notebook/ch08_gradient_descent.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: pylab import has clobbered these variables: ['random']\n",
      "`%matplotlib` prevents importing * from pylab and numpy\n"
     ]
    }
   ],
   "source": [
    "from collections import Counter\n",
    "from linear_algebra import distance, vector_subtract, scalar_multiply\n",
    "from functools import reduce\n",
    "import math, random\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%pylab inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def sum_of_squares(v):\n",
    "    \"\"\"computes the sum of squared elements in v\"\"\"\n",
    "    return sum(v_i ** 2 for v_i in v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "285"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vector = [i for i in range(10)]\n",
    "sum_of_squares(vector)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "285"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(np.square(vector))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def difference_quotient(f, x, h):\n",
    "    return (f(x + h) - f(x)) / h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_estimated_derivative():\n",
    "    def square(x):\n",
    "        return x * x\n",
    "\n",
    "    def derivative(x):\n",
    "        return 2 * x\n",
    "\n",
    "    derivative_estimate = lambda x: difference_quotient(square, x, h=0.00001)\n",
    "\n",
    "    # plot to show they're basically the same\n",
    "    x = range(-10,10)\n",
    "    plt.plot(x, list(map(derivative, x)), 'rx')           # red  x\n",
    "    plt.plot(x, list(map(derivative_estimate, x)), 'b+')  # blue +\n",
    "    plt.show()                                      # purple *, hopefully"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f870e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_estimated_derivative()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def partial_difference_quotient(f, v, i, h):\n",
    "\n",
    "    # add h to just the i-th element of v\n",
    "    w = [v_j + (h if j == i else 0)\n",
    "         for j, v_j in enumerate(v)]\n",
    "\n",
    "    return (f(w) - f(v)) / h\n",
    "\n",
    "def estimate_gradient(f, v, h=0.00001):\n",
    "    return [partial_difference_quotient(f, v, i, h)\n",
    "            for i, _ in enumerate(v)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def step(v, direction, step_size):\n",
    "    \"\"\"move step_size in the direction from v\"\"\"\n",
    "    return [v_i + step_size * direction_i\n",
    "            for v_i, direction_i in zip(v, direction)]\n",
    "\n",
    "def sum_of_squares_gradient(v):\n",
    "    return [2 * v_i for v_i in v]\n",
    "\n",
    "def safe(f):\n",
    "    \"\"\"define a new function that wraps f and return it\"\"\"\n",
    "    def safe_f(*args, **kwargs):\n",
    "        try:\n",
    "            return f(*args, **kwargs)\n",
    "        except:\n",
    "            return float('inf')         # this means \"infinity\" in Python\n",
    "    return safe_f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "using the gradient\n",
      "minimum v [-0.0002810713051264979, 0.0003613773923055063, -0.00020076521794750227]\n",
      "minimum value 2.4990137097254875e-07\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(\"using the gradient\")\n",
    "v = [random.randint(-10,10) for i in range(3)]\n",
    "tolerance = 0.0000001\n",
    "\n",
    "while True:\n",
    "    #print v, sum_of_squares(v)\n",
    "    gradient = sum_of_squares_gradient(v)   # compute the gradient at v\n",
    "    next_v = step(v, gradient, -0.0001)       # take a negative gradient step\n",
    "    if distance(next_v, v) < tolerance:     # stop if we're converging\n",
    "        break\n",
    "    v = next_v                              # continue if we're not\n",
    "\n",
    "print(\"minimum v\", v)\n",
    "print(\"minimum value\", sum_of_squares(v))\n",
    "print()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#\n",
    "#\n",
    "# minimize / maximize batch\n",
    "#\n",
    "#\n",
    "\n",
    "def minimize_batch(target_fn, gradient_fn, theta_0, tolerance=0.000001):\n",
    "    \"\"\"use gradient descent to find theta that minimizes target function\"\"\"\n",
    "\n",
    "    step_sizes = [100, 10, 1, 0.1, 0.01, 0.001, 0.0001, 0.00001]\n",
    "\n",
    "    theta = theta_0                           # set theta to initial value\n",
    "    target_fn = safe(target_fn)               # safe version of target_fn\n",
    "    value = target_fn(theta)                  # value we're minimizing\n",
    "\n",
    "    while True:\n",
    "        gradient = gradient_fn(theta)\n",
    "        next_thetas = [step(theta, gradient, -step_size)\n",
    "                       for step_size in step_sizes]\n",
    "\n",
    "        # choose the one that minimizes the error function\n",
    "        next_theta = min(next_thetas, key=target_fn)\n",
