w4_basisFunctions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "jupyter nbconvert --to slides --reveal-prefix \"http://cdn.jsdelivr.net/reveal.js/2.6.2\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import pods\n",
    "import mlai\n",
    "from matplotlib import pyplot as plt\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### MLAI Week 4: Basis Functions\n",
    "\n",
    "### Neil D. Lawrence\n",
    "\n",
    "### 20th October 2015"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Review\n",
    "- Last time: explored least squares for univariate and multivariate regression.\n",
    "- Introduced matrices, linear algebra and derivatives.\n",
    "- This time: introduce *basis functions* for non-linear regression models."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Nonlinear Regression\n",
    "\n",
    "-   Problem with Linear Regression—$\\mathbf{x}$ may not be linearly\n",
    "    related to $\\mathbf{y}$.\n",
    "\n",
    "-   Potential solution: create a feature space: define\n",
    "    $\\phi(\\mathbf{x})$ where $\\phi(\\cdot)$ is a\n",
    "    nonlinear function of $\\mathbf{x}$.\n",
    "\n",
    "-   Model for target is a linear combination of these nonlinear\n",
    "    functions\n",
    "    $$f(\\mathbf{x}) = \\sum_{j=1}^k w_j \\phi_j(\\mathbf{x})$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcQAAAG/CAYAAADGuqcdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nJEkA/C9ZL/z7iPxV+wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10909cda0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(figsize=(7, 7))\n",
    "\n",
    "loc =[[0, 1.4,],\n",
    "      [0, -0.7],\n",
    "      [0.75, -0.2]]\n",
    "text =['$\\phi(x) = 1$',\n",
    "       '$\\phi(x) = x$',\n",
    "       '$\\phi(x) = x^2$']\n",
    "mlai.plot_basis(mlai.polynomial, x_min=-1.3, x_max=1.3, fig=f, ax=ax, loc=loc, text=text)    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Quadratic Basis\n",
    "\n",
    "-   Basis functions can be global. E.g. quadratic basis:\n",
    "    $$\\boldsymbol{\\phi} = [1, x, x^2]$$\n",
    "    \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg height=\"504pt\" version=\"1.1\" viewBox=\"0 0 504 504\" width=\"504pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\" M0 504 L504 504 L504 0 L0 0 z \" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\" M63 441 L453.6 441 L453.6 50.4 L63 50.4 z \" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_1\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M63 148.05 L66.9455 148.05 L70.8909 148.05 L74.8364 148.05 L78.7818 148.05 L82.7273 148.05 L86.6727 148.05 L90.6182 148.05 L94.5636 148.05 L98.5091 148.05 L102.455 148.05 L106.4 148.05 L110.345 148.05 L114.291 148.05 L118.236 148.05 L122.182 148.05 L126.127 148.05 L130.073 148.05 L134.018 148.05 L137.964 148.05 L141.909 148.05 L145.855 148.05 L149.8 148.05 L153.745 148.05 L157.691 148.05 L161.636 148.05 L165.582 148.05 L169.527 148.05 L173.473 148.05 L177.418 148.05 L181.364 148.05 L185.309 148.05 L189.255 148.05 L193.2 148.05 L197.145 148.05 L201.091 148.05 L205.036 148.05 L208.982 148.05 L212.927 148.05 L216.873 148.05 L220.818 148.05 L224.764 148.05 L228.709 148.05 L232.655 148.05 L236.6 148.05 L240.545 148.05 L244.491 148.05 L248.436 148.05 L252.382 148.05 L256.327 148.05 L260.273 148.05 L264.218 148.05 L268.164 148.05 L272.109 148.05 L276.055 148.05 L280 148.05 L283.945 148.05 L287.891 148.05 L291.836 148.05 L295.782 148.05 L299.727 148.05 L303.673 148.05 L307.618 148.05 L311.564 148.05 L315.509 148.05 L319.455 148.05 L323.4 148.05 L327.345 148.05 L331.291 148.05 L335.236 148.05 L339.182 148.05 L343.127 148.05 L347.073 148.05 L351.018 148.05 L354.964 148.05 L358.909 148.05 L362.855 148.05 L366.8 148.05 L370.745 148.05 L374.691 148.05 L378.636 148.05 L382.582 148.05 L386.527 148.05 L390.473 148.05 L394.418 148.05 L398.364 148.05 L402.309 148.05 L406.255 148.05 L410.2 148.05 L414.145 148.05 L418.091 148.05 L422.036 148.05 L425.982 148.05 L429.927 148.05 L433.873 148.05 L437.818 148.05 L441.764 148.05 L445.709 148.05 L449.655 148.05 L453.6 148.05\" style=\"fill:none;stroke:#ff0000;stroke-linecap:square;stroke-width:3;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_2\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M63 372.645 L66.9455 370.08 L70.8909 367.516 L74.8364 364.951 L78.7818 362.387 L82.7273 359.822 L86.6727 357.258 L90.6182 354.693 L94.5636 352.129 L98.5091 349.564 L102.455 347 L106.4 344.435 L110.345 341.87 L114.291 339.306 L118.236 336.741 L122.182 334.177 L126.127 331.612 L130.073 329.048 L134.018 326.483 L137.964 323.919 L141.909 321.354 L145.855 318.79 L149.8 316.225 L153.745 313.66 L157.691 311.096 L161.636 308.531 L165.582 305.967 L169.527 303.402 L173.473 300.838 L177.418 298.273 L181.364 295.709 L185.309 293.144 L189.255 290.58 L193.2 288.015 L197.145 285.45 L201.091 282.886 L205.036 280.321 L208.982 277.757 L212.927 275.192 L216.873 272.628 L220.818 270.063 L224.764 267.499 L228.709 264.934 L232.655 262.37 L236.6 259.805 L240.545 257.24 L244.491 254.676 L248.436 252.111 L252.382 249.547 L256.327 246.982 L260.273 244.418 L264.218 241.853 L268.164 239.289 L272.109 236.724 L276.055 234.16 L280 231.595 L283.945 229.03 L287.891 226.466 L291.836 223.901 L295.782 221.337 L299.727 218.772 L303.673 216.208 L307.618 213.643 L311.564 211.079 L315.509 208.514 L319.455 205.95 L323.4 203.385 L327.345 200.82 L331.291 198.256 L335.236 195.691 L339.182 193.127 L343.127 190.562 L347.073 187.998 L351.018 185.433 L354.964 182.869 L358.909 180.304 L362.855 177.74 L366.8 175.175 L370.745 172.61 L374.691 170.046 L378.636 167.481 L382.582 164.917 L386.527 162.352 L390.473 159.788 L394.418 157.223 L398.364 154.659 L402.309 152.094 L406.255 149.53 L410.2 146.965 L414.145 144.4 L418.091 141.836 L422.036 139.271 L425.982 136.707 L429.927 134.142 L433.873 131.578 L437.818 129.013 L441.764 126.449 L445.709 123.884 L449.655 121.32 L453.6 118.755\" style=\"fill:none;stroke:#ff00ff;stroke-linecap:square;stroke-width:3;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\" M453.6 441 L453.6 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\" M63 441 L63 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\" M63 50.4 L453.6 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\" M63 441 L453.6 441\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 -4\" id=\"mc7db9fdffb\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"108.069230769\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_4\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 4\" id=\"m5a7d422ac3\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"108.069230769\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- −1 -->\n",
       "      <defs>\n",
       "       <path d=\" M10.5938 35.5 L73.1875 35.5 L73.1875 27.2031 L10.5938 27.2031 z \" id=\"BitstreamVeraSans-Roman-2212\"/>\n",
       "       <path d=\" M12.4062 8.29688 L28.5156 8.29688 L28.5156 63.9219 L10.9844 60.4062 L10.9844 69.3906 L28.4219 72.9062 L38.2812 72.9062 L38.2812 8.29688 L54.3906 8.29688 L54.3906 0 L12.4062 0 z \" id=\"BitstreamVeraSans-Roman-31\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(101.689543269 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"258.3\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"258.3\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0 -->\n",
       "      <defs>\n",
       "       <path d=\" M31.7812 66.4062 Q24.1719 66.4062 20.3281 58.9062 Q16.5 51.4219 16.5 36.375 Q16.5 21.3906 20.3281 13.8906 Q24.1719 6.39062 31.7812 6.39062 Q39.4531 6.39062 43.2812 13.8906 Q47.125 21.3906 47.125 36.375 Q47.125 51.4219 43.2812 58.9062 Q39.4531 66.4062 31.7812 66.4062 M31.7812 74.2188 Q44.0469 74.2188 50.5156 64.5156 Q56.9844 54.8281 56.9844 36.375 Q56.9844 17.9688 50.5156 8.26562 Q44.0469 -1.42188 31.7812 -1.42188 Q19.5312 -1.42188 13.0625 8.26562 Q6.59375 17.9688 6.59375 36.375 Q6.59375 54.8281 13.0625 64.5156 Q19.5312 74.2188 31.7812 74.2188\" id=\"BitstreamVeraSans-Roman-30\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(255.78046875 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"408.530769231\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"408.530769231\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(406.360456731 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_4\">\n",
       "     <!-- $x$ -->\n",
       "     <defs>\n",
       "      <path d=\" M7.8125 2.875 Q9.57812 1.51562 12.7969 1.51562 Q15.9219 1.51562 18.3125 4.51562 Q20.7031 7.51562 21.5781 11.0781 L26.125 28.8125 Q27.2031 33.6406 27.2031 35.4062 Q27.2031 37.8906 25.8125 39.75 Q24.4219 41.6094 21.9219 41.6094 Q18.75 41.6094 15.9688 39.625 Q13.1875 37.6406 11.2812 34.5938 Q9.375 31.5469 8.59375 28.4219 Q8.40625 27.7812 7.8125 27.7812 L6.59375 27.7812 Q5.8125 27.7812 5.8125 28.7188 L5.8125 29 Q6.78125 32.7188 9.125 36.25 Q11.4688 39.7969 14.8594 41.9844 Q18.2656 44.1875 22.125 44.1875 Q25.7812 44.1875 28.7344 42.2344 Q31.6875 40.2812 32.9062 36.9219 Q34.625 39.9844 37.2812 42.0781 Q39.9375 44.1875 43.1094 44.1875 Q45.2656 44.1875 47.5 43.4219 Q49.75 42.6719 51.1719 41.1094 Q52.5938 39.5469 52.5938 37.2031 Q52.5938 34.6719 50.9531 32.8281 Q49.3125 31 46.7812 31 Q45.1719 31 44.0938 32.0312 Q43.0156 33.0625 43.0156 34.625 Q43.0156 36.7188 44.4531 38.2969 Q45.9062 39.8906 47.9062 40.1875 Q46.0938 41.6094 42.9219 41.6094 Q39.7031 41.6094 37.3281 38.625 Q34.9688 35.6406 33.9844 31.9844 L29.5938 14.3125 Q28.5156 10.2969 28.5156 7.71875 Q28.5156 5.17188 29.9531 3.34375 Q31.3906 1.51562 33.7969 1.51562 Q38.4844 1.51562 42.1562 5.64062 Q45.8438 9.76562 47.0156 14.7031 Q47.2188 15.2812 47.7969 15.2812 L49.0312 15.2812 Q49.4219 15.2812 49.6562 15.0156 Q49.9062 14.75 49.9062 14.4062 Q49.9062 14.3125 49.8125 14.1094 Q48.3906 8.15625 43.8438 3.51562 Q39.3125 -1.125 33.5938 -1.125 Q29.9375 -1.125 26.9844 0.84375 Q24.0312 2.82812 22.7969 6.20312 Q21.2344 3.26562 18.4688 1.0625 Q15.7188 -1.125 12.5938 -1.125 Q10.4531 -1.125 8.17188 -0.359375 Q5.90625 0.390625 4.48438 1.95312 Q3.07812 3.51562 3.07812 5.90625 Q3.07812 8.25 4.70312 10.1719 Q6.34375 12.1094 8.79688 12.1094 Q10.4531 12.1094 11.5781 11.1094 Q12.7031 10.1094 12.7031 8.5 Q12.7031 6.39062 11.2969 4.82812 Q9.90625 3.26562 7.8125 2.875\" id=\"Cmmi10-78\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(252.5 474.875)scale(0.2 -0.2)\">\n",
       "      <use transform=\"translate(0.0 0.8125)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L4 0\" id=\"md7965d1ba0\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_10\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L-4 0\" id=\"md9a1c1a7cd\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- −2 -->\n",
       "      <defs>\n",
       "       <path d=\" M19.1875 8.29688 L53.6094 8.29688 L53.6094 0 L7.32812 0 L7.32812 8.29688 Q12.9375 14.1094 22.625 23.8906 Q32.3281 33.6875 34.8125 36.5312 Q39.5469 41.8438 41.4219 45.5312 Q43.3125 49.2188 43.3125 52.7812 Q43.3125 58.5938 39.2344 62.25 Q35.1562 65.9219 28.6094 65.9219 Q23.9688 65.9219 18.8125 64.3125 Q13.6719 62.7031 7.8125 59.4219 L7.8125 69.3906 Q13.7656 71.7812 18.9375 73 Q24.125 74.2188 28.4219 74.2188 Q39.75 74.2188 46.4844 68.5469 Q53.2188 62.8906 53.2188 53.4219 Q53.2188 48.9219 51.5312 44.8906 Q49.8594 40.875 45.4062 35.4062 Q44.1875 33.9844 37.6406 27.2188 Q31.1094 20.4531 19.1875 8.29688\" id=\"BitstreamVeraSans-Roman-32\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(46.31875 443.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"343.35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"343.35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- −1 -->\n",
       "      <g transform=\"translate(46.240625 346.109375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"245.7\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_14\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"245.7\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_7\">\n",
       "      <!-- 0 -->\n",
       "      <g transform=\"translate(53.9609375 248.459375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_15\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"148.05\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_16\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"148.05\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(54.659375 150.809375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_17\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_18\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 2 -->\n",
       "      <g transform=\"translate(54.371875 53.159375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_10\">\n",
       "     <!-- $\\phi(x)$ -->\n",
       "     <defs>\n",
       "      <path d=\" M19.3906 -19.4844 L23.875 -1.42188 Q18.7969 -1.42188 14.4062 0.953125 Q10.0156 3.32812 7.34375 7.5625 Q4.6875 11.8125 4.6875 16.7969 Q4.6875 22.2188 7.375 27.1406 Q10.0625 32.0781 14.7188 36.1094 Q19.3906 40.1406 24.75 42.3125 Q30.125 44.4844 35.4062 44.4844 L41.4062 68.7969 Q41.6094 69.3906 42.1875 69.3906 L43.4062 69.3906 Q43.7969 69.3906 44.0312 69.0469 Q44.2812 68.7031 44.2812 68.4062 L38.2812 44.4844 Q42.0469 44.4844 45.5625 43.1094 Q49.0781 41.75 51.7812 39.2812 Q54.5 36.8125 56 33.4688 Q57.5156 30.125 57.5156 26.3125 Q57.5156 20.9062 54.8281 15.9688 Q52.1562 11.0312 47.6094 7.07812 Q43.0625 3.125 37.4688 0.84375 Q31.8906 -1.42188 26.8125 -1.42188 L22.2188 -19.9219 Q22.0156 -20.5156 21.3906 -20.5156 L20.2188 -20.5156 Q19.8281 -20.5156 19.6094 -20.1406 Q19.3906 -19.7812 19.3906 -19.4844 M24.5156 1.21875 L34.7188 41.8906 Q32.1719 41.8906 29.3906 40.8594 Q26.6094 39.8438 24.1406 38.2812 Q21.6875 36.7188 19.6719 34.7188 Q17.2344 32.2812 15.375 28.9531 Q13.5312 25.6406 12.5781 21.9531 Q11.625 18.2656 11.625 14.7031 Q11.625 10.9375 13.2344 7.85938 Q14.8438 4.78125 17.7656 3 Q20.7031 1.21875 24.5156 1.21875 M27.4844 1.21875 Q31.25 1.21875 35.5 3.35938 Q39.75 5.51562 42.4844 8.40625 Q46.2344 12.1094 48.4062 17.5 Q50.5938 22.9062 50.5938 28.4219 Q50.5938 32.125 48.9688 35.2188 Q47.3594 38.3281 44.4219 40.1094 Q41.5 41.8906 37.7031 41.8906 z \" id=\"Cmmi10-c1\"/>\n",
       "      <path d=\" M6.5 -25 Q5.60938 -25 5.60938 -24.125 Q5.60938 -23.6875 5.8125 -23.4844 Q22.9062 -6.78125 22.9062 25 Q22.9062 56.7812 6 73.2969 Q5.60938 73.5312 5.60938 74.125 Q5.60938 74.4688 5.875 74.7344 Q6.15625 75 6.5 75 L7.42188 75 Q7.71875 75 7.90625 74.8125 Q15.0938 69.1406 19.875 61.0312 Q24.6562 52.9375 26.875 43.75 Q29.1094 34.5781 29.1094 25 Q29.1094 17.9219 27.9062 11.0625 Q26.7031 4.20312 24.0938 -2.45312 Q21.4844 -9.125 17.4844 -14.7656 Q13.4844 -20.4062 7.90625 -24.8125 Q7.71875 -25 7.42188 -25 z \" id=\"Cmr10-29\"/>\n",
       "      <path d=\" M31 -24.8125 Q25.4375 -20.4062 21.4062 -14.7188 Q17.3906 -9.03125 14.8125 -2.57812 Q12.25 3.85938 10.9844 10.8906 Q9.71875 17.9219 9.71875 25 Q9.71875 32.1719 10.9844 39.2031 Q12.25 46.2344 14.8594 52.7344 Q17.4844 59.2344 21.5312 64.8906 Q25.5938 70.5625 31 74.8125 Q31 75 31.5 75 L32.4219 75 Q32.7188 75 32.9531 74.7344 Q33.2031 74.4688 33.2031 74.125 Q33.2031 73.6875 33.0156 73.4844 Q28.125 68.7031 24.875 63.2344 Q21.625 57.7656 19.6406 51.5781 Q17.6719 45.4062 16.7969 38.7812 Q15.9219 32.1719 15.9219 25 Q15.9219 -6.78125 32.9062 -23.2969 Q33.2031 -23.5781 33.2031 -24.125 Q33.2031 -24.3594 32.9375 -24.6719 Q32.6719 -25 32.4219 -25 L31.5 -25 Q31 -25 31 -24.8125\" id=\"Cmr10-28\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(36.240625 265.2)rotate(-90.0)scale(0.2 -0.2)\">\n",
