{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "$$ \\LaTeX \\text{ command declarations here.}\n", "\\newcommand{\\R}{\\mathbb{R}}\n", "\\renewcommand{\\vec}[1]{\\mathbf{#1}}\n", "$$" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "%matplotlib inline\n", "from Lec04 import *" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# EECS 545: Machine Learning\n", "## Lecture 04: Linear Regression I\n", "* Instructor: **Jacob Abernethy**\n", "* Date: January 20, 2015\n", "\n", "\n", "*Lecture Exposition Credit: Benjamin Bray*" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Outline for this Lecture\n", "- Introduction to Regression\n", "- Solving Least Squares\n", " - Gradient Descent Method\n", " - Closed Form Solution\n", "\n", "\n", "## Reading List\n", "- Required: \n", " - **[PRML]**, §1.1: Polynomial Curve Fitting Example\n", " - **[PRML]**, §3.1: Linear Basis Function Models\n", "- Optional:\n", " - **[MLAPP]**, Chapter 7: Linear Regression" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "> In this lecture, we will cover linear regression. First some basic concepts and examples of linear regression will be introduced. Then we will show solving linear regression can be done by solving least squares problem. To solve least squares problem, two methods will be given: *gradient descent method* and *closed form solution*. Finally, the concept of Moore-Penrose pseudoinverse is introduced which is highly related to least squares." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Supervised Learning\n", "\n", "- Goal\n", " - Given data $X$ in feature sapce and the labels $Y$\n", " - Learn to predict $Y$ from $X$\n", "- Labels could be discrete or continuous\n", " - Discrete-valued labels: Classification\n", " - Continuous-valued labels: Regression\n", " \n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Notation\n", "\n", "- In this lecture, we will use\n", " - Let vector $\\vec{x}_n \\in \\R^D$ denote the $n\\text{th}$ data. $D$ denotes number of attributes in dataset.\n", " - Let vector $\\phi(\\vec{x}_n) \\in \\R^M$ denote features for data $\\vec{x}_n$. $\\phi_j(\\vec{x}_n)$ denotes the $j\\text{th}$ feature for data $x_n$.\n", " - Feature $\\phi(\\vec{x}_n)$ is the *artificial* features which represents the preprocessing step. $\\phi(\\vec{x}_n)$ is usually some combination of transformations of $\\vec{x}_n$. For example, $\\phi(\\vec{x})$ could be vector constructed by $[\\vec{x}_n^T, \\cos(\\vec{x}_n)^T, \\exp(\\vec{x}_n)^T]^T$. If we do nothing to $\\vec{x}_n$, then $\\phi(\\vec{x}_n)=\\vec{x}_n$.\n", " - Continuous-valued label vector $t \\in \\R^D$ (target values). $t_n \\in \\R$ denotes the target value for $i\\text{th}$ data." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Notation: Example\n", "- The table below is a dataset describing acceleration of the aircraft along a runway. Based on our notations above, we have $D=7$. Regardless of the header row, target value $t$ is the first column and $x_n$ denote the data on the $n\\text{th}$ row, $2\\text{th}$ to $7\\text{th}$ columns.\n", "- We could manipulate the data to have our own features. For example,\n", " - If we only choose the first three attributes as features, i.e. $\\phi(\\vec{x}_n)=\\vec{x}_n[1:3]$, then $M=3$\n", " - If we let $\\phi(\\vec{x}_n)=[\\vec{x}_n^T, \\cos(\\vec{x}_n)^T, \\exp(\\vec{x}_n)^T]$, then $M=3 \\times D=21$\n", " - We could also let $\\phi(\\vec{x}_n)=\\vec{x}_n$, then $M=D=7$. This will occur frequently in later lectures.\n", "
\n", "
(Example taken from [here](http://www.flightdatacommunity.com/linear-regression-applied-to-take-off/))
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Linear Regression " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "### Linear Regression (1D Inputs)\n", "\n", "- Consider 1D case (i.e. D=1)\n", " - Given a set of observations $x_1, \\dots, x_N \\in \\R^M$\n", " - and corresponding target values $t_1, \\dots, t_N$\n", "- We want to learn a function $y(x_n, \\vec{w}) \\approx t_n$ to predict future values.\n", "$$\n", "y(x_n, \\vec{w})\n", "= w_0 + w_1 x_n + w_2 x_n^2 + \\dots w_{M-1} x_n^{M-1}\n", "= \\sum_{k=0}^{M-1} w_k x_n^k\n", "= \\vec{w}^T\\phi(x_n)\n", "$$\n", "of which feature coefficient $\\vec{w}=[w_0, w_1, w_2, \\dots ,w_{M-1}]^T$, feature $\\phi(x_n)=[1, x_n, x_n^2, \\dots, x_n^{M-1}]$ (here we add a bias term $\\phi_0(x_n)=1$ to features)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": true, "slideshow": { "slide_type": "slide" } }, "source": [ "### Regression: Noisy Data" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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WSzKRw7ymZgPAfcH3FbyTg1izr1cS67an11fWVs/kigv8ggAct3l+wrott9Yi\n8puIrBKRJi44LnmQAdHRiAwNRbr1efayTAOio42M5ZWyv8BsFfYLzBVt5CcxMREBAQHYuXMn2rdv\nj5o1a6Jq1aqYOXMmzp49a/d9p0+fRp8+fXDLLbegatWqCA8Pz3d/Khas2QSgZNZtT6+vrK2eyV0X\n+P0KoK6qXhKRzgCWAbC7/s/48eNzHoeFhSEsLKy485GTgkNCMDwmBhMjIpCWcASoVRPDJ3zk0xeJ\nFJcB0dGI3L79xvVgC/EF5oo27Pn555+RlJSEtm3b4tFHH8WLL76IdevWoUyZMhgxYgTOnTsHADkj\nHrZef/11mEwm7N+/H1WqVMHy5csxfPhwpzMZJTY2FrGxsUbHKA6s2SVAdt1+9/VXcelYHKqGNMbw\nIlzs5k08ub6ytha/ItdsR+Zq5PcDoBWAtTbPX0MeF4zkek88gAA7rzk9N4WM9eG2DzVsTphmmbOM\njuLx7H3eXTG30hVt2M6rS0lJ0ZUrV2poaKgOGDBAVVVvvvlmnTdvnqqq7tixQ2vWrKn9+/dXVdVL\nly5p6dKl9dChQznt9ezZU4cOHapZWVl64sQJbdOmjdapU6fQuYxm7+8N3jFnmTWbrrP9+HatNbGW\nnkk/Y3QUl8nvc+kJ9ZW11b2crdmuKLyl8PfFImVhuVjktlz73GzzuCWAo/m057rfjqo+vfxp3XZs\nm0vbpPxlZmVqy1kt9T8//8foKB7P0zsa9erV04oVK6q/v79WrVpV77nnHv3Pf/6Tc2HJ0qVLNTg4\nWP39/fXhhx/W4cOH5xR0VdXIyEitUaOG3nTTTbpjxw7dv3+/tmjRQitXrqzNmzfXSZMmeWVB9/LO\nskfX7Ok7p+u0ndNc2iYVbMTaEdpvqfevgpGNtdU7a2txcbZmu+R21yLSCcAUWOZAz1bV90XkGWuI\nT0VkGIDnAGQAuAxghKrusNOWuiITAKz4YwVGrhuJ35/7HRXLVHRJm+SY/af3I2xuGHY/sxu3+N9i\ndByPxbUyvZO33+7aU2v2sYvHcNfMu7B10FY0rt7YJW2SYy5lXMId/7kDH3f+GF0adjE6jtNYW8mW\nszXbJZ1lV3JV4b145SJu/8/tmPfYPLQLaeeCZFRY42PHY8+pPfi217dGR/FYLOjeyds7y67kqpqt\nqui2qBta39Iab973pguSUWFtiNuAp1c8jX3P70OlspWMjuMU1lay5WzN9tnbXb++8XV0Cu3EjrKB\nxrYdi4TttdsFAAAgAElEQVTkBBw+d9joKETk4b7a/xWOXzyOV9q8YnSUEuvB+g/i/nr3Y+UfK42O\nQuRRfHJkeceJHej+VXfsf34/bqpwk4uSUVFkmbNQylTK6Bgei6Mf3okjy39zRc2+cPkCmk5vim96\nfYNWt7RyUTIqCl+p2aytZIsjy3moVLYS5jw2hx1lD+ALRZeIipdZzXj/wffZUfYArNlEN/LJkWUi\nb8HRD+/EkeW/sWaTJ2JtJVscWSYiIiIiKibuuoMfEVQV8cnxqH9TfaOjeIzg4OA878ZEni04ONjo\nCERuEXchjjWbSjxOwyC3OXbxGFp82gK/PfMbgvyDjI5D5HKchkG+xKxmNJnWBJMemuR1ay9zGgbZ\n4jQMWBZTf3nty8g0Z97wWkJ8PKLCwxHZrh2iwsOREB9vQEICgLpV6uLZFs9i5PqRRkchIoNFborE\n8YvH83yNddszmMSEKZ2m4IXVL+ByxmWj45DBEhISYDKZYDabjY7idj7RWX73h3dxKv0USpuun1WS\nEB+PqR06YPSCBYiKjcXoBQswtUMHFl4Djb13LHYm7sTGuI1GRyEig6w5vAYL9i5A9YrVb3iNdduz\nPNTgIdxV+y5M+HGC0VF8SkhICL7//nujYxRaQdMGFy5ciLvvvhuVK1dGUFAQunbtih