    "        next_value = target_fn(next_theta)\n",
    "\n",
    "        # stop if we're \"converging\"\n",
    "        if abs(value - next_value) < tolerance:\n",
    "            return theta\n",
    "        else:\n",
    "            theta, value = next_theta, next_value\n",
    "\n",
    "def negate(f):\n",
    "    \"\"\"return a function that for any input x returns -f(x)\"\"\"\n",
    "    return lambda *args, **kwargs: -f(*args, **kwargs)\n",
    "\n",
    "def negate_all(f):\n",
    "    \"\"\"the same when f returns a list of numbers\"\"\"\n",
    "    return lambda *args, **kwargs: [-y for y in f(*args, **kwargs)]\n",
    "\n",
    "def maximize_batch(target_fn, gradient_fn, theta_0, tolerance=0.000001):\n",
    "    return minimize_batch(negate(target_fn),\n",
    "                          negate_all(gradient_fn),\n",
    "                          theta_0,\n",
    "                          tolerance)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "using minimize_batch\n",
      "[7, -7, 4]\n",
      "minimum v [0.0009304595970494407, -0.0009304595970494407, 0.0005316911983139665]\n",
      "minimum value 2.0142056538473567e-06\n"
     ]
    }
   ],
   "source": [
    "print(\"using minimize_batch\")\n",
    "v = [random.randint(-10,10) for i in range(3)]\n",
    "print (v)\n",
    "v = minimize_batch(sum_of_squares, sum_of_squares_gradient, v)\n",
    "\n",
    "print(\"minimum v\", v)\n",
    "print(\"minimum value\", sum_of_squares(v))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#\n",
    "# minimize / maximize stochastic\n",
    "#\n",
    "\n",
    "def in_random_order(data):\n",
    "    \"\"\"generator that returns the elements of data in random order\"\"\"\n",
    "    indexes = [i for i, _ in enumerate(data)]  # create a list of indexes\n",
    "    random.shuffle(indexes)                    # shuffle them\n",
    "    for i in indexes:                          # return the data in that order\n",
    "        yield data[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def minimize_stochastic(target_fn, gradient_fn, x, y, theta_0, alpha_0=0.01):\n",
    "\n",
    "    data = list(zip(x, y))\n",
    "    theta = theta_0                             # initial guess\n",
    "    alpha = alpha_0                             # initial step size\n",
    "    min_theta, min_value = None, float(\"inf\")   # the minimum so far\n",
    "    iterations_with_no_improvement = 0\n",
    "\n",
    "    # if we ever go 100 iterations with no improvement, stop\n",
    "    while iterations_with_no_improvement < 100:\n",
    "        value = sum( target_fn(x_i, y_i, theta) for x_i, y_i in data )\n",
    "\n",
    "        if value < min_value:\n",
    "            # if we've found a new minimum, remember it\n",
    "            # and go back to the original step size\n",
    "            min_theta, min_value = theta, value\n",
    "            iterations_with_no_improvement = 0\n",
    "            alpha = alpha_0\n",
    "        else:\n",
    "            # otherwise we're not improving, so try shrinking the step size\n",
    "            iterations_with_no_improvement += 1\n",
    "            alpha *= 0.9\n",
    "\n",
    "        # and take a gradient step for each of the data points\n",
    "        for x_i, y_i in in_random_order(data):\n",
    "            gradient_i = gradient_fn(x_i, y_i, theta)\n",
    "            theta = vector_subtract(theta, scalar_multiply(alpha, gradient_i))\n",
    "\n",
    "    return min_theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def maximize_stochastic(target_fn, gradient_fn, x, y, theta_0, alpha_0=0.01):\n",
    "    return minimize_stochastic(negate(target_fn),\n",
    "                               negate_all(gradient_fn),\n",
    "                               x, y, theta_0, alpha_0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Test를 위한 예제들은 이후 ch14 등에서 제시됨, 현재 코드만으로는 활용에 대해 이해하기 어려우니 skip 할 것을 추천"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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Pq/d3lR4fKhT8jr6+sg8EyQ4FfZFFVi2wh0fz4Z87+vrKAvgdFc4pnVfv7xJxTx70ldMX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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f869cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Numpy version from : http://stackoverflow.com/questions/17784587/gradient-descent-using-python-and-numpy\n",
    "\n",
    "def genData(numPoints, bias, variance):\n",
    "    x = np.zeros(shape=(numPoints, 2))\n",
    "    y = np.zeros(shape=numPoints)\n",
    "    # basically a straight line\n",
    "    for i in range(0, numPoints):\n",
    "        # bias feature\n",
    "        x[i][0] = 1\n",
    "        x[i][1] = i\n",
    "        # our target variable\n",
    "        y[i] = (i+bias) + random.uniform(0, 1) * variance\n",
    "    return x, y\n",
    "\n",
    "# gen 100 points with a bias of 25 and 10 variance as a bit of noise\n",
    "x, y = genData(100, 25, 10)\n",
