYn0oQBNLPev1bFvXtmKNNhvQux9Je7f280LnlZaWrl69+u233xZrOgt99NFHM2fOBIDeveHrq3vB7NmirsYePRrff4+iRz+mbt+uN+9N7AQFaGKR524+J3zjbSGigRbALwbO9t7V+4033nB1dRVxRkvExcU1Cgs7M2OGoQt+/VW812cATk54553aNdGxsWjUSLzxiZWhAE3YTARqB4RZwbOClEHiTjEHaK3v+J07d+4n3h/+Sn0Wb+j6xM1t++bNFRUVWsdv3sSiRRzmGz4ctf4C0awZpTjsGQVowib4YbvRInXRH4V/tHbVG0st4gYEAz/rHJ88efL6YesrZBW661bqzeXLrufPh/zzn59//rnWmUmT8IyQbXHNUGtpCqU47BsFaMLmA+AKACAhL6FAXcBpFgdgFXC61pGNGzdGRka2a9fuaPHRWfcMNlyW2tGjWLr09TfeOHTo0IULF2oOf/cdWrTQu+RbJEuX4rvvAOTlIS+P2yykvllYnEn+59S0G33ZV09LIBGtBEYCewEXIDU1dcmSJfv27QPQ17NvJ7dOXKdmMHo0ABmwcuXK0aNH792718nJCUBwMPguo3nzTfToga5dZbIoqoO2Y/QGTdhogNyq3P43zG78I1QEMBGYAqhUqjFjxixdurTmt0Evhdfn9z9Pq0zj/QzGnDyJ6gI7AEBMTMzYsWOnTZtWWYkDB9C7N5wMLYsUhZMTVq7E22+HhSE8nOdEpF5RgCbMHKH8MvxLCSZ6AXAAnt2yZdCgQW3atKl9qpdnr5vlNyV4Bv3S0/H22wgLq33slVdeKSoqeu+9w3fuSPIMzZtj/fru3dG7tyTTkfpAAZqwOVly0kXu0tCpoTTTtfruO2RkvPXWW9rHXVp1du+8v3C/xsQWAhyUlmLkSCxbhgDtrRfnz19+9uysuLi/JXoSb+/jx3Hkp1SJpiOSowBNGORW5UZqyn1lCmmm27Fjx8+bN28fP/4QcE7fBcdLjv9Z9Kc0D/NIURFmz0Zb7bU5W7di+nSnDRs2vPvuu5cuGdmpXEyFhShYvg5btkgzHZEYBWgi1Ja8LT4Kn01unZh2PTXb7t27v/zyyw0bNiiVyhhgHHBe55ppQdO6une9UyFNTgGoqoJKhYAAdO2qdWb7dqxejaVLERAQ8MMPP4wZM+bKlSsSPFH37uizezK++QY/69YlEptHAZoIcrP85qmSUwC+B5g3hWX3008/ffHFF9u2bXNzcwMQBmwExgF6cwdrsteI1a3JmOJiDBmCw4d1z2g02LsXmzfDxQUAIiMjf/jhh9dee+3UqVO8H+r4cRw55YzNm3H9Ou+5iPQoQBMTitXFv+T/EuMU82nopwAOA3o29hCPRqOZP3/+1q1bExISandEigASgBmAbnv62SGzR/qOvFt5t0Rdwuux7t5F//544w2td2eNBuvXQybDsmWovf48Ojp627Zt7777bkJCAq9HAgBcuYKLF2u1aiospPaj9oQCNDFhZ/5Oea1/T3S72YkoJyfn+eefr6ys/OGHH5ydnbXOBgA7gZOA3nWEN8tv/l/q//F6spMnsWQJBmhviXvlCrKz9d8RHBy8a9euLVu2TJs2rbLSzK0aTdLuxLRrFwYP1rs5FrFFFKCJQV9mfHmy5OQLPi/09+Je9Qzgl19+6d+//z/+8Y+ZM2fKDLSAcwYWAE7AJ8Dxuqe6uHdZHrG8QlOxJnuNSqMS55kKCnDkCAAMGoQWLWqf2boV69YhPh6TJhm829XVdd26dXFxcc8888yZWkXTIlIq624F8OKLeO89DByI337jMR2RGAVookduVS6ACMeIVi6ttE7x+Pvz1atXhw4dunv37t27d3fTtymJrteABcB4ne1dlDJlubr8dsVtS59JrcZ336F3bxQXa525cgWDBuHPP/Hcc4JGGjt27Nq1a+fMmTNx4sSsLJET+MOGYdSouoc6d8a+fVo12sRG0VJvom1F5opMVebskNlDvIfovUDEFMe1a9cWLFiQkZExd+7cliytK8KBzcBvwPNAR2AqEPDg2WQTAiYA2Ja3bUvelvVR6818spUrkZeH33+vnVrOyoK/P5KSMG8emjVjGCwqKiohIeHXX38dMmRI165d33nnnQCdGmrzpKaislJnv3B3dzRpAgCbN+OXXzB9+oOvxOZobMrkyZNffPHF+n4K+3Sp9FL/6/3L1GUmLtNoKiyeq7Kycvv27QMHDhw6dOjRo0ctGUqt0WzXaA5oNBUazXGNRl3rVLm6XKPRfHzv4135uwSNdfeuZv58zalTek/++admxAhLnlSj0WjUavW2bdueeeaZ0aNHHz582NLhNJpNmzRr1hi9IjFR88ILmpdesnwuotfYsWOvX7/OaXCZxqZ+850yZUpaWtqGDSLsT0qq5Vflv5ny5tzQuZGOkRXqCg+Fh4nrAU9zX6LLysoOHjyYkJCQlJTUq1evMWPGNGjQwKyR9CgFZgDHgC7AQKADUL2cpqCq4L7qfiOnRpPvTn7N7zXdpA0A5OTgpZegVOKVVzBoEJRKAJWV2LcPCQk4fx4bNkC8JwWA06dPr1mz5sSJE08//fSgQYOeeOIJudycfKNaDY0GCpMrh8rLHzQH+fBDdOqEbt0elAQSi73++uvvv/9+bGwsj8ElDdAajaaoqMjd3d3QT0AmUYC2XHplulKm9HHw6Xe93/vB7z/l/lRaZVqYUmjKcgLwbt2e/cbl5+efOHHiyJEjR44cKSsr69q168CBA7Uaa4ioCjgE7AD+BkYDY4DLQAPAFUipSHGQOfgqfIfeGjpXPb7133kZp/cHRbTE5MnQaFBQAC8vALm5WLECb76J9HQkJGDgQI5dQ6uqqg4ePLhjx47ExMSgoKCnnnqqQ4cOrVq10q1gMeTOHX0pDiPOn8e2bfjjDzz/PCZMAICCgtoNpgkrrgFaihx0eXn5vHnzfvzxx+Tk5MrKSgcHh6ioqJdeemnGjBmOYm4HRPS4XXFbKVOGKkNnp81u5dLqOe/nPk3/dITviE4OnXY33F19jfDojFrtRnVVVlampqYmJyffuHHj2rVrly5dysvL8/T0bNOmTYcOHSZNmuTJPxAogKeBp2sd+Qk4XFJSXFoakl0Wn5/ful1oQkxC1YezsqICx/WqnP9Ur8zzmVNWnPVNebKsUDNz5a0nwmKeeAK+vggI4L5ZiUKh6N69e/fu3QGkp6f/9ddfmzdvnj17dmlpaUhISHx8fFxcXExMTGRkZEhIiN7XmkOHUFSEN98UPGXz5mjeHNX7KALIzsbo0cjPh6cnJk/GM89ApUJqKkJCRN2qi5hJijfoV199taCg4J///GfTpk09PDwKCwuvXLny5Zdfuri4rFmzhmmo//E36CpNlUKmyFJlKWVKL4XX+dLzrnLXGKeYX/J/CXAI6ODW4bP0z2KdYl/weWFcyrhObp1e9Xt1btrcTu6dnvF45mb5zXDHcEcZw391RUVFlZWVpaWlZWVlRUVFZWVlBQUF/2rY8JkzZ1zu3MnLy8vJycnKysrKyiouLnZwcJDL5WFhYdHR0dHR0XFxcfHx8T4+PiL/I1CrkZ8PAJWVCAwEgKQkpKejoKAULvKBA5yckPHBl+l/3y7LLVU0imm74f9OnEDKl9uGtL61prTbhvImmc7OgQcxdgc+L8W9dDRqBMcKlOUVzHRxbuqSParw96Fe/R3VFSsz/r0gaLqTumTz/c9nhMwphPqPgl2jvJ8HkFSa9ITrEwDuVd6LcIwQ+X/gQ/fu3bty5cq1a9du3bp1+/bt9PR0tVotk8nc3d39/f39/Px8fX19fHwuXGih0bgPGZLl7e3t4uLi7Ozs7u6uVCqrPzPMl5cHBwe4uyM1FR9+iLt3UVWF999Hz564dQtffglfX3h7o39/xMaipAQXLsDRER4eCA//Hw/lNp/iCAgIuHPnjta/LlVVVTExMbdvGyyH2r59+/r12j/BJyUlRUVF7dmzR9C83590DJXHF6ced2tcXuHlUKEI/XJz104/Hm37xR0Pd2XwbaRFlqpcffbdb7rybsrOlHTPdqq0WO8Gx5zy3UrS4xwWnC6KCnB620Xpd7swpXWc6lZaTF7ehmif7wryZ/l7NCyKL05NdI/TOKp8kaece1rt0KxkdOMSn2JlcLLsXlS5HF2Tv2j3T8XCBW9XNPB0qHTwjDrmnOdWmhbX4lr367Kh6c3f1TiWK1VorLqZ+v/t3XlwFGWfB/BfH3NfmVxAYpAICS+QEI4A61Eas7glKwjkfZVaXJF9fVlAa0W0KFEp8IjUiletr5wWr8gLLorIxvV4kZIlLpQoBhWNZIIYOZwkJJOZzPRMT9/7R9wsBbok1gvdE7+fvx4mXZ3vdD350un0PD08Gt9R/Dcfb//uiYJIrMaZZhNeg/XGsilme/JI4RDBVlP2qbdUkvyczNncsb9temHMo+ym5+erFTGt8wpD8ds0sinGAy/M2PqbmhOzFpazRwzVdVIboROj1MdHffJSzh1jDhfMGMp/x/HJo/pE0qnsn4+MGfH5yYdGH9fHJFyq3R62dxf53xMGbz115qkrEpOEdDrfKzpG8t+cVQsD93b9MHlodFG+pzz1xPyn6xpu+dcb/nz0tke2bStyhUJMIrHsyjfKxtv+KfbCgw9S2avL/uGN2UnNSW73nFW/+cMf6IEJ+0elP19Q+N79HY825lURUenZA2uzHv3T6ZsaXNetbazasoW2r2omQeANZeeMrcfvX7tkCb03dnnov8LL2+4nj4dGjly9mspeXfa7nXNWlb9VPtFec/QxVSX1VHjOmK8X3XTigb3Txs0aNm8erX440drBe3JdpaX0+9/Tl19SLEY33EBdXZRMUl4enVdcAlEHUT6Rm+htonYigShJlCRykfFbsbHROfo10k6onfl8fpy0iHx6HZ83TO24SYuPcJUphnJMbil2jCzRhWjLnFuL33yN4Zqk5iJnmWqoZ9Jfb3eUFCmtU7Xu0e4JsiEfSx+rcFUM0pOxljkzi9/cxrChdKjcVS4bcigd2uYYXvi/G/N6ulH8YrJnclBLtn0/97fFO98i+jz1eTlbnpSS3xrfLvhKOP4f8n/eVjEsJ9XWniv6Ogcxqfg3AemP39omXCXf6tNHHnGeKJFVnz7yyODHO5MHhsfWlnAy7wp+I+YwzInRWXRA/fKP8pmX0n+fo7l0P9fGueKxVMnwnf8eTPzls7u28JKDGIPywkWps4mWQUr9lkHlwdNld8qs3ZtkeU5MjfkqtzbCHXIld7M5oeHtRkG3l3jJPpT+O+ej1d3t631zz35d6NZayhjSdbvyL+uer2x98x+fOBBwRT32aHviKn+C49tE+nTtkMkF3w7/neHvmhg98606ss3tC6ztzDvRHN2pZ4eGxyhLKG/kvh+pnXC7XjqcM/5K99yOpgKn9n1ZbvaxpEcqesDlbIwdeWVcwB3p2fNI+dSQ7mMf/1s2O2mMdHOQyz0zLhb+TintGiRW/uW1WF3Z8doqb5q3cWKy/Gvu+5HsSaYiPs7b9NLBv7s5r/ObY3dP7UvPZHxBjxs37vHHH585c+a5L9bX1y9duvT/WaxAURThnMfL93jrrbeSyeR9993Xl+/bfPoHWUp7bXxU1VXO62DsHrnLzcsJ3SWQ7sl2M6o91tXtTqt+hldzVMnpU1IOO99FaU0RHDZWMVxee5ZdkaJCl57t5mWHpipej27vlmI2XvPY+Kiq27xen8Omd0ZlQU9zdt2te7LdpNq0hJTNioxMEd0puVwsS36bqIuSmGILs1jF6f5BYBlJIknyeXXWaVcUV7bHkOV0Z5eq60R2uzPocNrJHhFJlSQnIzPulGEnIjfJASPJqWw32bg8tywbnGJnFYVnqSibV12eH4Qfv1c8wZDd7nK68gOUTqc6IorXY7BOe1x1k07+lOhk02knI6p21ud2OEiMiGxCchikem18nkeWSY2Lbl5SZeIVu+52Cxxl2URfKp0QmECeXXe6u7uJRJFUNeBRWZcjrrq9XmIlMZkkmXMRUSBALEuiSDxPNhul0+f346+cSpQ4558BIpYoTqQREZGgCV7OaydStW4XF0gSxbW4n/MbZMS1eBEXIN1okeI5rkBKNJIUz3MFlJSRSsVdXEBxGKoj7tICimqojvggLSAmjbCe8PJ+f8CIKXE2FcgNGLLSrQhZScZIaInCLD/LGeFYotTnt3FGUyzhJL9Bhur4cc+y3Gkz+IjERETBx/psdkN1xPMUn65SzJWwa/5kigRd8HO+HL8uK91aKosLGN1qgkkGDDIEXSj1eWyc0ZxI5Xq8LGe0xZNuw8cbht3RHXD6IxKTYhI5Dp+YpogoDCGv0250q2Gb6lMVQ8k1XEaWnCQ9esbN+bmAHnMZbDrb59Pi0bAv4WMNo9XL5AcDLKu3h9vzXW4nq0c1snPBJBmaEs61e8U0JX3GMFdAShmnIm1uxmez6UqO7qIgyYZDOsVrWR1pJq/QV5jdp9//Mr6gDx8+PHPmzNzc3J5LHIIghEKh9vb2urq6ysrKfu3qjTfe6OzsvHB1YAAAU2T8HwknTZp06tSp/fv3t7S0RKPRYDB49913V1VV8Tw+JgMA8LMuU0XyPD91ap8u6AAAQA+sxQEAYFEoaAAAi0JBAwBYFAoaAMCiUNAAABaFggYAsCgUNACARaGgAQAsKsM+yxcIBFavXr179+6+bPzpp59m3IcV0+m03W7/ZWu3m0XTNE3TMm7l2FQq5T7ncVYZQRRFp9P5i5dTN0XPE81tNpvZQfrBMAyO48aN+6lnO1zg9OnTrkv39INL9KQWK6iqqjI7Qr8tWbLkiy++MDtF/3zwwQe1tbVmp+i3TJweNTU1kUjE7BT9s3nz5ldffdXsFP3T3t5+++23m53CMAwjk87UAAB+VVDQAAAWhYIGALAoFDQAgEWhoAEALGogF3TG3WNHRCzLZtY9dkTEcRzHcWan6DdMj8sjE6eHdY7z5XjklVkSiYTP5zM7Rf8IguDxeDLrRldN0yRJyrh7ijNxemRiZkmSGIbJuNvkLXKoB3JBAwBkNEucxgMAwIVQ0AAAFoWCBgCwKBQ0AIBFoaABACwKBQ0AYFEoaAAAixrIBb1x48Yrr7wyGAzeeuutHR0dZsfpk127dpWWlgaDwerq6qamJrPj9M+UKVNCoZDZKS7us88+mzBhQn5+/vz58yVJMjtOP2TKEe6RoZPZWr1h9oLUl0ooFHK73c3NzZFIZPr06YsWLTI70cW1trb6/f6DBw9qmvb000+PGTPG7ER9VV9fv2DBAiJqamoyO8tFKIoyZMiQ3bt3p1KpmTNnrly50uxEfZJBR7hHhk5mq/XGgD2D/uSTT6699tqSkpLs7Ox58+Y1NjaanejiDh06VFlZec0117Asu3Tp0mPHjsViMbND9cmhQ4c4jruED/7569m/f39ubu6sWbNcLteyZct27NhhdqI+yaAj3CNDJ7PlesPc/x8uNVVVGxsb582b98gjj5id5eIEQeh9oNG+ffuuuuoqc/P01+DBg61/frdp06Y5c+b0jLu6umw2m67r5kbqu4w4wj0yejJbpzcyb0Gvfnn77bfvvPPOQCDQ0NBgdpaL83g8Ho+HiOrq6hYuXLhu3TqzEw1A0Wi0dxEcn8+nKIogCFZYFmeAyejJbJ3eGFCXODZv3lxSUlJSUrJz586eV2bPnt3V1fXYY49NmzbN3Gw/57zM0Wj0tttuW758+a5du2pqasxO97MuPNSZIhgMCoLQM04kEjzPe71ecyMNVJkymS9kod4w9wT+0tmwYcPWrVt7xrFYjOM4RVHMjXRRsixPnjz5nnvukWXZ7Cy/REb8Ar53797x48f3jD/++OMRI0aYm6dfMuII98jQyWy13hhQZ9Dnys/Pf+aZZzo7Ow3DWL9+/ZQpU6y/QHtdXZ2mac8//7yu65IkZdYdYJmiqqqqvb29vr5e07QXX3zxjjvuMDvRwJShk9lqvWH1zvrFZs+e3dDQUFFRoarqxIkTt23bZnaii2toaGhoaHA6nb2vxGKxQCBgYqSBh+f5urq6BQsWRCKR6urqhx9+2OxEA1OGTmar9QYW7AcAsKgBe4kDACDToaABACwKBQ0AYFEoaAAAi0JBAwBYFAoaAMCiUNAAABaFggYAsCgUNACARaGgAQAsCgUNAGBRKGgAAItCQQMAWBQKGgDAolDQAAAWhYIGALAoFDQAgEWhoAEALAoFDQBgUSho+FXYsWPH9ddfHwgEhg4dunjx4o6OjvM22LNnj8/nO+/F7u7uvLy8y5UR4HwoaBj4nnrqqZUrV65cubK1tfXgwYOKotx8882apvVu0NraOnny5Pfff9/EkAAXQkHDgPLKK6/ceOONRKTr+sSJE999990zZ86sWbPmnXfemTp1qtvtLioqevnll8vLy0OhUGNjY1VV1dy5c6urq8Ph8OLFi3t28txzz11xxRXFxcUbNmww9d3Arx0KGgaU+fPnq6r6+uuvb9q0qbi4+JZbbjl48ODYsWNLS0t7t2EYZsuWLaNHjyaiQ4cOjR079vDhw71f/fDDD5999tm9e/d+9dVX+/fvv/xvAaAXb3YAgL8mhmHWr18/ffp0hmEOHDhAROFwuPc6cnNzc0VFRc94xYoVs2bN8vl8Dz30EMMwvXvYvXv3okWLRo0aRUSrVq2aMWPGZX8TAD/CGTQMNGVlZSNGjJgyZUphYSERFRcXt7S09HyptLRUFEVRFJcsWaKqKhHl5+ef285E1NbWNmzYsJ5x7wDAFChoGGg++uijs2fPHjhw4OjRo0R09dVXHz9+vKGhoXcDwzD27dvXM2bZ838ECgoKTp482TM+ffr0ZYkM8NNQ0DCgyLK8ePHidevW1dbW3nvvvUQ0aNCgFStWzJ49e8+ePaIohsPhhQsXhsPhn9tDTU3Nxo0bjx8/nk6na2trzzu/BricUNAwoKxZs2bSpEnXXXfdXXfdJcvy1q1biWj58uVPPvnkihUrBg8ePH369MrKyu3btxcUFPzkHqqqqh588MHq6upRo0ZNmzZt/Pjxl/fdK0jaAAAAdUlEQVQdAPwfxjAMszMAAMBPwBk0AIBFoaABACwKBQ0AYFEoaAAAi0JBAwBYFAoaAMCiUNAAABaFggYAsCgUNACARaGgAQAsCgUNAGBRKGgAAItCQQMAWBQKGgDAolDQAAAWhYIGALAoFDQAgEWhoAEALOp/ABrxYHaFEEWvAAAAAElFTkSuQmCC\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R \n",
    "\n",
    "xGrid <- seq(-3, 3, 0.005)\n",
    "\n",
    "pdfs <- c(dtransition(xGrid, 0))\n",