9//NFN6YqP\n13eWD507hBm/zMCHHT+84bU5ERGIOnIEftbnfgCijhzBnIgIt2akv1UsUxFTOk3BsNXDcC3rmtFx\niMjNrmRewfA1wzG181RUKFPhhtdZtz3PpIcmYerOqYi7EGd0FCqErKwstx5v0qRJGDlyJN58802c\nPn0ax44dw7Bhw7ByZeFvcuPu7AXx6s6yquLFNS/i9XtfR2DlwBteNycm5hTcbH4AzElJbslHeXu4\n0cMIDQjFp79+anQUInKzidsm4h83/wOdG3bO83XWbc9Tt0pdjL5nNF7f+LrRUUqE7777Ds2bN8dN\nN92Etm3bYu/evTmvffDBB2jQoAH8/f1x++23Y9myZTmvzZ07F23btsXIkSNRvXp1REVFYe7cubj3\n3nsxZswYBAQEIDQ0FGvXrs15T0pKCp5++mkEBgaiTp06iIiIyJnbazabMXr0aNSoUQMNGjTAqlWr\n7GZOSUlBZGQkpk+fjkcffRQVKlRAqVKl0KVLF7z//vsAgIEDB2LcuHE579m8eTPq1KmT8zwkJAQT\nJkxAs2bNUKlSJUyYMAE9evS47jgvvfQSXn755QKzu5pXr4ax/I/lOJ5yHMNbDs/zdVNQENKB6wpv\nOgBT4I0da3IfEcEXj34B/3L+RkchIjdKSE7A5O2T8evQX+3uw7rtmUa2HokLly8YHcPn7d69G4MH\nD8aqVavQokULzJ8/H4888ggOHTqEMmXKoEGDBvjxxx9x8803Y8mSJQgPD8eRI0dw8803AwB27NiB\nvn374vTp08jIyMDixYuxY8cODBw4EOfOncPMmTMxePBgJCYmAgCeeuop1K5dG3FxcUhLS0O3bt1Q\nt25dDBkyBJ9++ilWr16NPXv2oGLFivj3v/9tN/dPP/2Eq1ev4rHHHivUnzf3tI7FixdjzZo1qFat\nGk6dOoW33noL6enp8PPzg9lsxpIlS7B8+fICs7ucqnrUjyWSY9774T3dGLfR7utH4+J0VGiopgGq\ngKYBOio0VI/GxTl8DCIiR1nrl+F11J0/hanZqw+t1g+3fZjvPqzb5AqOfC6tHzGnf4qiXr16unHj\njf2X5557TseNG3fdtltvvVW3bNmSZzt33nmnrlixQlVV58yZo8HBwde9PmfOHG3YsGHO80uXLqmI\n6KlTp/TUqVNarlw5vXLlSs7rixYt0vbt26uqavv27XXmzJk5r61fv15NJpNmZWXdkGPBggVau3bt\nfP/MAwYM0IiIiJznsbGxWqdOnZzn9erV0zlz5lz3nnvvvVe//PLLnOM3aNBAVVX/+uuvPLO3a9cu\nz2Pb+zw4WrO9emT5tbav5ft6cEgIhsfEYGJEBMxJSTAFBmJ4dDSCQ0LclJCIiLJ1btjZ7vSLbKzb\n5C7qgSvLJSQkYN68eZg6dSoAy4BmRkYGkqzTkObNm4fJkyfj6NGjAID09HScPXs25/220xqy1apV\nK+dxhQqW6wTS0tJw7tw5ZGRkoHbt2jnHUlXUrVsXAJCUlHRde/mtL1+tWjWcPXsWZrMZJlPRZ/je\ncsst1z3v06cPFi1ahPDwcCxatAh9+/YFABw7dizf7K7m1Z1lRwSHhCBy/nyjYxARkYNYt6mkqlOn\nDt544w2MHTv2hteOHTuGoUOHYtOmTWjdujUAoHnz5tlneAAUvFpF7mOVL18e586dy/N9tWvXxvHj\nfy/vmJCQYLet1q1bo1y5cli2bJnd6Rp+fn64dOlSzvOTJ0/esE/uHD169MDo0aORmJiIb7/9Ftu3\nb3cou6t59QV+5DvUE/+JT0REeWLNdt61a9dw9erVnJ+srCwMGTIEM2bMwM6dOwFYRo5Xr16N9PR0\npKenw2QyoXr16jCbzfjiiy+wb9++Ih+/Vq1a6NixI0aMGIHU1FSoKuLi4rBlyxYAQM+ePfHxxx8j\nMTERFy5cwAcffGC3LX9/f0RFRWHYsGFYvnw5Ll++jMzMTKxduxavvWaZBXDnnXdi9erVuHDhAv76\n6y9MmTKlwIzVq1fH/fffj4EDB6J+/fq49dZbHcruauwsk+GuZF7B3bPuxpn0M0ZHISIiB/T9pi++\nj/e+dYI9SdeuXVGxYkVUqFABFStWRFRUFFq0aIFZs2bhhRdeQEBAABo1aoS5c+cCAG677TaMGjUK\nrVq1Qq1atbB//360bdu20Me1HYmdN28erl27hiZNmiAgIAA9evTAX3/9BQAYMmQIHnroITRr1gz/\n/Oc/8fjjj+fb7siRIzFp0iS8/fbbqFmzJurWrYtp06blXPTXv39/3HHHHahXrx46deqE3r17281l\nq2/fvti4cSP69et33fb8sruaV93uOiMrAz8n/Yx76tzj5lRU3EasHYH0jHR8+jCXkyPvxdtd32hL\nwha0rdsWJuHYjC/55sA3iIyNxO5ndt9wQzBPwNtdk60SdbvrT3Z+grc2v2V0DCoGkWGRWPHHCuw6\nucvoKETkIjtO7EDvr3sj/Vq60VHIxbo37o6afjUx85eZRkchKnZe01k+nX4a7259Fx91+sjoKFQM\nqpaviuh20XhxzYscDSDyAWY146W1L+HdB95F5XKVjY5DLiYimNJpCqI2R+HcpXNGxyEqVl7TWY74\nPgL97+iPxtUbGx2Fismg5oNwKeMSFu9bbHQUInLSgt8XwKxmPNnsSaOjUDG5vebt6NW0F8ZtGlfw\nzkRezPMmGuXht79+w/I/luPgCweNjkLFqJSpFGZ0m4G0a2lGRyEiJ6RdS8NrG1/D1z2+5lxlHxfV\nLgpf/+9ro2MQFSuPrGJR4eFIiI/Pef72lrfxVru3ULV8VQNTkTu0DGqJ9iHtjY5BRIWQu2Z/tusz\ntA9pj9Z1WhuYitwhoEIAhrYYanQMomLlkathpAGIDA3F8JgYBIeEIOVqCvzK+KGUqZTR8YiI7Cqp\nq2HkrtlZ5iykZ6TDv5y/0fGohOJqGGTL2dUwPLKzrADSAUzs1493cSIihyTEx2NORATMiYkwBQVh\ngAG3SC6pnWXWbPI09erVy/eOc1SyBAcH59wi3JajNdtj5yz7ATBb74VORJSfhPh4TO3QAVFHjsAP\nlo5b5PbtOSOdVPxYs8mT5NUxIs+RZ822OTvlaTxyzjJg+cWZAgONjkEGupZ1DVO2T4FZzUZHIQ83\nJyIip+gClo5b1JEjmBMRYWSsEoU1mwBg3p55OJl60ugY5OG8rWZ7ZGc5HcC40PoYEB1tdBQyUBlT\nGXy1/yvM/W2u0VHIw5kTE3OKbjaOdLoPazZl23tqL978/k2jY5CH87aa7ZGd5dceewg/P1fbI4fi\nyX1EBB91+ghvfP8GUq+mGh2HPJgpKAi57xHHkU73ebd3D6x6qhSq1g4wOgoZ7M373sSqw6t4N1bK\nl7fVbJd0lkWkk4gcFJFDIvKqnX0+FpHDIvKbiNyZX3u/dLuIQQ887Ypo5OVaBrXEg/UfxHtb3zM6\nCnmwAdHRiAwNzSm+2fPfONKZN1fX7GsDg9G22X2oUr5K8QQmr1GlfBW81e4tvLT2Ja5GQXZ5W812\nejUMETEBOATgAQBJAH4G0FtVD9rs0xnAC6raVUT+BWCKqray0562mNkCO4fs5GL2BABITEnEHTPu\nwC9DfkHITTzbQHnLWQ0jKQmmwECuhmFHcdTsah9Uw77n96FWpVpu+BOQp8syZ+GuT+/CG/e+gZ5N\nexodhzyUN9VsV3SWWwGIVNXO1uevAVBV/cBmnxkANqnqV9bnBwCEqeqpPNrTHxJ+QNu6bZ3KRb7l\n7S1vo2KZihjZeqTRUYjs8pLOsstr9ns/vIfX2r7mnj8AeYVN8ZswdedUfNPrG6OjENnlzqXjggAc\nt3l+AkDLAvZJtG67ofACYEeZbvDGvW9AxKP7IETewuU1++VWL7syH/mAdiHtEFYvzOgYRC7hkess\njx8/PudxWFgYwsLCDMtCnoEdZSrItaxrGLxiMKZ3mY7K5Sq75ZixsbGIjY11y7E82ftvv5/zmDWb\nsrFuU0Fe3/g6et/eG3fcfIdbjlfUmu2qaRjjVbWT9bkjp/QOArjf3ik9XhRARIU1+afJWB+3Hmv6\nrTEsgxdNw2DNJiJD7UzciccWP4Y/XvjDbQMcuTlas11xBd3PABqISLCIlAXQG8CKXPusAPCkNVgr\nAMl5FV0ioqI4e+ks3t36LiZ1nGR0FG/Amk1EhlJVvLz2ZbzT/h3DOsqF4fQ0DFXNEpEXAKyHpfM9\nW1UPiMgzlpf1U1VdLSJdRORPWFYIGejscalkS76SjKrlqxodgzxEYkoixtwzBrfVuM3oKB6PNZuM\nkHwlGVXKVeHUDAIAXLx6ES1qt8BTdz5ldBSHOD0Nw9V4So8Kkn4tHQ2mNsC2Qdu4lBx5FG+YhuFq\nrNnkiAfmPYBnWjzDpeTIo7hzGgaRW/mV9cOwu4dhTMwYo6MQEZEDIu6LwCsxr+ByxmWjoxAVGjvL\n5JVGtR6FX5J+weajm42OQkREBQirF4YWgS0w6SdeV0Deh51l8koVylTAhA4T8PK6l5FlzjI6DhER\nFeD/OvwfJm+fjKTUJKOjEBUKO8vktXo06YFKZSvh892fGx2FDLD12FakXUszOgYROaj+TfUx5K4h\nGLtxrNFRyAD/O/M/HLt4zOgYRcIL/MirHTp3CBVKV0CdKnWMjkJudDr9NJpOb4qtA7fi1uq3Gh0n\nBy/wI8pf6tVUHDh7AC2Dct80knyZWc1oPbs1ht09DE82e9LoODkcrdnsLBOR1xm6cigqla2ESQ95\n1vzHktpZBliziajojOr2OVqzPfJ210RE9uw+uRsr/liBgy8cNDoKWXF8g4jsSb2aisbTGmNpz6Vo\ndUsro+MUCecsE5HXUFW8uPZFRLeL5k1piIi8wHtb38MDIQ94bUcZ4MgyEXmRH479gLRraRjUfJDR\nUYiIqAAXr1zEnN/m4JehvxgdxSkcWSafkXYtDYOXD8a1rGtGR6Ficl/wfdgyYAtKmUoZHYWIXGD0\n+tGIuxBndAwqJlXKV8EfL/yBwMqBRkdxCjvL5DP8yvghKS0Jn+z8xOgoVIwql6tsdAQicpGACgEY\ntX6U0TGoGPlCzWZnmXyGiGDyQ5Px3tb3cCrtlNFxiIioACNbj8Tvp35HzJEYo6MQ2cXOMvmUxtUb\n46lmT+H1ja8bHYWIiApQvnR5TOo4CS+tfQkZWRlGxyHKEzvL5BUS4uMRFR6OyHbtEBUejoT4eLv7\njrt/HNb8uQY7E3e6MSEVF36BEnmfwtTsR259BHWq1OEUOh/hizWbNyUhj5cQH4+pHTog6sgR+AFI\nBxAZGorhMTEIDgnJ8z2L9y2GqqLPP/q4NSu5VkZWBu6edTcWPr4QTWo0MTpOgUrqTUlYs8lWUWr2\nH2f/wOzdszGhwwS3ZiXXG7x8MNrWbYuBzQcaHaVAvIMf+Yyo8HCMXrAAfjbb0gFM7NcPkfPnGxWL\n3ODDbR8iJi4Ga/qtgYjn90HZWSZizS7Jtp/Yjsf/+zgODDsA/3L+RscpEO/gRz7DnJh4XdEFAD8A\n5qQkI+KQm5xMPYn3tr6HbYO3eUVHmYgsWLNLpixzFoatHoYJD07wio5yYXDOMnk8U1AQ0nNtSwdg\nCvTudRspf69seAVD7hqCRtUaGR2FiAqBNbtk+mzXZ6hYpiL6/qOv0VFcjtMwyOMVZf4bebcfEn5A\n32/64sCwA6hUtpLRcRzGaRhErNkl0blL59BkehOsD1+PZrWaGR3HYZyzTD4lIT4ecyIiYE5Kgikw\nEAOiox0uuqqKVYdXoUvDLjAJT6Z4gxMpJ5CQnIA2ddsYHaVQ2FkmsnCmZgOWua/1b6qPmn41izEl\nucqljEvYGLcRD9/6sNFRCoWdZSIrs5pxz+x7MLTFUAxqPsjoOOTD2Fkmco1R60bh/JXz+OLRL4yO\nQj6MnWUiG7tO7kKXBV2w//n9qFaxmtFxyEexs0zkGilXU9BkWhMsfmIx2tZta3Qc8lGO1myek6YS\n4a7ad6FHkx4Yu3Gs0VGIiKgA/uX8MemhSXhu1XM+eZML8i7sLFOJ8Xb7t7Hq8CpsO77N6ChERFSA\nHk16IKhyED7a/pHRUaiEY2eZSowq5atgUsdJmLJjitFRKJfLGZfx1LKncDnjstFRiMhDiAimdZmG\n2btn41rWNaPjUC5j1o/BwbMHjY7hFpyzTCWKqiLTnIkypcoYHYVsvLHxDfx54U989cRXRkdxCucs\nE7leRlYGa7aHWXN4DYavGY69z+1FhTIVjI5TZLyDH1EeRIRF18PsP70fn+76FL8/+7vRUYjIA7Fm\ne5ZLGZcwbPUwzOg2w6s7yoXBaRhEZBizmvHsqmcRFRaF2pVrGx2HiIgKEL05Gq1uaYWOoR2NjuI2\nHFkmIsPM+nUWMs2ZeKbFM0ZHISKiAvz212+YvXs2fn+uZJ0JdKqzLCI3AfgKQDCAowB6qurFPPY7\nCuAiADOADFVt6cxxiVzl4pWLUCiqlq9qdJQSKUuz8NnDn6GUqZTRUUoM1m3yZlnmLJxMO4lb/G8x\nOkqJlHwlGZ90+QS1KtUyOopbOXWBn4h8AOCcqk4QkVcB3KSqr+WxXxyAFqp6wYE2ebEIuc0bG99A\nUloS7xJFLuENF/i5um6zZpM7rf1zLUauG4ndz+xGudLljI5DXs5dNyV5FMBc6+O5AB6zl8cFxyJy\nubH3jkXs0Vis/XOt0VGI3IV1m7zWQ6EPoXH1xojeEm10FCpBnC2ENVX1FACo6l8AatrZTwHEiMjP\nIjLEyWMSuUylspUw6+FZeOa7Z5ByNcXoOETuwLpNXit77eVPf/0Uu0/uNjoOlRAFzlkWkRgAN9tu\ngqWIvpnH7vbOxbVR1ZMiUgOW4ntAVbfaO+b48eNzHoeFhSEsLKygmERF9mD9B9Ghfge8tuE1TO86\n3eg45EViY2MRGxtrdIwbuLtus2aTO9WuXBsTOkzAoBWDsPPpnVxajhxW1Jrt7JzlAwDCVPWUiNQC\nsElVbyvgPZEAUlV1kp3XOf+N3C75SjLu+M8diOkfg1ur32p0HJ/13aHvYBITujTsYnSUYuElc5Zd\nWrdZs8kIqorOCzqjV9NeGNh8oNFxfFb8hXh8/b+vMabNGKOjFAt3zVleAWCA9fFTAJbnEaSiiFSy\nPvYD0BHAPiePS+RSVctXxZ5n97CjXIzOXjqLoSuHokq5KkZHKelYt8nriQi+euIrPHXnU0ZH8Vlm\nNWPQikFGx/AIzo4sBwD4L4A6ABJgWYIoWURqA5ilqt1EJATAt7Cc6isNYIGqvp9PmxylIPJBvb/u\njcDKgZj0UJ4nlXyCl4wsu7Rus2YT+aZPdn6C+b/Px4+DfvTZ5T0drdlOdZaLAwsvke9ZtHcRordE\n49ehv/r07VG9obPsaqzZRL7n4NmDaPt5W2wbvA2NqjUyOk6xcbRm8w5+RFSsjl88jpfWvoS14Wt9\nuqNMROQLMrIy0P/b/ohuF+3THeXC4BqaRHYcOHMAHDFzXvKVZLzT/h3cVfsuo6MQkQ87kXKCS4C6\nQOq1VHRt2BXP/vNZo6N4DE7DIMqDqqLlZy0x9K6hGNKCS8ySYzgNg8g4I9aOwPkr5zH3sbkF70wE\n962GQeSTRARzH5uLsRvH4tC5Q0bHISKiArzd/m1sP7EdX+37yugo5GPYWSayo0mNJogKi0Lvr3vj\nauZVo+MQEVE+/Mr6YcG/F2D4muGIvxBvdBzyIZyGQZQPVcUTS55AUOUgfNz5Y6PjeL2E+HjMiYiA\nOTERpqAgDIiORnBIiNGxXIbTMIiMN/mnyVi0bxG2DtqKsqXKGh3Hq7FmW/fztCLHwkueJvlKMjp8\n2QGr+q5CTb+aRsfxePP2zENSahJea/vaddsT4uMxtUMHRB05Aj8A6QAiQ0MxPCbGZ4ovO8tExsse\n5Hjun8/hwfoPGh3H4+09tRfRW6Lx1RNfQeTv8sWabbOfpxU5Fl7yRKp6XRGhvO09tRft57VH7FOx\naFqz6XWvRYWHY/SCBfCz2ZYOYGK/foicP9+tOYsLO8tEnoE12zEpV1Nw96y78ea9b6J/s/7Xvcaa\n/TfOWSZyAItuwVKupuCJJU9g8kOTb+goA4A5MfG6ogsAfgDMSUluyUdEJQdrdsFUFYNXDEa7eu1u\n6CgDrNm2eFMSInJadtFtX689wu8Iz3MfU1AQ0oEbRilMgYHuiEhERDam7JiCuAtx+LL7l3m+zpr9\nN44sE5HTpv88HccvHsfkTpPt7jMgOhqRoaFItz7Pnv82IDraLRmJiMji16Rf8f7W9/F1j69RvnT5\nPPdhzf4b5ywTFcH0n6ejaY2muL/e/UZH8QjJV5JxJfMKalWqle9+OVdWJyXBFBhYYq+s9iWs2eQN\nfkj4AXtO7cELLV8wOopHyDJnIe5CHBpWa5jvfqzZ1v08rcix8JI32BC3AeHfhGProK1oENDA6Djk\nIdhZJvJMxy8eR6vZrTC9y3Q82vhRo+OQh+AFfkTF6MH6D2J82Hh0W9gNFy5fMDoOERHlo06VOljW\naxmeXvk0fvvrN6PjkJdhZ5moiJ7957Po1KATeizpgWtZ14yOQ0RE+bg76G5M6zINjyx6BIkpiUbH\nIS/CzjKREz7s+CH8yvphzPoxRkdxmyxzFj7e8TH/gUBEXqdn0554/u7n8ejiR2FWs9Fx3Gbennn4\nK+0vo2N4Lc5ZJnLS5YzLOHvpLOpUqWN0lGKnqnjmu2cQdyEOq/ut5q1kc+GcZSLPp6o4fP4wGlVr\nZHQUt5j721yMix2HbYO2Icg/yOg4HoUX+BGRS6kqxsSMwQ/HfsCG/htQuVxloyN5HHaWiciTLP3f\nUgxfMxzfP/U9GldvbHQcj+NozeZNSYjIIW9veRvrj6xH7IBYdpSJiDzc2j/X4vnVz2Nd+Dp2lJ3E\nzjIRFWj+7/Mxf+98bB6wGQEVAoyOQ0RE+Thw5gCe/PZJLOu9DHfWutPoOF6P0zCoxMtZdD0xEaag\nIJcsuv7R9o8gELzU6iUXpTRW6tVUpGekF3jTkZKO0zCIil9x1Owfj/2IWbtmYdbDs1CmVBkXJTWO\nquLIhSO8D0ABOGeZyAEJ8fGY2qEDoo4cgR/+vp3n8JgYp4pvQnICOnzZAb1v742osCiIlKj+U4nF\nzjJR8Squmn054zJ6ft0TZjVjSY8lqFimossyk+fiTUmIHDAnIiKn6AKAH4CoI0cwJyLCqXaDqwZj\n66Ct+O7Qdxi2ehiyzFlOZyUiKumKq2ZXKFMB3/T8BgEVAtDxy4682RRdh51lKtHMiYk5RTebHwBz\nUpLTbdf0q4lNT23CwbMH0f2r7ki9mup0m9kS4uMRFR6OyHbtEBUejoT4eJe1nXwlGVcyr7isPSIi\nVynOml2mVBnMfWwuWga1xD2f34PD5w473aat4qrbWeYsnE4/7ZK2KG+8wI9KNFNQENKB64pvOgBT\nYKBL2q9SvgrWhq9F5KZIpF1Lc8kqEnmehty+3enTkACw//R+/Pu//8Yb976BJ5s96XRWIiJXKu6a\nbRITJj00CY2rN0ZSahIaVmvoknaLq26fv3we/b/tj8BKgZj1yCyXZKU8qKpH/VgiEbnH0bg4HRUa\nqmmAKqBpgI4KDdWjcXFGR7NrfL9+OXnVJvf4fv2canfh7wu1+oTqOmf3HBclLXms9cvwOurOH9Zs\ncidvrNmqxVO3f036VUM+CtERa0fotcxrLkxbcjhaszmyTCVacEgIhsfEYGJEBMxJSTAFBmK4C66s\nLk6uPg2ZejUVo9aPwsb4jYjpH8NlhojIY3ljzQZcW7fNasbHOz7GOz+8g2ldpqFn054uyUj2sbNM\nJV5wSAgi5893+3EzzZnYfXI37g66u1Dvc/VpyOgt0cgyZ2HX0F2oUr5KkdogInIXo2o2APx0/Ce0\nuqVVoVc4cmXd/u7Qd/j6f1/jp8E/cWk4N+HScUQGOXj2IB6Y9wC6NeyG9x58z+Gbfbh66aQscxZK\nmUoV+n10Iy4dR+S7rmZeRZvP26ByucqY0XUGbq1+q8PvdWXdVlWY1cy67QJuWTpORJ4QkX0ikiUi\nd+WzXycROSgih0TkVWeOSeQrGldvjP3P70dpU2nc+smteGfLO0i7llbg+3JOQ/brh8h27TCxXz+n\nLhJhwS1ZWLeJiqZc6XLY/vR2PHrro2jzeRsMWTEExy8ed+i9rqzbIsK67WZOjSyLyK0AzABmAhit\nqrvy2McE4BCABwAkAfgZQG9VPWinTY5SkM9w9E5Th88dRmRsJL6P/x4/Df4JITe5dv5dUmoSpmyf\ngscaP4bWdVq7tG36mzeMLLu6brNmky9xtGafv3weE7dNxMxfZ2L2I7PxWOPHXJoj/Vo6vvjtC5Qv\nXdIeTcEAACAASURBVB5P3/W0S9umvzlas52as6yqf1gPlt+BWgI4rKoJ1n0XA3gUQJ6dZSJfUZil\nghpWa4iFjy/EgTMHEFw12CXHT76SjJV/rMT8vfOxM3En+t/RH3Wq1HFJ2+S9WLeJ8laYmh1QIQDv\nPvAuXm71MkzimltWXM28ik1HN2HB3gVY+cdKtA9pj7Ftx7qkbXKOOy7wCwJge57iBCyFmMin2bvT\n1MSICLsXp9xW47Y8tx9NPoqPtn+E5rWao1G1RqjhVwM1KtZA2VJlUaFMhRv2X3ZwGfp/2x9h9cIw\n6M5B+LbXt7x9KxUG6zaVOEWp2TX9aua5XVXx0tqXcGu1W9G0ZlPc7HczavrVRIUyFfKsxUmpSWg0\ntRHuuPkO9Lm9Dz7s+KHdtsn9Cuwsi0gMgJttNwFQAG+o6sriCDV+/Picx2FhYQgLCyuOwxAVK1cu\nFVS+dHkEVQ7C2iNrMf2X6TiTfgZnLp3BfcH3YVXfVTfs36lBJ5x75RzKlipbtPDkkNjYWMTGxhod\n4wburtus2eQLXFmzszQLt1W/DbtO7sKifYtw5tIZnEk/A7Oakfxa8g37165UG0mjkuBfzr9o4ckh\nRa3ZBXaWVbVDUQLZSARQ1+b5LdZtdtkWXiJv5cqlgmpVqoUxbcY4vH/50uULfQwqvNwdw6ioKOPC\n2HB33WbNJl/gyppd2lQaz939nMP7iwg7ym5Q1Jrtmok2Fvbmv/0MoIGIBItIWQC9Aaxw4XGJPNKA\n6GhEhoYi3fo8e6mgAdHRRsYissW6TWTFmk32OLsaxmMApgKoDiAZwG+q2llEagOYpardrPt1AjAF\nls75bFV9P582eWU1+YycK6utd5qyd2U1+QYvWQ3DpXWbNZt8CWt2yeJozeZNSYiIXMQbOsuuxppN\nRN7KLTclISIiIiLyZewsExERERHZwc4yEREREZEd7CwTEREREdnBzjIRERERkR3uuN01EXmRnKWT\nEhNhCgri0klERB6MNbv4cek4IsqREB+PqR06IOrIEfjh70X5h8fEsPg6gEvHEZE7sWY7h0vHEVGh\nzYmIyCm6gOW2r1FHjmBORISRsYiIKA+s2e7BzjIR5TAnJuYU3Wx+AMxJSUbEISKifLBmuwc7y0SU\nwxQUhPRc29IBmAIDjYhDRET5YM12D3aWiSjHgOhoRIaG5hTf7PlvA6KjjYxFRER5YM12D17gR0TX\nybmyOikJpsBAXlldCLzAj4jcjTW76Byt2ewsExG5CDvLRETeg6thEBERERE5iZ1lIiIiIiI72Fkm\nIiIiIrKDnWUiIiIiIjvYWSYiIiIisoOdZSIiIiIiO9hZJiIiIiKyg51lIiIiIiI72FkmIiIiIrKD\nnWUiIiIiIjvYWSYiIiIisoOdZSIiIiIiO9hZJiIiIiKyg51lIiIiIiI72FkmIiIiIrKDnWUiIiIi\nIjuc6iyLyBMisk9EskTkrnz2Oyoie0Rkt4jsdOaYRERUdKzbRESF4+zI8l4A3QFsLmA/M4AwVW2u\nqi2dPKbHiI2NNTpCoXhTXmYtPt6U15uyepESW7e97fPkTXm9KSvgXXm9KSvgfXkd4VRnWVX/UNXD\nAKSAXcXZY3kib/tAeFNeZi0+3pTXm7J6i5Jct73t8+RNeb0pK+Bdeb0pK+B9eR3hrkKoAGJE5GcR\nGeKmYxIRUdGxbhMRAShd0A4iEgPgZttNsBTRN1R1pYPHaaOqJ0WkBizF94Cqbi18XCIiKgjrNhGR\n64iqOt+IyCYAo1R1lwP7RgJIVdVJdl53PhARkUFUtaDpDR7BVXWbNZuIvJkjNbvAkeVCyPNgIlIR\ngElV00TED0BHAFH2GvGWLxoiIh/gdN1mzSYiX+fs0nGPichxAK0AfCcia6zba4vId9bdbgawVUR2\nA9gOYKWqrnfmuEREVDSs20REheOSaRhERERERL7IY5YFEpFOInJQRA6JyKtG58mPiMwWkVMi8rvR\nWQoiIrfI/7N353E2lu8Dxz/3MdZhMJZsY4xBJWsbQg0tlkoqEjNCkRaSUr8ky6RNSYuEpOz6arFF\nsiuVpbRYElkGQ2RnrDPn+v3xHNNgzmxnec6Zud6v13k58yz3c51x5jr3uZ97MWapMWajMWa9MeYp\nu2PKiDGmoDFmtWshhPWuvpIBzRjjMMasM8bMsTuWzATTQhPGmOLGmM+NMX+63r8N7I7JHWNMDdfv\ndJ3r32OB/rfmKc3ZvqE52/c0Z/tGbs7ZAdGybIxxAFuAW4G9wFrgQRHZbGtgbhhjmgAngUkiUsfu\neDJijCkHlBOR34wxRYFfgHsC9XcLVn9JETlljMkH/AA8JSIBmySMMX2B64AwEWljdzwZMcZsB64T\nkSN2x5IZY8wEYIWIfGqMCQGKiMhxm8PKlCuf7QEaiMhuu+PxBc3ZvqM52/c0Z/tGbs7ZgdKyfCOw\nVUQSROQ88Blwj80xueWaPing37gAIvKPiPzmen4S+BOoaG9UGRORU66nBbEGodr/jc4NY0wloDXw\nsd2xZFFQLDRhjAkDmorIpwAikhwMSdflNmBbbq0ou2jO9hHN2b6lOds3cnvODpT/gIpA2iD3EODJ\nIRgZY6oA9YDV9kaSMdctsl+Bf4BFIrLW7pgy8A7wHAH84XCJYFloIgo4aIz51HWb7CNjTGG7g8qi\nDsB0u4PwMc3ZfqA52yc