       "      <use xlink:href=\"#Cmmi10-c1\"/>\n",
       "      <use transform=\"translate(59.521484375 0.0)\" xlink:href=\"#Cmr10-28\"/>\n",
       "      <use transform=\"translate(98.33984375 0.0)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "      <use transform=\"translate(155.419921875 0.0)\" xlink:href=\"#Cmr10-29\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_11\">\n",
       "    <!-- $\\phi(x) = 1$ -->\n",
       "    <defs>\n",
       "     <path d=\" M7.51562 13.2812 Q6.6875 13.2812 6.14062 13.9062 Q5.60938 14.5469 5.60938 15.2812 Q5.60938 16.1094 6.14062 16.6875 Q6.6875 17.2812 7.51562 17.2812 L70.3125 17.2812 Q71.0469 17.2812 71.5781 16.6875 Q72.125 16.1094 72.125 15.2812 Q72.125 14.5469 71.5781 13.9062 Q71.0469 13.2812 70.3125 13.2812 z  M7.51562 32.7188 Q6.6875 32.7188 6.14062 33.2969 Q5.60938 33.8906 5.60938 34.7188 Q5.60938 35.4531 6.14062 36.0781 Q6.6875 36.7188 7.51562 36.7188 L70.3125 36.7188 Q71.0469 36.7188 71.5781 36.0781 Q72.125 35.4531 72.125 34.7188 Q72.125 33.8906 71.5781 33.2969 Q71.0469 32.7188 70.3125 32.7188 z \" id=\"Cmr10-3d\"/>\n",
       "     <path d=\" M9.28125 0 L9.28125 3.51562 Q21.7812 3.51562 21.7812 6.6875 L21.7812 59.1875 Q16.6094 56.6875 8.6875 56.6875 L8.6875 60.2031 Q20.9531 60.2031 27.2031 66.6094 L28.6094 66.6094 Q28.9531 66.6094 29.2656 66.3281 Q29.5938 66.0625 29.5938 65.7188 L29.5938 6.6875 Q29.5938 3.51562 42.0938 3.51562 L42.0938 0 z \" id=\"Cmr10-31\"/>\n",
       "    </defs>\n",
       "    <g transform=\"translate(223.7 108.99)scale(0.2 -0.2)\">\n",
       "     <use xlink:href=\"#Cmmi10-c1\"/>\n",
       "     <use transform=\"translate(59.521484375 0.0)\" xlink:href=\"#Cmr10-28\"/>\n",
       "     <use transform=\"translate(98.33984375 0.0)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     <use transform=\"translate(155.419921875 0.0)\" xlink:href=\"#Cmr10-29\"/>\n",
       "     <use transform=\"translate(211.796875 0.0)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(295.85546875 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_12\">\n",
       "    <!-- $\\phi(x) = x$ -->\n",
       "    <g transform=\"translate(223.0 314.055)scale(0.2 -0.2)\">\n",
       "     <use xlink:href=\"#Cmmi10-c1\"/>\n",
       "     <use transform=\"translate(59.521484375 0.0)\" xlink:href=\"#Cmr10-28\"/>\n",
       "     <use transform=\"translate(98.33984375 0.0)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     <use transform=\"translate(155.419921875 0.0)\" xlink:href=\"#Cmr10-29\"/>\n",
       "     <use transform=\"translate(211.796875 0.0)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(295.85546875 0.0)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p41d7ee9abc\">\n",
       "   <rect height=\"390.6\" width=\"390.6\" x=\"63.0\" y=\"50.4\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pods.notebook.display_plots('polynomial_basis{num_basis}.svg', directory='./diagrams', num_basis=(1,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Functions Derived from Quadratic Basis\n",
    "\n",
    "$$f(x) = {\\color{\\redColor}w_0} + {\\color{\\magentaColor}w_1x} + {\\color{\\blueColor}w_2 x^2}$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg height=\"504pt\" version=\"1.1\" viewBox=\"0 0 504 504\" width=\"504pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\" M0 504 L504 504 L504 0 L0 0 z \" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\" M63 441 L453.6 441 L453.6 50.4 L63 50.4 z \" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_1\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M89.04 32.8064 L92.4594 36.8017 L95.8788 40.6955 L99.2982 44.4881 L102.718 48.1792 L106.137 51.769 L109.556 55.2574 L112.976 58.6444 L116.395 61.93 L119.815 65.1143 L123.234 68.1972 L126.653 71.1788 L130.073 74.059 L133.492 76.8378 L136.912 79.5152 L140.331 82.0913 L143.75 84.5659 L147.17 86.9393 L150.589 89.2112 L154.008 91.3818 L157.428 93.451 L160.847 95.4188 L164.267 97.2853 L167.686 99.0504 L171.105 100.714 L174.525 102.277 L177.944 103.737 L181.364 105.097 L184.783 106.355 L188.202 107.512 L191.622 108.568 L195.041 109.522 L198.461 110.375 L201.88 111.126 L205.299 111.776 L208.719 112.325 L212.138 112.772 L215.558 113.118 L218.977 113.362 L222.396 113.506 L225.816 113.547 L229.235 113.488 L232.655 113.327 L236.074 113.065 L239.493 112.701 L242.913 112.236 L246.332 111.669 L249.752 111.002 L253.171 110.233 L256.59 109.362 L260.01 108.39 L263.429 107.317 L266.848 106.142 L270.268 104.866 L273.687 103.489 L277.107 102.01 L280.526 100.43 L283.945 98.7484 L287.365 96.9656 L290.784 95.0813 L294.204 93.0957 L297.623 91.0087 L301.042 88.8204 L304.462 86.5307 L307.881 84.1396 L311.301 81.6471 L314.72 79.0533 L318.139 76.3581 L321.559 73.5615 L324.978 70.6636 L328.398 67.6643 L331.817 64.5636 L335.236 61.3616 L338.656 58.0581 L342.075 54.6534 L345.495 51.1472 L348.914 47.5397 L352.333 43.8308 L355.753 40.0205 L359.172 36.1088 L362.592 32.0958 L366.011 27.9814 L369.43 23.7657 L372.85 19.4486 L376.269 15.0301 L379.688 10.5102 L383.108 5.88897 L386.527 1.16637 L388.063 -1\" style=\"fill:none;stroke:#0000ff;stroke-linecap:square;stroke-width:3;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_2\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M89.04 162 L92.4594 162 L95.8788 162 L99.2982 162 L102.718 162 L106.137 162 L109.556 162 L112.976 162 L116.395 162 L119.815 162 L123.234 162 L126.653 162 L130.073 162 L133.492 162 L136.912 162 L140.331 162 L143.75 162 L147.17 162 L150.589 162 L154.008 162 L157.428 162 L160.847 162 L164.267 162 L167.686 162 L171.105 162 L174.525 162 L177.944 162 L181.364 162 L184.783 162 L188.202 162 L191.622 162 L195.041 162 L198.461 162 L201.88 162 L205.299 162 L208.719 162 L212.138 162 L215.558 162 L218.977 162 L222.396 162 L225.816 162 L229.235 162 L232.655 162 L236.074 162 L239.493 162 L242.913 162 L246.332 162 L249.752 162 L253.171 162 L256.59 162 L260.01 162 L263.429 162 L266.848 162 L270.268 162 L273.687 162 L277.107 162 L280.526 162 L283.945 162 L287.365 162 L290.784 162 L294.204 162 L297.623 162 L301.042 162 L304.462 162 L307.881 162 L311.301 162 L314.72 162 L318.139 162 L321.559 162 L324.978 162 L328.398 162 L331.817 162 L335.236 162 L338.656 162 L342.075 162 L345.495 162 L348.914 162 L352.333 162 L355.753 162 L359.172 162 L362.592 162 L366.011 162 L369.43 162 L372.85 162 L376.269 162 L379.688 162 L383.108 162 L386.527 162 L389.947 162 L393.366 162 L396.785 162 L400.205 162 L403.624 162 L407.044 162 L410.463 162 L413.882 162 L417.302 162 L420.721 162 L424.141 162 L427.56 162\" style=\"fill:none;stroke:#ff0000;stroke-linecap:square;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_3\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M89.04 290.34 L92.4594 288.875 L95.8788 287.409 L99.2982 285.944 L102.718 284.478 L106.137 283.013 L109.556 281.547 L112.976 280.082 L116.395 278.616 L119.815 277.151 L123.234 275.685 L126.653 274.22 L130.073 272.755 L133.492 271.289 L136.912 269.824 L140.331 268.358 L143.75 266.893 L147.17 265.427 L150.589 263.962 L154.008 262.496 L157.428 261.031 L160.847 259.565 L164.267 258.1 L167.686 256.635 L171.105 255.169 L174.525 253.704 L177.944 252.238 L181.364 250.773 L184.783 249.307 L188.202 247.842 L191.622 246.376 L195.041 244.911 L198.461 243.445 L201.88 241.98 L205.299 240.515 L208.719 239.049 L212.138 237.584 L215.558 236.118 L218.977 234.653 L222.396 233.187 L225.816 231.722 L229.235 230.256 L232.655 228.791 L236.074 227.325 L239.493 225.86 L242.913 224.395 L246.332 222.929 L249.752 221.464 L253.171 219.998 L256.59 218.533 L260.01 217.067 L263.429 215.602 L266.848 214.136 L270.268 212.671 L273.687 211.205 L277.107 209.74 L280.526 208.275 L283.945 206.809 L287.365 205.344 L290.784 203.878 L294.204 202.413 L297.623 200.947 L301.042 199.482 L304.462 198.016 L307.881 196.551 L311.301 195.085 L314.72 193.62 L318.139 192.155 L321.559 190.689 L324.978 189.224 L328.398 187.758 L331.817 186.293 L335.236 184.827 L338.656 183.362 L342.075 181.896 L345.495 180.431 L348.914 178.965 L352.333 177.5 L355.753 176.035 L359.172 174.569 L362.592 173.104 L366.011 171.638 L369.43 170.173 L372.85 168.707 L376.269 167.242 L379.688 165.776 L383.108 164.311 L386.527 162.845 L389.947 161.38 L393.366 159.915 L396.785 158.449 L400.205 156.984 L403.624 155.518 L407.044 154.053 L410.463 152.587 L413.882 151.122 L417.302 149.656 L420.721 148.191 L424.141 146.725 L427.56 145.26\" style=\"fill:none;stroke:#ff00ff;stroke-linecap:square;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_4\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M89.04 123.498 L92.4594 127.27 L95.8788 130.964 L99.2982 134.582 L102.718 138.123 L106.137 141.587 L109.556 144.974 L112.976 148.283 L116.395 151.516 L119.815 154.672 L123.234 157.751 L126.653 160.753 L130.073 163.678 L133.492 166.526 L136.912 169.297 L140.331 171.991 L143.75 174.608 L147.17 177.148 L150.589 179.612 L154.008 181.998 L157.428 184.307 L160.847 186.539 L164.267 188.694 L167.686 190.773 L171.105 192.774 L174.525 194.698 L177.944 196.546 L181.364 198.316 L184.783 200.01 L188.202 201.626 L191.622 203.165 L195.041 204.628 L198.461 206.013 L201.88 207.322 L205.299 208.554 L208.719 209.708 L212.138 210.786 L215.558 211.786 L218.977 212.71 L222.396 213.557 L225.816 214.327 L229.235 215.019 L232.655 215.635 L236.074 216.174 L239.493 216.636 L242.913 217.021 L246.332 217.329 L249.752 217.559 L253.171 217.713 L256.59 217.79 L260.01 217.79 L263.429 217.713 L266.848 217.559 L270.268 217.329 L273.687 217.021 L277.107 216.636 L280.526 216.174 L283.945 215.635 L287.365 215.019 L290.784 214.327 L294.204 213.557 L297.623 212.71 L301.042 211.786 L304.462 210.786 L307.881 209.708 L311.301 208.554 L314.72 207.322 L318.139 206.013 L321.559 204.628 L324.978 203.165 L328.398 201.626 L331.817 200.01 L335.236 198.316 L338.656 196.546 L342.075 194.698 L345.495 192.774 L348.914 190.773 L352.333 188.694 L355.753 186.539 L359.172 184.307 L362.592 181.998 L366.011 179.612 L369.43 177.148 L372.85 174.608 L376.269 171.991 L379.688 169.297 L383.108 166.526 L386.527 163.678 L389.947 160.753 L393.366 157.751 L396.785 154.672 L400.205 151.516 L403.624 148.283 L407.044 144.974 L410.463 141.587 L413.882 138.123 L417.302 134.582 L420.721 130.964 L424.141 127.27 L427.56 123.498\" style=\"fill:none;stroke:#0000ff;stroke-linecap:square;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\" M453.6 441 L453.6 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\" M63 441 L63 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\" M63 50.4 L453.6 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\" M63 441 L453.6 441\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 -4\" id=\"mc7db9fdffb\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"128.1\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_6\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 4\" id=\"m5a7d422ac3\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"128.1\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- −1 -->\n",
       "      <defs>\n",
       "       <path d=\" M10.5938 35.5 L73.1875 35.5 L73.1875 27.2031 L10.5938 27.2031 z \" id=\"BitstreamVeraSans-Roman-2212\"/>\n",
       "       <path d=\" M12.4062 8.29688 L28.5156 8.29688 L28.5156 63.9219 L10.9844 60.4062 L10.9844 69.3906 L28.4219 72.9062 L38.2812 72.9062 L38.2812 8.29688 L54.3906 8.29688 L54.3906 0 L12.4062 0 z \" id=\"BitstreamVeraSans-Roman-31\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(121.7203125 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"258.3\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"258.3\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0 -->\n",
       "      <defs>\n",
       "       <path d=\" M31.7812 66.4062 Q24.1719 66.4062 20.3281 58.9062 Q16.5 51.4219 16.5 36.375 Q16.5 21.3906 20.3281 13.8906 Q24.1719 6.39062 31.7812 6.39062 Q39.4531 6.39062 43.2812 13.8906 Q47.125 21.3906 47.125 36.375 Q47.125 51.4219 43.2812 58.9062 Q39.4531 66.4062 31.7812 66.4062 M31.7812 74.2188 Q44.0469 74.2188 50.5156 64.5156 Q56.9844 54.8281 56.9844 36.375 Q56.9844 17.9688 50.5156 8.26562 Q44.0469 -1.42188 31.7812 -1.42188 Q19.5312 -1.42188 13.0625 8.26562 Q6.59375 17.9688 6.59375 36.375 Q6.59375 54.8281 13.0625 64.5156 Q19.5312 74.2188 31.7812 74.2188\" id=\"BitstreamVeraSans-Roman-30\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(255.78046875 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"388.5\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"388.5\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(386.3296875 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_4\">\n",
       "     <!-- $x$ -->\n",
       "     <defs>\n",
       "      <path d=\" M7.8125 2.875 Q9.57812 1.51562 12.7969 1.51562 Q15.9219 1.51562 18.3125 4.51562 Q20.7031 7.51562 21.5781 11.0781 L26.125 28.8125 Q27.2031 33.6406 27.2031 35.4062 Q27.2031 37.8906 25.8125 39.75 Q24.4219 41.6094 21.9219 41.6094 Q18.75 41.6094 15.9688 39.625 Q13.1875 37.6406 11.2812 34.5938 Q9.375 31.5469 8.59375 28.4219 Q8.40625 27.7812 7.8125 27.7812 L6.59375 27.7812 Q5.8125 27.7812 5.8125 28.7188 L5.8125 29 Q6.78125 32.7188 9.125 36.25 Q11.4688 39.7969 14.8594 41.9844 Q18.2656 44.1875 22.125 44.1875 Q25.7812 44.1875 28.7344 42.2344 Q31.6875 40.2812 32.9062 36.9219 Q34.625 39.9844 37.2812 42.0781 Q39.9375 44.1875 43.1094 44.1875 Q45.2656 44.1875 47.5 43.4219 Q49.75 42.6719 51.1719 41.1094 Q52.5938 39.5469 52.5938 37.2031 Q52.5938 34.6719 50.9531 32.8281 Q49.3125 31 46.7812 31 Q45.1719 31 44.0938 32.0312 Q43.0156 33.0625 43.0156 34.625 Q43.0156 36.7188 44.4531 38.2969 Q45.9062 39.8906 47.9062 40.1875 Q46.0938 41.6094 42.9219 41.6094 Q39.7031 41.6094 37.3281 38.625 Q34.9688 35.6406 33.9844 31.9844 L29.5938 14.3125 Q28.5156 10.2969 28.5156 7.71875 Q28.5156 5.17188 29.9531 3.34375 Q31.3906 1.51562 33.7969 1.51562 Q38.4844 1.51562 42.1562 5.64062 Q45.8438 9.76562 47.0156 14.7031 Q47.2188 15.2812 47.7969 15.2812 L49.0312 15.2812 Q49.4219 15.2812 49.6562 15.0156 Q49.9062 14.75 49.9062 14.4062 Q49.9062 14.3125 49.8125 14.1094 Q48.3906 8.15625 43.8438 3.51562 Q39.3125 -1.125 33.5938 -1.125 Q29.9375 -1.125 26.9844 0.84375 Q24.0312 2.82812 22.7969 6.20312 Q21.2344 3.26562 18.4688 1.0625 Q15.7188 -1.125 12.5938 -1.125 Q10.4531 -1.125 8.17188 -0.359375 Q5.90625 0.390625 4.48438 1.95312 Q3.07812 3.51562 3.07812 5.90625 Q3.07812 8.25 4.70312 10.1719 Q6.34375 12.1094 8.79688 12.1094 Q10.4531 12.1094 11.5781 11.1094 Q12.7031 10.1094 12.7031 8.5 Q12.7031 6.39062 11.2969 4.82812 Q9.90625 3.26562 7.8125 2.875\" id=\"Cmmi10-78\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(252.5 474.875)scale(0.2 -0.2)\">\n",
       "      <use transform=\"translate(0.0 0.8125)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L4 0\" id=\"md7965d1ba0\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_12\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L-4 0\" id=\"md9a1c1a7cd\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- −4 -->\n",
       "      <defs>\n",