    "pdfs <- c(pdfs, dtransition(xGrid, 0.5))\n",
    "pdfs <- c(pdfs, dtransition(xGrid, -0.5))\n",
    "pdfs <- c(pdfs, dtransition(xGrid, 1.1))\n",
    "pdfs <- c(pdfs, dtransition(xGrid, -1.1))\n",
    "\n",
    "allPDFs <- matrix(pdfs, ncol=5)\n",
    "matplot(xGrid, allPDFs, type=\"l\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## And vectorise the transition simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1] 0.5320334\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "\n",
    "lambda <- 5\n",
    "\n",
    "rtransition <- function(x) { \n",
    "\n",
    "    E <- rexp(1, lambda)\n",
    "    \n",
    "    probJumpLeft <- (1 / (x+1)^2) / \n",
    "        ((1 / (x+1)^2) + (1 / (x-1)^2))\n",
    "\n",
    "    if (x > 1) {\n",
    "        return( x - E )\n",
    "    }\n",
    "    if (x < -1) {\n",
    "        return( x + E )\n",
    "    }\n",
    "    \n",
    "    if (runif(1) < probJumpLeft) {\n",
    "        return( x - rexp(1, lambda) )\n",
    "    } else {\n",
    "        return( x + rexp(1, lambda) )\n",
    "    }\n",
    "}\n",
    "\n",
    "rtransition(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "cell_style": "split"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1] -1.3049987  0.1950013  1.6950013\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "\n",
    "rtransitionVectorised <- function(x) { \n",
    "    \n",
    "    R <- length(x)\n",
    "    \n",
    "    Es <- rexp(R, lambda)\n",
    "    \n",
    "    probJumpLeft <- (1 / (x+1)^2) / \n",
    "        ((1 / (x+1)^2) + (1 / (x-1)^2))\n",
    "    \n",
    "    jumpLeft <- (runif(R) < probJumpLeft)\n",
    "    jumpLeft[which(x < -1)] <- FALSE\n",
    "    jumpLeft[which(x > 1)] <- TRUE\n",
    "    \n",
    "    jumpSizes <- (-1)^jumpLeft * Es\n",
    "    \n",
    "    return(x + jumpSizes)\n",
    "}\n",
    "\n",
    "rtransitionVectorised(c(-1.5, 0, 1.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Simulate the chain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "%%R \n",
    "R <- 1000; N <- 5000\n",
    "\n",
    "X <- matrix(rep(NA, N*R), nrow=N, ncol=R)\n",
    "\n",
    "X[1,] <- rtransitionVectorised(rep(0, R))\n",
    "for (n in 2:N)\n",
    "    X[n,] <- rtransitionVectorised(X[n-1,])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R \n",
    "# What's the distribution of X_N?\n",