0Z/tGrs7ZgVJZVj7mup33BdDH1VoRsETEKSL1gUpAA2NMTbtjSo8x5k5g\nv6sVyJD58sGBoLGIXIvVsvKk6/Z0IAoBrgVGueI9Bbxgb0iZM8bkB9oAn9sdiwpumrO9T3O2T+Xq\nnB0oleVEoHKanyu5tikvcPUd+gKYLCKz7Y4nq1y3cJYBLe2OxY3GQBtXn7LpQDNjzCSbY8qQiOxz\n/fsvMBPrdnog2gPsFpGfXT9/gZWIA10r4BfX7zc305ztQ5qzfUZztu/k6pwdKJXltUA1Y0ykMaYA\n8CAQ6KNUg+VbKcAnwCYRec/uQDJjjCltjCnuel4YuB0IyIEtIvKiiFQWkapY79mlIvKQ3XG5Y4wp\n4mqtwvy30MQGe6NKn4jsB3YbY2q4Nt0KbLIxpKzqSO7vggGas31Nc7YPaM72ndyes725gl+OiUiK\nMaYXsBCrAj9eRP60OSy3jDHTgBiglDFmFzD4Qqf2QGOMaQzEAutdfcoEeFFEFtgbmVvlgYmu0akO\n4H8iMt/mmHKLK4CZxlqeOASYGuALTTwFTHXdJtsOdLM5ngwZa9W724BH7Y7F1zRn+47mbJWG5mwf\nyk7ODoip45RSSimllApEgdINQymllFJKqYCjlWWllFJKKaXc0MqyUkoppZRSbmhlWSmllFJKKTe0\nsqyUUkoppZQbWllWSimllFLKDa0sK6WUUkop5YZWlpVSSimllHJDK8tKKaWUUkq5oZVlpZRSSiml\n3NDKssqUMWawMWayDdftYoz53t/XDdQ40jLGNDHG/JnFYwMufqWU72jODow40tKcHdy0spwHGWOe\nNMasNcacMcZ8csm+W4wxu9M5TbJ5jeeMMVuMMUnGmJ3GmNeMMQVyEG62rptJTMuNMaeNMceNMQeM\nMV8aY67wdxzeICIrReTq7Jzis2CUUj5ljJlsjNlnjDlqjNlsjHkkzT7N2T6Owxs0Zwc3rSznTYnA\nUGB8OvsMHv6RGmNGAt2BOKAY0Aq4FZiRwTlefS+6KU+AJ0QkDKgBlADe8eZ1lVLKB14HokSkBNAG\neMUYU9+1T3O2Uj6mleU8SERmicgc4HDa7caYIsB8oIIx5oTr23w51+6CxpiJrm3rjTHXple2MaYa\n8DjQSUTWiIhTRP4E7gdaGmNiXMd9aoz50BgzzxhzAogxxoQbY+YYY44ZY1YB0ZeUfZUxZqEx5pAx\n5k9jTPs0+y4rz83LN67fwVHgS6CW6/wwY8wkV+vFDmPMADev7wNjzPBLts02xvRxPd9hjHnWGPO7\nMeaIMWZ62tYZY0wPY8xWY8xBY8wsY0z5NPucxpjHXa07x4wxLxtjqhpjfnC1KH1mjAlxHXtRa5Ix\n5v+MMX+7/n82GGPaunn9SqkgIyKbROSM68cLleNozdmas5V/aGVZpRKRU1gtCntFpJiIhInIP67d\ndwPTgOLAXGCUm2JuBXaLyC+XlL0HWAXcnmZzR2CoiBQDfgA+BE4BVwCPAA9fOND1obAQmAKUBh4E\nPjTGXOWmvJUZvVZjTGmsD4N1rk0fYLWoVMFK2g8ZY7qlc+pE17UvlFPK9ZqnpjmmPXAHEAXUBbq6\njm0OvAa0A8oDu4DPLin/DqA+0BB4HhgLdAIigNqu13hB2takv4HGrhaYeGCKyfrtSqVUgDPGjDLG\nJAF/AnuB+ZqzNWcr/9DKssqqlSLyrYgIMBmo4+a40sA+N/v2ufZfMFtEVrmenwfuAwaKyBkR2YiV\n5C64C9ghIpPE8jtWK0P79MoTkXNuYhhpjDkM/Ir1gfOssW7/dQBeEJFTIpIAvA10vvRkEVkLHDPG\n3Ora9CCwXEQOpjnsPRHZ72oJmQvUc23vBIwXkd9F5DzQH2hkjKmc5txhIpLkatnZACwUkQQROQF8\ng5WULyMiX4rIftfzz4GtwI1ufgdKqSAjIk8CRYEmwFfA2UxO0ZyN5mzlHVpZVln1T5rnp4BCJv0+\nZgexvoGnp7xr/wVpB6WUAfIBe9JsS0jzPBJoaIw57HocwUpkab+JpzfI5VK9RSRcRCJEpLOIHML6\nMAjBajVIe+2KbsqYhNW3D9e/l44635/m+SmsDziACmlfk4gkAYcuuc6BNM9PX1LW6TRlXcQY85Ax\n5lfXbcQjwDVc/CGnlApyrkrnj1itlo9ncrjm7P9ozlYe0cqyupSnI3CXAhHGmOvTbjTGRGDdplrs\n5lr/AslYHwIXpP32vhurNSDc9SjpuuXYywuxH8RqJYlMsy0SayBkeqYA9xhj6gBXAbOyeJ29aa9h\njAkFSnHxh022uVo5PsIaCFNSREoCG3H19VNK5Toh/Nc/WHO2RXO28hmtLOdBxph8xphCWK0CIcaY\ngsaYfK7d+4FSxpiwzIpJb6OIbMXqszXVGNPAGOMwxlwDfIF1e2qZm/OcWLcWhxhjChtjagJd0hzy\nNVDDGBNnjAkxxuQ3xlxvjLkyq6/bHde1ZwCvGmOKGmMigb5c3vpw4fhE4GfX/i9FJLPboRdMB7oZ\nY+oYYwpi9YVbJSJZaV3JSCjgBA66ft/dcA2CUUoFN2NMGWNMB2NMqOvvuwVWV4ILlVjN2ZqzlY9p\nZTlvegnrVtP/AbGu5wMAROQvrASx3XXrrJybMty2CLj61n2M9W3+BNZo7aVYgyQyOr831oCNfcAn\nrseFMk9iDaR4EOvb/l7gDaBgxi81azEDT2H9HrYD3wFTROTTDI6fiJXcJmX1GiKyBBiI9QGTiDWY\n5MG0h2Qj3rTl/onVX28V1q3Xa8hksIxSKmgIVpeL3VgzGL0J9BGReaA5G83Zyg+M1fffw0KMGY/V\nmX+/iKQ7iMAY8z7WqN0koKuI/ObxhZWyiTGmKTBZRKrYHYtS2aU5W+U1mrOVJ7zVsvwp0MLdTmNM\nKyBaRKoDPYExXrquUn5njMkP9AHG2R2LUjmkOVvlGZqzlae8UlkWkZXAkQwOuQfXrQ8RWQ0U1/kE\nVTByzRF6BGtE93s2h6NUjmjOVnmF5mzlDSF+uk5FLp4iJtG1bX/6hysVmERkM26mAlIqF9GcrXIF\nzdnKG3SAn1JKKaWUUm74q2U5kYvnYqyEm/kQjTGejzhUSimbiEhumCtVc7ZSKk/ISs72Zsuywf2E\n2nOAhwCMMQ2BoxeWeUyPiATFY/DgwTk+t+1nbXlu4XP8nPgzySnJ2T4/6VwS6/au81u8wfS71Vhz\nT7zBFKtI0NUZNWdn4zFo6SC6zOzCt39/y+nzp7N9fnJKMqt2r8LpdObK938wxRps8QZTrMEWb1Z5\npWXZGDMNiMGaGH0XMBgoYOVQ+UhE5htjWhtj/saahqibN64bzGZ2mOnR+UXyF6F++XSXnOfomaOU\nKFTCo/KVUrmX5uzsi28W79H5+Rz5aFCpQbr7Tpw9QeH8hQlx+Otmr1IqO7zylykinbJwTK/MjsmN\nEo4mEFkiMvMDvejh2Q9z8txJRrYayZWlPV4sSSmVy2jOds+OnP3xuo8Z+8tYRrYaye3Rt/v12kqp\nzOkAPw/ExMS43Xcg6QD3z7if+2bch1Oc/gsK+F+7/3Fn9Ttp/Elj4pfHczbZWtkzo3gDjcbqO8EU\nbzDFqgJfRu+ns8lneXHJi1w/7np2H/N0NePsebrh07x9x9s8+vWjxH4Vy/6TVo+XYHr/B1OsEFzx\nBlOsEHzxZoVXVvDzJmOMBFpM2fX5xs/p/U1vutXrxpCYIRQMyc7qnt6z5/geen/Tm62HtvJZu8+o\nVVaXnlfKl4wxSO4Y4JdluSFn/7rvV7rM6kLVklUZc9cYyhV1t2K0b506f4qXV7zMp799yti7xtL2\nqra2xKFUXpHVnK2VZS86k3yGXvN78f2u75nYdiINKzW0OyREhEm/TyK0QCjtarazO5ygVaVKFRIS\nEuwOQwWIyMhIdu7cedl2rSwHFxFh1NpRvLziZd6+423i6sRhjP3/fav2rGLx9sW8dPNLdocStDRn\nq7Q8zdlaWfaizQc388bKNxjZaiTFChazOxzlRa4/KLvDUAHC3ftBK8vB5fT503Sb3Y1Xm79KdHi0\n3eEoL9KcrdLyNGdrZVmpLNDEq9LSyvJ/NGerQKQ5W6Xlac7WAX552Knzp+wOQSmlVBZpzlbKHlpZ\nzqMOnz7MVR9cxfcJ39sdilJKqUyICLdPvp1Ra0bZHYpSeY5WlnNARBiyfAiTfp9kdyg5Fl44nPFt\nxnP/jPuZt2We3eEopZRPfb3la5765im7w8gxYwxT7p3Cu6vfJX55vHYxUMqPtLKcTSJC/yX9+erP\nr2hZraXd4Xjk9ujbmdtxLg/PeZjZm2fbHY7KoWLFihEWFkZYWBj58uWjSJEiqdumT5/ulxhWrVpF\nq1atKFGiBKVLl6ZRo0ZMnjzZL9dWKjNfbvqSR+Y8QlyduMv2nToFCxfCW29B587QqBHUqAGlS0NY\nGJQtC5Urw403QocO0L8/fPUV7N3r/9cRVTKKld1W8tXmrxiwdIBWmIOU5uzgo2trZoOI8Pyi51m6\ncynLuiyjVJFSPrtWwo4dTBg4EGdiIo6KFek6dCiRUVE+KXt8r3E88nUPHMbB3Vfe7ZVrKP85ceJE\n6vOqVasyfvx4mjVr5vb4lJQU8uXL57Xrr1y5klatWhEfH8+0adMoWbIk69at4+2336Zz587ZLs/p\ndOJw6Pd45R1fbPqCXvN7sSB2AfXL1wfgyBGYNg3mzIEff4R69eC666BZM+jZE8qUgfBwKFAAzp6F\n06etyvHOnbB1K4wfDz16QIkScM890LYtVKqwg8lD/JOzJ7z4CV1WdCPEEcLLzV72yjWU/2jODkIi\nElAPK6TAFL88XuqMriOHTx326XV2bt8uz0ZHy0kQATkJ8mx0tOzcvt1nZc/+caYs2rbIC9HnToH8\nvkyrSpUqsmTJkou2vfTSS9KhQwfp2LGjhIWFycSJEyUuLk7i4+NTj1m8eLFUqVIl9ec9e/bIvffe\nK2XKlJGqVavKqFGj3F6zYcOG0rdvX7f7P/74Y4mJiUn9OTk5WYwxkpCQICIicXFx8uSTT0rLli2l\naNGi8uqrr0rFihUvKmPGjBly7bXXioiI0+mUV199VaKjo6VMmTLSsWNHOXr0aBZ+O97j7v3g2m57\nHvXnI5D/NuZvmS9l3yorv+77VURE1q4V6dxZpHhxkQ4dRL74QiSnbx2nU+S330SGDBG5+uozUjTf\nXunPYNlFJb/k7J83rJFxv4zzuPzcKpDfl2lpzvYPT3N2Lv8q4D2HTx9m6Y6lLIxbSMnCJX16rQkD\nBxK/bRuhrp9Dgfht25gwcKDPyv511BfcVvU2j8tXgWnWrFnExcVx7NgxHnjggXSPubAYg4hw1113\n0aBBA/bt28eiRYsYPnw4y5Ytu+yckydPsmbNGu6///4Mr3/pQg+X/jx9+nTi4+M5ceIE/fr1o0CB\nAqxYseKi/XFx1i30ESNG8M0337By5Ur27NlD0aJF6d27d+a/BJWniAhjfxnLrA6zcByoR9u2Vgtw\nnTpW6/Bnn8H990Px4jkr3xioWxcGD4YO1z7C4pQWHKc0dfmdh/mM9tuK+DRnf/36e3S/trvH5avA\npDk7sGhlOYvCC4ezrMsyrih6hc+v5UxMTE2MF4QCTi90kvNl2XndkOVDMPHmsseQ5UOydLy747yh\nSZMmtG7dGoBChQpleOyPP/7IiRMn+L//+z/y5ctH1apVefjhh/nss88uO/bw4cMAlC9fPlvxyCV9\nLe+9915uvPFGAAoUKECHDh2YNm0aAEePHuXbb7/lwQcfBGDs2LG89tprlCtXjgIFCjBw4EA+//zz\nbF1f5X7GGD65YyafDm3EHXdATIxVSe7Xz+pm4U3OxEQasJ4P6M0uKnMDa2nLt0z79nE2bvS8bM3Z\nvuFpzvZl3tacHVi0z3I2+GsZVEfFiiTBRQkyCXBUqBDQZed1Q2KGMCRmiM+O90RERESWj921axcJ\nCQmEh4cDVpJ0Op3p9qm7cMy+ffuoWrWq1+Lr1KkTzZs358MPP+TLL7+kYcOGqcl9165d3H333al9\n5EQEh8PBgQMHKFu2bI5jULnL559Dnz6Gtm3hr79y3oKcFWnzalGS6MfbdGUUcZXHERPTmPbtIT4+\nZ5V0zdm+ozlbc3ZWactyAOo6dCiDo6NJcv2cBAyOjqbr0KF+LXvx9sWcPHfS42sq+136RS80NJRT\np/5b4GDfvn2pzyMiIqhRowaHDx/m8OHDHDlyhGPHjjFr1qzLyi1atCgNGjTgyy+/dHvt9K6V2S2+\n2rVrU65cORYsWMD06dPp1KnTRfEtWrTooviSkpKCJukq3zp1Ch5+GAYOtCrMH37o24oypJ9X34iu\nyNgvGrN5szVQ8JprrIGB2Z3AIjs5e/PBzWw+uNmDV6IChebswKKVZTfOJp+17dqRUVH0XrSI4bGx\nDG7WjOGxsfRetMgrI6uzU/aMjTOI/SqWFGeKx9dVgaVevXrMmzePo0ePsm/fPkaOHJm6r1GjRhQo\nUIARI0Zw9uxZUlJS2LBhA+vWrUu3rDfffJOPP/6Yd999lyNHjgDw66+/EhsbC0DdunX5448/2Lhx\nI6dPn+bll7M2er9Tp0688847rFq1inbt2qVu79mzJ/3792f37t0AHDhwgLlz5+bo96Byj2RnMhv/\nTKZhQ2sGi59/hsaN/XPtjPJqqVLw7ruwaBGMHg233grbtnmn7EutTVzLndPu5OCpg158dSoQaM62\nWVZGAfrzQQCMYD1z/ozcNP4m+fbvb+0OxVZnk89KzIQY6fdtP7tDsV0gvC+zIioqKt2R1d26dbto\n2+nTp6Vdu3YSFhYm9erVk3feeUeioqJS9+/du1c6dOgg5cqVk/DwcGncuLEsX77c7XVXr14tLVq0\nkOLFi0upUqWkUaNGMnXq1NT9Q4cOlVKlSklkZKRMmTJFHA5H6sjqzp07XzTK+4IdO3aIw+GQe++9\n96LtTqdThg8fLtWrV5ewsDCpXr26DBo0KOu/JC9w935AZ8Owzf2vfyBFSpyQMWOsmSoC0fnzIsOH\ni5QuLfLJJ76J84VFL0jTT5rKmfNnvF94EAmU92VmNGf7h6c520h27wn5mDFG7IxJROgxtweHTx/m\niwe+wGHyduP74dOHafhxQwY0HUCXel3sDsc2xpjLBjiovMvd+8G13T+DGwKE3TkboOfrKxj/Wi1m\nflGAu1sUszWWrNiwATp2hKuvho8+suZr9hanOGk3ox2lCpdiXJtx3is4yGjOVml5mrPzdk0wHR+s\n+YA1iWuYdO+kPF9RBmsWkJkdZtJvUT/W7Uv/lo5SStmlz+CdfPxmNF/NOxYUFWWAWrVg7Vq44gq4\n/nr44w/vle0wDia2ncgPu39g3C95t7KslDdpbTCNH3b9wCvfv8KsB2dRtEBRu8MJGNeUvYaP7/6Y\n8ynn7Q5FKaVSvTzsBKNGwbiZf9Lm5pyP7LdDoUIwciS8/LLVj3nKFO+VXaxgMWZ2mEn+fPm9V6hS\neZh2w3AREZpNbMZzNz3HnTXu9Pv1VWDTW3oqLe2G8R+7cvbIkTDotcN0enscozr9n9+v703r18O9\n91qLpLz+OuT2lYP9QXO2SsvTnK2V5TTOJp+lYEhBW66tApsmXpWWVpb/Y0fO/uQTq0V2ydIUIqsI\nIY7gXzLg0CG47z4oVQomT4bQS1ciUdmiOVulpX2WvUgrykopFdgWLoQXX7T+ja6aL1dUlMGqJC9c\nCGFh1mqDBw7YHZFS6gKtLKscS3Ym2x2CUioP+f13iIuDL76AGjXsjsb7ChaETz+FVq2gSRPYscO7\n5WvOVipntLLsgYQdO4iPi2Nws2bEx8WR4O3MFsC+2foNbaa3wSlOu0NRSuUB+/bB3XdbfZWbNMlZ\nGcGQs42xupg89RQ0beq9mTL+OfkPdUbX4fDpw94pUKk8JM/2WRYRxv86ns51Oueo+0XCjh2MvP12\n4rdtI5T/liD11kp7ge58ynmafNqE2NqxPNXgKbvD8Tnt/6bS0j7L//FHzj5/3poxolztTbz3RknK\nFyuf7TKCMWf/739WpXnePGuKOU89veBpEk8kMqPdjMuWK85tNGertHJln+VHHwU3qzR6zdhfxjL6\n59E5Pn/CwIGpSRcgFIjfto0JAwd6Jb5Alz9ffqbeN5Wh3w1lw4ENdoejlMrFXngBUkKOs7RyDGdT\nzuaojGDM2R06wLhxcOed8NNPnpf3xm1vsPngZib+PtHzwpTKQwKyshwZaU2jc+ON1qjnpCTvlr/l\n0BZeWvoSU++bmuNBfc7ERC4drBwKOPfu9Ti+YFEtvBpv3vYmnb7sxJnkM0FxizO3qlKlCkWK4Rjm\n6gAAIABJREFUFKF48eKEh4fTpEkTxo4dm6WWlYSEBBwOB06ndqlRgefzz+HLr5wcbNmS91q9Q5US\nVXJUTrDm7DZtYOJEuOce+P57z8oqFFKIafdN47lFz/H34b+B4Oiakhtpzg4yWVkT258PXOt3JyeL\nzJsnctddIuHhIr16iaxfn+7S3tlyPuW83DjuRnl/1fselTMkNlZOgkiax0mQIbGxngcZRJxOp7Sb\n0U6emtpLno2OTv2dnAR5Njpadm7fbneIXoGbdeVFRHZu3y5DYmNlUEyMDImNzfZr9vR8EZEqVarI\n0qVLRUTk+PHjMnfuXImKipJu3bpleu6OHTvE4XBIcnJytq+bV7l7P7i2255H/fnI6G/DUzt2iJQu\nLdLxvbel/Yz24nQ6c1xWsOfsRYtEypQR+fFHz8t656d3pNmEZrJz+/Zcm7d9mbO9UYbmbP/yNGfb\nnmgvCyidF5SQIDJwoEiFCiKNG4tMnixy+nT2f1kiIi8vf1lun3S7pDhTclaAS25OMtl16NQh6d+h\nXVB/EGXG3R+ap+8Db72PqlSpIkuWLLlo25o1a8ThcMjGjRtl3rx5Ur9+fQkLC5PKlSvLkCFDUo+r\nXLmyOBwOKVq0qBQrVkxWrVol27Ztk+bNm0upUqWkTJkyEhsbK8eOHctWTLmZVpZ9X1lOThZp0kSk\nx/9tkYpvV5SDSQc9Ki835Oz5860K89q1npWT4kyRvw/9HfRfIDLiq5ztrTI0Z/tXQFSWgZbAZmAL\n8H/p7L8FOAqscz1eyqAsty/23DmRr74SadHCam145hmRv/7K+i/L6XRKjzk9ZM+xPVk/KQOp3yyb\nNcvxt9PcYlBMzEUJ98JjULNmdofmFe7el55+2Hjrwyq9xCtiJdUxY8bIihUrZMOGDSIisn79eilX\nrpzMnj1bRER27twpDofjola7v//+WxYvXiznz5+XgwcPyi233CJ9+/bNVky5WbBXlv2Vsz3x+usi\nMTEi7/zwvnyz9RuvlJkbcvbs2SJly4r8/rvnZeXmvO2rnO2tMjRn+5enOdvj2dyNMQ7gA+BWYC+w\n1hgzW0Q2X3LodyLSxpNr5c9v9WW+917Yvt0a+NC0KVxzDTz2GLRtCwUKZBgrH939kSchXCQyKorB\nU6Z4rbxg5qhYkSS4qE9gEuCoUMGmiPzD036Qvu5HWaFCBQ4fPszNN9+cuq1WrVo8+OCDrFixgjZt\n/vuTFJHUEfLR0dFER0cDUKpUKfr27cvLL7/slZiUvfyZs3Nq3ToYMQJ+/hkqV+7ttXJzQ85u0wZO\nn7bmYl6xAqpVy3lZeTFveyPn+jJva84OTN4Y4HcjsFVEEkTkPPAZcE86x3l1npqqVeH112H3bujZ\nE8aMgcqVoX9/70/krjLXdehQBkdHc2Es5oVpmboOHWpnWD534cMmrex82Hh6fmYSExMJDw9nzZo1\nNG/enLJly1KiRAnGjh3LwYMH3Z534MABOnbsSKVKlShRogRxcXEZHq+Cii05O6vOn4euXa3KcuXK\ndkQQ+Dp0gEGD4I47wJP6WV7M297Iub7M25qzA5M3KssVgd1pft7j2napRsaY34wx84wxNb1wXcBq\nSe7QAZYutb5lnztnzaLRsiXMmgXJumCRX0RGRdF70SKGx8YyMOYW3ujYIaDnL/UWTz9sfPlhtXbt\nWvbu3UuTJk3o1KkTbdu2JTExkaNHj9KzZ88Lt9DTnW/1xRdfxOFwsHHjRo4ePcqUKVNSj1dBz9ac\nnZm33oJKlSA21l9XDE49e0KPHlaF+ciRnJWRNm8PuKUpw2Njc33e9kbO9VXe1pwduDzuhpFFvwCV\nReSUMaYVMAtwu1jpkCFDUp/HxMQQExOTpYtceSW8/Ta88oq1HOrw4dCrFzzyCHTvDhERHr0GlYkL\ntzgHLRvE0TNHc3XCvSD1w2bgQJx79+KoUIHeQ4dm+bV7en56Tpw4wYoVK3j66afp3Lkz11xzDSdP\nnqRkyZLkz5+fNWvWMG3aNFq0aAFAmTJlcDgcbNu2jerVq6eWUaJECYoVK0ZiYiJvvfVWjuPJzZYv\nX87y5cvtDsMX/JKzL7Vly3/dL3L5mhle8cILsH+/1TVxwQIoVCj7ZURGRXFTfBdeXPoiPz0ygRCH\nv6oF9vBGzvV23tac7T85ztlZ6dic0QNoCCxI8/MLpDNg5JJzdgDhbvblsPt2+l77Yo7EPXJEwsNF\n2rSxpqPT2VZ869CpQ1J+eHn5PuF7u0PxGm+/L72tSpUqUqRIEQkLC5MSJUrITTfdJKNHj04dAPLl\nl19KZGSkhIWFyd133y29e/eWzp07p54/ePBgKVOmjJQsWVJWr14tGzdulOuuu06KFSsm9evXlxEj\nRkhERIRdLy/guHs/EAQD/AI1Zzud1oC+XgO3yaJti7xSZl6QkiLSvr3IAw9Yz3PC6XTKrRNvlWEr\nh3k3OBtpztacnZanOdvj5a6NMfmAv7AGi+wD1gAdReTPNMdcISL7Xc9vBGaISBU35YmnMV2w+eBm\nmnzShF8e/YXS+SP57DOrb/O//1q3rx5+GMpnf9VUlQVf/fkV/Zf05/fHfqdQSA6aOwKMLp2q0grm\n5a4DNWdPmAAjP3ByrPPVjGj5Fm2utGVsYVA6cwZatLCWxH777ZyVsePIDm4YdwM/PvIjNUq5vYkQ\nNDRnq7RsX+5aRFKAXsBCYCPwmYj8aYzpaYx51HVYO2PMBmPMr8C7QAdPr5sZpzjpMbcHg28ZTGSJ\nSEJDre4Ya9fCl19CQgLUrAnt28PixaAL4XjXfVffR+2ytXn1u1ftDkUplUYg5uzjx+HFF6FWtw+5\nrmI9rShnU6FC1hidefNg9OiclRFVMoqBNw+k59c9tZKp1CU8bln2Nm+1UoxeO5rJf0zm+27fk8+R\nL91jjh+HqVOt1uZTp+DRR61R2GXKeHx5Bew9sZe6Y+ry0yM/US3cg/mNAoC2Uqi0grll2du8kbOf\nfx42Jxxi1fVXs/7x9VxR9AovRZe3bNsGTZrAp59ag9yzK8WZQsPxDenbsC+danfyfoB+pDlbpeVp\nzs6VleULlbTlXZZzTdlrMj1eBFavtirNs2ZB69bWvM1Nm+ogE0/9dfAvapSqke7o3WCiiVelpZXl\n/3ias7dsgZtuEiq90Iqnbn2Ah+s/7MXo8p4ffrAG/C1ZArVrZ//8Xcd2UaZIGQrnL+z94PxIc7ZK\nSyvL6fhm6zf8+s+vvNj0xWyfe+QITJpkVZyNsabneeghKFnSo5BUkNPEq9LSyvJ/PM3Zd98NtW44\nTGLtp5nYdmLQf7EOBNOmwYABsGZN3r1TqjlbpaWVZR8Rge++g7FjYf58a3XAxx6DBg20tTkv0sSr\n0tLK8n88ydmLF8Pjj8PGjRmvvqqy78UXYeVK63ecF3+3mrNVWlpZ9oN//7X6gH30EYSGWpXm2FgI\nC7M7MuUvmnhVWlpZ/k9Oc7YI3HCD1V/5gQd8EFge53Ra3THKlrU+u/JaI4/mbJWW7bNh5AVlylgJ\nfcsWa6GTJUsgMtIaELhund3RBRen6LQjSilr4SgRaNfO7khyJ4cDpkyBVatg1Kicl6M5WymtLGeL\nwwG3324l+U2brArzffdZy2uPHw9Jly4Wry6y69gubhx3I2eSz9gdilLKRufPW31q33jDyqvKN4oV\nswatDx1qdSvMLqc4aTS+EVsObfF+cEoFkVyRpkSE+Vvnp9vEnrBjB/FxcQxu1oz4uDgSduzwyjXL\nl7eS/bZtMGQIzJ4NlStD796wYYNXLpHrVC5emcrFKzNs5TC7Q1EBICEhAYfDgVMnOc9zPvkEipY5\nTJOY0+nu91Xezouio61B6x06wO7d2TvXYRx0uKYDT8x7Qrs0qLyds7OyzJ8/H+Rgicqpf0yVemPq\nyfmU8xdt37l9uzwbHS0nrbt9chLk2eho2bl9e7avkRW7dokMGiRSoYJI48YikyaJnD7tk0sFrYSj\nCVJqWCnZdnib3aFkS07el/5UpUoVWbJkid1hZMvOnTvF4XBISgZr9E6dOlWuv/56KVq0qFSoUEFa\nt24tK1eu9GOU6XP3fiAIlrv29iO7fxunT4tcUf6cFO91m+w+tvuy/f7O23nFsGEi11+f/c+k8ynn\npfaHtWXGhhm+CcxHNGd7X17O2UHfsnzszDGeW/Qco+8cTYgj5KJ9EwYOJH7bNkJdP4cC8du2MWHg\nQJ/EEhEB8fHW6oD9+lkLnkREwLPPwl9/+eSSQady8cr0u6kffRb0sTsUlU0pKSl+vd6IESN45pln\neOmllzhw4AC7du3iySefZO7cudkuy9+xK/fGjROc5X5mYMeWVAqrdNl+f+ftvOK55yAqyrr7mR0h\njhBGtR7FMwuf4eS5k74JTvmE5mzvCfrKcvyKeFpVa0XDSg0v2+dMTExNuBeEAs69e30aU0iINdXc\nggXWYicFCsDNN0Pz5jBjBpw759PLB7xnGj3DlkNbmPtX9v+AVPZ9/fXX1K9fn5IlS9KkSRPWr1+f\num/YsGFUq1aNsLAwatWqxaxZs1L3TZw4kSZNmvDMM89QunRp4uPjmThxIk2bNuW5554jPDyc6Oho\nFixYkHrO8ePH6d69OxUqVCAiIoKBAwem3r51Op3069ePMmXKUK1aNebNm+c25uPHjzN48GA+/PBD\n7rnnHgoXLky+fPlo3bo1b7zxBgDdunVj0KBBqeesWLGCiIiI1