       "       <path d=\" M37.7969 64.3125 L12.8906 25.3906 L37.7969 25.3906 z  M35.2031 72.9062 L47.6094 72.9062 L47.6094 25.3906 L58.0156 25.3906 L58.0156 17.1875 L47.6094 17.1875 L47.6094 0 L37.7969 0 L37.7969 17.1875 L4.89062 17.1875 L4.89062 26.7031 z \" id=\"BitstreamVeraSans-Roman-34\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(45.878125 443.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-34\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"385.2\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_14\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"385.2\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- −3 -->\n",
       "      <defs>\n",
       "       <path d=\" M40.5781 39.3125 Q47.6562 37.7969 51.625 33 Q55.6094 28.2188 55.6094 21.1875 Q55.6094 10.4062 48.1875 4.48438 Q40.7656 -1.42188 27.0938 -1.42188 Q22.5156 -1.42188 17.6562 -0.515625 Q12.7969 0.390625 7.625 2.20312 L7.625 11.7188 Q11.7188 9.32812 16.5938 8.10938 Q21.4844 6.89062 26.8125 6.89062 Q36.0781 6.89062 40.9375 10.5469 Q45.7969 14.2031 45.7969 21.1875 Q45.7969 27.6406 41.2812 31.2656 Q36.7656 34.9062 28.7188 34.9062 L20.2188 34.9062 L20.2188 43.0156 L29.1094 43.0156 Q36.375 43.0156 40.2344 45.9219 Q44.0938 48.8281 44.0938 54.2969 Q44.0938 59.9062 40.1094 62.9062 Q36.1406 65.9219 28.7188 65.9219 Q24.6562 65.9219 20.0156 65.0312 Q15.375 64.1562 9.8125 62.3125 L9.8125 71.0938 Q15.4375 72.6562 20.3438 73.4375 Q25.25 74.2188 29.5938 74.2188 Q40.8281 74.2188 47.3594 69.1094 Q53.9062 64.0156 53.9062 55.3281 Q53.9062 49.2656 50.4375 45.0938 Q46.9688 40.9219 40.5781 39.3125\" id=\"BitstreamVeraSans-Roman-33\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(46.11875 387.959375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-33\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_15\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"329.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_16\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"329.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_7\">\n",
       "      <!-- −2 -->\n",
       "      <defs>\n",
       "       <path d=\" M19.1875 8.29688 L53.6094 8.29688 L53.6094 0 L7.32812 0 L7.32812 8.29688 Q12.9375 14.1094 22.625 23.8906 Q32.3281 33.6875 34.8125 36.5312 Q39.5469 41.8438 41.4219 45.5312 Q43.3125 49.2188 43.3125 52.7812 Q43.3125 58.5938 39.2344 62.25 Q35.1562 65.9219 28.6094 65.9219 Q23.9688 65.9219 18.8125 64.3125 Q13.6719 62.7031 7.8125 59.4219 L7.8125 69.3906 Q13.7656 71.7812 18.9375 73 Q24.125 74.2188 28.4219 74.2188 Q39.75 74.2188 46.4844 68.5469 Q53.2188 62.8906 53.2188 53.4219 Q53.2188 48.9219 51.5312 44.8906 Q49.8594 40.875 45.4062 35.4062 Q44.1875 33.9844 37.6406 27.2188 Q31.1094 20.4531 19.1875 8.29688\" id=\"BitstreamVeraSans-Roman-32\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(46.31875 332.159375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_17\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"273.6\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_18\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"273.6\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- −1 -->\n",
       "      <g transform=\"translate(46.240625 276.359375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_19\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"217.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_20\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"217.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0 -->\n",
       "      <g transform=\"translate(53.9609375 220.559375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_21\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"162.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_22\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"162.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(54.659375 164.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_7\">\n",
       "     <g id=\"line2d_23\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"106.2\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_24\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"106.2\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 2 -->\n",
       "      <g transform=\"translate(54.371875 108.959375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_8\">\n",
       "     <g id=\"line2d_25\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_26\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 3 -->\n",
       "      <g transform=\"translate(54.2015625 53.159375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-33\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_13\">\n",
       "     <!-- $f(x)$ -->\n",
       "     <defs>\n",
       "      <path d=\" M9.71875 -16.5 Q11.5781 -17.9219 14.2031 -17.9219 Q17.7812 -17.9219 20.0156 -10.0156 Q20.9531 -6.20312 25.0938 15.0938 L29.6875 39.5938 L21.0938 39.5938 Q20.125 39.5938 20.125 40.9219 Q20.5156 43.1094 21.3906 43.1094 L30.3281 43.1094 L31.5 49.6094 Q32.0781 52.6406 32.5625 54.8125 Q33.0625 56.9844 33.625 58.8281 Q34.1875 60.6875 35.2969 62.9844 Q37.0156 66.2656 39.9219 68.3906 Q42.8281 70.5156 46.1875 70.5156 Q48.3906 70.5156 50.4688 69.7031 Q52.5469 68.8906 53.8594 67.2812 Q55.1719 65.6719 55.1719 63.4844 Q55.1719 60.9375 53.4844 59.0469 Q51.8125 57.1719 49.4219 57.1719 Q47.7969 57.1719 46.6562 58.1719 Q45.5156 59.1875 45.5156 60.7969 Q45.5156 62.9844 47 64.625 Q48.4844 66.2656 50.6875 66.5 Q48.8281 67.9219 46.0938 67.9219 Q44.5781 67.9219 43.2344 66.5 Q41.8906 65.0938 41.5 63.4844 Q40.875 60.9375 38.7188 49.7031 L37.5 43.1094 L47.7969 43.1094 Q48.7812 43.1094 48.7812 41.7969 Q48.7344 41.5469 48.5781 40.9375 Q48.4375 40.3281 48.1719 39.9531 Q47.9062 39.5938 47.5156 39.5938 L36.8125 39.5938 L32.1719 15.1875 Q31.2969 9.85938 30.1406 4.35938 Q29 -1.125 26.9219 -6.90625 Q24.8594 -12.7031 21.625 -16.6094 Q18.4062 -20.5156 14.0156 -20.5156 Q10.6406 -20.5156 7.98438 -18.5781 Q5.32812 -16.6562 5.32812 -13.4844 Q5.32812 -10.9375 6.95312 -9.04688 Q8.59375 -7.17188 11.0781 -7.17188 Q12.75 -7.17188 13.8594 -8.17188 Q14.9844 -9.1875 14.9844 -10.7969 Q14.9844 -12.9375 13.3906 -14.7188 Q11.8125 -16.5 9.71875 -16.5\" id=\"Cmmi10-66\"/>\n",
       "      <path d=\" M6.5 -25 Q5.60938 -25 5.60938 -24.125 Q5.60938 -23.6875 5.8125 -23.4844 Q22.9062 -6.78125 22.9062 25 Q22.9062 56.7812 6 73.2969 Q5.60938 73.5312 5.60938 74.125 Q5.60938 74.4688 5.875 74.7344 Q6.15625 75 6.5 75 L7.42188 75 Q7.71875 75 7.90625 74.8125 Q15.0938 69.1406 19.875 61.0312 Q24.6562 52.9375 26.875 43.75 Q29.1094 34.5781 29.1094 25 Q29.1094 17.9219 27.9062 11.0625 Q26.7031 4.20312 24.0938 -2.45312 Q21.4844 -9.125 17.4844 -14.7656 Q13.4844 -20.4062 7.90625 -24.8125 Q7.71875 -25 7.42188 -25 z \" id=\"Cmr10-29\"/>\n",
       "      <path d=\" M31 -24.8125 Q25.4375 -20.4062 21.4062 -14.7188 Q17.3906 -9.03125 14.8125 -2.57812 Q12.25 3.85938 10.9844 10.8906 Q9.71875 17.9219 9.71875 25 Q9.71875 32.1719 10.9844 39.2031 Q12.25 46.2344 14.8594 52.7344 Q17.4844 59.2344 21.5312 64.8906 Q25.5938 70.5625 31 74.8125 Q31 75 31.5 75 L32.4219 75 Q32.7188 75 32.9531 74.7344 Q33.2031 74.4688 33.2031 74.125 Q33.2031 73.6875 33.0156 73.4844 Q28.125 68.7031 24.875 63.2344 Q21.625 57.7656 19.6406 51.5781 Q17.6719 45.4062 16.7969 38.7812 Q15.9219 32.1719 15.9219 25 Q15.9219 -6.78125 32.9062 -23.2969 Q33.2031 -23.5781 33.2031 -24.125 Q33.2031 -24.3594 32.9375 -24.6719 Q32.6719 -25 32.4219 -25 L31.5 -25 Q31 -25 31 -24.8125\" id=\"Cmr10-28\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(35.878125 264.1)rotate(-90.0)scale(0.2 -0.2)\">\n",
       "      <use xlink:href=\"#Cmmi10-66\"/>\n",
       "      <use transform=\"translate(48.876953125 0.0)\" xlink:href=\"#Cmr10-28\"/>\n",
       "      <use transform=\"translate(87.6953125 0.0)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "      <use transform=\"translate(144.775390625 0.0)\" xlink:href=\"#Cmr10-29\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_14\">\n",
       "    <!-- $w_0 = [ 1.95181505]$ -->\n",
       "    <defs>\n",
       "     <path d=\" M4.20312 15.1875 Q4.20312 21.1875 8.15625 25.7969 Q12.1094 30.4219 18.3125 33.5 L14.5938 35.8906 Q11.1875 38.1406 9.03125 41.875 Q6.89062 45.6094 6.89062 49.7031 Q6.89062 54.5 9.40625 58.3438 Q11.9219 62.2031 16.0938 64.4062 Q20.2656 66.6094 25 66.6094 Q29.4375 66.6094 33.5625 64.7969 Q37.7031 62.9844 40.3594 59.6094 Q43.0156 56.25 43.0156 51.6094 Q43.0156 48.25 41.4219 45.3594 Q39.8438 42.4844 37.0781 40.1875 Q34.3281 37.8906 31.2031 36.2812 L36.9219 32.625 Q40.875 30.0312 43.2812 25.8281 Q45.7031 21.625 45.7031 17 Q45.7031 11.5781 42.7969 7.125 Q39.8906 2.6875 35.1094 0.234375 Q30.3281 -2.20312 25 -2.20312 Q19.8281 -2.20312 15.0156 -0.09375 Q10.2031 2 7.20312 5.98438 Q4.20312 9.96875 4.20312 15.1875 M9.625 15.1875 Q9.625 11.2344 11.7969 7.95312 Q13.9688 4.6875 17.5312 2.82812 Q21.0938 0.984375 25 0.984375 Q30.8125 0.984375 35.5469 4.375 Q40.2812 7.76562 40.2812 13.375 Q40.2812 15.2812 39.5156 17.1562 Q38.7656 19.0469 37.4219 20.5781 Q36.0781 22.125 34.4219 23.0938 L21 31.7812 Q17.875 30.125 15.25 27.5781 Q12.6406 25.0469 11.125 21.875 Q9.625 18.7031 9.625 15.1875 M16.5 45.7031 L28.6094 37.8906 Q32.8125 40.3281 35.5 43.7969 Q38.1875 47.2656 38.1875 51.6094 Q38.1875 54.9844 36.2969 57.7812 Q34.4219 60.5938 31.3906 62.1562 Q28.375 63.7188 24.9062 63.7188 Q21.875 63.7188 18.7969 62.5469 Q15.7188 61.375 13.7188 59.0469 Q11.7188 56.7344 11.7188 53.6094 Q11.7188 48.9219 16.5 45.7031\" id=\"Cmr10-38\"/>\n",
       "     <path d=\" M8.40625 5.51562 Q8.40625 7.76562 10.0625 9.375 Q11.7188 10.9844 13.9219 10.9844 Q15.2812 10.9844 16.5938 10.25 Q17.9219 9.51562 18.6562 8.20312 Q19.3906 6.89062 19.3906 5.51562 Q19.3906 3.32812 17.7656 1.65625 Q16.1562 0 13.9219 0 Q11.7188 0 10.0625 1.65625 Q8.40625 3.32812 8.40625 5.51562\" id=\"Cmmi10-3a\"/>\n",
       "     <path d=\" M9.28125 0 L9.28125 3.51562 Q21.7812 3.51562 21.7812 6.6875 L21.7812 59.1875 Q16.6094 56.6875 8.6875 56.6875 L8.6875 60.2031 Q20.9531 60.2031 27.2031 66.6094 L28.6094 66.6094 Q28.9531 66.6094 29.2656 66.3281 Q29.5938 66.0625 29.5938 65.7188 L29.5938 6.6875 Q29.5938 3.51562 42.0938 3.51562 L42.0938 0 z \" id=\"Cmr10-31\"/>\n",
       "     <path d=\" M2.20312 -25 L2.20312 -21 L11.9219 -21 L11.9219 71 L2.20312 71 L2.20312 75 L15.9219 75 L15.9219 -25 z \" id=\"Cmr10-5d\"/>\n",
       "     <path d=\" M25 -2.20312 Q12.75 -2.20312 8.32812 7.875 Q3.90625 17.9688 3.90625 31.8906 Q3.90625 40.5781 5.48438 48.2344 Q7.07812 55.9062 11.7812 61.25 Q16.5 66.6094 25 66.6094 Q31.5938 66.6094 35.7812 63.375 Q39.9844 60.1562 42.1875 55.0469 Q44.3906 49.9531 45.1875 44.1094 Q46 38.2812 46 31.8906 Q46 23.2969 44.4062 15.7969 Q42.8281 8.29688 38.1875 3.04688 Q33.5469 -2.20312 25 -2.20312 M25 0.390625 Q30.5625 0.390625 33.2969 6.09375 Q36.0312 11.8125 36.6719 18.75 Q37.3125 25.6875 37.3125 33.5 Q37.3125 41.0156 36.6719 47.3594 Q36.0312 53.7188 33.3125 58.8594 Q30.6094 64.0156 25 64.0156 Q19.3438 64.0156 16.6094 58.8281 Q13.875 53.6562 13.2344 47.3281 Q12.5938 41.0156 12.5938 33.5 Q12.5938 27.9375 12.8594 23 Q13.1406 18.0625 14.3125 12.8125 Q15.4844 7.5625 18.0938 3.96875 Q20.7031 0.390625 25 0.390625\" id=\"Cmr10-30\"/>\n",
       "     <path d=\" M11.2812 4.20312 Q14.0156 0.984375 20.7969 0.984375 Q24.6094 0.984375 27.9062 3.5625 Q31.2031 6.15625 33.0156 9.90625 Q35.1094 14.1562 35.6875 18.9688 Q36.2812 23.7812 36.2812 30.9062 Q34.5781 26.8594 31.375 24.2656 Q28.1719 21.6875 24.0312 21.6875 Q18.2188 21.6875 13.6406 24.8281 Q9.07812 27.9844 6.64062 33.125 Q4.20312 38.2812 4.20312 44.0938 Q4.20312 50.0938 6.9375 55.2969 Q9.67188 60.5 14.5 63.5469 Q19.3438 66.6094 25.4844 66.6094 Q31.5469 66.6094 35.625 63.3125 Q39.7031 60.0156 41.8438 54.8125 Q44 49.6094 44.8438 43.7969 Q45.7031 37.9844 45.7031 32.3281 Q45.7031 24.6094 42.875 16.5781 Q40.0469 8.54688 34.4062 3.17188 Q28.7656 -2.20312 20.7969 -2.20312 Q14.8906 -2.20312 10.7812 0.578125 Q6.6875 3.375 6.6875 8.98438 Q6.6875 11.0312 8.07812 12.4219 Q9.46875 13.8125 11.5312 13.8125 Q13.5312 13.8125 14.9219 12.4219 Q16.3125 11.0312 16.3125 8.98438 Q16.3125 7.03125 14.8906 5.60938 Q13.4844 4.20312 11.5312 4.20312 z  M24.4219 24.3125 Q28.5156 24.3125 31.125 27.0625 Q33.7344 29.8281 34.9062 33.9219 Q36.0781 38.0312 36.0781 42.0938 L36.0781 44 L36.0781 44.3906 Q36.0781 51.9062 33.875 57.8125 Q31.6875 63.7188 25.4844 63.7188 Q21.5312 63.7188 19.0625 61.9844 Q16.6094 60.25 15.4375 57.375 Q14.2656 54.5 13.9375 51.2188 Q13.625 47.9531 13.625 44.0938 Q13.625 38.4219 14.1562 34.4219 Q14.7031 30.4219 17.0938 27.3594 Q19.4844 24.3125 24.4219 24.3125\" id=\"Cmr10-39\"/>\n",
       "     <path d=\" M10.7969 11.625 Q10.7969 13.7188 11.3594 16.4531 Q11.9219 19.1875 12.5938 21.3125 Q13.2812 23.4375 14.9062 27.8281 Q16.5469 32.2344 16.7031 32.625 Q18.0156 36.2812 18.0156 38.625 Q18.0156 41.6094 15.8281 41.6094 Q11.8594 41.6094 9.29688 37.5312 Q6.73438 33.4531 5.51562 28.4219 Q5.32812 27.7812 4.6875 27.7812 L3.51562 27.7812 Q2.6875 27.7812 2.6875 28.7188 L2.6875 29 Q4.29688 34.9688 7.60938 39.5781 Q10.9375 44.1875 16.0156 44.1875 Q19.5781 44.1875 22.0469 41.8438 Q24.5156 39.5 24.5156 35.8906 Q24.5156 34.0312 23.6875 31.9844 Q22.5156 28.9531 20.9219 24.7031 Q19.3438 20.4531 18.4531 16.9844 Q17.5781 13.5312 17.5781 10.2969 Q17.5781 6.39062 19.625 3.95312 Q21.6875 1.51562 25.5938 1.51562 Q30.9062 1.51562 34.4219 9.625 Q34.2812 10.2031 34.2812 11.2812 Q34.2812 13.9688 35.1094 17.3906 L40.7188 39.8906 Q41.0625 41.2188 42.2812 42.1562 Q43.5 43.1094 44.9219 43.1094 Q46.0938 43.1094 47 42.3281 Q47.9062 41.5469 47.9062 40.2812 Q47.9062 39.7031 47.7969 39.5 L42.1875 17.1875 Q41.2188 13.1875 41.2188 10.0156 Q41.2188 6.25 42.9375 3.875 Q44.6719 1.51562 48.3906 1.51562 Q54.8281 1.51562 58.9844 10.0156 Q60.5469 13.0312 62.4688 18.9375 Q64.4062 24.8594 64.4062 27.9844 Q64.4062 30.8594 63.4688 32.5469 Q62.5469 34.2344 60.9062 36.1875 Q59.2812 38.1406 59.2812 39.1094 Q59.2812 41.0156 60.9062 42.6406 Q62.5469 44.2812 64.5 44.2812 Q66.8906 44.2812 67.9375 42.0781 Q69 39.8906 69 37.1094 Q69 33.7344 67.875 28.5312 Q66.75 23.3438 65.0312 18.0938 Q63.3281 12.8438 61.8125 9.71875 Q56.3438 -1.125 48.1875 -1.125 Q44 -1.125 40.5312 0.484375 Q37.0625 2.09375 35.4062 5.60938 Q33.7969 2.73438 31.1719 0.796875 Q28.5625 -1.125 25.3906 -1.125 Q18.7969 -1.125 14.7969 2.01562 Q10.7969 5.17188 10.7969 11.625\" id=\"Cmmi10-77\"/>\n",
       "     <path d=\" M7.51562 13.2812 Q6.6875 13.2812 6.14062 13.9062 Q5.60938 14.5469 5.60938 15.2812 Q5.60938 16.1094 6.14062 16.6875 Q6.6875 17.2812 7.51562 17.2812 L70.3125 17.2812 Q71.0469 17.2812 71.5781 16.6875 Q72.125 16.1094 72.125 15.2812 Q72.125 14.5469 71.5781 13.9062 Q71.0469 13.2812 70.3125 13.2812 z  M7.51562 32.7188 Q6.6875 32.7188 6.14062 33.2969 Q5.60938 33.8906 5.60938 34.7188 Q5.60938 35.4531 6.14062 36.0781 Q6.6875 36.7188 7.51562 36.7188 L70.3125 36.7188 Q71.0469 36.7188 71.5781 36.0781 Q72.125 35.4531 72.125 34.7188 Q72.125 33.8906 71.5781 33.2969 Q71.0469 32.7188 70.3125 32.7188 z \" id=\"Cmr10-3d\"/>\n",