    "\n",
    "plt.plot(x[:,1]+1,y,\"ro\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 29.22828174,   1.01274694])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# m denotes the number of examples here, not the number of features\n",
    "def gradientDescent(x, y, theta, alpha, m, numIterations):\n",
    "    xTrans = x.transpose()\n",
    "    theta_list = []\n",
    "    for i in range(0, numIterations):\n",
    "        hypothesis = np.dot(x, theta)\n",
    "        loss = hypothesis - y\n",
    "        \n",
    "        # avg cost per example (the 2 in 2*m doesn't really matter here.\n",
    "        # But to be consistent with the gradient, I include it)\n",
    "        cost = np.sum(loss ** 2) / (2 * m)\n",
    "        \n",
    "        gradient = np.dot(xTrans, loss) / m\n",
    "        # update\n",
    "        theta = theta - alpha * gradient\n",
    "        if i % 2500 == 0:\n",
    "            theta_list.append(theta)\n",
    "    return theta,np.array(theta_list)\n",
    "\n",
    "m, n = np.shape(x)\n",
    "numIterations= 50000\n",
    "alpha = 0.0005\n",
    "x.transpose()\n",
    "theta = np.ones(n)\n",
    "theta,theta_list = gradientDescent(x, y, theta, alpha, m, numIterations)\n",
    "\n",
    "theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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jNWYMr1oVoyNHWNupE95tOhBb5Xna/7md6Jun6Rd9ic7RceyvXROTBDjV90Xc\nf9yQpdzM03j06tUrUIAsbCrglm4q4CqKQnw8zJmjrX++dCmMGFH2pvqEh4WxZvYsQu6f4VjjmyTU\nlIxpMZrEqT/x6YUrGADRJiZ89NprXOr5Fo4+/tw4s50OMRd5PTKKAHNjQiqZMzLqGjI1/9N4CtIF\nXNhUwC3dVMBVlApu3z5tqo+jo9aVXKtWSdcofzK3NO+YmXL+7jGuN4smpRJMOga3Yq3pql+FriEh\nHG/dmjW9XyVRtqTGqU1UuxXM8BthVE6V/FpPR6xZbR7Urom+lHnOf503ciRTf/zxsUFOS0aMyJLr\nuKSogFu6qQXoFaWCunVLW6v22DGt+7hv35KuUf6ltzQ/vBrCRjtY3h7a3IGFh+DlS1pe2mjTO8zo\n15+lw9xpecaPqof2Mijagxdux7GjTi3WNWyPYcvGjFmwMN+t04JO41GKhq+vL25ubgQHB5d0VZ6Z\nWoBeUcohKcHLC1q31tIynjlTdoJt+qLr/+nRCb2GIbSeDL4N4M2f4Y+18Mol8G3bFrf33PnY5X8Q\ncp1Xfp7O1IMrGHzzApeEOfNTYEj4TT4L+JMPN/yER+/e+V68Pb+5jpWiodPpOHz48GPbHR0diyzY\nBgYG0q1bN8zNzWnQoAELFiwokuuoFq6ilDOhodqqPtHR2gLx9vYlXaP8Cw8LY96g7lD/Kr8PB5sz\ncOpbaHAPIqpVY8mgvvxt8woW/vuwOvATQ2+GYvBIsrVWPcL1TOhxPY7DxDMfHltHdsns2fnqEh49\nfz5z/fwev4ebLdexUn688cYbDB48mCNHjhAaGoqjoyPt27enf//+hXod1cJVlHIiJQU+/xw6dYI+\nfbSMUWUp2B6/epyXVnRnZ++r2NyD75fBl7shTNeOCe8sYGG/dwm/FMaIn99nwbFNNH0Qi2cNW/RP\n/cOwA7+za6Azc3v0ILx27QJ1CWcsGjBiBHN79GDJiBGlZsBURebj44O19b95lXQ6HUuXLqVdu3ZY\nWFjg4uJCUlJSxv5du3ZhZ2eHhYUFjo6OBAUF5Vp2eHg4b7zxBgCNGjXC0dGRs2fPFvp7UC1cRSkH\nTp0CNzct7/HJk1BWFpRJlansurCLz49+TmRcJI4XK7F6J9w1qcb3/foxyvIFTP13YLfvG4ZERXLa\nzJiDlrb4NTJFv0FDpmQa9JTeep03ciQJOQx6epou4YY6XakYIKVklX2FoM2bN3PgwAEMDAxwcHBg\n7dq1jBt+3UsTAAAgAElEQVQ3Dn9/f8aOHcvu3buxt7dn/fr1ODs7c+HCBSpXrvxYue+//z7r1q1j\n/vz5hISE4Ofnx0cffVTo9VcBV1HKsIQEmDsXfvhBS2Lh6lp6pvrkNZf1wqV/+PCL0RwzCaKKngEr\nao2gv28tDkZEMOWtt0m8dhGr4BN87PMLhimSX2pZsbhHf975ajk908rIrXzVJVxw3sK7UMpxkk6F\nUk5uJk+ejKWlJQADBgwgICAAAE9PTyZMmEDHjh0BcHV1ZeHChfj5+dG1a9fHyunXrx+jRo1iyZIl\npKamMmfOHDp06FDo9VUBV1HKqAMHtLSMDg7aoKjimuqTn6QQOc5l9fNj1O4t/Hj+R74++gW9ox9x\n9GeollqFb3slsKtmRwzDInhp72K63L7Hnjq1CNCvjb95Fd47eOjJc2X9/DK6fnNaR1Z1CedfUQfK\nwpIebAGMjIyIiooCtC5iLy8vPDw8AG1d3eTkZCJzuK0QGxvLSy+9xIoVK3BxceH69esMHjwYS0tL\nJkyYUKj1VQFXUcqYW7fgv/8FX1/45ht46aXiu/aTAl267EvSRZtBYuMQOv/YiZeu1OLi5lTONrPn\nmwH9iLt4ko7+B3g3KpIgcxNOte/KCWNj9KJvo2dlxXv5XPIu88Ao1SVcsVlbWzNz5kxmzJjxxGND\nQ0PR19dnxIgRAFhZWTF8+HD27NmjAq6iVFRSwvr18OGHWpaoM2fAOPvooCL2pECXLn0ua6AlLHGA\n/baw5AC88YMlPh1eYk6vR9S7eIyxOz/FKEWyp6YVn7V9gcV+R+mRj3qoubLlW1JSEomJiRmv9Z8y\n0bebmxuDBg2iV69edOrUiYSEBHx8fOjevTvG2f6nadq0KVJKNm7cyLBhw7hx4wY///wzvXr1KpT3\nkpkapawoZUBoqNaSXboUdu3S/i3uYAv5C3RSSi43rsSLI+HNgfD23wKvw8/j33wKP5lb0ebEJj4/\nsIYmD2LwN6iNzb0UxoRewaRx/rt81VzZ8q1fv34YGRlhaGiIkZFRjksSZh9AlZm9vT2enp5MnDiR\n6tWr07RpU9atW5fjsaampmzdupUvvviC6tWr06FDB9q2bcvMmTML7f1k1FmldlSU0islBf7v/2DR\nIpg2DaZMgRwGWRaK/NybzSvt4Uyvtfxy7hc+9/2MJudu8frBVMKavMT5OzexjwhkSFQEZ82MOdH2\nBaIuhbEw9BLGQDAw3cSEFq1bY2Rrm6+FAkp7vuOipFI7lm4ql7KilEF//61N9TE3h2+/hcaNi+5a\nOQWwKdbWmNnZYXTvXkYABh477uNmOiy+cOXns2t5M7gqTW634ndpRo2QEwy5EYJxyiO2WNlQ4613\nuBr4N6kREdwzM0NfSpJv3uTemTN4xMc/deAszCXvyhIVcEs3FXAVpQy5f1+b6uPlBZ99Bm++WXRT\nfdKDVuDBg/xw82ZGyzUcWA4ZGZsyB0LQ7uXG3brMn83uklDlKu9eacK1h+25de0MfW7+g+PtO+yq\na0lM3yFM/Go5V69eybFFKlq1wn3HjlK7UEBppAJu6ZbX9/PEe7hCiNVCiBtCiNOZtlkIIQ4IIc4L\nIfYLIcwy7ZshhLgohAgWQvQpnLegKBXDwYNa/uOICAgKgtGjizbYevTuzdQff6RNpmALsBZyTI+4\ndvZskqqlcP11I8LrBTH8mi0dT3bhwd/XmfDbagZFneFs1ZokPqrMa5dvcOW3fVy9eiXXwVY3/PzU\n4CelwsjPoKk1QPa05x8Bv0kpmwGHgRkAQoiWwFCgBfAysELkdWdbURQAbt/WWrJubvD117BhA9Su\nXbTXzBwE9SDLIKRUeCwQBtWHfVX3sXJCH4xXxNHyRH26HNnLtMD9JBqZ8mm3V3juejzTQy5hmZKc\nJUjnNtgqXgg1+EmpMJ4YcKWUvkBsts2vAulDvtYBA9OeOwMbpZQpUsrLwEWgU+FUVVHKHym1BeFb\nt4bq1bWpPi+/XDzXzhwERwNz+TfopqY9TxWwsymMcdbj1zY96eTfBCe/O8z02UjTBzH4DhuP+Osf\n7rXtwG0/31xbq7mNKm7YuTNzbW0z9qV3NY9WWaGUcuhZ5+HWllLeAJBSXhdCpP8Wrwccz3RcRNo2\nRVGyuXxZWxQ+MhJ27oTnnitYefkZZZxZehA0BhoCk4BFQLilJWYvPMeQu0dpaFKFSjFt6XQ2mCGR\nv3POzIi9lk2wXL+cEd27ZRlstQQyykuX3lrNLd3i+8