    "hist(X[N,], 40)\n",
    "# library(ks)# plot(kde(X[N,]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R \n",
    "# What does one sample path look like?\n",
    "plot(X[,1], type=\"l\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Compare histogram of $X_N$ to that of all $X_i$'s "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "library(ks); plot(kde(X[N,]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "library(ks); plot(kde(as.vector(X)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## How does this compare with a different starting position?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "%%R \n",
    "R <- 1000; N <- 500\n",
    "X <- matrix(rep(NA, R*N), nrow=N, ncol=R)\n",
    "\n",
    "X[1,] <- rtransitionVectorised(rep(100, R))\n",
    "\n",
    "for (n in 2:N)\n",
    "    X[n,] <- rtransitionVectorised(X[n-1,])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "# Plot one sample path\n",
    "plot(X[,1], type=\"l\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "# Plot histograms for X_N and all X_i's\n",
    "#plot(kde(X[N,]))\n",
    "library(ks); plot(kde(as.vector(X)))\n",
    "#plot(kde(as.vector(X[1000:N,])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Markov chain Monte Carlo\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "\n",
    "_Input_: $f_X$, $R$, $q$, $X_0$\n",
    "\n",
    "To generate the $r$-th random variable:   \n",
    "$\\quad$ Make a proposal $Y$ from the distribution $q(\\,\\cdot\\, \\mid X_{r-1})$  \n",
    "$\\quad$ With probability $\\alpha(X_{r-1}, Y)$:  \n",
    "$\\quad$ $\\quad$ We accept the proposal, so $X_r \\gets Y$  \n",
    "$\\quad$ Otherwise:  \n",
    "$\\quad$ $\\quad$ We reject and stay where we are, so $X_r \\gets X_{r-1}$  \n",
    "Return $(X_1, \\dots, X_R)$\n",
    "\n",
    "Here we use\n",
    "$$\\alpha(X,Y) := \\min\\Bigl\\{ \\frac{ f_X(Y) \\, q(X_{r-1} \\mid Y)\n",
    "}{ f_X(X_{r-1}) \\, q(Y \\mid X_{r-1}) } , 1 \\Bigr\\} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "\n",
    "_Input_: $f_X$, $R$, $q$, $X_0$\n",
    "\n",
    "For $r = 1$ to $R$  \n",
    "$\\quad$ $Y \\sim q(\\,\\cdot\\, \\mid X_{r-1})$  \n",
    "$\\quad$ $U \\sim \\mathsf{Unif}(0,1)$  \n",
    "$\\quad$ If \n",
    "$U \\le \\alpha(X_{r-1}, Y) = \\min\\bigl\\{  \\frac{ f_X(Y) \\, q(X_{r-1} \\mid Y) \n",