       "     <path d=\" M8.6875 11.375 Q9.71875 8.45312 11.8438 6.04688 Q13.9688 3.65625 16.875 2.3125 Q19.7812 0.984375 22.9062 0.984375 Q30.125 0.984375 32.8594 6.59375 Q35.5938 12.2031 35.5938 20.2188 Q35.5938 23.6875 35.4688 26.0469 Q35.3594 28.4219 34.8125 30.6094 Q33.8906 34.125 31.5625 36.7656 Q29.25 39.4062 25.875 39.4062 Q22.5156 39.4062 20.0938 38.375 Q17.6719 37.3594 16.1562 35.9844 Q14.6562 34.625 13.4844 33.1094 Q12.3125 31.5938 12.0156 31.5 L10.8906 31.5 Q10.6406 31.5 10.2656 31.8125 Q9.90625 32.125 9.90625 32.4219 L9.90625 65.8281 Q9.90625 66.0625 10.2188 66.3281 Q10.5469 66.6094 10.8906 66.6094 L11.1875 66.6094 Q17.9219 63.375 25.4844 63.375 Q32.9062 63.375 39.7969 66.6094 L40.0938 66.6094 Q40.4375 66.6094 40.7188 66.3594 Q41.0156 66.1094 41.0156 65.8281 L41.0156 64.8906 Q41.0156 64.4062 40.8281 64.4062 Q37.4062 59.8594 32.25 57.3125 Q27.0938 54.7812 21.5781 54.7812 Q17.5781 54.7812 13.375 55.9062 L13.375 37.0156 Q16.7031 39.7031 19.3125 40.8438 Q21.9219 42 25.9844 42 Q31.5 42 35.8594 38.8125 Q40.2344 35.6406 42.5781 30.5312 Q44.9219 25.4375 44.9219 20.125 Q44.9219 14.1094 41.9688 8.98438 Q39.0156 3.85938 33.9375 0.828125 Q28.8594 -2.20312 22.9062 -2.20312 Q17.9688 -2.20312 13.8438 0.328125 Q9.71875 2.875 7.34375 7.17188 Q4.98438 11.4688 4.98438 16.3125 Q4.98438 18.5625 6.4375 19.9688 Q7.90625 21.3906 10.1094 21.3906 Q12.3125 21.3906 13.7969 19.9375 Q15.2812 18.5 15.2812 16.3125 Q15.2812 14.1562 13.7969 12.6719 Q12.3125 11.1875 10.1094 11.1875 Q9.76562 11.1875 9.32812 11.25 Q8.89062 11.3281 8.6875 11.375\" id=\"Cmr10-35\"/>\n",
       "     <path d=\" M11.8125 -25 L11.8125 75 L25.4844 75 L25.4844 71 L15.8281 71 L15.8281 -21 L25.4844 -21 L25.4844 -25 z \" id=\"Cmr10-5b\"/>\n",
       "    </defs>\n",
       "    <g transform=\"translate(182.5 139.68)scale(0.2 -0.2)\">\n",
       "     <use xlink:href=\"#Cmmi10-77\"/>\n",
       "     <use transform=\"translate(71.484375 -25.509375000000006)scale(0.7)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(139.04296875 0.0)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(223.1015625 0.0)\" xlink:href=\"#Cmr10-5b\"/>\n",
       "     <use transform=\"translate(250.787109375 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(300.787109375 0.0)\" xlink:href=\"#Cmmi10-3a\"/>\n",
       "     <use transform=\"translate(329.330078125 0.0)\" xlink:href=\"#Cmr10-39\"/>\n",
       "     <use transform=\"translate(379.330078125 0.0)\" xlink:href=\"#Cmr10-35\"/>\n",
       "     <use transform=\"translate(429.330078125 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(479.330078125 0.0)\" xlink:href=\"#Cmr10-38\"/>\n",
       "     <use transform=\"translate(529.330078125 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(579.330078125 0.0)\" xlink:href=\"#Cmr10-35\"/>\n",
       "     <use transform=\"translate(629.330078125 0.0)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(679.330078125 0.0)\" xlink:href=\"#Cmr10-35\"/>\n",
       "     <use transform=\"translate(729.330078125 0.0)\" xlink:href=\"#Cmr10-5d\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_15\">\n",
       "    <!-- $w_1 = [ 0.66320798]$ -->\n",
       "    <defs>\n",
       "     <path d=\" M4.98438 0 L4.98438 2.6875 Q4.98438 2.9375 5.17188 3.21875 L20.7031 20.4062 Q24.2188 24.2188 26.4062 26.7969 Q28.6094 29.3906 30.75 32.7656 Q32.9062 36.1406 34.1562 39.625 Q35.4062 43.1094 35.4062 47.0156 Q35.4062 51.125 33.8906 54.8594 Q32.375 58.5938 29.375 60.8438 Q26.375 63.0938 22.125 63.0938 Q17.7812 63.0938 14.3125 60.4688 Q10.8438 57.8594 9.42188 53.7188 Q9.8125 53.8125 10.5 53.8125 Q12.75 53.8125 14.3281 52.2969 Q15.9219 50.7812 15.9219 48.3906 Q15.9219 46.0938 14.3281 44.5 Q12.75 42.9219 10.5 42.9219 Q8.15625 42.9219 6.5625 44.5469 Q4.98438 46.1875 4.98438 48.3906 Q4.98438 52.1562 6.39062 55.4375 Q7.8125 58.7344 10.4688 61.2969 Q13.1406 63.875 16.4844 65.2344 Q19.8281 66.6094 23.5781 66.6094 Q29.2969 66.6094 34.2188 64.1875 Q39.1562 61.7656 42.0312 57.3438 Q44.9219 52.9375 44.9219 47.0156 Q44.9219 42.6719 43.0156 38.7656 Q41.1094 34.8594 38.125 31.6562 Q35.1562 28.4688 30.5156 24.4062 Q25.875 20.3594 24.4219 19 L13.0938 8.10938 L22.7031 8.10938 Q29.7812 8.10938 34.5469 8.21875 Q39.3125 8.34375 39.5938 8.59375 Q40.7656 9.85938 42 17.8281 L44.9219 17.8281 L42.0938 0 z \" id=\"Cmr10-32\"/>\n",
       "     <path d=\" M9.51562 7.71875 Q11.8594 4.29688 15.8125 2.64062 Q19.7812 0.984375 24.3125 0.984375 Q30.125 0.984375 32.5625 5.9375 Q35.0156 10.8906 35.0156 17.1875 Q35.0156 20.0156 34.5 22.8438 Q33.9844 25.6875 32.7656 28.125 Q31.5469 30.5625 29.4219 32.0312 Q27.2969 33.5 24.2188 33.5 L17.5781 33.5 Q16.7031 33.5 16.7031 34.4219 L16.7031 35.2969 Q16.7031 36.0781 17.5781 36.0781 L23.0938 36.5312 Q26.6094 36.5312 28.9219 39.1562 Q31.25 41.7969 32.3281 45.5781 Q33.4062 49.3594 33.4062 52.7812 Q33.4062 57.5625 31.1562 60.6406 Q28.9062 63.7188 24.3125 63.7188 Q20.5156 63.7188 17.0469 62.2812 Q13.5781 60.8438 11.5312 57.9062 Q11.7188 57.9531 11.8594 57.9844 Q12.0156 58.0156 12.2031 58.0156 Q14.4531 58.0156 15.9688 56.4531 Q17.4844 54.8906 17.4844 52.6875 Q17.4844 50.5312 15.9688 48.9688 Q14.4531 47.4062 12.2031 47.4062 Q10.0156 47.4062 8.45312 48.9688 Q6.89062 50.5312 6.89062 52.6875 Q6.89062 56.9844 9.46875 60.1562 Q12.0625 63.3281 16.1406 64.9688 Q20.2188 66.6094 24.3125 66.6094 Q27.3438 66.6094 30.7031 65.7031 Q34.0781 64.7969 36.8125 63.1094 Q39.5469 61.4219 41.2812 58.7812 Q43.0156 56.1562 43.0156 52.7812 Q43.0156 48.5781 41.1406 45.0156 Q39.2656 41.4531 35.9844 38.8594 Q32.7188 36.2812 28.8125 35.0156 Q33.1562 34.1875 37.0625 31.7344 Q40.9688 29.2969 43.3281 25.4844 Q45.7031 21.6875 45.7031 17.2812 Q45.7031 11.7656 42.6719 7.29688 Q39.6562 2.82812 34.7188 0.3125 Q29.7812 -2.20312 24.3125 -2.20312 Q19.625 -2.20312 14.9062 -0.40625 Q10.2031 1.375 7.20312 4.9375 Q4.20312 8.5 4.20312 13.4844 Q4.20312 15.9688 5.85938 17.625 Q7.51562 19.2812 10.0156 19.2812 Q11.625 19.2812 12.9688 18.5312 Q14.3125 17.7812 15.0625 16.4062 Q15.8281 15.0469 15.8281 13.4844 Q15.8281 11.0312 14.1094 9.375 Q12.4062 7.71875 10.0156 7.71875 z \" id=\"Cmr10-33\"/>\n",
       "     <path d=\" M25 -2.20312 Q18.7969 -2.20312 14.6406 1.09375 Q10.5 4.39062 8.21875 9.64062 Q5.95312 14.8906 5.07812 20.6562 Q4.20312 26.4219 4.20312 32.3281 Q4.20312 40.2344 7.28125 48.1875 Q10.3594 56.1562 16.3281 61.375 Q22.3125 66.6094 30.5156 66.6094 Q33.9375 66.6094 36.8906 65.3125 Q39.8438 64.0156 41.5312 61.5 Q43.2188 58.9844 43.2188 55.4219 Q43.2188 53.375 41.8281 51.9844 Q40.4375 50.5938 38.375 50.5938 Q36.4219 50.5938 35 52 Q33.5938 53.4219 33.5938 55.4219 Q33.5938 57.375 35 58.7812 Q36.4219 60.2031 38.375 60.2031 L38.9219 60.2031 Q37.6406 62.0156 35.3281 62.8594 Q33.0156 63.7188 30.5156 63.7188 Q27.4844 63.7188 24.9219 62.3906 Q22.3594 61.0781 20.3125 58.8281 Q18.2656 56.5938 16.8906 53.875 Q15.5312 51.1719 14.7656 47.7031 Q14.0156 44.2344 13.8125 41.2031 Q13.625 38.1875 13.625 33.5938 Q15.375 37.7031 18.5938 40.3125 Q21.8281 42.9219 25.875 42.9219 Q30.3281 42.9219 33.9844 41.1094 Q37.6406 39.3125 40.2812 36.1094 Q42.9219 32.9062 44.3125 28.7969 Q45.7031 24.7031 45.7031 20.5156 Q45.7031 14.6562 43.0938 9.34375 Q40.4844 4.04688 35.7344 0.921875 Q31 -2.20312 25 -2.20312 M25 0.984375 Q28.8594 0.984375 31.2031 2.73438 Q33.5469 4.5 34.6406 7.39062 Q35.75 10.2969 36.0156 13.25 Q36.2812 16.2188 36.2812 20.5156 Q36.2812 26.1719 35.7344 30.1719 Q35.2031 34.1875 32.8125 37.2344 Q30.4219 40.2812 25.4844 40.2812 Q21.4375 40.2812 18.8281 37.5469 Q16.2188 34.8125 15.0156 30.6406 Q13.8125 26.4688 13.8125 22.6094 Q13.8125 21.2969 13.9219 20.6094 Q13.9219 20.4531 13.8906 20.3594 Q13.875 20.2656 13.8125 20.125 Q13.8125 15.8281 14.6875 11.4219 Q15.5781 7.03125 18.0625 4 Q20.5625 0.984375 25 0.984375\" id=\"Cmr10-36\"/>\n",
       "     <path d=\" M17.3906 2.59375 Q17.3906 8.15625 18.3594 13.4688 Q19.3438 18.7969 21.2188 24 Q23.0938 29.2031 25.75 34.2031 Q28.4219 39.2031 31.5938 43.6094 L40.7188 56.2969 L29.2969 56.2969 Q11.5312 56.2969 10.9844 55.8125 Q9.67188 54.2031 8.5 46.5781 L5.60938 46.5781 L8.89062 67.5781 L11.8125 67.5781 L11.8125 67.2812 Q11.8125 65.4844 17.9219 64.9375 Q24.0312 64.4062 29.8906 64.4062 L48.4844 64.4062 L48.4844 61.8125 Q48.4844 61.7188 48.4531 61.6719 Q48.4375 61.625 48.3906 61.5312 L34.625 42.1875 Q29.5469 34.6719 28.2656 25.4844 Q27 16.3125 27 2.59375 Q27 0.640625 25.5781 -0.78125 Q24.1719 -2.20312 22.2188 -2.20312 Q20.2188 -2.20312 18.7969 -0.78125 Q17.3906 0.640625 17.3906 2.59375\" id=\"Cmr10-37\"/>\n",
       "    </defs>\n",
       "    <g transform=\"translate(182.5 256.86)scale(0.2 -0.2)\">\n",
       "     <use xlink:href=\"#Cmmi10-77\"/>\n",
       "     <use transform=\"translate(71.484375 -25.509375000000006)scale(0.7)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(139.04296875 0.0)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(223.1015625 0.0)\" xlink:href=\"#Cmr10-5b\"/>\n",
       "     <use transform=\"translate(250.787109375 0.0)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(300.787109375 0.0)\" xlink:href=\"#Cmmi10-3a\"/>\n",
       "     <use transform=\"translate(329.330078125 0.0)\" xlink:href=\"#Cmr10-36\"/>\n",
       "     <use transform=\"translate(379.330078125 0.0)\" xlink:href=\"#Cmr10-36\"/>\n",
       "     <use transform=\"translate(429.330078125 0.0)\" xlink:href=\"#Cmr10-33\"/>\n",
       "     <use transform=\"translate(479.330078125 0.0)\" xlink:href=\"#Cmr10-32\"/>\n",
       "     <use transform=\"translate(529.330078125 0.0)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(579.330078125 0.0)\" xlink:href=\"#Cmr10-37\"/>\n",
       "     <use transform=\"translate(629.330078125 0.0)\" xlink:href=\"#Cmr10-39\"/>\n",
       "     <use transform=\"translate(679.330078125 0.0)\" xlink:href=\"#Cmr10-38\"/>\n",
       "     <use transform=\"translate(729.330078125 0.0)\" xlink:href=\"#Cmr10-5d\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_16\">\n",
       "    <!-- $w_2 = [ 1.31695415]$ -->\n",
       "    <defs>\n",
       "     <path d=\" M2.78125 16.5 L2.78125 20.0156 L33.6875 66.1094 Q34.0312 66.6094 34.7188 66.6094 L36.1875 66.6094 Q37.3125 66.6094 37.3125 65.4844 L37.3125 20.0156 L47.125 20.0156 L47.125 16.5 L37.3125 16.5 L37.3125 6.6875 Q37.3125 4.64062 40.2344 4.07812 Q43.1719 3.51562 47.0156 3.51562 L47.0156 0 L19.4844 0 L19.4844 3.51562 Q23.3438 3.51562 26.2656 4.07812 Q29.2031 4.64062 29.2031 6.6875 L29.2031 16.5 z  M6.10938 20.0156 L29.7812 20.0156 L29.7812 55.4219 z \" id=\"Cmr10-34\"/>\n",
       "    </defs>\n",
       "    <g transform=\"translate(280.15 228.96)scale(0.2 -0.2)\">\n",
       "     <use xlink:href=\"#Cmmi10-77\"/>\n",
       "     <use transform=\"translate(71.484375 -25.509375000000006)scale(0.7)\" xlink:href=\"#Cmr10-32\"/>\n",
       "     <use transform=\"translate(139.04296875 0.0)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(223.1015625 0.0)\" xlink:href=\"#Cmr10-5b\"/>\n",
       "     <use transform=\"translate(250.787109375 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(300.787109375 0.0)\" xlink:href=\"#Cmmi10-3a\"/>\n",
       "     <use transform=\"translate(329.330078125 0.0)\" xlink:href=\"#Cmr10-33\"/>\n",
       "     <use transform=\"translate(379.330078125 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(429.330078125 0.0)\" xlink:href=\"#Cmr10-36\"/>\n",
       "     <use transform=\"translate(479.330078125 0.0)\" xlink:href=\"#Cmr10-39\"/>\n",
       "     <use transform=\"translate(529.330078125 0.0)\" xlink:href=\"#Cmr10-35\"/>\n",
       "     <use transform=\"translate(579.330078125 0.0)\" xlink:href=\"#Cmr10-34\"/>\n",
       "     <use transform=\"translate(629.330078125 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(679.330078125 0.0)\" xlink:href=\"#Cmr10-35\"/>\n",
       "     <use transform=\"translate(729.330078125 0.0)\" xlink:href=\"#Cmr10-5d\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p41d7ee9abc\">\n",
       "   <rect height=\"390.6\" width=\"390.6\" x=\"63.0\" y=\"50.4\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pods.notebook.display_plots('polynomial_function{func_num}.svg', directory='./diagrams', func_num=(1,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcYAAAG/CAYAAADCT3cgAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10989a860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(figsize=(7, 7))\n",
    "\n",
    "loc = [[-1.25, -0.4],\n",
    "       [0., 1.25],\n",
    "       [1.25, -0.4]]\n",
    "text = ['$\\phi_1(x) = e^{-(x + 1)^2}$',\n",
    "        '$\\phi_2(x) = e^{-2x^2}$', \n",
    "        '$\\phi_3(x) = e^{-2(x-1)^2}$']\n",
    "mlai.plot_basis(mlai.radial, x_min=-2, x_max=2, fig=f, ax=ax, loc=loc, text=text)    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Radial Basis Functions\n",
    "\n",
    "-   Or they can be local. E.g. radial (or Gaussian) basis\n",
    "    $$\\phi_j(x) = \\exp\\left(-\\frac{(x-\\mu_j)^2}{\\ell^2}\\right)$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg height=\"504pt\" version=\"1.1\" viewBox=\"0 0 504 504\" width=\"504pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\" M0 504 L504 504 L504 0 L0 0 z \" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\" M63 441 L453.6 441 L453.6 50.4 L63 50.4 z \" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_1\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M63 232.485 L66.9455 230.217 L70.8909 227.679 L74.8364 224.86 L78.7818 221.758 L82.7273 218.374 L86.6727 214.714 L90.6182 210.792 L94.5636 206.63 L98.5091 202.256 L102.455 197.707 L106.4 193.027 L110.345 188.267 L114.291 183.484 L118.236 178.741 L122.182 174.106 L126.127 169.648 L130.073 165.439 L134.018 161.548 L137.964 158.042 L141.909 154.985 L145.855 152.432 L149.8 150.432 L153.745 149.022 L157.691 148.229 L161.636 148.07 L165.582 148.547 L169.527 149.651 L173.473 151.36 L177.418 153.642 L181.364 156.454 L185.309 159.743 L189.255 163.449 L193.2 167.508 L197.145 171.85 L201.091 176.406 L205.036 181.103 L208.982 185.874 L212.927 190.653 L216.873 195.38 L220.818 200.001 L224.764 204.467 L228.709 208.739 L232.655 212.784 L236.6 216.577 L240.545 220.101 L244.491 223.345 L248.436 226.305 L252.382 228.982 L256.327 231.384 L260.273 233.52 L264.218 235.405 L268.164 237.055 L272.109 238.488 L276.055 239.723 L280 240.778 L283.945 241.673 L287.891 242.427 L291.836 243.057 L295.782 243.58 L299.727 244.01 L303.673 244.362 L307.618 244.648 L311.564 244.878 L315.509 245.061 L319.455 245.207 L323.4 245.322 L327.345 245.413 L331.291 245.483 L335.236 245.537 L339.182 245.578 L343.127 245.61 L347.073 245.633 L351.018 245.651 L354.964 245.664 L358.909 245.674 L362.855 245.682 L366.8 245.687 L370.745 245.691 L374.691 245.693 L378.636 245.695 L382.582 245.697 L386.527 245.698 L390.473 245.698 L394.418 245.699 L398.364 245.699 L402.309 245.7 L406.255 245.7 L410.2 245.7 L414.145 245.7 L418.091 245.7 L422.036 245.7 L425.982 245.7 L429.927 245.7 L433.873 245.7 L437.818 245.7 L441.764 245.7 L445.709 245.7 L449.655 245.7 L453.6 245.7\" style=\"fill:none;stroke:#ff0000;stroke-linecap:square;stroke-width:3;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_2\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M63 245.667 L66.9455 245.655 L70.8909 245.638 L74.8364 245.616 L78.7818 245.587 L82.7273 245.548 L86.6727 245.497 L90.6182 245.432 L94.5636 245.347 L98.5091 245.239 L102.455 245.101 L106.4 244.927 L110.345 244.71 L114.291 244.439 L118.236 244.105 L122.182 243.696 L126.127 243.198 L130.073 242.596 L134.018 241.874 L137.964 241.016 L141.909 240.002 L145.855 238.815 L149.8 237.433 L153.745 235.839 L157.691 234.015 L161.636 231.942 L165.582 229.608 L169.527 227 L173.473 224.112 L177.418 220.939 L181.364 217.484 L185.309 213.757 L189.255 209.773 L193.2 205.555 L197.145 201.134 L201.091 196.547 L205.036 191.842 L208.982 187.071 L212.927 182.292 L216.873 177.569 L220.818 172.972 L224.764 168.57 L228.709 164.433 L232.655 160.633 L236.6 157.233 L240.545 154.297 L244.491 151.879 L248.436 150.022 L252.382 148.765 L256.327 148.13 L260.273 148.13 L264.218 148.765 L268.164 150.022 L272.109 151.879 L276.055 154.297 L280 157.233 L283.945 160.633 L287.891 164.433 L291.836 168.57 L295.782 172.972 L299.727 177.569 L303.673 182.292 L307.618 187.071 L311.564 191.842 L315.509 196.547 L319.455 201.134 L323.4 205.555 L327.345 209.773 L331.291 213.757 L335.236 217.484 L339.182 220.939 L343.127 224.112 L347.073 227 L351.018 229.608 L354.964 231.942 L358.909 234.015 L362.855 235.839 L366.8 237.433 L370.745 238.815 L374.691 240.002 L378.636 241.016 L382.582 241.874 L386.527 242.596 L390.473 243.198 L394.418 243.696 L398.364 244.105 L402.309 244.439 L406.255 244.71 L410.2 244.927 L414.145 245.101 L418.091 245.239 L422.036 245.347 L425.982 245.432 L429.927 245.497 L433.873 245.548 L437.818 245.587 L441.764 245.616 L445.709 245.638 L449.655 245.655 L453.6 245.667\" style=\"fill:none;stroke:#ff00ff;stroke-linecap:square;stroke-width:3;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\" M453.6 441 L453.6 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\" M63 441 L63 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\" M63 50.4 L453.6 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\" M63 441 L453.6 441\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 -4\" id=\"mc7db9fdffb\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"160.65\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_4\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 4\" id=\"m5a7d422ac3\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"160.65\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- −1 -->\n",