uWAaisUErFIKV84gPt/8fTmV7HZNsfnfav\nB/BGpu2rgEG5lCkVpSJKTpZy6VIpa9SQ8tNPpUxKKniZl0ND5Qe2tjJeazTLeJAf2NrKy6GhOR7r\nPmKEnPz883KMiUmWc95tbiPnr/6P/M8rfeTcli1lYPUqMrRaJflZk0by15qWj5XtPmJExvmXQX6Q\ntj+nOqQfP6dHD+k+YkSOdVOeLO1vZ77+duf2UH9/i05e38+ztnBvCCEspZQ3hBB1gJtp2yMA60zH\n1U/bliN3d/eM505OTjg5OT1jdRSlbAgIgLffhmrVwM+v8Kb65DcDVPbpP8GAi4kJVvaNSapZGdNQ\nPdrO/Yl3b91hd11LDg5w5V5iAtM2bsyx7Mzd0umt5CVAiLk5tv36ZWmtqnSLz8bb2xtvb++SroZS\nGHKLxDLrryEbICjT68+A6WnPpwOL0p63BPyBKoAOuETa1KMcyiyuHxyKUuISEqScNk3K2rWlXL1a\nytTUwi1/jpOTlGkty8yPOT16ZDkuc4tUgjxRE/nOC13l17bW8qahkN51jOXcFxxkaPA/+So7e3np\nLVv3ESMK9w0qGaigLdw//vhDNm/evKSr8UR5fT/5mRa0ATgGNBVCXBFCjEG71dNbCHEe6JX2Ginl\nOWATcA7YA7yTVgFFqbAOHYK2beHKFTh9Gt56q/Cn+uQ31WFqRARGwBftWuLesiVVU6vw4ZljJOhX\nwXvRCrpHxeN+7Ci65s3yVfbo+fPVoCelUOl0Og4fPvzYdkdHR4KDg4vkmnPmzKFt27ZUrlyZTz75\n5LH9GzZswMbGBlNTUwYNGsSdO3ee7UK5ReKiflAGf2EpytO4fVvKN9+UskEDKXftKtpr5ece7uUL\nF+Rk+05ye30LGWuA/MGmtpzbyk5GP6FF+qSy1b3Z4kU5b+Ha2NjIQ4cOFes1vby85L59++TAgQPl\nvHnzsuw7c+aMNDU1lb6+vjIhIUG+8cYbcvjw4bmWldf3owKuohSy1FQp16+X0tJSysmTpYyLK57r\n5hb45o0YK7+2tZY3jLQu42lt2svjZtVzDcxPU7ZS/CpqwPX29pb169fPctySJUtk27Ztpbm5uRw+\nfLhMTEzM2L9z507Zvn17aW5uLrt06SJPnz79xGuPHDnysYD78ccfyxGZfpCGhITIKlWqyPj4+BzL\nyOv7UcvzKUohSp/qExEBO3ZAp2KchZ55UNKRHXvxevVVnCPPMyrpEZvr2uDZtD3vB/jzY+VotnR3\nZG9cXL6n4agBT0ppkD2P0ubNmzlw4AAGBgY4ODiwdu1axo0bh7+/P2PHjmX37t3Y29uzfv16nJ2d\nuXDhApWfcvWPs2fP0qVLl4zXjRo1wsDAgAsXLmBnZ/dUZamAqyiF4NEj+PJLWLgQPvgApk4tmlV9\n8ppre+d2DP/3hiv2wcfpeiuWK3Vr812T53C5eIIPL4RklLHs6lWWdOvG3O3bC7+CSrng7V04gwyc\nnIp2CM/kyZOxtLQEYMCAAQQEBADg6enJhAkT6NixIwCurq4sXLgQPz8/unbt+lTXiI+Px8zMLMu2\natWqERcX99T1VQFXUQoofaqPqSkcPw5NmhSsvNyCao5L0vn5Yd3vVars/YXB167Sq5ohuyyb8ufg\nAbz7ySIuvvoGjtEpWcpXuYqVJynqQFlY0oMtgJGREVFRUQCEh4fj5eWFh4cHoN06TU5OJvIZ/rs3\nMTHh3r17WbbdvXsXU1PTpy5LBVxFeUb378O8ebBmjbZe7ZgxeY8+zk8mqNyC6qSDB7PMtf3bxJR9\n9awYefsy5muXs7mODXMdXsTuHQdm959KdMQtVr7zIUHnzuWa/UlRyitra2tmzpzJjBkzClxWq1at\nCAwMzHgdEhJCcnIyTZs2feqyVMBVlGdw6BCMHw/29tqqPpl+aOcox/Vmjxx5bL3ZvBJYJF+5gkeT\n5rR8cINut2Jp+DAWT9vOxNW5SnKt2iyY8TW2tk2yXOs2MJscltlT03aUUiwpKYnExMSM1/r6Txeq\n3NzcGDRoEL169aJTp04kJCTg4+ND9+7dMTbOnvEbUlJSSElJITU1leTkZBITE6lcuTJ6enqMGDEC\nBwcHjh49Svv27ZkzZw6DBw/OsZwnym00VVE/KMWj5BQlN7dvSzl6tJTW1lLu3Jn/87IniLgMckoO\naRCnde4ssyeY+LGutTbK2FBIH0tj+WEbO/nBS31l/1f05U4dMjWPdIvp15oF0tXSUo0wLgeoAKOU\n9fT0pJ6enhRCSD09PTl79mzp7e0tra2tM47T6XRZRjO7u7tLV1fXjNf79++Xzz33nLSwsJBWVlZy\n6NChuY4sHj16dMa10h/r1q3L2P/TTz/JBg0aSBMTE/naa6/J2NjYXOuf1/ejFqBXlHyQEjZuhP/+\nF15/XRsc9TS3cOb26MG8TOn55gFTebyrd5SNDV6XLxNqaMKe+nXoE3OVGknJbKprQ3i9Nshq/iQl\nX6HSKVhy4/Hzl4wYQWpERJZrZalDDgkFlLJFLUBfuuX1/aguZUV5gvBwbarP1auwbRt07vz0ZWRe\nmQdyXm+2koAGpmb8Vt+C7rdisU66yypbB+pVMyDQ6ij1h9kw2HoqBz9bSUjibozJmu0m+1J46r6t\nopQu+VmAXlEqpEePYPly7T5tly5w6tSzBVvgsRSI6evNAmyrXYeVjepz11Aw6OZFjlrYsKiHKxE6\nI47ZH+PKKGt0si8miwM5+NlKRs+fj22/firdoqKUMapLWVFycPq0NtXHyAi+/RaaNXvyOZD3SOSM\nfZGR3NavQu3wEJyjtS7jn+vacKG+PY7GqYSb/c229vq822Ma3ao58t1L/R9bR/a1779n21tvPbZ9\n0sGDGVOI1mZaY/ZJa+MqZYfqUi7d8vp+VMBVlEwePID582HVKvjf/7SFBvTy2Q+U45SeTEEw8f59\nPP/zDta/76L7zWh216nNH3VaUrdOM5pF/c4vHe9wvUNjpjlOp3/T/ugJPeaNHMnUH3/M8V5t+qhm\nFVQrFhVwSzd1D1dR8uH337WpPu3bay3cxAdhzB/1hNZqpn05TekZGxLCzD4v4yATGBwVQXtTQ3bU\nasLBLq68GBvGEKt7vF9vM/WcO1L/GLRan4T/75toN78NDXW6LOvNkqnc1MhIlW5RUcoYFXCVCi8m\nBj78EA4ehK+/hgED8k5AAeS4r1KtWhnBMdTAiK31Lel5J5L/3bjIxro2fPzC6zS0qMnI2DOcsDnH\nuDYnGdjmdb60nsmeIeNzvJYaAKUo5YfqUlYqLClh0yaYMgUGD9a6kNOn+uTVlQvkuG9Uw4b0TJVY\nyzi63YplT51aHKnVkqotuvPi+d/ocM6PzxxSWdXJgCk9pjKp0yQsTSyf2G2cVze1UvGoLuXSTXUp\nK0o2V67AO+9oq/ts3fr46OO8unKRMsu+/ea1uGJRma9vXOWSaVW212zMzs6jqM0DZnpv5W64D9N6\nwzkHmHIc3jveiQXzF+TrWg11OiYdPMiSTPdq87O6j6IopU+BAq4QYgowFm2WQxAwBu1vxc9AQ+Ay\nMFRKebdg1VSUwvHoEXz1lTYw6v33tWBbpcrjxz2pK/e8QVX2W1nSJe4GLR9Gc1q/IbOffxX9F/rR\n22ctTY5s4Jt6D+n7ehwGqfDhMRh6FhJTYcmIBk91LXWvVlHA19cXNzc3goODS7oqz+yZ5+EKIayA\nSUAHKWVbtODtAnwE/CalbAYcBgqePVpRCkFQEDg4aEH26FGYNSvnYAuPz5tNAOY00lGzUhXa/7Ef\nSx5SMzWBVQ06Mm/4/7jRqQ+zr51hWfI5bk5x5OXpVTgzoT2tTtflj2/hjSAt2OY0H1bNm1WUf+l0\nOg7nkBHN0dGxyILtnDlzaNu2LZUrV+aTTz7Jsm/Pnj107doVCwsLrKysGDduHAkJ2WfB588z38NN\nC7jHgfZAHLAV+BL4CugupbwhhKgDeEspm+dwvrqHoBSLhw+1Fq2np5aScezY/E31SR+JnHA+BJu7\nkQyKuEaoqQHbajYmrll3Yhq3ZMixDXQ8dw6fRnVZ1/IuJxvepG/jl5nbx532ddrnez6smjer5Fd5\nv4er0+lYvXo1PXv2LLZr/vDDD9SuXZuVK1diZ2fHnDlzMvZt3LiR6tWr061bNxITE3FxccHGxoYV\nK1bkWFae309uSZbz8wDeQwu2N4Af0rbFZjsmJpdzc03+rCiF5fffpWzSRMohQ6SMjMz/eRHnL8qv\neveSf9WsKq+a6MnPm+jkuG6vypFzNsuFi76QETY2Uo4eLY//sFy2cjWVFtOR7/dFnjP7dxEBRSkK\nVIDFCzIvSpDO29tb1q9fP8txS5YskW3btpXm5uZy+PDhMjExMWP/zp07Zfv27aW5ubns0qWLPH36\n9BOvPXLkSDlv3rw8j9m6dats27Ztrvvz+n4K0qVsDryKdq/WCjAWQowAsv9sKp0/o5RyLTZWyxTl\n6gpLlsDmzVC3bt7npCQmsm7yf9muq4Vx2yZUvxDIqgYdWDTsU85N92BwfWPW/TCNGSQR+ssyXnnx\nBr3PT2fI5ThClsOy/dDirrac3trZs4vnjSpKBSKyLTi9efNmDhw4QFhYGIGBgaxduxYAf39/xo4d\ni6enJzExMYwfPx5nZ2eSk5MLXAcfHx9atWr1TOcWZNDUi0ColDIGQAixDXAAbgghLOW/Xco3cyvA\n3d0947mTkxNOTk4FqI5S0YWHhbFm1mzOBDRhf9h7DBpSibNnq1GtWt7n+W7czD+L5tDv0gWamBjw\naw1bfu/Zn3869aez2TUmrPgKs0v12IJkuH1Vjt75nKpHzPio2wzsvo/H/Y8/spSXMZpZUQqBt7c3\n3jms/lSUsge2ZyWLuNt68uTJWKYtRj1gwAACAgIA8PT0ZMKECXTs2BEAV1dXFi5ciJ+fH127dn3m\n6x08eJAffviBkydPPtP5BQm4V4DOQoiqQCLQC/gTiAdGA58BbwLbcysgc8BVlPzILVdxeFgYC3uM\n4lr4NMJpxK/0Y++xm8RGH6Ratcfvhd4MDWPLOxPodOoPGiYmcrxuA+bbvcSDdoNJMoxizOGtLP1k\nHb7VqhHqsYzZP00jtOltrOLgi21wMsWcPgNfZG1dH5WYQilS2Rsj8+bNK/JrFnWgLCzpwRbAyMiI\nqKgoAMLDw/Hy8sLDwwPQ3k9ycjKRBfgh7Ofnx4gRI9iyZQu2trbPVMYzB1wp5UkhxC+AP5Cc9u93\ngCmwSQjxFhAODH3WaygVV06BFXLO8PTOvoNMdDnCifBtTOZLfmUwVUimcwgsmT07Y0rNo6QkNs2Y\nhfHWtXSPuoWFZU08G7SndvOX8OvuhPX5HfxvzVTqxsYCcMsQ9tvd5Yuzb9O7Wio/bgWHq1r9+hPK\nktmztRHGfn6PJ6ZQI4wVpcRYW1szc+ZMZswonEky/v7+DBw4kLVr1xaoJ7ZA83CllPPQ1tLOLAat\nu1lRnkn2tIrBwKTt2xEGBmyIjs6Sq3hoSFW6v5CCSLbjKI4053xGOelduye2/MrZBTN5+cI/NDQ1\nYGvNxhzv8TKh9kN49Lwh7b9bzraJCzitp0fdxERCzOELB9jQBoacg76bjNkeFpeljioxhaIUnaSk\nJBITEzNe6+s/Xahyc3Nj0KBB9OrVi06dOpGQkICPjw/du3fH2Dh7mhlISUkhJSWF1NRUkpOTSUxM\npHLlyujp6XHmzBlefvllPDw8eOWVVwr2xnIbTVXUD0rxKDmlZLmPGCHjtcyL8jLID0DGg5yTtk2C\nfICBnMl8WZObsl/TpXLOGyMzzpEg/Svry0+sreXJtFHGnzXRyfEv9JHu41bJrj8dlYt37JBXnJzk\nnUqVZGLaOWPqIQe+jqwxDflxT2SUiXbdQTY2WcqWafVxHzGipD8qpQKiAoxS1tPTk3p6elIIIfX0\n9OTs2bOlt7e3tLa2zjhOp9NlGc3s7u4uXV1dM17v379fPvfcc9LCwkJaWVnJoUOHyvj4+ByvOXr0\n6IxrpT/WrVsnpZRyzJgxslKlStLU1FSamJhIExMT2bp161zrn9f3o3IpK6XO3B49mJc2SGQeMBWt\nRZn+/BRdccOT1pzhMybx44ieWs7hXr2wv/8QI/2HdL+t5TL2rt6IpjU7sLfTK0TXDufNPTsZ4+3N\nbRMTtiUnM+HePY40gcVd4JI5NPaDjX9DnaT8rz2rKMWpvM/DLetULmWlTMmc6jCVfwckvYYZL/A5\n0bzCV0yiD78yxdoas4gIdvfuzgc3IrhsYsAWs0b46ZxItenHafuqWB3bziyPETjdu0c88A3glpKI\nbxtY5wD6qfDhUS31YmQqvGNpSZuWLbN0D9dX3caKohSQauEqpU7me7hLgA+AfQxmMsvpyXbqMIPo\nuoY0rFGTV66HYvXwIRvqNuRCDR3NLLqzqdvzNLt7hHG7d+EYFEQ02kg+A2CGAZjYwzedweAWLDsG\nA0Ig/edo+io9KnexUlqpFm7pltf3owKuUiqlj1K+ci6Og6fHYfRIx3e8zc3aoehVSabnrRj21qnJ\nb9V1tDdrSbCuLyc6JjP88A6sjx1jWFRURhCdC4wzheWd4Ss7GBiitWirXwcPtK5q1VWslBUq4JZu\nKuAqZc6jR/DNN+DuDm/22oRtwEcMjAgn3KQKm2s2Rt+yMRbGvdjUqxFtr//BuF27cDhzBgEctLLi\nhL4+U65cIaw2jHCAq83ANRCEHyy88283dTiwCgi3tMT2xRdVDmOl1FMBt3RT93CVUiO3xBWZnTkD\nk966Sqe4yezV24/VngdsqNOQWe260bt+X261cuBsj6o0+H0f+2bNwDI+PuPcBOBoDydavDsAu20f\nEFn5Nu1uN+L1LXH871IEt4HZwHy0oFsTeGBry3zVqlUUpYipFq5SKPITSLPPr83ehXs/LolFwxfR\n+vRK+tyKYr9lDfbXtKV97TakNh7ON45VeKFtLcbfuUPnlStJ/fVXTklJm7g4DIF7AkY6WXJlkCX3\n5QOmOkxlVLtRVNWvmmU1nnvVqqEvJUZxcWplHqXMUS3c0k11KStF6kmBNN28kSOZ+uOPj6VB/Ozl\nl2kQFcUrF09zzbgKG2s2RtZpQNe6w/ihgy2hDpUZZ1mLEX/8gbmHB0RHw3/+A2+9Rfi9e3w3Zwb+\n0h+/hpHo6jVlVo9ZODdzppJepWL+JBSl6KmAW7qpLmWlSK2dPTsj2ELanNmQkCxpFQFSIyIyjrmr\nL9hsWZc2ibFMOLKPDXUbMr2tA32r2vHI0oFtw+tyr5klHwGd1qxBfP89dOyo3dR96SWoVInb92+z\nNsiLVe1+p3P9zux0+J4uDboU75tXFEXJJxVwlWeSuQs57Nw5sidLy2nFHL26ddhZqyayyiO634rF\ntFIi31i3omX1RlSJ7cj1PjYsqXeXN3dv5IP/BuFmY4NRYCC8+SYcPw6NGwMQGhvKF8e/YEPQBga3\nGIzPaB+a12xePG9cUZQS4evri5ubG8HBwSVdlWeXWwqqon5QilOLKXm7HBoqP7C1zUh3OCst1WFu\nqQ/P//ab9OrcQV4z1pN+tavK/7ZoJSf06CVX2y2X7zv/IC027pCjP/hAnmzWTKamnZ8Ccvvzz0uZ\nkJBx3T8j/pRDNw+VNT6rIT86+JGMvPcUK8orSjlBBUjtmNMC9EVp9uzZsk2bNlJfXz/PBejHjBkj\nhRAyJCQk12Py+n5UC1d5atm7kN8m68jfBGBeIxuaJ8Rz0tKE+gn3uVGnIdPsuvCKgR0pMYM4+MpD\n/KpdYcKuTcx86xAf3L+f5RqVgFNGRgwwNGTfxb0sPraYSzGXmNJ5CqsGrMLUwLQ437KiKOVYkyZN\nWLx4MStXrsz1mKNHjxIaGlqgtYL1nvlMpcLKfC8WoCEwGRhVuxbudnbsa1CTjyMuY/K3L1/Xa8ma\nrv2pZTqFmzafMnbSYKJW1OHF8F0cGz+O8Tt34nb/PknZrhFbCU43e0jblW356NBHvGX3FiHvhTDl\nhSkq2CpKBeTj44O1tXXGa51Ox9KlS2nXrh0WFha4uLiQlPTvX5Jdu3ZhZ2eHhYUFjo6OBAUF5Vq2\nq6srffv2xcTEJMf9jx49YtKkSXz11VfpPQTPRLVwK6D8TOHJS+ZcxwAXTQw5YV6NL2NvcS3iHj/V\naMK+bi3pF/8Clkl1WN2/OVf1zXltly//nbaaaX17Y/Lzz1yuWpX6Dx8SCyxHayE/MoCv7OHTLpVo\nZytY+uJSejfqXaBflYqilA/Z/w5s3ryZAwcOYGBggIODA2vXrmXcuHH4+/szduxYdu/ejb29PevX\nr8fZ2ZkLFy5QuXLlp77uF198gZOTE61bty5Q/VXArWBynMLj5/dU6QxHz5/PJ8d9afcwiSZJd6j/\n4CHXDSz5oMMLDKjWnFYhLfCvb8Bw50ZU9alB++U3OHFxALW4jQT+ko947vffqWxkxGdpc2Pjqlem\nR61QztS4jO3Demwc6kG/jgOK8qNQFCUXhfX7tqhnHk2ePBlLS0sABgwYQEBAAACenp5MmDCBjh07\nAloLduHChfj5+dG1a9enusbVq1fx9PTk77//LnB9CxRwhRBmaJnxWqMt7PIWcAH4Ga2n8TIwVEp5\nt2DVVApLfqfw5ESmpOC9fBlx333JjIhrHLCsgUftVthaW9Dtvgtmjh2Y1uwqlnciqbbLBDOXRux7\n6EJ7AjLyGgtgV/36PNeqFQ2B15d9zJJjS/j1n18Z2XYkP3eegs5CJaFQlJJUVqbopgdbACMjI6Ki\noq/pP8AAACAASURBVAAIDw/Hy8sLDw8PQBscnJycTGS2mRP5MWXKFObMmZNrd/PTKGgLdzmwR0r5\nuhBCH+3v98fAb1LKz4UQ04EZwEcFvI7yDHLqOs5+/xWyTuHJfM49MzP0pcT0ZhS1YqMZEBGOiZE+\n26s3YZuTNcNSezKywUB+flGPwfUScLE0pNWksyQeeJ45fEJnvKmCJPOP5QRAWNXlSPgRPj/6OX9F\n/sW7z73LxUkXqWFUo5g+GUVRyjNra2tmzpzJjBkzClzWoUP/3959h+d8vQ8cf5/sgSyyKojYYsaM\nkRit2EWt2iNqpJSKVtX6tbQVo6hVlFit0W+NoEI1UTM1a8QWESKIGAkyPOf3xyeISEiCDDmv68rl\nGZ95nnjunHWfv9izZw++vr5PX6tbty4zZ86kS5cumTpWlgOuEKIQ0EBK2RtASpkE3BVCtAU8kjfz\nB4JQATfbpdd0LCpWfK7/leT39Bwdn9vnqj6stbPj/YR7OD18xEr7Ygyr7MaH1u/R+aAbIbY6ug2s\nSPHiCXzq7MzoeGvWddlKx4ObqctYjEkkAviRZ6vxPE29WPckqzb25/O6n7O241pMDU2zvXwURcm9\nEhISiI+Pf/rcwCBzocrb25v27dvTpEkTatWqRVxcHMHBwXh4eGBunrrKAUlJSSQlJaHT6UhMTCQ+\nPh5DQ0P09PQ4d+4cOp0O0GrKDg4OBAQEULly5Uzf1+uMUnYGbgkhlgghDgshfhZCmAF2Usqo5Iu7\nDti+xjmULEqv6ThJCMa7uBCX/PqTNIy9v/mGpWO+ovndaHY6WmGnD2UMkpj1XkUWNKmNq2kf+oR7\nsbVsA1rNd+V8lZts+dqXLtNmUGHgEkxcSvPx1SmU+a4940o6EQcUBfoBnSzN+aCjM86jC3KlgwPj\nGo8jdEgon9T4RAVbRVFe0LJlS8zMzDA1NcXMzIyJEye+sM3LBlK6ubmxcOFCfHx8sLa2pkyZMvj7\n+6e7vbe3N2ZmZvz2229MnjwZMzMzViR3sRUuXBhbW1tsbW2xs7NDCIGNjQ3GxsaZvq8s51IWQrgB\n+4G6UsqDQogZwH3AR0ppnWK7aCnlC22FQgg5fvz4p889PT3x9PTM0rXkZ+mNOB7fqBETg4Je2H58\no0b0Xbz4aSJ/PUdHmrdrzcVZM6l/KIRIcwNWWZfmhpMxH+vcKHyoKPtqJDChX2Vcb17He/NmOgYF\nYRYfzx0KIZDsKtKRCnOG4NKx+nPXFHvjMgfL3ePEe1epW8wdX3df6herr0YcK0omBAUFEZTi//LE\niRNVLuVc7K0sXiCEsAP2SSlLJj+vj9Z07AJ4SimjhBD2wN9SyvJp7K8+8Cx6EtBizp/n3smTzI6N\nfWHRgKVjx6a5UMDUbt0Yv2IFD25E8ufoL3lvyx8Uux/LKvvi7La1p7W9OeX/fZ8ggxhCuhXhn5ql\n6LZjB0kBAUwJC9P6exHcwZIfGcaZqjb8esgHvRRtJZdiLjF933RWHl9Ju3LtGOk+kvJFXvgVUBQl\nC9TiBbnbW1stSAgRDHhLKc8KIcYDZslv3ZZS/pA8aMpKSvlCH676wLMmZT/rVGAkaSymbmuLTd26\n6I4cYXJ4+LNgXNKZRr16oFu1lIaXwtlub83/LF0oUd2cpmHN0LNpwFyjYP5pV4ai1yOx2bSJFcHB\nFE5IIBowBBKw5Gu+4yp2OBafxld/L386nejQtUP47fVj+8XteFf3ZmjtoTgWdMz+QlKUd5gKuLnb\n2wy4VdC+4w2Bi0AftKx8awAntBjQSUp5J4191QeeBSmXuBuPNiAJtIKezbMBSnHAcCcnLKpVo9DN\nSIrE3KZVRBhRpvosty7LzbIF6WLuin1UJ843tOBbxzOEOVpS9swZqhw/TvGwMBwePaLBpUsUv32b\nQ87l+NVpPiv3lqNZyVlUrHKePt9+Q7ESJdh2YRt+e/04G32W4XWG0796fwoZF8qZAlKUd5wKuLnb\nW1ueT0p5DKiZxltNX+e4SvpSTuvRg6cjjpfyLNgC6OmDV9IjHPf9RYkHD1hpX5zBrjVp7GJAy6Od\nMLSuze5OhehrfgOT0EP83+q1dN21i8eJifxlY0PzAgUwtLaGSZPYVKArgz43o0VJuLwWrKwmkPg4\nkd9O/Ibf/LYA+Lr70tm1M0b6RtlcIoqiKHmDyjSVx6RMq9ibZ7VcHVp7/gHrgtw0MaB+dAw6w8fM\ntKmIQ0193r/kjsdeK67eu8r3PRPZV/463e0M6Lx4NT/MnftcX2+L6Gj83dxo/suf+HwqOHkSVq4E\nDw+4H3+f6fsWMmP/DEpbl+aHpj/gVcpLDYRSFEV5hddqUn6tE6smjSxJPb82FPjeuhDVC5jRPvom\nUeb6rLQsy9USBnQrYIvx4U5cLPKAiNIXWdyhPhUuX2ZAQADN/vmHOR06UO7YMTqmWl9Sh6B1mWmE\n3B7OwIEwZgzEJEYy88BMFh1eRNOSTfF198XN0S1HykBR8jPVpJy7vbU+3NehPvCsu3zpEsu+GIlp\n6Cnq3bhCydgHrHIoxk4bR7xK38X1eDcKnnjEooYP+G9SB47ejmZwQAD9N2+mTETE0+PcNzLippUV\njlFRmCS/dpqy9ONnLts4svXvUhjYhzJ171T+OP0H3Sp1Y0TdESr1oqLkIBVwczcVcPOgtFIsmt29\ng6m+pOK1yzS4dIWddpasLVgG27qSNg8aYHLqA/41P8kmz0T216tENSsrhpYsyfGRI/ly+fLnmo0T\ngZ9btaLVrFnMfv99xlwIZzZfMpOhVC78EwM3lGfl5ZUcuHqAITWHMKTmEJV6UVFyARVwczcVcPOI\ntObX3gIWmOlTzcqaOndvc8NMn+WWZQkraUafqsWwDmiDTb0anGxnwpwitzn98CG97e3p7+BAKTMz\nuH+f6BkzSPz2W2wTE58OtHoyX7e4szP/W3eNgf11WBiGUa7lZK7WvM69x/cY6T6SXlV6qWxQipKL\n5NeAu3v3bry9vQlN1QWW26iAmweknl/7qR4csLPBIvEhLg8estKhGDss3+P9Ctepfq4jMZcrUuz/\nGvJr7XiW3rtJ1QIF+MTBgTaFC2OkpwehoTB3rjbaqXFjrnfowIKAAHSRkeg5OtL7m2+wsnFm9Gj4\n38aHtPp6GUEJU7E2tWaU+yg+LPch+nr6OV0siqKk8q4HXGdnZxYvXkzjxo2z7Zzjxo1j/fr1hIaG\nMnbsWMaNG/fc+7du3WLYsGFs3rwZfX19WrRowfLly9M81lubFqRkTHrpF1O+fiIsDP+wMM5Ym+Nq\nYohe9F0eGupYaumKtWsMHRPLUzuoDYWP/MuROhf54rvyJJS+Sh9zew6Uqk5JU1NISoING2DOHDhx\nAry94b//oGhR7IHxXbs+vaaNG2HgiGgc2sxFN3QOkQVrsLjeYhoUa6BGHCuKkq+ULl0aPz8/5s+f\nn+b77du3p3bt2kRERGBqasqJEyeydB4VcDMhvcD5qn3SWrWn3S+/8Effvky8cIH7pnpssLYmpoA+\n6Cew29yZpeVc6F3sId77OmOzNg5pe4jlXY+yvFlzqp05Q51z51jRoweGenpw4wZMnw4LFoCTEwwZ\nAh06QBrJtSMjoe+IS+xjBro+K6ji+iHL3XdSoUiFt1NoiqIob0BwcDDdu3fnypUrgFYT9vHxYdmy\nZYSHh+Pl5YW/vz9GRlougICAAMaOHUtYWBgVK1Zk3rx5VKpUKc1j9+jRA+DpggUpbd++nYiICIKD\ng59WRqpUqZKle1ABN4PSC5xP+kHTk96qPd69utPtURwnCptR4sFD4kzMGVCkLJ7VLvNhWD3u766B\ny+MgttffzQ9DGqFz6E7fP/8keOBAlpib82lgIIYHDmi12c2btQC7YQNUq5bmdeh0MG7eYabu9UOv\nbCCDavdnRL3jvFfovTddVIqiKG9F6ta3tWvXEhgYiLGxMe7u7ixdupQBAwZw5MgR+vXrx+bNm3Fz\nc2PFihW0adOGs2fPYmhomKlz7t+/nzJlytCzZ0+2bt2Ki4sLfn5+NGzYMNPXrwJuBqUXOKeOHcv4\nNP4qeiL1gu9Hrcy4ZmrM3MgIdtpZMt2yMuaVbtFZ34FKf/fG/uhZkuwPs3CAEWve70TRyEiqnz1L\n6Z07Mbh7l/UeHnxZrRqFO3aEu3dh8GCYPRusrNI8v5SSxcGB+P7hR5zpaT5t/xnjWy1QqRcVRUmX\nmPhmupXk+LfbTzxs2DDs7OwAaN26NUePHgVg4cKFDBw4kBo1agBaDXbSpEns37+fBg0aZOocERER\nbN++ncWLF7N06VLWrVtH27ZtuXDhAtbW1q8+QAoq4GZQ6sAJWtDVXbv2wrap+2YvmwiO2lhT7e4d\ndAaJBJqX4scypelTKopPD32AyTpbCuvvYLXHeqZ824ZEW3eqnT/P3+XLU69NG+2gFy7AvHmwdCnE\nxsK330KzZjy3TE8KiY8TWXlsNV9u8uPWLR1divmycFgXTI1U6kVFUV7ubQfKN+VJsAUwMzMjMjIS\ngMuXL7Ns2TJmz54NaBWPxMRErqXxff0qpqamlChRgt69ewPQuXNnJk2axJ49e2jdunWmjqUCbgal\nTKn4RByg5/j8ajhPmp6/vniBg7ZWNH8cj6keXDQpwFyb8tSteZZ6UaXx2tWRi3v2U8Lub/x9qrKq\nSW8sw8L4vm5d+pQvj4GentYOvHUr/PQThIRAr17avyVLpnud9+Pvs+jwIn7YNYPY8FJUuPM9ByZ7\nUby4GgilKEr+4OTkxJgxYxg9evRrH6ty5coEBAQ891pWB5amXT1SXtD7m28Y7+JCXPLzJ3NZe3/z\nzbONdDpWe/ei8cNoMBLcN4YfbSrzWXMXLJsUYWTcx5RZ+gVVt93jWp0VLPixNI2nf8rhQoXotWUL\nO9u0wbtiRQzu3IFp06BMGfj6a61/9vJlmDo13WAbeT+S0TtGU+JHZ+Zu2kfSyv+xuOFODqxoroKt\noih5SkJCAvHx8U9/Hj9+nKn9vb29mT9/PiEhIQDExcWxZcsW4uLi0tw+KSmJR48eodPpSExMJD4+\nHp1OB0C7du2IiYlh+fLl6HQ61q1bx9WrV6lXr16m70vVcDOouLMzn27fztSxY9Fdu4aeoyPtBgxg\n6dixGF46TeG792gWHsYHJrDUshwzSpalV+WLfHasGrr17pxxusBWjzDut07Ap5EPDY4fZ/Ly5TQI\nCWFG165aP/CRI9C/P/z+O7RqBStWQO3a8JK/pk7fOs3UvVP5PfR33At2w3hZCA3rlMQvCDLZvaAo\nipIrtGzZEtCagoUQjBkzhiZNmjy3zctqmW5ubixcuBAfHx/Onz+Pqakp9evXx8PDI83tvb298ff3\nf3rMyZMns2TJEnr27ImVlRUbN25k0KBBDBkyhHLlyrFx48ZM999CPk58kZUpPildOHaYPzq0pf69\n25SMe8hKx2IEGJegqud52l0vg25nH2xizmNpuIP1Ta2Y3LsnscbGfP6//9F3yxaK3rr19Fjrypfn\nI0tLiIiAgQO1oGtr+9Lz7wnfw5S9U9gfsZ+e5QZzduUQTh0szM8/Q6NGWS4WRVFyuXc98UVepzJN\npZLmFJ8UqQ7TpdNxYtUvREz1o87pcwTZWfCrSXlkhdv0t4vHfFdvoiMqcMVoB1Us/mVFW0/WeHpi\nc/48Y7y8uPTdd4xaseK5fmAdcMneHpd587RarUH6jQ6PdY/ZeGYjfnv9iIqLYkSdz+FobyZ+bUb/\n/jB2LJiqLIyK8k5TATd3e6sBVwihBxwEIqSUbYQQVsBqoDgQBnSSUt5NY78c+8Andu/OyJUrXxgA\nNbVbtzSn+EQfO8i+r7+icnAQMcYSf+sKHLYpSM+ax3E99QHx/3TCLH4/243/Yn3rMlz6qAOPTEyo\