    "}{ f_X(X_{r-1}) \\, q(Y \\mid X_{r-1}) } , 1 \\bigr\\} $  \n",
    "$\\quad$ $\\quad$ $X_r \\gets Y$  \n",
    "$\\quad$ Else  \n",
    "$\\quad$ $\\quad$ $X_r \\gets X_{r-1}$  \n",
    "$\\quad$ End If  \n",
    "End For  \n",
    "Return $(X_1, \\dots, X_R)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Example: sampling from $Z \\mid Z > 5$ \n",
    "\n",
    "Will propose jumps which are Laplace distributed (i.e. double exponential distributed)\n",
    "\n",
    "$$ X \\sim \\mathsf{Laplace}(\\mu, \\lambda) \\quad \\Rightarrow \\quad f_X(x) = \\frac{1}{2\\lambda} \\exp \\,\\Bigl\\{ \\frac{| x - \\mu | }{\\lambda} \\Bigr\\} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1440x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = np.linspace(-5,5, 500)\n",
    "plt.plot(xs, stats.laplace.pdf(xs), 'r');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1440x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "zs = np.linspace(3, 8, 500)\n",
    "plt.plot(zs, (zs > 5) * stats.norm.pdf(zs) / (stats.norm.sf(5)));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "_Input_:\n",
    "$$f_X(x) \\propto 1\\{x > 5\\} f_Z(x) , \\quad R = 10^6, \\quad X_0 = 5.01, \\quad\n",
    "q(x_r \\mid x_{r-1}) = \\frac{1}{2\\lambda} \\exp \\,\\Bigl\\{ -\\frac{| x_r - x_{r-1} | }{\\lambda} \\Bigr\\}$$\n",
    "Note: $q(x_r \\mid x_{r-1}) = q(x_{r-1} \\mid x_r)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "For $r = 1$ to $R$  \n",
    "$\\quad$ $Y \\sim \\mathsf{Laplace}(X_{r-1}, \\lambda)$   \n",
    "$\\quad$ $U \\sim \\mathsf{Unif}(0,1)$  \n",
    "$\\quad$ If \n",
    "$U \\le \\frac{ f_X(Y) q(X_{r-1} \\mid Y)\n",
    "}{ f_X(X_{r-1}) q(Y \\mid X_{r-1}) } = \\frac{ f_X(Y) }{ f_X(X_{r-1}) } = 1\\{Y > 5\\} \\mathrm{e}^{ \\frac12 (X_{r-1}^2 - Y^2) } $  \n",
    "$\\quad$ $\\quad$ $X_r \\gets Y$  \n",
    "$\\quad$ Else  \n",
    "$\\quad$ $\\quad$ $X_r \\gets X_{r-1}$  \n",
    "$\\quad$ End If  \n",
    "End For  \n",
    "Return $(X_1, \\dots, X_R)$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "To generate the $r$-th random variable:   \n",
    "$\\quad$ Make a proposal $Y$ from the distribution $\\mathsf{Laplace}(X_{r-1}, \\lambda)$  \n",
    "$\\quad$ Three scenarios:  \n",
    "$\\quad$ $\\quad$ a) $Y$ is not valid ($f_X(Y) = 0$, e.g. $Y \\le 5$)  \n",
    "$\\quad$ $\\quad$ $\\quad$ We reject and stay where we are, so $X_r \\gets X_{r-1}$  \n",