       "      <defs>\n",
       "       <path d=\" M10.5938 35.5 L73.1875 35.5 L73.1875 27.2031 L10.5938 27.2031 z \" id=\"BitstreamVeraSans-Roman-2212\"/>\n",
       "       <path d=\" M12.4062 8.29688 L28.5156 8.29688 L28.5156 63.9219 L10.9844 60.4062 L10.9844 69.3906 L28.4219 72.9062 L38.2812 72.9062 L38.2812 8.29688 L54.3906 8.29688 L54.3906 0 L12.4062 0 z \" id=\"BitstreamVeraSans-Roman-31\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(154.2703125 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"258.3\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"258.3\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0 -->\n",
       "      <defs>\n",
       "       <path d=\" M31.7812 66.4062 Q24.1719 66.4062 20.3281 58.9062 Q16.5 51.4219 16.5 36.375 Q16.5 21.3906 20.3281 13.8906 Q24.1719 6.39062 31.7812 6.39062 Q39.4531 6.39062 43.2812 13.8906 Q47.125 21.3906 47.125 36.375 Q47.125 51.4219 43.2812 58.9062 Q39.4531 66.4062 31.7812 66.4062 M31.7812 74.2188 Q44.0469 74.2188 50.5156 64.5156 Q56.9844 54.8281 56.9844 36.375 Q56.9844 17.9688 50.5156 8.26562 Q44.0469 -1.42188 31.7812 -1.42188 Q19.5312 -1.42188 13.0625 8.26562 Q6.59375 17.9688 6.59375 36.375 Q6.59375 54.8281 13.0625 64.5156 Q19.5312 74.2188 31.7812 74.2188\" id=\"BitstreamVeraSans-Roman-30\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(255.78046875 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"355.95\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"355.95\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(353.7796875 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_4\">\n",
       "     <!-- $x$ -->\n",
       "     <defs>\n",
       "      <path d=\" M7.8125 2.875 Q9.57812 1.51562 12.7969 1.51562 Q15.9219 1.51562 18.3125 4.51562 Q20.7031 7.51562 21.5781 11.0781 L26.125 28.8125 Q27.2031 33.6406 27.2031 35.4062 Q27.2031 37.8906 25.8125 39.75 Q24.4219 41.6094 21.9219 41.6094 Q18.75 41.6094 15.9688 39.625 Q13.1875 37.6406 11.2812 34.5938 Q9.375 31.5469 8.59375 28.4219 Q8.40625 27.7812 7.8125 27.7812 L6.59375 27.7812 Q5.8125 27.7812 5.8125 28.7188 L5.8125 29 Q6.78125 32.7188 9.125 36.25 Q11.4688 39.7969 14.8594 41.9844 Q18.2656 44.1875 22.125 44.1875 Q25.7812 44.1875 28.7344 42.2344 Q31.6875 40.2812 32.9062 36.9219 Q34.625 39.9844 37.2812 42.0781 Q39.9375 44.1875 43.1094 44.1875 Q45.2656 44.1875 47.5 43.4219 Q49.75 42.6719 51.1719 41.1094 Q52.5938 39.5469 52.5938 37.2031 Q52.5938 34.6719 50.9531 32.8281 Q49.3125 31 46.7812 31 Q45.1719 31 44.0938 32.0312 Q43.0156 33.0625 43.0156 34.625 Q43.0156 36.7188 44.4531 38.2969 Q45.9062 39.8906 47.9062 40.1875 Q46.0938 41.6094 42.9219 41.6094 Q39.7031 41.6094 37.3281 38.625 Q34.9688 35.6406 33.9844 31.9844 L29.5938 14.3125 Q28.5156 10.2969 28.5156 7.71875 Q28.5156 5.17188 29.9531 3.34375 Q31.3906 1.51562 33.7969 1.51562 Q38.4844 1.51562 42.1562 5.64062 Q45.8438 9.76562 47.0156 14.7031 Q47.2188 15.2812 47.7969 15.2812 L49.0312 15.2812 Q49.4219 15.2812 49.6562 15.0156 Q49.9062 14.75 49.9062 14.4062 Q49.9062 14.3125 49.8125 14.1094 Q48.3906 8.15625 43.8438 3.51562 Q39.3125 -1.125 33.5938 -1.125 Q29.9375 -1.125 26.9844 0.84375 Q24.0312 2.82812 22.7969 6.20312 Q21.2344 3.26562 18.4688 1.0625 Q15.7188 -1.125 12.5938 -1.125 Q10.4531 -1.125 8.17188 -0.359375 Q5.90625 0.390625 4.48438 1.95312 Q3.07812 3.51562 3.07812 5.90625 Q3.07812 8.25 4.70312 10.1719 Q6.34375 12.1094 8.79688 12.1094 Q10.4531 12.1094 11.5781 11.1094 Q12.7031 10.1094 12.7031 8.5 Q12.7031 6.39062 11.2969 4.82812 Q9.90625 3.26562 7.8125 2.875\" id=\"Cmmi10-78\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(252.5 474.875)scale(0.2 -0.2)\">\n",
       "      <use transform=\"translate(0.0 0.8125)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L4 0\" id=\"md7965d1ba0\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_10\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L-4 0\" id=\"md9a1c1a7cd\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- −2 -->\n",
       "      <defs>\n",
       "       <path d=\" M19.1875 8.29688 L53.6094 8.29688 L53.6094 0 L7.32812 0 L7.32812 8.29688 Q12.9375 14.1094 22.625 23.8906 Q32.3281 33.6875 34.8125 36.5312 Q39.5469 41.8438 41.4219 45.5312 Q43.3125 49.2188 43.3125 52.7812 Q43.3125 58.5938 39.2344 62.25 Q35.1562 65.9219 28.6094 65.9219 Q23.9688 65.9219 18.8125 64.3125 Q13.6719 62.7031 7.8125 59.4219 L7.8125 69.3906 Q13.7656 71.7812 18.9375 73 Q24.125 74.2188 28.4219 74.2188 Q39.75 74.2188 46.4844 68.5469 Q53.2188 62.8906 53.2188 53.4219 Q53.2188 48.9219 51.5312 44.8906 Q49.8594 40.875 45.4062 35.4062 Q44.1875 33.9844 37.6406 27.2188 Q31.1094 20.4531 19.1875 8.29688\" id=\"BitstreamVeraSans-Roman-32\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(46.31875 443.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"343.35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"343.35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- −1 -->\n",
       "      <g transform=\"translate(46.240625 346.109375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"245.7\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_14\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"245.7\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_7\">\n",
       "      <!-- 0 -->\n",
       "      <g transform=\"translate(53.9609375 248.459375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_15\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"148.05\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_16\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"148.05\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(54.659375 150.809375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_17\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_18\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 2 -->\n",
       "      <g transform=\"translate(54.371875 53.159375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_10\">\n",
       "     <!-- $\\phi(x)$ -->\n",
       "     <defs>\n",
       "      <path d=\" M19.3906 -19.4844 L23.875 -1.42188 Q18.7969 -1.42188 14.4062 0.953125 Q10.0156 3.32812 7.34375 7.5625 Q4.6875 11.8125 4.6875 16.7969 Q4.6875 22.2188 7.375 27.1406 Q10.0625 32.0781 14.7188 36.1094 Q19.3906 40.1406 24.75 42.3125 Q30.125 44.4844 35.4062 44.4844 L41.4062 68.7969 Q41.6094 69.3906 42.1875 69.3906 L43.4062 69.3906 Q43.7969 69.3906 44.0312 69.0469 Q44.2812 68.7031 44.2812 68.4062 L38.2812 44.4844 Q42.0469 44.4844 45.5625 43.1094 Q49.0781 41.75 51.7812 39.2812 Q54.5 36.8125 56 33.4688 Q57.5156 30.125 57.5156 26.3125 Q57.5156 20.9062 54.8281 15.9688 Q52.1562 11.0312 47.6094 7.07812 Q43.0625 3.125 37.4688 0.84375 Q31.8906 -1.42188 26.8125 -1.42188 L22.2188 -19.9219 Q22.0156 -20.5156 21.3906 -20.5156 L20.2188 -20.5156 Q19.8281 -20.5156 19.6094 -20.1406 Q19.3906 -19.7812 19.3906 -19.4844 M24.5156 1.21875 L34.7188 41.8906 Q32.1719 41.8906 29.3906 40.8594 Q26.6094 39.8438 24.1406 38.2812 Q21.6875 36.7188 19.6719 34.7188 Q17.2344 32.2812 15.375 28.9531 Q13.5312 25.6406 12.5781 21.9531 Q11.625 18.2656 11.625 14.7031 Q11.625 10.9375 13.2344 7.85938 Q14.8438 4.78125 17.7656 3 Q20.7031 1.21875 24.5156 1.21875 M27.4844 1.21875 Q31.25 1.21875 35.5 3.35938 Q39.75 5.51562 42.4844 8.40625 Q46.2344 12.1094 48.4062 17.5 Q50.5938 22.9062 50.5938 28.4219 Q50.5938 32.125 48.9688 35.2188 Q47.3594 38.3281 44.4219 40.1094 Q41.5 41.8906 37.7031 41.8906 z \" id=\"Cmmi10-c1\"/>\n",
       "      <path d=\" M6.5 -25 Q5.60938 -25 5.60938 -24.125 Q5.60938 -23.6875 5.8125 -23.4844 Q22.9062 -6.78125 22.9062 25 Q22.9062 56.7812 6 73.2969 Q5.60938 73.5312 5.60938 74.125 Q5.60938 74.4688 5.875 74.7344 Q6.15625 75 6.5 75 L7.42188 75 Q7.71875 75 7.90625 74.8125 Q15.0938 69.1406 19.875 61.0312 Q24.6562 52.9375 26.875 43.75 Q29.1094 34.5781 29.1094 25 Q29.1094 17.9219 27.9062 11.0625 Q26.7031 4.20312 24.0938 -2.45312 Q21.4844 -9.125 17.4844 -14.7656 Q13.4844 -20.4062 7.90625 -24.8125 Q7.71875 -25 7.42188 -25 z \" id=\"Cmr10-29\"/>\n",
       "      <path d=\" M31 -24.8125 Q25.4375 -20.4062 21.4062 -14.7188 Q17.3906 -9.03125 14.8125 -2.57812 Q12.25 3.85938 10.9844 10.8906 Q9.71875 17.9219 9.71875 25 Q9.71875 32.1719 10.9844 39.2031 Q12.25 46.2344 14.8594 52.7344 Q17.4844 59.2344 21.5312 64.8906 Q25.5938 70.5625 31 74.8125 Q31 75 31.5 75 L32.4219 75 Q32.7188 75 32.9531 74.7344 Q33.2031 74.4688 33.2031 74.125 Q33.2031 73.6875 33.0156 73.4844 Q28.125 68.7031 24.875 63.2344 Q21.625 57.7656 19.6406 51.5781 Q17.6719 45.4062 16.7969 38.7812 Q15.9219 32.1719 15.9219 25 Q15.9219 -6.78125 32.9062 -23.2969 Q33.2031 -23.5781 33.2031 -24.125 Q33.2031 -24.3594 32.9375 -24.6719 Q32.6719 -25 32.4219 -25 L31.5 -25 Q31 -25 31 -24.8125\" id=\"Cmr10-28\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(36.240625 265.2)rotate(-90.0)scale(0.2 -0.2)\">\n",
       "      <use xlink:href=\"#Cmmi10-c1\"/>\n",
       "      <use transform=\"translate(59.521484375 0.0)\" xlink:href=\"#Cmr10-28\"/>\n",
       "      <use transform=\"translate(98.33984375 0.0)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "      <use transform=\"translate(155.419921875 0.0)\" xlink:href=\"#Cmr10-29\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_11\">\n",
       "    <!-- $\\phi_1(x) = e^{-(x + 1)^2}$ -->\n",
       "    <defs>\n",
       "     <path d=\" M7.51562 23 Q6.6875 23 6.14062 23.625 Q5.60938 24.2656 5.60938 25 Q5.60938 25.7344 6.14062 26.3594 Q6.6875 27 7.51562 27 L36.9219 27 L36.9219 56.5 Q36.9219 57.2812 37.5 57.7812 Q38.0938 58.2969 38.9219 58.2969 Q39.6562 58.2969 40.2812 57.7812 Q40.9219 57.2812 40.9219 56.5 L40.9219 27 L70.3125 27 Q71.0469 27 71.5781 26.3594 Q72.125 25.7344 72.125 25 Q72.125 24.2656 71.5781 23.625 Q71.0469 23 70.3125 23 L40.9219 23 L40.9219 -6.5 Q40.9219 -7.28125 40.2812 -7.78125 Q39.6562 -8.29688 38.9219 -8.29688 Q38.0938 -8.29688 37.5 -7.78125 Q36.9219 -7.28125 36.9219 -6.5 L36.9219 23 z \" id=\"Cmr10-2b\"/>\n",
       "     <path d=\" M4.98438 0 L4.98438 2.6875 Q4.98438 2.9375 5.17188 3.21875 L20.7031 20.4062 Q24.2188 24.2188 26.4062 26.7969 Q28.6094 29.3906 30.75 32.7656 Q32.9062 36.1406 34.1562 39.625 Q35.4062 43.1094 35.4062 47.0156 Q35.4062 51.125 33.8906 54.8594 Q32.375 58.5938 29.375 60.8438 Q26.375 63.0938 22.125 63.0938 Q17.7812 63.0938 14.3125 60.4688 Q10.8438 57.8594 9.42188 53.7188 Q9.8125 53.8125 10.5 53.8125 Q12.75 53.8125 14.3281 52.2969 Q15.9219 50.7812 15.9219 48.3906 Q15.9219 46.0938 14.3281 44.5 Q12.75 42.9219 10.5 42.9219 Q8.15625 42.9219 6.5625 44.5469 Q4.98438 46.1875 4.98438 48.3906 Q4.98438 52.1562 6.39062 55.4375 Q7.8125 58.7344 10.4688 61.2969 Q13.1406 63.875 16.4844 65.2344 Q19.8281 66.6094 23.5781 66.6094 Q29.2969 66.6094 34.2188 64.1875 Q39.1562 61.7656 42.0312 57.3438 Q44.9219 52.9375 44.9219 47.0156 Q44.9219 42.6719 43.0156 38.7656 Q41.1094 34.8594 38.125 31.6562 Q35.1562 28.4688 30.5156 24.4062 Q25.875 20.3594 24.4219 19 L13.0938 8.10938 L22.7031 8.10938 Q29.7812 8.10938 34.5469 8.21875 Q39.3125 8.34375 39.5938 8.59375 Q40.7656 9.85938 42 17.8281 L44.9219 17.8281 L42.0938 0 z \" id=\"Cmr10-32\"/>\n",
       "     <path d=\" M20.125 -1.125 Q15.3281 -1.125 11.7344 1.35938 Q8.15625 3.85938 6.26562 8.07812 Q4.39062 12.3125 4.39062 17 Q4.39062 24.125 7.98438 30.3906 Q11.5781 36.6719 17.7188 40.4219 Q23.875 44.1875 30.9062 44.1875 Q35.2031 44.1875 38.4062 41.9688 Q41.6094 39.75 41.6094 35.5938 Q41.6094 29.8281 37.0156 27.0156 Q32.4219 24.2188 26.6562 23.5625 Q20.9062 22.9062 13.9219 22.9062 L13.625 22.9062 Q12.0156 16.9375 12.0156 12.7031 Q12.0156 8.25 14.0625 4.875 Q16.1094 1.51562 20.3125 1.51562 Q26.2656 1.51562 31.75 4.25 Q37.25 6.98438 40.7188 11.8125 Q41.0156 12.1094 41.5 12.1094 Q42 12.1094 42.5 11.5469 Q43.0156 10.9844 43.0156 10.5 Q43.0156 10.1094 42.8281 9.90625 Q39.1562 4.78125 32.9531 1.82812 Q26.7656 -1.125 20.125 -1.125 M14.2031 25.4844 Q20.0156 25.4844 24.9531 26.0156 Q29.8906 26.5625 33.8906 28.7344 Q37.8906 30.9062 37.8906 35.5 Q37.8906 37.3594 36.8594 38.75 Q35.8438 40.1406 34.2031 40.875 Q32.5625 41.6094 30.8125 41.6094 Q26.5625 41.6094 23.1719 39.3125 Q19.7812 37.0156 17.5781 33.3281 Q15.375 29.6406 14.2031 25.4844\" id=\"Cmmi10-65\"/>\n",
       "     <path d=\" M9.28125 0 L9.28125 3.51562 Q21.7812 3.51562 21.7812 6.6875 L21.7812 59.1875 Q16.6094 56.6875 8.6875 56.6875 L8.6875 60.2031 Q20.9531 60.2031 27.2031 66.6094 L28.6094 66.6094 Q28.9531 66.6094 29.2656 66.3281 Q29.5938 66.0625 29.5938 65.7188 L29.5938 6.6875 Q29.5938 3.51562 42.0938 3.51562 L42.0938 0 z \" id=\"Cmr10-31\"/>\n",
       "     <path d=\" M10.2031 23 Q9.375 23 8.82812 23.625 Q8.29688 24.2656 8.29688 25 Q8.29688 25.7344 8.82812 26.3594 Q9.375 27 10.2031 27 L67.5781 27 Q68.3594 27 68.875 26.3594 Q69.3906 25.7344 69.3906 25 Q69.3906 24.2656 68.875 23.625 Q68.3594 23 67.5781 23 z \" id=\"Cmsy10-a1\"/>\n",
       "     <path d=\" M7.51562 13.2812 Q6.6875 13.2812 6.14062 13.9062 Q5.60938 14.5469 5.60938 15.2812 Q5.60938 16.1094 6.14062 16.6875 Q6.6875 17.2812 7.51562 17.2812 L70.3125 17.2812 Q71.0469 17.2812 71.5781 16.6875 Q72.125 16.1094 72.125 15.2812 Q72.125 14.5469 71.5781 13.9062 Q71.0469 13.2812 70.3125 13.2812 z  M7.51562 32.7188 Q6.6875 32.7188 6.14062 33.2969 Q5.60938 33.8906 5.60938 34.7188 Q5.60938 35.4531 6.14062 36.0781 Q6.6875 36.7188 7.51562 36.7188 L70.3125 36.7188 Q71.0469 36.7188 71.5781 36.0781 Q72.125 35.4531 72.125 34.7188 Q72.125 33.8906 71.5781 33.2969 Q71.0469 32.7188 70.3125 32.7188 z \" id=\"Cmr10-3d\"/>\n",
       "    </defs>\n",
       "    <g transform=\"translate(65.0375 284.76)scale(0.2 -0.2)\">\n",