nfvgw1SMjGTp6tJZZ6uJFttavT//ISAzQFhCY7uREl+Dglwb6R0mPWHZsGdP2TcPC2IJR9UZRUa8d\ngwbq8+ABLFoElSu/6VJSFCU3UgE3d3vbAXc44AYUSg64PwDRUsopQogvACsp5Zdp7PdGPvCMNg2n\n3O7SqVMsu3HjhW3GN2rExJ07AUi6G8PuyRMwWbmckjF3+NXBifUmJXHzCqNDSS/u+9XAOiyW99jK\nwVJXGdO2GVc8PGh05Ag9AwLYFRPDsMBA7Vru3YPly2HOHBIeP2a7hQVHTUxIKlbspU3Ztx/eZt6/\n85gdMhs3Rzd83X2p6+DB1KmCGTO0Gq2PD+irlMeKkm+ogJu7vfTzkVJm+QcoCmwHPIGNya+dBuyS\nH9sDp9PZV76usIsX5ecuLjIWpAQZC/JzFxcZdvHiS7f7OnlbmeInFuSEjz+Wp1b+Iv+sVlbGGAu5\nvpil/Ki0u2zXurzcPLSI3Oc2RG41XCcPDzgkjwSGyqZ9+8oqCxZIy1Wr5Phu3eQ1a+vnjvdTq1ZS\nDh4spZWVlB99JOXff0up0736vmLC5LCtw6TV91ay9/re8kTUCSmllPv3S+nqKmWLFlKGhb128SmK\nkgclf3e+7nd3tl93fvGyz+e1arhCiLXAJMAC+FxqNdwYKaVVim1uSylfGM71Jv7CymjTcOrtLgMz\ngW/Q5tUeMobdRQrT9s4d7prAUusK/GthQTOPozQ7W5P4nf2xjL2IPVs5WTaGEcOHcqliReoaGWHz\n+++wcCHLY2JeuL77RkYU/PJLGDAA3nt1CsUjkUfw2+vHtgvb6FetH8NqD+O9Qu9x/z6MGQNr18KM\nGdC580sHLiuK8g5TNdzc7a2sFiSEaAlESSmPCiE8X7Jpup/qhAkTnj729PTE0/Nlh3nRq7I/PWlG\nvhAQ8Nx2xQEfAVOK2dH0YRxlY+PAxJw+RV1xrXcCN/0HdNrRgZip3tizBTPTUaxp68bPrXpz39SU\nEqdPs7VPH+yMjKBWLSZevUpcqsD/CPixfXvGTpz40nuQUrLj4g6m7J1C6M1QhtUexryW87AwsQAg\nIEBbi6BJEzh5Uk31UZT8JigoiKCgoJy+DOUNeJ15uPWANkKIFoApUFAIsRy4LoSwk1JGCSHsgRc7\nS5OlDLhZ8bLsT6nXl32y3RkLI8LMzKh9+y5u8hEzLasQXz2G/hWvMO3fWkT7L8A+6V+K8ju+5RL5\nrU1rNtWfy/uHDjFt3jxqHTnC9I8/1oKtlLBvHyPv30fo6ZGo02FIilHPkyene+2JjxNZc3INfnv9\nSNIl4evuS9dKXTHS11IvXr8Ow4bBoUPwyy9awFUUJf9JXRmZ+Io/4pXc641MCxJCePCsSXkK2qCp\nH97WoKknNdeY8+e5d/Iks2NjX5jes3Ts2KfNyKFGsMvGimb373PPBJZaVWBvAUs6dgmj+KXSOK7t\ngHE0OLCVn8wP4PBBPRa2bMl9Y2OKBQSwdNs2nO/ceXr8oRs3UmzfPi3lYmwsDB7MlUaN+GXq1KdJ\nMdIbDBWbEMuiw4uYsX8GzpbOjKo3iualmj+dcC2lFmBHj4a+fWHcODAzy1IxKYryDlJNyrlbdi9A\n/z2wRgjRF627tNObPHjqObShQNcCBSjn6oq5iwufJgc6vfAwjtgWxFD3mJJxD4k3L0TPAtUoXe84\njUrCRye7o/MrT5jNUUpFT+WcqxmjW7Vidb0BtD5wgB/nzMHz6FGuSMki4LKdHTVq12ZckSIUatgQ\n6taF776DDz4APT2c4KWrBl2Pvc7sA7NZcGgBjZ0bs7bjWmq9V+u5bc6d07p7Y2MhMBCqVn2TJaco\nipJ37d69G29vb0JDQ3P6UrIuvdFUb/uHLI6Sm9CtW9ojjLt1k/LxY3lmjb/cVqOivG0s5IZiFrJj\nqXqyWePKcs2XRvJgs6YysNAf8uj7R2XUb1Hy5r1HctzRo9JmxQpZ2t9f+nXqJHdbWMheBgbPjXxe\nZG8vH3h4SFm4sJSjRkmZahT0y5y+eVr239BfWn5vKQcHDJbno8+/sE1CgpSTJ0tpYyPl9OlSJiVl\nqWgURckHeMdHKZcoUUL+9ddf2XrOsWPHykqVKkkDAwM5ceLEF96fNWuWdHZ2lhYWFrJmzZpy9+7d\n6R7rZZ9PnsulnNZAqSQjcDy0hzBrU+INdWyzLs9XZTzx8tqH780oZGBPknaW4ZDVARptqsD1yrZM\ni4xk49EztLCxYU716oROmsT9mzfZ0aoV3gMG8NPs2VQ9coRaUVF0L1IE4969teHBGUzltCd8D357\n/dh7ZS+Daw7mrM9ZipgXeWG7kBDw9gZHRzh4EEqUeO0iUhRFUTKhdOnS+Pn5MX/+/BfeCwkJYfTo\n0ezevZuqVasyf/582rVrR1RUVKZXDcrRgDuxe3eaDhjAjp9/Rnf1KvcsLDCQErN799JNYvFkoJQJ\ncKCIOfpISsY+5FGSjp6OjXCucYxe9mH0CKrAnWm/cVjvOHdLnSX6g8sYffYpbePvIM/eYYCDAzNK\nlcLG0FA78JPm4MOHYc4c6u3YAa1ba5klaj3f9JsendQ9Tb14PfY6I+qMYFWHVZgZvtgJGxurLQS0\nerW2MFDXrmqqj6IoSnqCg4Pp3r07V65cAcDZ2RkfHx+WLVtGeHg4Xl5e+Pv7Y5S85ndAQABjx44l\nLCyMihUrMm/ePCpVqpTmsXv06AHAijS6BcPCwnB1daVqch9fz549GTJkCDdu3HhuPd6MyNGA22nl\nSn5YvZo5SUncQpsbOwGeDYDav//5/MY6HV71qrN11580vXGbaFN9lhlX4ka5h/TpdYehR0xwXDYB\ng/tG2LCF8vTiUaUSjPLxIbxsWVobGTG/eHEaWFg8/5dJfLw2yXXOHLh2TVtA4MyZVy4g8MSjpEcs\nP7acafumUdC4IL7uvrQv3x4DvbSLd8sWGDQIPD3hxAmwsclyESqKouQbqWuUa9euJTAwEGNjY9zd\n3Vm6dCkDBgzgyJEj9OvXj82bN+Pm5saKFSto06YNZ8+exfBJJSuDmjdvjp+fHyEhIdSoUYPFixdT\ntWrVTAdbyOGAuwaYk5SEOTCVZ4koSP534oULTB07luFfj+LA+K8ptW0b5gaP2W9TkcmlqtG85T7a\nF3xIiVOf8XikM+GmIbjen0l8wass82rGwlaz0NfpKHrmDEG9emGduqCvXIH5858lI/7ii1cuIJBS\nzMMY5h3UUi9Wd6jO/Fbz8SjukW4zQ1QUfPaZ1oy8aBG8/37Wyk1RFOWtelPNbW95JPSwYcOeBr7W\nrVtz9OhRABYuXMjAgQOpUaMGoNVgJ02axP79+2nQoEGmzlGwYEHat29P/fr1AbC0tGTr1q1Zut4c\nXYBex7MAm/IxwGPgeBFzGu/YSEL1qpw5/B+9HZrwQ4vieE23wr9vW5qv3ETZPxdS4v2m1AmvQ2C3\nSPp93YHSq1ZyrFQpFvv5caBPH9yvXHkWbKWEv/6C9u2hShWtbXfXLti+HT78MEPB9vKdywz/czgu\ns1w4E32GwO6BbP54M54lPNMMtlLCkiVQqRI4OcHx4yrYKoqSiz03LPU1ft6ylLVMMzMzYmNjAbh8\n+TLTpk3D2toaa2trrKysiIiI4FpyUqTMWLRoEUuWLCE0NJSEhASWL19Oy5YtuX79eqaPlaM1XD2e\nJaR48jiqoD6XzAtQPeYeN031WW5cmRsVDOjW5TRDLj7CePWn6NaWIamnPdV2lCWupCH+16+z8NQh\n9D78EEd/f0507UrR2Nhn83K/+UZbQGDZMpg7V8v2P2SI9rxAgQxf77Hrx/Db68eWc1voV60f/w36\nj6KFir50n/Pn4ZNP4M4d2LYNqlXLenkpiqIor+bk5MSYMWMYPXr0ax/r2LFjtG7dGhcXFwCaNWuG\ng4MDe/fupX379pk6Vs724QJDDAz4QS8JNxsLomLjeGAk+bNQMXwL2NK41T7aWIejv6sdNoNGUER3\nnKKsw5gD9LvajIf3xrH7QDztihTBv1w56hQqRLijI4sfPHiagGJEjx44+vnBb79B06Ywbx40bJjh\nJhOZnHrRb68fJ2+eZFjtYfzU4icsTSxful9iojYYaupU+OorGDo0wy3ViqIo+VpCQgLx8fFPnxtk\n8svT29ub9u3b06RJE2rVqkVcXBzBwcF4eHhgbp56ngskJSWRlJSETqcjMTGR+Ph4DA0N0dPTo2bN\nmkyePBkfHx+cnZ3Zvn07586dw9XVNdP3laMh4K96Vfkk/Ar6t24TZl6Iyea1sa90lub1wqgQZE75\n+VPhUSFOsBUv+pJg9ZhfmjVjYatBmD56hNOOHYT5+mKZom+2uLMz45csgQ0btExQffpo826OH8/Q\nAgJPJOmSnqZeTHicwMi6I/m40scYGxi/ct9//9VOaW+vPX7ZmvaKoijK81q2bAloFR4hBGPGjKFJ\nqvy2L5uS4+bmxsKFC/Hx8eH8+fOYmppSv359PDw80tze29sbf3//p8ecPHkyS5YsoWfPnvTs2ZOL\nFy/i6enJnTt3KFq0KD///DNlypTJ9H29kdSOWSGEkBuLFWKZSVUi7Y35uPMRGpbuxuM173PT3wC7\nxGCKsoVCHKeXWzXiW7Vie40atP/nHz7ZtIlaoaFMSLF+LaAlIF64EBYs0KKcjw+0awfJw8QzIjYh\nlsWHFzNj/wyKWxZnlPsompdujp54dXd3bKy2Ru2vv2q1248/VlN9FEV5s1Rqx9wtu1M7ZtiuTsZ8\n3sqCQkc/5M6UsTyyMcWhnwMbQ/vhfSqE+c2bs6iFL3EPH/Llpk0smjoVi7g44NkiBUgJe/ZoU3r+\n/BM6dYLNm7UBUZkQFRvF7BAt9aJHcQ9Wf7Sa2kVrZ3j/rVu1qT4NG2pTfQoXztTpFUVRlHdcjgbc\nj8+sIW6JAYZd7ai03h7zKgXYHhPDr6admV54CB8FB/Prt99ie/o0s1JcbBzwrbMzI11dtVFIDx7A\n4MFa/6zly/tWUzsbfZZpe6ex5tQaulTswt6+eyltUzrD+9+4oU312b9fq1yr0ceKoihKWnK0Sfn6\nqusUbleYKJHEL5GRLIqMpLChIe2NjLjetSvfnTr1NAnGcCcnLKpVo2hUFLXu3qVWVBT69etrGe37\nowAACsJJREFUzcZNm4Je5mY47buyD7+9fuwO382gGoMYUmsItuYZS3QBWsV62TIYNQp69oQJEyCN\nvnhFUZQ3SjUp524v+3xyNOBuvnWLn69dI/juXToVKcIAR0fcChYEni3Bp7t2DX17ewY2bozt779r\nC8T27atlg8pk4mGd1BFwNoApe6Zw7f41RtQdQZ+qfTA3ylykvHBBm+oTE6PVaqtXz9TuiqIoWaYC\nbu6WawNuzYMHGeDgQBdbWwqkNew7OhoWL9aaiosU0ebOdu4MJiaZOtejpEes+G8F0/ZNw9zQHF93\nXzpU6JBu6sX0JCbC9Ong56etVztsmJrqoyhK9lIBN3fLtYOmQtzc0n7j0CFtSs/69dCmjZbhP4ML\nCKQU8zCG+QfnMytkFtXsqzG3xdx0s0G9yqFD0L+/FvdDQqBkyUwfQlEURcnHck/9LD4e1qzRRhtH\nRmqDoM6dy9Jw3/C74czYNwP/Y/60LtuawO6BVLJLe5WIV4mLg3HjYOVKrWbbvbua6qMoSt5mYmIS\nJYTIfPZ95ZVMTEyi0nsvywFXCFEUWAbYoaVCXiilnCWEsAJWA8WBMKCTlPJuugcKD9cWEFi8WJvK\nM3q0toCAvn6mr+m/qP+epl7sU7UPxwYew8nCKQt3p9m2Tesqrl9fy5tR5MXlbBVFUfKchw8f2uf0\nNeRHWe7DFULYA/ZSyqNCiALAIaAt0AeIllJOEUJ8AVhJKb9MY38p27aFf/7Rqo1DhkAWMndIKdl5\naSd+e/04fuM4Q2sN5ZMan7wy9eLL3LgBw4fDvn1a93GzZlk+lKIoyhv1JvpwlZyR5RqulPI6cD35\ncawQIhQoihZ0n+TP8geCgBcCLgDNm2sLv2diAYEnknRJrD25lqn7pvIw8SEj3UeyocuGDKVeTE/K\nqT49emi1WjXVR1EURXkT3sgoZSFECbTA6gpckVJapXjvtpTSOo19sjRKLi4hjsVHtNSLToWcGFVv\nFC1Kt8hQ6sWXuXhRm+oTHa1N9UlvPJeiKEpOUjXcvOu1B00lNyevA4Yl13RTR9F0o+qECROePvb0\n9MTT0zPd89yIu8HsA7OZf0hb5P23Dr9lKvViepKStKk+U6Zo688PH66m+iiKknsEBQURFBSU05eh\nvAGvVcMVQhgAAcBWKeXM5NdCAU8pZVRyP+/fUsryaeyboRruuehzTNs3jdUnV9O5Ymc+r/t5plIv\nvsyhQ9qqPjY22rit5OUOFUVRci1Vw827Xq8dFn4BTj0Jtsk2Ar2TH/cCNmTlwPsj9tN+dXvcf3HH\n1tyWMz5nmN9q/hsJtnFxMHIktGih5UEODFTBVlEURXm7XmeUcj1gF3AcrdlYAl8BIcAawAm4jDYt\n6E4a+79Qw9VJHZvPbmbK3ilE3ItgRJ0R9K3WN9OpF18mMFCb6uPuDjNmqKk+iqLkLaqGm3flaGrH\nJ+eOT4pn5fGV+O31w8zQDF93Xz6q8FGmUy++zM2bMGKENgtp/nzw8npjh1YURck2KuDmXTk6POjO\nozssOLiAWSGzqGRbiTkt5tCoRKMspV5Mj5TazKORI6FbN22t2izMQlIURVGU15KjAddllgstS7dk\na7etVLar/MaPf+mS1nwcFQUBAVCz5hs/haIoiqJkSI42KYffCX+t1IvpSUqCH3+E778HX1+tKdnQ\n8I2fRlEUJdupJuW8K0druG8j2B45oq3qY2kJ+/dDqVJv/BSKoiiKkmmvOy0o13jwQKvNennBp5/C\njh0q2CqKoii5xzsRcHfsgEqV4OpVLf9x795qCT1FURQld8nTSQyjo7X+2eBgmDtXS2ShKIqiKLlR\nnqzhSqktCF+xIlhba1N9VLBVFEVRcrM8V8MNC9Om+kRGwqZNaqqPoiiKkjfkmRruk1V9atQAT084\neFAFW0VRFCXvyBM13KNHtak+hQqpqT6KoihK3pSra7gPHmhr1DZrBkOGwF9/qWCrKIqi5E25NuDu\n2AGVK0N4OPz3H/Tpo6b6KIqiKHlXrmtSjo6Gzz+Hv/+GefPU6GNFURTl3ZBrarhSwqpV4OoKFhZw\n8qQKtoqiKMq7463VcIUQXsCPaEF9sZTyh/S2vXwZBg2CiAhYvx5q135bV6UoiqIoOeOt1HCFEHrA\nT0AzoCLQVQhRLvV2jx9rq/q4uUH9+nDoUP4MtkFBQTl9CbmGKotnVFk8o8pCeRe8rSblWsA5KeVl\nKWUi8BvQNvVGderAhg2wdy989VX+XUJPfZk8o8riGVUWz6iyUN4FbyvgvgdcSfE8Ivm15wwcCDt3\nQpkyb+kqFEVRFCWXyNFRyv365eTZFUVRFCX7CCnlmz+oEHWACVJKr+TnXwIy5cApIcSbP7GiKEo+\nIKVUWQnyoLcVcPWBM0ATIBIIAbpKKUPf+MkURVEUJQ94K03KUsrHQggfIJBn04JUsFUURVHyrbdS\nw1UURVEU5Xk5kmlKCOElhDgthDgrhPgiJ64hpwghigohdgohTgohjgshhia/biWECBRCnBFCbBNC\nWOT0tWYHIYSeEOKwEGJj8vN8WQ4AQggLIcRaIURo8u9H7fxYHkKI4UKIE0KI/4QQK4UQRvmpHIQQ\ni4UQUUKI/1K8lu79CyFGCyHOJf/efJAzV61kRLYH3IwmxXiHJQEjpJQVgbrAkOT7/xLYIaUsC+wE\nRufgNWanYcCpFM/zazkAzAS2SCnLA1WA0+Sz8hBCOAKfAtWllJXRur26kr/KYQna92NKad6/EKIC\n0AkoDzQH5gqhlnnJrXKihpuhpBjvKinldSnl0eTHsUAoUBStDPyTN/MHPsyZK8w+QoiiQAtgUYqX\n8105AAghCgENpJRLAKSUSVLKu+TP8tAHzIUQBoApcJV8VA5Syt1ATKqX07v/NsBvyb8vYcA5tO9Y\nJRfKiYCboaQY+YEQogRQFdgP2Ekpo0ALyoBtzl1ZtpkB+AIpBxLkx3IAcAZuCSGWJDex/yyEMCOf\nlYeU8howDQhHC7R3pZQ7yGflkAbbdO4/9ffpVfLp92lekGtWC8pvhBAFgHXAsOSaburRa+/0aDYh\nREsgKrm2/7ImsHe6HFIwAKoDc6SU1YE4tGbE/PZ7YYlWmysOOKLVdLuRz8ohA/L7/edJORFwrwLF\nUjwvmvxavpHcVLYOWC6l3JD8cpQQwi75fXvgRk5dXzapB7QRQlwEfgUaCyGWA9fzWTk8EQFckVIe\nTH7+O1oAzm+/F02Bi1LK21LKx8AfgDv5rxxSS+/+rwJOKbbLd9+neUlOBNx/gVJCiOJCCCOgC7Ax\nB64jJ/0CnJJSzkzx2kagd/LjXsCG1Du9S6SUX0kpi0kpS6L9DuyUUvYANpGPyuGJ5ObCK0KIJ5nF\nmwAnyWe/F2hNyXWEECbJg3+aoA2qy2/lIHi+5Se9+98IdEkeye0MlEJLNKTkQjkyDzd5rdyZPEuK\n8X22X0QOEULUA3YBx9GahSTwFdp/kjVof61eBjpJKe/k1HVmJyGEB/C5lLKNEMKa/FsOVdAGkBkC\nF4E+aAOI8lV5CCHGo/0RlggcAfoDBckn5SCEWAV4AjZAFDAeWA+sJY37F0KMBvqhldcwKWVgDly2\nkgEq8YWiKIqiZAM1aEpRFEVRsoEKuIqiKIqSDVTAVRRFUZRsoAKuoiiKomQDFXAVRVEUJRuogKso\niqIo2UAFXEVRFEXJBirgKoqiKEo2+H8z7sKl1f/SRgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112329d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_predict_step= np.dot(x, theta_list.transpose())\n",
    "plt.plot(x[:,1],y,\"ro\")\n",
    "for i in range (0,20,2):\n",
    "    plt.plot(x[:,1],y_predict_step[:,i], label='Line %d'%i)\n",
    "\n",
    "plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1124879b0>]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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vQ2Jj+Sn6eNbvTWfuX9o4rvNcXJVx5iqf9XDKaDtCkbqhoC815msMfxDQr+dQ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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112487a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_predict= np.dot(x, theta)\n",
    "\n",
    "plt.plot(x[:,1],y,\"ro\")\n",
    "plt.plot(x[:,1],y_predict, lw=3)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3.0
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}