J+joqJ48803qVu3LkWLFuXNN9+k\nffv2F12nT58+PP3005nGrrzj7FmIf+00RW9/l6caPJXuMXbl7dzOGGtszcqV1r/Z0TSyKc2qNGPo\niqG+CU5dRHN2AObsrDQ/+/NBNm6drN+/Xsq8WUYOnDyQ7v4hsbGpt/IkzS29IbGxWb6Gt5w9K/K/\n/4k0ayZyxRUiL7wgsi24eiJ41U+7f5L1+9fbHUaWZed9aQd3t/TWrVsnZcuWlbVr14rT6ZRJkyZJ\nlSpV5Ny5cyIi8sUXX8g///wjIiIzZsyQ0NDQ1J8nTJggISEhMmrUKElJSZEzZ87IhAkTJH/+/DJ+\n/HhxOp0yevRoqVChQur12rZtK48//ricPn1a/v33X2nQoIF89NFHIiIyevRoufrqqyUxMVGOHDki\nzZo1c3tLb8GCBZI/f/4Mb/d17dpVBg4cmPrz8uXLJSIi4qLfSf369SUxMVHOnDkjCQkJEhoaKidP\nnhQRkZSUFClfvrysWbMm09gv5e79gHbDyNB7H5yVQlcvlWU7lrk9JpDydm60aZNI6dIiP/+cvfP2\nndgnX//1tW+C8gHN2Zqz0/I0Z9ueaC8LKBtv8B5zesgHqz9wuz9Q+75t3izyzDNWwmrRQmTmTJHz\n5zM/T9knK+/LSz7fc/TIKXeJ9/HHH5dBgwZdtO3KK6+U7777Lt1y6tWrJ3PmzBERK/FGRkZetH/C\nhAlSvXr11J9PnTolxhjZv3+/7N+/XwoWLChnzpxJ3T99+nRp3ry5iIg0b95cxo4dm7pv4cKFbhPv\n1KlTpXz58hm+5qwk3gkTJlx0TtOmTWXy5Mmp169WrZqIiPzzzz/pxt6sWbN0r62V5ezn7LNnRUqV\nPyG3vfZShscFat7OTT7/XKRKFZHDh+2OxHf8lbNzmrc1ZwdXzr64k2+Q+fDODzG4n0s6MiqK3osW\nMXzgQJx79+KoUIHeQ4cSGRXlxygvd+WV8Pbb8Oqr8Pnn1tzNvXrBI49A9+5WP2cVfCQA79gnJCQw\nadIkRo4cCVhfjs+fP89e1y3tSZMm8c4777Bz504AkpKSOHjwYOr5Eem8GcuVK5f6vHDhwgCcPHmS\nQ4cOcf78ecqXL596LRGhcuXKAOzdu/ei8iIjI93GXapUKQ4ePIjT6cThwdxilSpd3Ce2Y8eOTJ8+\nnbi4OKZPn06nTp0A2LVrV4axK89NmgTX1S7KnOdeyvC4QM3buUm7dvDDD9ClizW1XF6dvk9ztubs\nrArqyvKlA/rSExkVxeApU/wQTfYVKgSdO1uP9eutVZbq1YPGja1VAlu0gHz57I5SBbOIiAgGDBhA\n//79L9u3a9cuHn30UZYtW0ajRo0AqF+//oXWQsBa3Sg71ypUqBCHDh1K97zy5cu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RAAAg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\nn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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "The expression for the first polynomial is y=-0.129\n", "The expression for the second polynomial is y=-0.231+0.598x^1\n", "The expression for the third polynomial is y=0.085-0.781x^1+1.584x^2-0.097x^3\n" ] } ], "source": [ "regression_example_draw(degree1=0,degree2=1,degree3=3, ifprint=True)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "> **Remark**\n", "> - In the above image, we try to predict the true sinusoidal curve hidden in the data.\n", "> - $0\\text{th}$ order means $\\phi(x_n)=1$\n", "> - $1\\text{th}$ order means $\\phi(x_n)=[1, x_n]^T$\n", "> - $3\\text{th}$ order means $\\phi(x_n)=[1, x_n, x_n^2, x_n^3]^T$\n", "> - As the degree M increases, the learned curve matches the observed data better\n", "> - In real world, the model for data label is usually $$t = z + \\epsilon$$ The true state/label is $z$, but it's corrupted by some noise $\\epsilon$ so what we observe is actually $z + \\epsilon$. Many machine tasks are essentially to infer $z$ of new observations based on $z + \\epsilon$ of training dataset." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Basis Functions\n", "- For feature basis function, we used polynomial functions for example above. \n", "- In fact, we have multiple choices for basis function $\\phi_j(\\vec{x})$.\n", "- Different basis functions will produce different features, thus may have different performances in prediction." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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iCxaIFBW53FxedbW8tWePTE9Pl5AlS+S4tWvloe3b5feiIqlxwfhuoGx92eFn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SyF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\nNLMFufx74fe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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "basis_function_plot()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Linear Regression (General Case)\n", "- The function $y(\\vec{x}_n, \\vec{w})$ is linear in parameters $\\vec{w}$.\n", " - **Goal:** Find the best value for the weights $\\vec{w}$.\n", " - For simplicity, add a **bias term** $\\phi_0(\\vec{x}_n) = 1$.\n", "$$\n", "\\begin{align}\n", "y(\\vec{x}_n, \\vec{w})\n", "&= w_0 \\phi_0(\\vec{x}_n)+w_1 \\phi_1(\\vec{x}_n)+ w_2 \\phi_2(\\vec{x}_n)+\\dots +w_{M-1} \\phi_{M-1}(\\vec{x}_n) \\\\\n", "&= \\sum_{j=0}^{M-1} w_j \\phi_j(\\vec{x}_n) \\\\\n", "&= \\vec{w}^T \\phi(\\vec{x}_n)\n", "\\end{align}\n", "$$\n", "of which $\\phi(\\vec{x}_n) = [\\phi_0(\\vec{x}_n),\\phi_1(\\vec{x}_n),\\phi_2(\\vec{x}_n), \\dots, \\phi_{M-1}(\\vec{x}_n)]^T$\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Least Squares\n", "### Least Squares: Objective Function\n", "- We will find the solution $\\vec{w}$ to linear regression by minimizing a cost/objective function.\n", "- When the objective function is sum of squared errors (sum differences between target $t$ and prediction $y$ over entire training data), this approach is also called **least squares**.\n", "- The objective function is \n", "$$\n", "E(\\vec{w}) \n", "= \\frac12 \\sum_{n=1}^N (y(\\vec{x}_n, \\vec{w}) - t_n)^2\n", "= \\frac12 \\sum_{n=1}^N \\left( \\sum_{j=0}^{M-1} w_j\\phi_j(\\vec{x}_n) - t_n \\right)^2\n", "= \\frac12 \\sum_{n=1}^N \\left( \\vec{w}^T \\phi(\\vec{x}_n) - t_n \\right)^2\n", "$$\n", "
\n", "
(Minizing the objective function is equivalent to minimizing sum of squares of green segments.)
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### How to Minimize the Objective Function?\n", "- We will solve the least square problem in two approaches:\n", " - **Gradient Descent** Method: approach the solution step by step. We will show two ways when iterate:\n", " - **Batch Gradient Descent**\n", " - **Stochastic Gradient Descent**\n", " - **Closed Form** Solution" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Method I: Gradient Descent—Gradient Calculation\n", "- To minimize the objective function, take derivative w.r.t coefficient vector $\\vec{w}$:\n", "$$\n", "\\begin{align}\n", "\\nabla_\\vec{w} E(\\vec{w})\n", "&= \\frac{\\partial}{\\partial \\vec{w}} \\left[ \\frac12 \\sum_{n=1}^N \\left( \\vec{w}^T \\phi(\\vec{x}_n) - t_n \\right)^2 \\right] \\\\\n", "&= \\sum_{n=1}^N \\frac{\\partial}{\\partial \\vec{w}} \\left[ \\frac12 \\left( \\vec{w}^T\\phi(x_n) - t_n \\right)^2 \\right] \\\\\n", "\\text{Applying chain rule:}\n", "&= \\sum_{n=1}^N \\left[ \\frac12 \\cdot 2 \\cdot \\left( \\vec{w}^T\\phi(x_n) - t_n \\right) \\cdot \\frac{\\partial}{\\partial \\vec{w}} \\vec{w}^T\\phi(x_n) \\right] \\\\\n", "&= \\sum_{n=1}^N \\left( \\vec{w}^T\\phi(x_n) - t_n \\right)\\phi(x_n)\n", "\\end{align}\n", "$$\n", "- Since we are taking derivative of a scalar $E(\\vec{w})$ w.r.t a vector $\\vec{w}$, the derivative $\\nabla_\\vec{w} E(\\vec{w})$ will be a vector.\n", "- For details about matrix/vector derivative, please refer to appendix attached in the end of the slide." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Method I-1: Gradient Descent—Batch Gradient Descent\n", "- **Input**: Given dataset $\\{(\\vec{x}_n, t_n)\\}_{n=1}^N$\n", "- **Initialize**: $\\vec{w}_0$, learning rate $\\eta$\n", "- **Repeat** until convergence:\n", " - $\\nabla_\\vec{w} E(\\vec{w}_\\text{old}) = \\sum_{n=1}^N \\left( \\vec{w}_\\text{old}^T\\phi(\\vec{x}_n) - t_n \\right)\\phi(\\vec{x}_n)$\n", " - $\\vec{w}_\\text{new} = \\vec{w}_\\text{old}-\\eta \\nabla_\\vec{w} E(\\vec{w}_\\text{old})$\n", "- **End**\n", "- **Output**: $\\vec{w}_\\text{final}$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Method I-2: Gradient Descent—Stochastic Gradient Descent\n", "**Main Idea:** Instead of computing batch gradient (over entire training data), just compute gradient for individual training sample and update.\n", "\n", "- **Input**: Given dataset $\\{(\\vec{x}_n, t_n)\\}_{n=1}^N$\n", "- **Initialize**: $\\vec{w}_0$, learning rate $\\eta$\n", "- **Repeat** until convergence:\n", " - Random shuffle $\\{(\\vec{x}_n, t_n)\\}_{n=1}^N$\n", " - **For** $n=1,\\dots,N$ **do**:\n", " - $\\nabla_{\\vec{w}}E(\\vec{w}_\\text{old} | \\vec{x}_n) = \\left( \\vec{w}_\\text{old}^T\\phi(\\vec{x}_n) - t_n \\right)\\phi(\\vec{x}_n)$\n", " - $\\vec{w}_\\text{new} = \\vec{w}_\\text{old}-\\eta \\nabla_{\\vec{w}}E(\\vec{w}_\\text{old} | \\vec{x}_n)$\n", " - **End**\n", "- **End**\n", "- **Output**: $\\vec{w}_\\text{final}$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "> **Remark **\n", "> - The derivative is computed as followimg:\n", " $$\n", " \\begin{align}\n", " \\nabla_\\vec{w}E(\\vec{w} | \\vec{x}_n)\n", " &= \\frac{\\partial}{\\partial \\vec{w}} \\left[ \\frac12 \\left( \\vec{w}^T \\phi(\\vec{x}_n) - t_n \\right)^2 \\right] \\\\\n", " &= \\left( \\vec{w}^T\\phi(\\vec{x}_n) - t_n \\right)\\phi(\\vec{x}_n)\n", " \\end{align}\n", " $$\n", "\n", "> - This derivative w.r.t individual sample is just one summand of gradient in batch gradient descent.\n", "> - Sometimes, stochastic gradient descent has faster convergence than batch gradient descent." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Method II: Closed Form Solution\n", "**Main Idea:** Compute gradient and set to gradient to zero, solving in closed form.\n", "- Objective Function $E(\\vec{w}) = \\frac12 \\sum_{n=1}^N \\left( \\vec{w}^T \\phi(\\vec{x}_n) - t_n \\right)^2 = \\frac12 \\sum_{n=1}^N \\left( \\phi(\\vec{x}_n)^T \\vec{w} - t_n \\right)^2$\n", "- Let $\\vec{e} = [\\phi(\\vec{x}_1)^T \\vec{w} - t_1,\\quad \\phi(\\vec{x}_2)^T \\vec{w} - t_2, \\quad \\dots ,\\quad \\phi(\\vec{x}_N)^T \\vec{w} - t_N]^T$, we have $$E(\\vec{w}) = \\frac12 \\vec{e}^T \\vec{e} = \\frac12 \\parallel \\vec{e}\\ \\parallel^2$$\n", "- Look at $\\vec{e}$:\n", "$$\n", "\\vec{e} = \n", "\\begin{bmatrix}\n", "\\phi(\\vec{x}_1)^T\\\\ \n", "\\vdots\\\\\n", "\\phi(\\vec{x}_N)^T\n", "\\end{bmatrix}\n", "\\vec{w}-\n", "\\begin{bmatrix}\n", "t_1\\\\ \n", "\\vdots\\\\\n", "t_N\n", "\\end{bmatrix}\n", "\\triangleq \\Phi \\vec{w}-\\vec{t}\n", "$$\n", "Here $\\Phi \\in \\R^{N \\times M}$ is called **design matrix**. Each row represents one sample. Each column represents one feature\n", "$$\\Phi = \\begin{bmatrix}\n", "\\phi(\\vec{x}_1)^T\\\\ \n", "\\phi(\\vec{x}_2)^T\\\\ \n", "\\vdots\\\\\n", "\\phi(\\vec{x}_N)^T\n", "\\end{bmatrix}\n", "= \\begin{bmatrix}\n", "\\phi_0(\\vec{x}_1) & \\phi_1(\\vec{x}_1) & \\cdots & \\phi_{M-1}(\\vec{x}_1) \\\\\n", "\\phi_0(\\vec{x}_2) & \\phi_1(\\vec{x}_2) & \\cdots & \\phi_{M-1}(\\vec{x}_2) \\\\\n", "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n", "\\phi_0(\\vec{x}_N) & \\phi_1(\\vec{x}_N) & \\cdots & \\phi_{M-1}(\\vec{x}_N) \\\\\n", "\\end{bmatrix}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "- From $E(\\vec{w}) = \\frac12 \\left \\| \\vec{e} \\right \\|^2 = \\frac12 \\vec{e}^T \\vec{e}$ and $\\vec{e}=\\Phi \\vec{w}-\\vec{t}$, we have\n", "$$\n", "\\begin{align}\n", "E(\\vec{w}) \n", "& = \\frac12 \\left \\| \\Phi \\vec{w}-\\vec{t} \\right \\|^2=\\frac12 (\\Phi \\vec{w}-\\vec{t})^T (\\Phi \\vec{w}-\\vec{t}) \\\\\n", "& = \\frac12 \\vec{w}^T \\Phi^T \\Phi \\vec{w} - \\vec{t}^T \\Phi \\vec{w} + \\frac12 \\vec{t}^T \\vec{t}\n", "\\end{align}\n", "$$\n", "- So the derivative is $$\\nabla_\\vec{w} E(\\vec{w}) = \\Phi^T\\Phi\\vec{w} - \\Phi^T \\vec{t}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "> **Remark**\n", "> - Note that this derivative can simply be obtained from the derivative we just derived in gradient descent method, which is\n", "$$\n", "\\begin{align}\n", "\\nabla_\\vec{w} E(\\vec{w}) \n", "&= \\sum_{n=1}^N \\left( \\vec{w}^T\\phi(x_n) - t_n \\right)\\phi(x_n) & \\text{non-matrix/vector form} \\\\\n", "&= \\Phi^T\\Phi \\vec{w} - \\Phi^T \\vec{t} & \\text{matrix/vector form} \n", "\\end{align}\n", "$$\n", "> - For gradient descent and closed form solution we obtain the derivative from two different perspectives, **non-matrix/vector** form and **matrix/vector** form. You could choose whichever you like when in use. We suggest matrix/vector form which is more neat.\n", "> - From $\\vec{e}=\\Phi \\vec{w}-\\vec{t}$, we know that $\\Phi \\vec{w}$ is the prediction of training samples and $\\vec{e}$ is just the residual error. What we are doing in least squares is just making prediction of training samples as close to training labels as possible, i.e. $\\Phi \\vec{w} \\approx \\vec{t}$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "- To minimize $E(\\vec{w})$, we need to let $\\nabla_\\vec{w} E(\\vec{w}) = \\Phi^T\\Phi \\vec{w} - \\Phi^T \\vec{t} = 0$, which is also $\\Phi^T\\Phi \\vec{w} = \\Phi^T \\vec{t}$\n", "- When $\\Phi^T \\Phi$ is **invertible** ($\\Phi$ has *linearly independent columns*), we simply have \n", "$$\n", "\\begin{align}\n", "\\hat{\\vec{w}}\n", "&=(\\Phi^T \\Phi)^{-1} \\Phi^T \\vec{t} \\\\\n", "&\\triangleq \\Phi^\\dagger \\vec{t}\n", "\\end{align}\n", "$$\n", "of which $\\Phi^\\dagger$ is called the **Moore-Penrose Pseudoinverse** of $\\Phi$.\n", "- We will talk about case $\\Phi^T \\Phi$ is **non-invertible** later." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "> **Remark**\n", "> - A common mistake is writing $\\hat{\\vec{w}}=(\\Phi^T \\Phi)^{-1} \\Phi^T \\vec{t}=\\Phi^{-1}(\\Phi^T)^{-1}\\Phi^T \\vec{t}=\\Phi^{-1} \\vec{t}$. This is wrong because $\\Phi$ is not necessarily square and invertible.\n", "> - When $\\Phi$ is square and invertible, you are free to write that." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Digression: Moore-Penrose Pseudoinverse\n", "- When we have a matrix $A$ that is non-invertible or *not even square*, we might want to invert anyway\n", "- For these situations we use $A^\\dagger$, the **Moore-Penrose Pseudoinverse** of $A$\n", "- In general, we can get $A^\\dagger$ by SVD: if we write $A \\in \\R^{m \\times n} = U_{m \\times m} \\Sigma_{m \\times n} V_{n \\times n}^T$ then $A^\\dagger \\in \\R^{n \\times m} = V \\Sigma^\\dagger U^T$, where $\\Sigma^\\dagger \\in \\R^{n \\times m}$ is obtained by taking reciprocals of *non-zero entries* of $\\Sigma^T$.