       "     <use transform=\"translate(0.0 0.20234375000001137)\" xlink:href=\"#Cmmi10-c1\"/>\n",
       "     <use transform=\"translate(59.521484375 -25.307031249999994)scale(0.7)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(109.521484375 0.20234375000001137)\" xlink:href=\"#Cmr10-28\"/>\n",
       "     <use transform=\"translate(148.33984375 0.20234375000001137)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     <use transform=\"translate(205.419921875 0.20234375000001137)\" xlink:href=\"#Cmr10-29\"/>\n",
       "     <use transform=\"translate(261.796875 0.20234375000001137)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(345.85546875 0.20234375000001137)\" xlink:href=\"#Cmmi10-65\"/>\n",
       "     <use transform=\"translate(392.33984375 42.71796875000001)scale(0.7)\" xlink:href=\"#Cmsy10-a1\"/>\n",
       "     <use transform=\"translate(446.7197265625 42.71796875000001)scale(0.7)\" xlink:href=\"#Cmr10-28\"/>\n",
       "     <use transform=\"translate(473.892578125 42.71796875000001)scale(0.7)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     <use transform=\"translate(526.1396484375 42.71796875000001)scale(0.7)\" xlink:href=\"#Cmr10-2b\"/>\n",
       "     <use transform=\"translate(584.9806640625 42.71796875000001)scale(0.7)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(619.9806640625 42.71796875000001)scale(0.7)\" xlink:href=\"#Cmr10-29\"/>\n",
       "     <use transform=\"translate(647.153515625 77.36140625000002)scale(0.49)\" xlink:href=\"#Cmr10-32\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_12\">\n",
       "    <!-- $\\phi_2(x) = e^{-2x^2}$ -->\n",
       "    <g transform=\"translate(199.7 123.6375)scale(0.2 -0.2)\">\n",
       "     <use transform=\"translate(0.0 0.08484375000000455)\" xlink:href=\"#Cmmi10-c1\"/>\n",
       "     <use transform=\"translate(59.521484375 -25.42453125)scale(0.7)\" xlink:href=\"#Cmr10-32\"/>\n",
       "     <use transform=\"translate(109.521484375 0.08484375000000455)\" xlink:href=\"#Cmr10-28\"/>\n",
       "     <use transform=\"translate(148.33984375 0.08484375000000455)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     <use transform=\"translate(205.419921875 0.08484375000000455)\" xlink:href=\"#Cmr10-29\"/>\n",
       "     <use transform=\"translate(261.796875 0.08484375000000455)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(345.85546875 0.08484375000000455)\" xlink:href=\"#Cmmi10-65\"/>\n",
       "     <use transform=\"translate(392.33984375 42.600468750000005)scale(0.7)\" xlink:href=\"#Cmsy10-a1\"/>\n",
       "     <use transform=\"translate(446.7197265625 42.600468750000005)scale(0.7)\" xlink:href=\"#Cmr10-32\"/>\n",
       "     <use transform=\"translate(481.7197265625 42.600468750000005)scale(0.7)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     <use transform=\"translate(521.67578125 72.36140625)scale(0.49)\" xlink:href=\"#Cmr10-32\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p41d7ee9abc\">\n",
       "   <rect height=\"390.6\" width=\"390.6\" x=\"63.0\" y=\"50.4\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pods.notebook.display_plots('radial_basis{num_basis}.svg', directory='./diagrams', num_basis=(1,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Functions Derived from Radial Basis\n",
    "\n",
    "$$f(x) = {\\color{\\redColor}w_1 e^{-2(x+1)^2}}  + {\\color{\\magentaColor}w_2e^{-2x^2}} + {\\color{\\blueColor}w_3 e^{-2(x-1)^2}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg height=\"504pt\" version=\"1.1\" viewBox=\"0 0 504 504\" width=\"504pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\" M0 504 L504 504 L504 0 L0 0 z \" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\" M63 441 L453.6 441 L453.6 50.4 L63 50.4 z \" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_1\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M63 213.169 L66.9455 212.372 L70.8909 211.479 L74.8364 210.487 L78.7818 209.392 L82.7273 208.196 L86.6727 206.9 L90.6182 205.507 L94.5636 204.024 L98.5091 202.459 L102.455 200.823 L106.4 199.129 L110.345 197.393 L114.291 195.631 L118.236 193.862 L122.182 192.106 L126.127 190.382 L130.073 188.711 L134.018 187.112 L137.964 185.602 L141.909 184.197 L145.855 182.912 L149.8 181.756 L153.745 180.737 L157.691 179.858 L161.636 179.12 L165.582 178.52 L169.527 178.051 L173.473 177.703 L177.418 177.466 L181.364 177.327 L185.309 177.272 L189.255 177.288 L193.2 177.361 L197.145 177.481 L201.091 177.64 L205.036 177.833 L208.982 178.057 L212.927 178.316 L216.873 178.617 L220.818 178.97 L224.764 179.391 L228.709 179.899 L232.655 180.515 L236.6 181.263 L240.545 182.17 L244.491 183.261 L248.436 184.562 L252.382 186.096 L256.327 187.886 L260.273 189.949 L264.218 192.298 L268.164 194.942 L272.109 197.883 L276.055 201.116 L280 204.631 L283.945 208.41 L287.891 212.428 L291.836 216.654 L295.782 221.05 L299.727 225.574 L303.673 230.178 L307.618 234.809 L311.564 239.413 L315.509 243.934 L319.455 248.315 L323.4 252.499 L327.345 256.432 L331.291 260.063 L335.236 263.346 L339.182 266.24 L343.127 268.711 L347.073 270.732 L351.018 272.284 L354.964 273.358 L358.909 273.952 L362.855 274.072 L366.8 273.735 L370.745 272.963 L374.691 271.786 L378.636 270.24 L382.582 268.366 L386.527 266.209 L390.473 263.816 L394.418 261.236 L398.364 258.517 L402.309 255.707 L406.255 252.85 L410.2 249.99 L414.145 247.165 L418.091 244.408 L422.036 241.75 L425.982 239.214 L429.927 236.82 L433.873 234.583 L437.818 232.51 L441.764 230.609 L445.709 228.881 L449.655 227.323 L453.6 225.93\" style=\"fill:none;stroke:#0000ff;stroke-linecap:square;stroke-width:3;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_2\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M63 210.248 L66.9455 208.953 L70.8909 207.502 L74.8364 205.892 L78.7818 204.119 L82.7273 202.185 L86.6727 200.093 L90.6182 197.853 L94.5636 195.474 L98.5091 192.975 L102.455 190.375 L106.4 187.701 L110.345 184.981 L114.291 182.248 L118.236 179.538 L122.182 176.889 L126.127 174.342 L130.073 171.936 L134.018 169.713 L137.964 167.71 L141.909 165.963 L145.855 164.504 L149.8 163.361 L153.745 162.555 L157.691 162.102 L161.636 162.011 L165.582 162.284 L169.527 162.915 L173.473 163.892 L177.418 165.196 L181.364 166.802 L185.309 168.681 L189.255 170.799 L193.2 173.119 L197.145 175.6 L201.091 178.203 L205.036 180.887 L208.982 183.614 L212.927 186.345 L216.873 189.046 L220.818 191.686 L224.764 194.238 L228.709 196.68 L232.655 198.991 L236.6 201.159 L240.545 203.172 L244.491 205.026 L248.436 206.717 L252.382 208.247 L256.327 209.619 L260.273 210.84 L264.218 211.917 L268.164 212.86 L272.109 213.679 L276.055 214.384 L280 214.987 L283.945 215.499 L287.891 215.93 L291.836 216.29 L295.782 216.589 L299.727 216.835 L303.673 217.036 L307.618 217.199 L311.564 217.33 L315.509 217.435 L319.455 217.519 L323.4 217.584 L327.345 217.636 L331.291 217.676 L335.236 217.707 L339.182 217.73 L343.127 217.748 L347.073 217.762 L351.018 217.772 L354.964 217.78 L358.909 217.785 L362.855 217.789 L366.8 217.792 L370.745 217.795 L374.691 217.796 L378.636 217.797 L382.582 217.798 L386.527 217.799 L390.473 217.799 L394.418 217.799 L398.364 217.8 L402.309 217.8 L406.255 217.8 L410.2 217.8 L414.145 217.8 L418.091 217.8 L422.036 217.8 L425.982 217.8 L429.927 217.8 L433.873 217.8 L437.818 217.8 L441.764 217.8 L445.709 217.8 L449.655 217.8 L453.6 217.8\" style=\"fill:none;stroke:#ff0000;stroke-linecap:square;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_3\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M63 217.781 L66.9455 217.774 L70.8909 217.765 L74.8364 217.752 L78.7818 217.735 L82.7273 217.713 L86.6727 217.684 L90.6182 217.647 L94.5636 217.598 L98.5091 217.536 L102.455 217.458 L106.4 217.359 L110.345 217.234 L114.291 217.08 L118.236 216.889 L122.182 216.655 L126.127 216.37 L130.073 216.026 L134.018 215.614 L137.964 215.123 L141.909 214.544 L145.855 213.865 L149.8 213.076 L153.745 212.165 L157.691 211.123 L161.636 209.939 L165.582 208.605 L169.527 207.115 L173.473 205.464 L177.418 203.651 L181.364 201.677 L185.309 199.547 L189.255 197.27 L193.2 194.86 L197.145 192.333 L201.091 189.713 L205.036 187.024 L208.982 184.297 L212.927 181.567 L216.873 178.868 L220.818 176.241 L224.764 173.726 L228.709 171.362 L232.655 169.19 L236.6 167.248 L240.545 165.57 L244.491 164.188 L248.436 163.127 L252.382 162.408 L256.327 162.046 L260.273 162.046 L264.218 162.408 L268.164 163.127 L272.109 164.188 L276.055 165.57 L280 167.248 L283.945 169.19 L287.891 171.362 L291.836 173.726 L295.782 176.241 L299.727 178.868 L303.673 181.567 L307.618 184.297 L311.564 187.024 L315.509 189.713 L319.455 192.333 L323.4 194.86 L327.345 197.27 L331.291 199.547 L335.236 201.677 L339.182 203.651 L343.127 205.464 L347.073 207.115 L351.018 208.605 L354.964 209.939 L358.909 211.123 L362.855 212.165 L366.8 213.076 L370.745 213.865 L374.691 214.544 L378.636 215.123 L382.582 215.614 L386.527 216.026 L390.473 216.37 L394.418 216.655 L398.364 216.889 L402.309 217.08 L406.255 217.234 L410.2 217.359 L414.145 217.458 L418.091 217.536 L422.036 217.598 L425.982 217.647 L429.927 217.684 L433.873 217.713 L437.818 217.735 L441.764 217.752 L445.709 217.765 L449.655 217.774 L453.6 217.781\" style=\"fill:none;stroke:#ff00ff;stroke-linecap:square;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_4\">\n",
       "    <path clip-path=\"url(#p41d7ee9abc)\" d=\" M63 217.8 L66.9455 217.8 L70.8909 217.8 L74.8364 217.8 L78.7818 217.8 L82.7273 217.8 L86.6727 217.8 L90.6182 217.8 L94.5636 217.8 L98.5091 217.8 L102.455 217.8 L106.4 217.8 L110.345 217.8 L114.291 217.8 L118.236 217.8 L122.182 217.799 L126.127 217.799 L130.073 217.799 L134.018 217.798 L137.964 217.797 L141.909 217.796 L145.855 217.795 L149.8 217.792 L153.745 217.789 L157.691 217.785 L161.636 217.78 L165.582 217.772 L169.527 217.762 L173.473 217.748 L177.418 217.73 L181.364 217.707 L185.309 217.676 L189.255 217.636 L193.2 217.584 L197.145 217.519 L201.091 217.435 L205.036 217.33 L208.982 217.199 L212.927 217.036 L216.873 216.835 L220.818 216.589 L224.764 216.29 L228.709 215.93 L232.655 215.499 L236.6 214.987 L240.545 214.384 L244.491 213.679 L248.436 212.86 L252.382 211.917 L256.327 210.84 L260.273 209.619 L264.218 208.247 L268.164 206.717 L272.109 205.026 L276.055 203.172 L280 201.159 L283.945 198.991 L287.891 196.68 L291.836 194.238 L295.782 191.686 L299.727 189.046 L303.673 186.345 L307.618 183.614 L311.564 180.887 L315.509 178.203 L319.455 175.6 L323.4 173.119 L327.345 170.799 L331.291 168.681 L335.236 166.802 L339.182 165.196 L343.127 163.892 L347.073 162.915 L351.018 162.284 L354.964 162.011 L358.909 162.102 L362.855 162.555 L366.8 163.361 L370.745 164.504 L374.691 165.963 L378.636 167.71 L382.582 169.713 L386.527 171.936 L390.473 174.342 L394.418 176.889 L398.364 179.538 L402.309 182.248 L406.255 184.981 L410.2 187.701 L414.145 190.375 L418.091 192.975 L422.036 195.474 L425.982 197.853 L429.927 200.093 L433.873 202.185 L437.818 204.119 L441.764 205.892 L445.709 207.502 L449.655 208.953 L453.6 210.248\" style=\"fill:none;stroke:#0000ff;stroke-linecap:square;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\" M453.6 441 L453.6 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\" M63 441 L63 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\" M63 50.4 L453.6 50.4\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\" M63 441 L453.6 441\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 -4\" id=\"mc7db9fdffb\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"160.65\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_6\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 4\" id=\"m5a7d422ac3\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"160.65\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- −1 -->\n",
       "      <defs>\n",
       "       <path d=\" M10.5938 35.5 L73.1875 35.5 L73.1875 27.2031 L10.5938 27.2031 z \" id=\"BitstreamVeraSans-Roman-2212\"/>\n",
       "       <path d=\" M12.4062 8.29688 L28.5156 8.29688 L28.5156 63.9219 L10.9844 60.4062 L10.9844 69.3906 L28.4219 72.9062 L38.2812 72.9062 L38.2812 8.29688 L54.3906 8.29688 L54.3906 0 L12.4062 0 z \" id=\"BitstreamVeraSans-Roman-31\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(154.2703125 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"258.3\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"258.3\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0 -->\n",
       "      <defs>\n",
       "       <path d=\" M31.7812 66.4062 Q24.1719 66.4062 20.3281 58.9062 Q16.5 51.4219 16.5 36.375 Q16.5 21.3906 20.3281 13.8906 Q24.1719 6.39062 31.7812 6.39062 Q39.4531 6.39062 43.2812 13.8906 Q47.125 21.3906 47.125 36.375 Q47.125 51.4219 43.2812 58.9062 Q39.4531 66.4062 31.7812 66.4062 M31.7812 74.2188 Q44.0469 74.2188 50.5156 64.5156 Q56.9844 54.8281 56.9844 36.375 Q56.9844 17.9688 50.5156 8.26562 Q44.0469 -1.42188 31.7812 -1.42188 Q19.5312 -1.42188 13.0625 8.26562 Q6.59375 17.9688 6.59375 36.375 Q6.59375 54.8281 13.0625 64.5156 Q19.5312 74.2188 31.7812 74.2188\" id=\"BitstreamVeraSans-Roman-30\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(255.78046875 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"355.95\" xlink:href=\"#mc7db9fdffb\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"355.95\" xlink:href=\"#m5a7d422ac3\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(353.7796875 452.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_4\">\n",
       "     <!-- $x$ -->\n",
       "     <defs>\n",
       "      <path d=\" M7.8125 2.875 Q9.57812 1.51562 12.7969 1.51562 Q15.9219 1.51562 18.3125 4.51562 Q20.7031 7.51562 21.5781 11.0781 L26.125 28.8125 Q27.2031 33.6406 27.2031 35.4062 Q27.2031 37.8906 25.8125 39.75 Q24.4219 41.6094 21.9219 41.6094 Q18.75 41.6094 15.9688 39.625 Q13.1875 37.6406 11.2812 34.5938 Q9.375 31.5469 8.59375 28.4219 Q8.40625 27.7812 7.8125 27.7812 L6.59375 27.7812 Q5.8125 27.7812 5.8125 28.7188 L5.8125 29 Q6.78125 32.7188 9.125 36.25 Q11.4688 39.7969 14.8594 41.9844 Q18.2656 44.1875 22.125 44.1875 Q25.7812 44.1875 28.7344 42.2344 Q31.6875 40.2812 32.9062 36.9219 Q34.625 39.9844 37.2812 42.0781 Q39.9375 44.1875 43.1094 44.1875 Q45.2656 44.1875 47.5 43.4219 Q49.75 42.6719 51.1719 41.1094 Q52.5938 39.5469 52.5938 37.2031 Q52.5938 34.6719 50.9531 32.8281 Q49.3125 31 46.7812 31 Q45.1719 31 44.0938 32.0312 Q43.0156 33.0625 43.0156 34.625 Q43.0156 36.7188 44.4531 38.2969 Q45.9062 39.8906 47.9062 40.1875 Q46.0938 41.6094 42.9219 41.6094 Q39.7031 41.6094 37.3281 38.625 Q34.9688 35.6406 33.9844 31.9844 L29.5938 14.3125 Q28.5156 10.2969 28.5156 7.71875 Q28.5156 5.17188 29.9531 3.34375 Q31.3906 1.51562 33.7969 1.51562 Q38.4844 1.51562 42.1562 5.64062 Q45.8438 9.76562 47.0156 14.7031 Q47.2188 15.2812 47.7969 15.2812 L49.0312 15.2812 Q49.4219 15.2812 49.6562 15.0156 Q49.9062 14.75 49.9062 14.4062 Q49.9062 14.3125 49.8125 14.1094 Q48.3906 8.15625 43.8438 3.51562 Q39.3125 -1.125 33.5938 -1.125 Q29.9375 -1.125 26.9844 0.84375 Q24.0312 2.82812 22.7969 6.20312 Q21.2344 3.26562 18.4688 1.0625 Q15.7188 -1.125 12.5938 -1.125 Q10.4531 -1.125 8.17188 -0.359375 Q5.90625 0.390625 4.48438 1.95312 Q3.07812 3.51562 3.07812 5.90625 Q3.07812 8.25 4.70312 10.1719 Q6.34375 12.1094 8.79688 12.1094 Q10.4531 12.1094 11.5781 11.1094 Q12.7031 10.1094 12.7031 8.5 Q12.7031 6.39062 11.2969 4.82812 Q9.90625 3.26562 7.8125 2.875\" id=\"Cmmi10-78\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(252.5 474.875)scale(0.2 -0.2)\">\n",