    "$\\quad$ $\\quad$ b) $Y$ is valid are more likely than $X_{r-1}$ ($\\frac{ f_X(Y) }{ f_X(X_{r-1}) } \\ge 1$)  \n",
    "$\\quad$ $\\quad$ $\\quad$ We accept the proposal, so $X_r \\gets Y$  \n",
    "$\\quad$ $\\quad$ c) $Y$ is valid but less likely than $X_{r-1}$ ($\\frac{ f_X(Y) }{ f_X(X_{r-1}) } < 1$)  \n",
    "$\\quad$ $\\quad$ $\\quad$ We accept with probability $\\frac{ f_X(Y) }{ f_X(X_{r-1}) }$, and reject otherwise.  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Into R land"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "For $r = 1$ to $R$  \n",
    "$\\quad$ $Y \\sim \\mathsf{Laplace}(X_{r-1}, \\lambda)$   \n",
    "$\\quad$ $U \\sim \\mathsf{Unif}(0,1)$  \n",
    "$\\quad$ If \n",
    "$U \\le \\frac{ f_X(Y) q(X_{r-1} \\mid Y)\n",
    "}{ f_X(X_{r-1}) q(Y \\mid X_{r-1}) } = \\frac{ f_X(Y) }{ f_X(X_{r-1}) } = 1\\{Y > 5\\} \\mathrm{e}^{ \\frac12 (X_{r-1}^2 - Y^2) } $  \n",
    "$\\quad$ $\\quad$ $X_r \\gets Y$  \n",
    "$\\quad$ Else  \n",
    "$\\quad$ $\\quad$ $X_r \\gets X_{r-1}$  \n",
    "$\\quad$ End If  \n",
    "End For  \n",
    "Return $(X_1, \\dots, X_R)$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "%%R \n",
    "\n",
    "lambda <- 10\n",
    "Xstart <- 5.01\n",
    "R <- 5 * 10^6\n",
    "Xs <- rep(NA, R)\n",
    "\n",
    "Xs[1] <- Xstart\n",
    "\n",
    "for (r in 2:R) {\n",
    "    # Generate proposal\n",
    "    U1 <- (runif(1) < 0.5)\n",
    "    sign <- (-1)^U1\n",
    "    Y <- Xs[r-1] + sign * rexp(1, lambda)\n",
    "    \n",
    "    # Calculate acceptance probability.\n",
    "    alpha <- (Y > 5) * exp(0.5 * (Xs[r-1]^2 - Y^2))\n",
    "    # Transition with this probability\n",
    "    U <- runif(1)\n",
    "    if (U < alpha) {\n",
    "        Xs[r] <- Y\n",
    "    } else {\n",
    "        Xs[r] <- Xs[r-1]\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## The histogram of the samples against the desired density"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R\n",
    "hist(Xs, 40, prob=T, ylim=c(0, 5.5))\n",
    "zs <- seq(4.9, 7, 0.005)\n",
    "lines(zs, (zs > 5) * dnorm(zs) / (1-pnorm(5)), col=\"red\");"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.7"
  },
  "rise": {
   "autolaunch": false,
   "enable_chalkboard": true
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}