\n", "- Particularly, when $A$ has linearly independent columns then $A^\\dagger = (A^T A)^{-1} A^T$. When $A$ is invertible, then $A^\\dagger = A^{-1}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "> **Remark**\n", ">- The solution to $A\\vec{x}=\\vec{b}$ is $\\hat{\\vec{x}}=A^{\\dagger}b$ in the sense that $\\left \\| A\\vec{x} - \\vec{b} \\right \\|^2$ is minimized. Meanwhile, among all the minimizers that achieve the same minimal value of $\\left \\| A\\vec{x} - \\vec{b} \\right \\|_2^2$, $\\hat{\\vec{x}}$ has the smallest norm $\\left \\| \\vec{x} \\right \\|_2$.\n", ">- $Ax$ **doesn't necessarily** equal $\\vec{b}$:\n", "> - When $\\vec{b}$ is in the column space of $A$, $A\\vec{x} = \\vec{b}$.\n", "> - When $\\vec{b}$ is **not** in the column space of $A$, $A\\vec{x} \\neq \\vec{b}$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Back to Closed Form Solution\n", "- From previous derivation, we have $\\hat{w}=(\\Phi^T \\Phi)^{-1} \\Phi^T \\vec{t} \\triangleq \\Phi^\\dagger \\vec{t}$.\n", "- What if $\\Phi^T \\Phi$ is non-invertible? This corresponds to the case where $\\Phi$ doesn't have linearly independent columns. For dataset, this means the feature vector of certain sample is the linear combination of feature vectors of some other samples.\n", "- We could still resolve this using pseudoinverse.\n", "- To make $\\nabla_\\vec{w} E(\\vec{w}) = \\Phi^T\\Phi \\vec{w} - \\Phi^T \\vec{t} = 0$, we have\n", "$$ \\hat{\\vec{w}} = (\\Phi^T\\Phi)^\\dagger \\Phi^T \\vec{t} = \\Phi^\\dagger \\vec{t}$$\n", "of which $(\\Phi^T\\Phi)^\\dagger\\Phi^T = \\Phi^\\dagger$. This is left as an exercise. (**Hint:** use SVD)\n", "- Now we could conclude the optimal $\\vec{w}$ in the sense that minimizes sum of squared errors is $$\\boxed{\\hat{\\vec{w}} = \\Phi^\\dagger \\vec{t}}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ ">**Remark**\n", "> - In the above derivation, we get $\\hat{\\vec{w}} = \\Phi^\\dagger \\vec{t}$ by letting the derivative $\\nabla_\\vec{w} E(\\vec{w})=0$. Actually, $\\hat{\\vec{w}}=\\Phi^\\dagger \\vec{t}$ is just the solution to $\\Phi \\vec{w} = \\vec{t}$ in the sense that $\\left \\| \\Phi\\vec{w} - \\vec{t} \\right \\|_2^2$ is minimized. Meanwhile, among all the minimizers that achieve the same minimal value of $\\left \\| \\Phi\\vec{w} - \\vec{t} \\right \\|_2^2$, $\\hat{\\vec{w}}$ has the smallest norm $\\left \\| \\vec{w} \\right \\|_2$.\n", "> - Now that we have neat and closed form solution to least squares, why do we introduce gradient descent method?\n", "> - $\\hat{w}=(\\Phi^T \\Phi)^{-1} \\Phi^T \\vec{t}$ involves matrix inversion. It could be quite computationally expensive when the dimension $M$ is high. On the other hand, gradient descent involves nothing but matrix multiplication which can be computed much faster." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "## Appendix I: Differential Operation in Matrix Calculus\n", "### General Gradient\n", "- Suppose that $f : \\R^{m \\times n} \\rightarrow \\R$, that is, the function $f$\n", " - takes as input a matrix $A \\in \\R^{m\\times n} = [a_{ij}]$\n", " - returns a real value\n", "\n", "- Then, the **gradient** of $f$ with respect to $A$ is:\n", "\n", "$$\n", "\\nabla_A f(A) \\in \\R^{m \\times n} = \\begin{bmatrix}\n", "%\n", "\\frac{\\partial f}{\\partial a_{11}}\n", " & \\frac{\\partial f}{\\partial a_{12}} \n", " & \\cdots \n", " & \\frac{\\partial f}{\\partial a_{1n}} \\\\ \n", "%\n", "\\frac{\\partial f}{\\partial a_{21}}\n", " & \\frac{\\partial f}{\\partial a_{22}} \n", " & \\cdots \n", " & \\frac{\\partial f}{\\partial a_{2n}} \\\\\n", "%\n", "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n", "%\n", "\\frac{\\partial f}{\\partial a_{m1}}\n", " & \\frac{\\partial f}{\\partial a_{m2}} \n", " & \\cdots \n", " & \\frac{\\partial f}{\\partial a_{mn}} \\\\ \n", "\\end{bmatrix}\n", "$$\n", "\n", "$$\n", "[\\nabla_A f(A)]_{ij} = \\frac{\\partial f}{\\partial a_{ij}}\n", "$$\n", "\n", "- Note that the size of $\\nabla_A f(A)$ is always the same as the size of $A$. \n", "- In particular, if $A$ is a vector $\\vec{x} \\in \\R^n$,\n", "$$\n", "\\nabla_\\vec{x} f(\\vec{x}) = \\begin{bmatrix}\n", " \\frac{\\partial f}{\\partial \\vec{x}_1} \\\\\n", " \\frac{\\partial f}{\\partial \\vec{x}_2} \\\\\n", " \\cdots \\\\\n", " \\frac{\\partial f}{\\partial \\vec{x}_n}\n", "\\end{bmatrix}\n", "$$\n", "\n", "- The gradient is a **linear operator** from $\\R^n \\mapsto \\R^n$:\n", " - $ \\nabla_x (f + g) = \\nabla_x f + \\nabla_x g $\n", " - $ \\forall\\, c \\in \\R, \\nabla_x (c f) = c \\nabla_x f$\n", "\n", "### Gradient of Linear Functions\n", "- The gradient of the **linear function** $f(\\vec{x}) = \\sum_{k=1}^n b_k x_k = \\vec{b}^T \\vec{x}$ is\n", "$$\n", "\\frac{\\partial f}{\\partial x_k}\n", "= \\frac{\\partial}{\\partial x_k} \\sum_{k=1}^n b_k x_k\n", "= \\sum_{k=1}^n \\frac{\\partial}{\\partial x_k} b_k x_k\n", "= b_k\n", "$$\n", "- Compact form\n", "$$\n", "\\nabla_\\vec{x} \\vec{b}^T \\vec{x} = \\vec{b}\n", "$$\n", "- Similarly,\n", "$$\n", "\\nabla_\\vec{x} \\vec{x}^T \\vec{b} = \\vec{b}\n", "$$\n", "\n", "### Gradient of Quadratic Forms\n", "- Every symmetric $A \\in \\R^{n \\times n}$ corresponds to a **quadratic form**:\n", " $$ f(x) = \\sum_{i=1}^n \\sum_{j=1}^n x_i A_{ij} x_j = \\vec{x}^T A \\vec{x} $$\n", "- The partial derivatives are\n", " $$ \n", " \\frac{\\partial f}{\\partial x_k}\n", " = 2 \\sum_{j=1}^n A_{ij} x_j\n", " = 2 [A\\vec{x}]_i\n", " $$\n", "- Compact form $$\\nabla_\\vec{x} \\vec{x}^T A \\vec{x} = 2 A \\vec{x}$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "## Appendix II: Geometric Interpretation of Least Squares\n", "- Assume number of samples is much larger than number of features, i.e. $N>>M$ ($\\Phi \\in \\R^{N \\times M}$ is a *tall* matrix).\n", "- The columns of $\\Phi$ construct a $M$-dimensional subspace $\\mathcal{S}$.\n", "- Target $\\mathbf t = (t_1, \\dots, t_N)$ is a vector in $N$-dimensional space\n", "- Prediction $\\vec{y}(\\mathcal{D}, \\vec{w}) = \\Phi \\vec{w} \\in \\mathcal{S}$.\n", "- The prediction $\\hat{\\vec{y}} (\\mathcal{D}, \\hat{\\vec{w}})$ we get via least squares is the projection of $\\vec{t}$ onto $\\mathcal{S}$. In other words, of all $\\vec{w} \\in \\R^{M}$, $\\hat{\\vec{w}}$ has smallest $\\left \\| \\Phi \\vec{w} \\right \\|$. \n", "
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