       "      <use transform=\"translate(0.0 0.8125)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L4 0\" id=\"md7965d1ba0\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_12\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L-4 0\" id=\"md9a1c1a7cd\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"441.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- −4 -->\n",
       "      <defs>\n",
       "       <path d=\" M37.7969 64.3125 L12.8906 25.3906 L37.7969 25.3906 z  M35.2031 72.9062 L47.6094 72.9062 L47.6094 25.3906 L58.0156 25.3906 L58.0156 17.1875 L47.6094 17.1875 L47.6094 0 L37.7969 0 L37.7969 17.1875 L4.89062 17.1875 L4.89062 26.7031 z \" id=\"BitstreamVeraSans-Roman-34\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(45.878125 443.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-34\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"385.2\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_14\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"385.2\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- −3 -->\n",
       "      <defs>\n",
       "       <path d=\" M40.5781 39.3125 Q47.6562 37.7969 51.625 33 Q55.6094 28.2188 55.6094 21.1875 Q55.6094 10.4062 48.1875 4.48438 Q40.7656 -1.42188 27.0938 -1.42188 Q22.5156 -1.42188 17.6562 -0.515625 Q12.7969 0.390625 7.625 2.20312 L7.625 11.7188 Q11.7188 9.32812 16.5938 8.10938 Q21.4844 6.89062 26.8125 6.89062 Q36.0781 6.89062 40.9375 10.5469 Q45.7969 14.2031 45.7969 21.1875 Q45.7969 27.6406 41.2812 31.2656 Q36.7656 34.9062 28.7188 34.9062 L20.2188 34.9062 L20.2188 43.0156 L29.1094 43.0156 Q36.375 43.0156 40.2344 45.9219 Q44.0938 48.8281 44.0938 54.2969 Q44.0938 59.9062 40.1094 62.9062 Q36.1406 65.9219 28.7188 65.9219 Q24.6562 65.9219 20.0156 65.0312 Q15.375 64.1562 9.8125 62.3125 L9.8125 71.0938 Q15.4375 72.6562 20.3438 73.4375 Q25.25 74.2188 29.5938 74.2188 Q40.8281 74.2188 47.3594 69.1094 Q53.9062 64.0156 53.9062 55.3281 Q53.9062 49.2656 50.4375 45.0938 Q46.9688 40.9219 40.5781 39.3125\" id=\"BitstreamVeraSans-Roman-33\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(46.11875 387.959375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-33\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_15\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"329.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_16\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"329.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_7\">\n",
       "      <!-- −2 -->\n",
       "      <defs>\n",
       "       <path d=\" M19.1875 8.29688 L53.6094 8.29688 L53.6094 0 L7.32812 0 L7.32812 8.29688 Q12.9375 14.1094 22.625 23.8906 Q32.3281 33.6875 34.8125 36.5312 Q39.5469 41.8438 41.4219 45.5312 Q43.3125 49.2188 43.3125 52.7812 Q43.3125 58.5938 39.2344 62.25 Q35.1562 65.9219 28.6094 65.9219 Q23.9688 65.9219 18.8125 64.3125 Q13.6719 62.7031 7.8125 59.4219 L7.8125 69.3906 Q13.7656 71.7812 18.9375 73 Q24.125 74.2188 28.4219 74.2188 Q39.75 74.2188 46.4844 68.5469 Q53.2188 62.8906 53.2188 53.4219 Q53.2188 48.9219 51.5312 44.8906 Q49.8594 40.875 45.4062 35.4062 Q44.1875 33.9844 37.6406 27.2188 Q31.1094 20.4531 19.1875 8.29688\" id=\"BitstreamVeraSans-Roman-32\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(46.31875 332.159375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_17\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"273.6\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_18\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"273.6\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- −1 -->\n",
       "      <g transform=\"translate(46.240625 276.359375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_19\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"217.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_20\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"217.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0 -->\n",
       "      <g transform=\"translate(53.9609375 220.559375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_21\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"162.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_22\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"162.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(54.659375 164.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_7\">\n",
       "     <g id=\"line2d_23\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"106.2\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_24\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"106.2\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 2 -->\n",
       "      <g transform=\"translate(54.371875 108.959375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_8\">\n",
       "     <g id=\"line2d_25\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"63.0\" xlink:href=\"#md7965d1ba0\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_26\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"453.6\" xlink:href=\"#md9a1c1a7cd\" y=\"50.4\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 3 -->\n",
       "      <g transform=\"translate(54.2015625 53.159375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-33\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_13\">\n",
       "     <!-- $f(x)$ -->\n",
       "     <defs>\n",
       "      <path d=\" M9.71875 -16.5 Q11.5781 -17.9219 14.2031 -17.9219 Q17.7812 -17.9219 20.0156 -10.0156 Q20.9531 -6.20312 25.0938 15.0938 L29.6875 39.5938 L21.0938 39.5938 Q20.125 39.5938 20.125 40.9219 Q20.5156 43.1094 21.3906 43.1094 L30.3281 43.1094 L31.5 49.6094 Q32.0781 52.6406 32.5625 54.8125 Q33.0625 56.9844 33.625 58.8281 Q34.1875 60.6875 35.2969 62.9844 Q37.0156 66.2656 39.9219 68.3906 Q42.8281 70.5156 46.1875 70.5156 Q48.3906 70.5156 50.4688 69.7031 Q52.5469 68.8906 53.8594 67.2812 Q55.1719 65.6719 55.1719 63.4844 Q55.1719 60.9375 53.4844 59.0469 Q51.8125 57.1719 49.4219 57.1719 Q47.7969 57.1719 46.6562 58.1719 Q45.5156 59.1875 45.5156 60.7969 Q45.5156 62.9844 47 64.625 Q48.4844 66.2656 50.6875 66.5 Q48.8281 67.9219 46.0938 67.9219 Q44.5781 67.9219 43.2344 66.5 Q41.8906 65.0938 41.5 63.4844 Q40.875 60.9375 38.7188 49.7031 L37.5 43.1094 L47.7969 43.1094 Q48.7812 43.1094 48.7812 41.7969 Q48.7344 41.5469 48.5781 40.9375 Q48.4375 40.3281 48.1719 39.9531 Q47.9062 39.5938 47.5156 39.5938 L36.8125 39.5938 L32.1719 15.1875 Q31.2969 9.85938 30.1406 4.35938 Q29 -1.125 26.9219 -6.90625 Q24.8594 -12.7031 21.625 -16.6094 Q18.4062 -20.5156 14.0156 -20.5156 Q10.6406 -20.5156 7.98438 -18.5781 Q5.32812 -16.6562 5.32812 -13.4844 Q5.32812 -10.9375 6.95312 -9.04688 Q8.59375 -7.17188 11.0781 -7.17188 Q12.75 -7.17188 13.8594 -8.17188 Q14.9844 -9.1875 14.9844 -10.7969 Q14.9844 -12.9375 13.3906 -14.7188 Q11.8125 -16.5 9.71875 -16.5\" id=\"Cmmi10-66\"/>\n",
       "      <path d=\" M6.5 -25 Q5.60938 -25 5.60938 -24.125 Q5.60938 -23.6875 5.8125 -23.4844 Q22.9062 -6.78125 22.9062 25 Q22.9062 56.7812 6 73.2969 Q5.60938 73.5312 5.60938 74.125 Q5.60938 74.4688 5.875 74.7344 Q6.15625 75 6.5 75 L7.42188 75 Q7.71875 75 7.90625 74.8125 Q15.0938 69.1406 19.875 61.0312 Q24.6562 52.9375 26.875 43.75 Q29.1094 34.5781 29.1094 25 Q29.1094 17.9219 27.9062 11.0625 Q26.7031 4.20312 24.0938 -2.45312 Q21.4844 -9.125 17.4844 -14.7656 Q13.4844 -20.4062 7.90625 -24.8125 Q7.71875 -25 7.42188 -25 z \" id=\"Cmr10-29\"/>\n",
       "      <path d=\" M31 -24.8125 Q25.4375 -20.4062 21.4062 -14.7188 Q17.3906 -9.03125 14.8125 -2.57812 Q12.25 3.85938 10.9844 10.8906 Q9.71875 17.9219 9.71875 25 Q9.71875 32.1719 10.9844 39.2031 Q12.25 46.2344 14.8594 52.7344 Q17.4844 59.2344 21.5312 64.8906 Q25.5938 70.5625 31 74.8125 Q31 75 31.5 75 L32.4219 75 Q32.7188 75 32.9531 74.7344 Q33.2031 74.4688 33.2031 74.125 Q33.2031 73.6875 33.0156 73.4844 Q28.125 68.7031 24.875 63.2344 Q21.625 57.7656 19.6406 51.5781 Q17.6719 45.4062 16.7969 38.7812 Q15.9219 32.1719 15.9219 25 Q15.9219 -6.78125 32.9062 -23.2969 Q33.2031 -23.5781 33.2031 -24.125 Q33.2031 -24.3594 32.9375 -24.6719 Q32.6719 -25 32.4219 -25 L31.5 -25 Q31 -25 31 -24.8125\" id=\"Cmr10-28\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(35.878125 264.1)rotate(-90.0)scale(0.2 -0.2)\">\n",
       "      <use xlink:href=\"#Cmmi10-66\"/>\n",
       "      <use transform=\"translate(48.876953125 0.0)\" xlink:href=\"#Cmr10-28\"/>\n",
       "      <use transform=\"translate(87.6953125 0.0)\" xlink:href=\"#Cmmi10-78\"/>\n",
       "      <use transform=\"translate(144.775390625 0.0)\" xlink:href=\"#Cmr10-29\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_14\">\n",
       "    <!-- $w_0 = [ 0.61179831]$ -->\n",
       "    <defs>\n",
       "     <path d=\" M4.20312 15.1875 Q4.20312 21.1875 8.15625 25.7969 Q12.1094 30.4219 18.3125 33.5 L14.5938 35.8906 Q11.1875 38.1406 9.03125 41.875 Q6.89062 45.6094 6.89062 49.7031 Q6.89062 54.5 9.40625 58.3438 Q11.9219 62.2031 16.0938 64.4062 Q20.2656 66.6094 25 66.6094 Q29.4375 66.6094 33.5625 64.7969 Q37.7031 62.9844 40.3594 59.6094 Q43.0156 56.25 43.0156 51.6094 Q43.0156 48.25 41.4219 45.3594 Q39.8438 42.4844 37.0781 40.1875 Q34.3281 37.8906 31.2031 36.2812 L36.9219 32.625 Q40.875 30.0312 43.2812 25.8281 Q45.7031 21.625 45.7031 17 Q45.7031 11.5781 42.7969 7.125 Q39.8906 2.6875 35.1094 0.234375 Q30.3281 -2.20312 25 -2.20312 Q19.8281 -2.20312 15.0156 -0.09375 Q10.2031 2 7.20312 5.98438 Q4.20312 9.96875 4.20312 15.1875 M9.625 15.1875 Q9.625 11.2344 11.7969 7.95312 Q13.9688 4.6875 17.5312 2.82812 Q21.0938 0.984375 25 0.984375 Q30.8125 0.984375 35.5469 4.375 Q40.2812 7.76562 40.2812 13.375 Q40.2812 15.2812 39.5156 17.1562 Q38.7656 19.0469 37.4219 20.5781 Q36.0781 22.125 34.4219 23.0938 L21 31.7812 Q17.875 30.125 15.25 27.5781 Q12.6406 25.0469 11.125 21.875 Q9.625 18.7031 9.625 15.1875 M16.5 45.7031 L28.6094 37.8906 Q32.8125 40.3281 35.5 43.7969 Q38.1875 47.2656 38.1875 51.6094 Q38.1875 54.9844 36.2969 57.7812 Q34.4219 60.5938 31.3906 62.1562 Q28.375 63.7188 24.9062 63.7188 Q21.875 63.7188 18.7969 62.5469 Q15.7188 61.375 13.7188 59.0469 Q11.7188 56.7344 11.7188 53.6094 Q11.7188 48.9219 16.5 45.7031\" id=\"Cmr10-38\"/>\n",
       "     <path d=\" M8.40625 5.51562 Q8.40625 7.76562 10.0625 9.375 Q11.7188 10.9844 13.9219 10.9844 Q15.2812 10.9844 16.5938 10.25 Q17.9219 9.51562 18.6562 8.20312 Q19.3906 6.89062 19.3906 5.51562 Q19.3906 3.32812 17.7656 1.65625 Q16.1562 0 13.9219 0 Q11.7188 0 10.0625 1.65625 Q8.40625 3.32812 8.40625 5.51562\" id=\"Cmmi10-3a\"/>\n",
       "     <path d=\" M9.51562 7.71875 Q11.8594 4.29688 15.8125 2.64062 Q19.7812 0.984375 24.3125 0.984375 Q30.125 0.984375 32.5625 5.9375 Q35.0156 10.8906 35.0156 17.1875 Q35.0156 20.0156 34.5 22.8438 Q33.9844 25.6875 32.7656 28.125 Q31.5469 30.5625 29.4219 32.0312 Q27.2969 33.5 24.2188 33.5 L17.5781 33.5 Q16.7031 33.5 16.7031 34.4219 L16.7031 35.2969 Q16.7031 36.0781 17.5781 36.0781 L23.0938 36.5312 Q26.6094 36.5312 28.9219 39.1562 Q31.25 41.7969 32.3281 45.5781 Q33.4062 49.3594 33.4062 52.7812 Q33.4062 57.5625 31.1562 60.6406 Q28.9062 63.7188 24.3125 63.7188 Q20.5156 63.7188 17.0469 62.2812 Q13.5781 60.8438 11.5312 57.9062 Q11.7188 57.9531 11.8594 57.9844 Q12.0156 58.0156 12.2031 58.0156 Q14.4531 58.0156 15.9688 56.4531 Q17.4844 54.8906 17.4844 52.6875 Q17.4844 50.5312 15.9688 48.9688 Q14.4531 47.4062 12.2031 47.4062 Q10.0156 47.4062 8.45312 48.9688 Q6.89062 50.5312 6.89062 52.6875 Q6.89062 56.9844 9.46875 60.1562 Q12.0625 63.3281 16.1406 64.9688 Q20.2188 66.6094 24.3125 66.6094 Q27.3438 66.6094 30.7031 65.7031 Q34.0781 64.7969 36.8125 63.1094 Q39.5469 61.4219 41.2812 58.7812 Q43.0156 56.1562 43.0156 52.7812 Q43.0156 48.5781 41.1406 45.0156 Q39.2656 41.4531 35.9844 38.8594 Q32.7188 36.2812 28.8125 35.0156 Q33.1562 34.1875 37.0625 31.7344 Q40.9688 29.2969 43.3281 25.4844 Q45.7031 21.6875 45.7031 17.2812 Q45.7031 11.7656 42.6719 7.29688 Q39.6562 2.82812 34.7188 0.3125 Q29.7812 -2.20312 24.3125 -2.20312 Q19.625 -2.20312 14.9062 -0.40625 Q10.2031 1.375 7.20312 4.9375 Q4.20312 8.5 4.20312 13.4844 Q4.20312 15.9688 5.85938 17.625 Q7.51562 19.2812 10.0156 19.2812 Q11.625 19.2812 12.9688 18.5312 Q14.3125 17.7812 15.0625 16.4062 Q15.8281 15.0469 15.8281 13.4844 Q15.8281 11.0312 14.1094 9.375 Q12.4062 7.71875 10.0156 7.71875 z \" id=\"Cmr10-33\"/>\n",
       "     <path d=\" M10.7969 11.625 Q10.7969 13.7188 11.3594 16.4531 Q11.9219 19.1875 12.5938 21.3125 Q13.2812 23.4375 14.9062 27.8281 Q16.5469 32.2344 16.7031 32.625 Q18.0156 36.2812 18.0156 38.625 Q18.0156 41.6094 15.8281 41.6094 Q11.8594 41.6094 9.29688 37.5312 Q6.73438 33.4531 5.51562 28.4219 Q5.32812 27.7812 4.6875 27.7812 L3.51562 27.7812 Q2.6875 27.7812 2.6875 28.7188 L2.6875 29 Q4.29688 34.9688 7.60938 39.5781 Q10.9375 44.1875 16.0156 44.1875 Q19.5781 44.1875 22.0469 41.8438 Q24.5156 39.5 24.5156 35.8906 Q24.5156 34.0312 23.6875 31.9844 Q22.5156 28.9531 20.9219 24.7031 Q19.3438 20.4531 18.4531 16.9844 Q17.5781 13.5312 17.5781 10.2969 Q17.5781 6.39062 19.625 3.95312 Q21.6875 1.51562 25.5938 1.51562 Q30.9062 1.51562 34.4219 9.625 Q34.2812 10.2031 34.2812 11.2812 Q34.2812 13.9688 35.1094 17.3906 L40.7188 39.8906 Q41.0625 41.2188 42.2812 42.1562 Q43.5 43.1094 44.9219 43.1094 Q46.0938 43.1094 47 42.3281 Q47.9062 41.5469 47.9062 40.2812 Q47.9062 39.7031 47.7969 39.5 L42.1875 17.1875 Q41.2188 13.1875 41.2188 10.0156 Q41.2188 6.25 42.9375 3.875 Q44.6719 1.51562 48.3906 1.51562 Q54.8281 1.51562 58.9844 10.0156 Q60.5469 13.0312 62.4688 18.9375 Q64.4062 24.8594 64.4062 27.9844 Q64.4062 30.8594 63.4688 32.5469 Q62.5469 34.2344 60.9062 36.1875 Q59.2812 38.1406 59.2812 39.1094 Q59.2812 41.0156 60.9062 42.6406 Q62.5469 44.2812 64.5 44.2812 Q66.8906 44.2812 67.9375 42.0781 Q69 39.8906 69 37.1094 Q69 33.7344 67.875 28.5312 Q66.75 23.3438 65.0312 18.0938 Q63.3281 12.8438 61.8125 9.71875 Q56.3438 -1.125 48.1875 -1.125 Q44 -1.125 40.5312 0.484375 Q37.0625 2.09375 35.4062 5.60938 Q33.7969 2.73438 31.1719 0.796875 Q28.5625 -1.125 25.3906 -1.125 Q18.7969 -1.125 14.7969 2.01562 Q10.7969 5.17188 10.7969 11.625\" id=\"Cmmi10-77\"/>\n",
       "     <path d=\" M11.8125 -25 L11.8125 75 L25.4844 75 L25.4844 71 L15.8281 71 L15.8281 -21 L25.4844 -21 L25.4844 -25 z \" id=\"Cmr10-5b\"/>\n",
       "     <path d=\" M17.3906 2.59375 Q17.3906 8.15625 18.3594 13.4688 Q19.3438 18.7969 21.2188 24 Q23.0938 29.2031 25.75 34.2031 Q28.4219 39.2031 31.5938 43.6094 L40.7188 56.2969 L29.2969 56.2969 Q11.5312 56.2969 10.9844 55.8125 Q9.67188 54.2031 8.5 46.5781 L5.60938 46.5781 L8.89062 67.5781 L11.8125 67.5781 L11.8125 67.2812 Q11.8125 65.4844 17.9219 64.9375 Q24.0312 64.4062 29.8906 64.4062 L48.4844 64.4062 L48.4844 61.8125 Q48.4844 61.7188 48.4531 61.6719 Q48.4375 61.625 48.3906 61.5312 L34.625 42.1875 Q29.5469 34.6719 28.2656 25.4844 Q27 16.3125 27 2.59375 Q27 0.640625 25.5781 -0.78125 Q24.1719 -2.20312 22.2188 -2.20312 Q20.2188 -2.20312 18.7969 -0.78125 Q17.3906 0.640625 17.3906 2.59375\" id=\"Cmr10-37\"/>\n",
       "     <path d=\" M9.28125 0 L9.28125 3.51562 Q21.7812 3.51562 21.7812 6.6875 L21.7812 59.1875 Q16.6094 56.6875 8.6875 56.6875 L8.6875 60.2031 Q20.9531 60.2031 27.2031 66.6094 L28.6094 66.6094 Q28.9531 66.6094 29.2656 66.3281 Q29.5938 66.0625 29.5938 65.7188 L29.5938 6.6875 Q29.5938 3.51562 42.0938 3.51562 L42.0938 0 z \" id=\"Cmr10-31\"/>\n",
       "     <path d=\" M2.20312 -25 L2.20312 -21 L11.9219 -21 L11.9219 71 L2.20312 71 L2.20312 75 L15.9219 75 L15.9219 -25 z \" id=\"Cmr10-5d\"/>\n",
       "     <path d=\" M25 -2.20312 Q12.75 -2.20312 8.32812 7.875 Q3.90625 17.9688 3.90625 31.8906 Q3.90625 40.5781 5.48438 48.2344 Q7.07812 55.9062 11.7812 61.25 Q16.5 66.6094 25 66.6094 Q31.5938 66.6094 35.7812 63.375 Q39.9844 60.1562 42.1875 55.0469 Q44.3906 49.9531 45.1875 44.1094 Q46 38.2812 46 31.8906 Q46 23.2969 44.4062 15.7969 Q42.8281 8.29688 38.1875 3.04688 Q33.5469 -2.20312 25 -2.20312 M25 0.390625 Q30.5625 0.390625 33.2969 6.09375 Q36.0312 11.8125 36.6719 18.75 Q37.3125 25.6875 37.3125 33.5 Q37.3125 41.0156 36.6719 47.3594 Q36.0312 53.7188 33.3125 58.8594 Q30.6094 64.0156 25 64.0156 Q19.3438 64.0156 16.6094 58.8281 Q13.875 53.6562 13.2344 47.3281 Q12.5938 41.0156 12.5938 33.5 Q12.5938 27.9375 12.8594 23 Q13.1406 18.0625 14.3125 12.8125 Q15.4844 7.5625 18.0938 3.96875 Q20.7031 0.390625 25 0.390625\" id=\"Cmr10-30\"/>\n",
       "     <path d=\" M7.51562 13.2812 Q6.6875 13.2812 6.14062 13.9062 Q5.60938 14.5469 5.60938 15.2812 Q5.60938 16.1094 6.14062 16.6875 Q6.6875 17.2812 7.51562 17.2812 L70.3125 17.2812 Q71.0469 17.2812 71.5781 16.6875 Q72.125 16.1094 72.125 15.2812 Q72.125 14.5469 71.5781 13.9062 Q71.0469 13.2812 70.3125 13.2812 z  M7.51562 32.7188 Q6.6875 32.7188 6.14062 33.2969 Q5.60938 33.8906 5.60938 34.7188 Q5.60938 35.4531 6.14062 36.0781 Q6.6875 36.7188 7.51562 36.7188 L70.3125 36.7188 Q71.0469 36.7188 71.5781 36.0781 Q72.125 35.4531 72.125 34.7188 Q72.125 33.8906 71.5781 33.2969 Q71.0469 32.7188 70.3125 32.7188 z \" id=\"Cmr10-3d\"/>\n",
       "     <path d=\" M25 -2.20312 Q18.7969 -2.20312 14.6406 1.09375 Q10.5 4.39062 8.21875 9.64062 Q5.95312 14.8906 5.07812 20.6562 Q4.20312 26.4219 4.20312 32.3281 Q4.20312 40.2344 7.28125 48.1875 Q10.3594 56.1562 16.3281 61.375 Q22.3125 66.6094 30.5156 66.6094 Q33.9375 66.6094 36.8906 65.3125 Q39.8438 64.0156 41.5312 61.5 Q43.2188 58.9844 43.2188 55.4219 Q43.2188 53.375 41.8281 51.9844 Q40.4375 50.5938 38.375 50.5938 Q36.4219 50.5938 35 52 Q33.5938 53.4219 33.5938 55.4219 Q33.5938 57.375 35 58.7812 Q36.4219 60.2031 38.375 60.2031 L38.9219 60.2031 Q37.6406 62.0156 35.3281 62.8594 Q33.0156 63.7188 30.5156 63.7188 Q27.4844 63.7188 24.9219 62.3906 Q22.3594 61.0781 20.3125 58.8281 Q18.2656 56.5938 16.8906 53.875 Q15.5312 51.1719 14.7656 47.7031 Q14.0156 44.2344 13.8125 41.2031 Q13.625 38.1875 13.625 33.5938 Q15.375 37.7031 18.5938 40.3125 Q21.8281 42.9219 25.875 42.9219 Q30.3281 42.9219 33.9844 41.1094 Q37.6406 39.3125 40.2812 36.1094 Q42.9219 32.9062 44.3125 28.7969 Q45.7031 24.7031 45.7031 20.5156 Q45.7031 14.6562 43.0938 9.34375 Q40.4844 4.04688 35.7344 0.921875 Q31 -2.20312 25 -2.20312 M25 0.984375 Q28.8594 0.984375 31.2031 2.73438 Q33.5469 4.5 34.6406 7.39062 Q35.75 10.2969 36.0156 13.25 Q36.2812 16.2188 36.2812 20.5156 Q36.2812 26.1719 35.7344 30.1719 Q35.2031 34.1875 32.8125 37.2344 Q30.4219 40.2812 25.4844 40.2812 Q21.4375 40.2812 18.8281 37.5469 Q16.2188 34.8125 15.0156 30.6406 Q13.8125 26.4688 13.8125 22.6094 Q13.8125 21.2969 13.9219 20.6094 Q13.9219 20.4531 13.8906 20.3594 Q13.875 20.2656 13.8125 20.125 Q13.8125 15.8281 14.6875 11.4219 Q15.5781 7.03125 18.0625 4 Q20.5625 0.984375 25 0.984375\" id=\"Cmr10-36\"/>\n",
       "     <path d=\" M11.2812 4.20312 Q14.0156 0.984375 20.7969 0.984375 Q24.6094 0.984375 27.9062 3.5625 Q31.2031 6.15625 33.0156 9.90625 Q35.1094 14.1562 35.6875 18.9688 Q36.2812 23.7812 36.2812 30.9062 Q34.5781 26.8594 31.375 24.2656 Q28.1719 21.6875 24.0312 21.6875 Q18.2188 21.6875 13.6406 24.8281 Q9.07812 27.9844 6.64062 33.125 Q4.20312 38.2812 4.20312 44.0938 Q4.20312 50.0938 6.9375 55.2969 Q9.67188 60.5 14.5 63.5469 Q19.3438 66.6094 25.4844 66.6094 Q31.5469 66.6094 35.625 63.3125 Q39.7031 60.0156 41.8438 54.8125 Q44 49.6094 44.8438 43.7969 Q45.7031 37.9844 45.7031 32.3281 Q45.7031 24.6094 42.875 16.5781 Q40.0469 8.54688 34.4062 3.17188 Q28.7656 -2.20312 20.7969 -2.20312 Q14.8906 -2.20312 10.7812 0.578125 Q6.6875 3.375 6.6875 8.98438 Q6.6875 11.0312 8.07812 12.4219 Q9.46875 13.8125 11.5312 13.8125 Q13.5312 13.8125 14.9219 12.4219 Q16.3125 11.0312 16.3125 8.98438 Q16.3125 7.03125 14.8906 5.60938 Q13.4844 4.20312 11.5312 4.20312 z  M24.4219 24.3125 Q28.5156 24.3125 31.125 27.0625 Q33.7344 29.8281 34.9062 33.9219 Q36.0781 38.0312 36.0781 42.0938 L36.0781 44 L36.0781 44.3906 Q36.0781 51.9062 33.875 57.8125 Q31.6875 63.7188 25.4844 63.7188 Q21.5312 63.7188 19.0625 61.9844 Q16.6094 60.25 15.4375 57.375 Q14.2656 54.5 13.9375 51.2188 Q13.625 47.9531 13.625 44.0938 Q13.625 38.4219 14.1562 34.4219 Q14.7031 30.4219 17.0938 27.3594 Q19.4844 24.3125 24.4219 24.3125\" id=\"Cmr10-39\"/>\n",
       "    </defs>\n",
       "    <g transform=\"translate(60.4375 240.12)scale(0.2 -0.2)\">\n",
       "     <use xlink:href=\"#Cmmi10-77\"/>\n",
       "     <use transform=\"translate(71.484375 -25.509375000000006)scale(0.7)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(139.04296875 0.0)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(223.1015625 0.0)\" xlink:href=\"#Cmr10-5b\"/>\n",
       "     <use transform=\"translate(250.787109375 0.0)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(300.787109375 0.0)\" xlink:href=\"#Cmmi10-3a\"/>\n",
       "     <use transform=\"translate(329.330078125 0.0)\" xlink:href=\"#Cmr10-36\"/>\n",
       "     <use transform=\"translate(379.330078125 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(429.330078125 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(479.330078125 0.0)\" xlink:href=\"#Cmr10-37\"/>\n",
       "     <use transform=\"translate(529.330078125 0.0)\" xlink:href=\"#Cmr10-39\"/>\n",
       "     <use transform=\"translate(579.330078125 0.0)\" xlink:href=\"#Cmr10-38\"/>\n",
       "     <use transform=\"translate(629.330078125 0.0)\" xlink:href=\"#Cmr10-33\"/>\n",
       "     <use transform=\"translate(679.330078125 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(729.330078125 0.0)\" xlink:href=\"#Cmr10-5d\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_15\">\n",
       "    <!-- $w_1 = [ 0.58133213]$ -->\n",
       "    <defs>\n",
       "     <path d=\" M4.98438 0 L4.98438 2.6875 Q4.98438 2.9375 5.17188 3.21875 L20.7031 20.4062 Q24.2188 24.2188 26.4062 26.7969 Q28.6094 29.3906 30.75 32.7656 Q32.9062 36.1406 34.1562 39.625 Q35.4062 43.1094 35.4062 47.0156 Q35.4062 51.125 33.8906 54.8594 Q32.375 58.5938 29.375 60.8438 Q26.375 63.0938 22.125 63.0938 Q17.7812 63.0938 14.3125 60.4688 Q10.8438 57.8594 9.42188 53.7188 Q9.8125 53.8125 10.5 53.8125 Q12.75 53.8125 14.3281 52.2969 Q15.9219 50.7812 15.9219 48.3906 Q15.9219 46.0938 14.3281 44.5 Q12.75 42.9219 10.5 42.9219 Q8.15625 42.9219 6.5625 44.5469 Q4.98438 46.1875 4.98438 48.3906 Q4.98438 52.1562 6.39062 55.4375 Q7.8125 58.7344 10.4688 61.2969 Q13.1406 63.875 16.4844 65.2344 Q19.8281 66.6094 23.5781 66.6094 Q29.2969 66.6094 34.2188 64.1875 Q39.1562 61.7656 42.0312 57.3438 Q44.9219 52.9375 44.9219 47.0156 Q44.9219 42.6719 43.0156 38.7656 Q41.1094 34.8594 38.125 31.6562 Q35.1562 28.4688 30.5156 24.4062 Q25.875 20.3594 24.4219 19 L13.0938 8.10938 L22.7031 8.10938 Q29.7812 8.10938 34.5469 8.21875 Q39.3125 8.34375 39.5938 8.59375 Q40.7656 9.85938 42 17.8281 L44.9219 17.8281 L42.0938 0 z \" id=\"Cmr10-32\"/>\n",
       "     <path d=\" M8.6875 11.375 Q9.71875 8.45312 11.8438 6.04688 Q13.9688 3.65625 16.875 2.3125 Q19.7812 0.984375 22.9062 0.984375 Q30.125 0.984375 32.8594 6.59375 Q35.5938 12.2031 35.5938 20.2188 Q35.5938 23.6875 35.4688 26.0469 Q35.3594 28.4219 34.8125 30.6094 Q33.8906 34.125 31.5625 36.7656 Q29.25 39.4062 25.875 39.4062 Q22.5156 39.4062 20.0938 38.375 Q17.6719 37.3594 16.1562 35.9844 Q14.6562 34.625 13.4844 33.1094 Q12.3125 31.5938 12.0156 31.5 L10.8906 31.5 Q10.6406 31.5 10.2656 31.8125 Q9.90625 32.125 9.90625 32.4219 L9.90625 65.8281 Q9.90625 66.0625 10.2188 66.3281 Q10.5469 66.6094 10.8906 66.6094 L11.1875 66.6094 Q17.9219 63.375 25.4844 63.375 Q32.9062 63.375 39.7969 66.6094 L40.0938 66.6094 Q40.4375 66.6094 40.7188 66.3594 Q41.0156 66.1094 41.0156 65.8281 L41.0156 64.8906 Q41.0156 64.4062 40.8281 64.4062 Q37.4062 59.8594 32.25 57.3125 Q27.0938 54.7812 21.5781 54.7812 Q17.5781 54.7812 13.375 55.9062 L13.375 37.0156 Q16.7031 39.7031 19.3125 40.8438 Q21.9219 42 25.9844 42 Q31.5 42 35.8594 38.8125 Q40.2344 35.6406 42.5781 30.5312 Q44.9219 25.4375 44.9219 20.125 Q44.9219 14.1094 41.9688 8.98438 Q39.0156 3.85938 33.9375 0.828125 Q28.8594 -2.20312 22.9062 -2.20312 Q17.9688 -2.20312 13.8438 0.328125 Q9.71875 2.875 7.34375 7.17188 Q4.98438 11.4688 4.98438 16.3125 Q4.98438 18.5625 6.4375 19.9688 Q7.90625 21.3906 10.1094 21.3906 Q12.3125 21.3906 13.7969 19.9375 Q15.2812 18.5 15.2812 16.3125 Q15.2812 14.1562 13.7969 12.6719 Q12.3125 11.1875 10.1094 11.1875 Q9.76562 11.1875 9.32812 11.25 Q8.89062 11.3281 8.6875 11.375\" id=\"Cmr10-35\"/>\n",
       "    </defs>\n",
       "    <g transform=\"translate(182.5 148.05)scale(0.2 -0.2)\">\n",
       "     <use xlink:href=\"#Cmmi10-77\"/>\n",
       "     <use transform=\"translate(71.484375 -25.509375000000006)scale(0.7)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(139.04296875 0.0)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(223.1015625 0.0)\" xlink:href=\"#Cmr10-5b\"/>\n",
       "     <use transform=\"translate(250.787109375 0.0)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(300.787109375 0.0)\" xlink:href=\"#Cmmi10-3a\"/>\n",
       "     <use transform=\"translate(329.330078125 0.0)\" xlink:href=\"#Cmr10-35\"/>\n",
       "     <use transform=\"translate(379.330078125 0.0)\" xlink:href=\"#Cmr10-38\"/>\n",
       "     <use transform=\"translate(429.330078125 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(479.330078125 0.0)\" xlink:href=\"#Cmr10-33\"/>\n",
       "     <use transform=\"translate(529.330078125 0.0)\" xlink:href=\"#Cmr10-33\"/>\n",
       "     <use transform=\"translate(579.330078125 0.0)\" xlink:href=\"#Cmr10-32\"/>\n",
       "     <use transform=\"translate(629.330078125 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(679.330078125 0.0)\" xlink:href=\"#Cmr10-33\"/>\n",
       "     <use transform=\"translate(729.330078125 0.0)\" xlink:href=\"#Cmr10-5d\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"text_16\">\n",
       "    <!-- $w_2 = [-1.07800849]$ -->\n",
       "    <defs>\n",
       "     <path d=\" M2.78125 16.5 L2.78125 20.0156 L33.6875 66.1094 Q34.0312 66.6094 34.7188 66.6094 L36.1875 66.6094 Q37.3125 66.6094 37.3125 65.4844 L37.3125 20.0156 L47.125 20.0156 L47.125 16.5 L37.3125 16.5 L37.3125 6.6875 Q37.3125 4.64062 40.2344 4.07812 Q43.1719 3.51562 47.0156 3.51562 L47.0156 0 L19.4844 0 L19.4844 3.51562 Q23.3438 3.51562 26.2656 4.07812 Q29.2031 4.64062 29.2031 6.6875 L29.2031 16.5 z  M6.10938 20.0156 L29.7812 20.0156 L29.7812 55.4219 z \" id=\"Cmr10-34\"/>\n",
       "     <path d=\" M10.2031 23 Q9.375 23 8.82812 23.625 Q8.29688 24.2656 8.29688 25 Q8.29688 25.7344 8.82812 26.3594 Q9.375 27 10.2031 27 L67.5781 27 Q68.3594 27 68.875 26.3594 Q69.3906 25.7344 69.3906 25 Q69.3906 24.2656 68.875 23.625 Q68.3594 23 67.5781 23 z \" id=\"Cmsy10-a1\"/>\n",
       "    </defs>\n",
       "    <g transform=\"translate(296.8625 240.12)scale(0.2 -0.2)\">\n",
       "     <use xlink:href=\"#Cmmi10-77\"/>\n",
       "     <use transform=\"translate(71.484375 -25.509375000000006)scale(0.7)\" xlink:href=\"#Cmr10-32\"/>\n",
       "     <use transform=\"translate(139.04296875 0.0)\" xlink:href=\"#Cmr10-3d\"/>\n",
       "     <use transform=\"translate(223.1015625 0.0)\" xlink:href=\"#Cmr10-5b\"/>\n",
       "     <use transform=\"translate(250.787109375 0.0)\" xlink:href=\"#Cmsy10-a1\"/>\n",
       "     <use transform=\"translate(328.47265625 0.0)\" xlink:href=\"#Cmr10-31\"/>\n",
       "     <use transform=\"translate(378.47265625 0.0)\" xlink:href=\"#Cmmi10-3a\"/>\n",
       "     <use transform=\"translate(407.015625 0.0)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(457.015625 0.0)\" xlink:href=\"#Cmr10-37\"/>\n",
       "     <use transform=\"translate(507.015625 0.0)\" xlink:href=\"#Cmr10-38\"/>\n",
       "     <use transform=\"translate(557.015625 0.0)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(607.015625 0.0)\" xlink:href=\"#Cmr10-30\"/>\n",
       "     <use transform=\"translate(657.015625 0.0)\" xlink:href=\"#Cmr10-38\"/>\n",
       "     <use transform=\"translate(707.015625 0.0)\" xlink:href=\"#Cmr10-34\"/>\n",
       "     <use transform=\"translate(757.015625 0.0)\" xlink:href=\"#Cmr10-39\"/>\n",
       "     <use transform=\"translate(807.015625 0.0)\" xlink:href=\"#Cmr10-5d\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p41d7ee9abc\">\n",
       "   <rect height=\"390.6\" width=\"390.6\" x=\"63.0\" y=\"50.4\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pods.notebook.display_plots('radial_function{func_num}.svg', directory='./diagrams', func_num=(1,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Basis Function Models\n",
    "\n",
    "-   The *prediction function* is now defined as\n",
    "    $$f(\\mathbf{x}_i) = \\sum_{j=1}^m w_j \\phi_{i, j}$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Vector Notation\n",
    "\n",
    "-   Write in vector notation,\n",
    "    $$f(\\mathbf{x}_i) = \\mathbf{w}^\\top \\boldsymbol{\\phi}_i$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Log Likelihood for Basis Function Model\n",
    "\n",
    "-   The likelihood of a single data point is\n",
    "    $$p\\left(y_i|x_i\\right)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\exp\n",
    "        \\left(-\\frac{\\left(y_i-\\mathbf{w}^{\\top}\\boldsymbol{\\phi}_i\\right)^{2}}{2\\sigma^2}\\right).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Log Likelihood for Basis Function Model\n",
    "\n",
    "-   Leading to a log likelihood for the data set of\n",
    "    $$L(\\mathbf{w},\\sigma^2)= -\\frac{n}{2}\\log \\sigma^2\n",
    "          -\\frac{n}{2}\\log 2\\pi -\\frac{\\sum\n",
    "            _{i=1}^{n}\\left(y_i-\\mathbf{w}^{\\top}\\boldsymbol{\\phi}_i\\right)^{2}}{2\\sigma^2}.$$\n",
    "\n",
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Objective Function\n",
    "\n",
    "- And a corresponding *objective function* of the form\n",
    "    $$E(\\mathbf{w},\\sigma^2)= \\frac{n}{2}\\log\n",
    "          \\sigma^2 + \\frac{\\sum\n",
    "            _{i=1}^{n}\\left(y_i-\\mathbf{w}^{\\top}\\boldsymbol{\\phi}_i\\right)^{2}}{2\\sigma^2}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Expand the Brackets\n",
    "\n",
    "$$\\begin{align}\n",
    "  E(\\mathbf{w},\\sigma^2) =  &\\frac{n}{2}\\log \\sigma^2 + \\frac{1}{2\\sigma^2}\\sum _{i=1}^{n}y_i^{2}-\\frac{1}{\\sigma^2}\\sum _{i=1}^{n}y_i\\mathbf{w}^{\\top}\\boldsymbol{\\phi}_i\\\\ &+\\frac{1}{2\\sigma^2}\\sum _{i=1}^{n}\\mathbf{w}^{\\top}\\boldsymbol{\\phi}_i\\boldsymbol{\\phi}_i^{\\top}\\mathbf{w} +\\text{const}.\\end{align}$$\n",
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    " ### Expand the Brackets\n",
    " $$ \\begin{align} E(\\mathbf{w}, \\sigma^2) = & \\frac{n}{2}\\log \\sigma^2 + \\frac{1}{2\\sigma^2}\\sum _{i=1}^{n}y_i^{2}-\\frac{1}{\\sigma^2}\n",
    "  \\mathbf{w}^\\top\\sum_{i=1}^{n}\\boldsymbol{\\phi}_i y_i\\\\ & +\\frac{1}{2\\sigma^2} \\mathbf{w}^{\\top}\\left[\\sum\n",
    "    _{i=1}^{n}\\boldsymbol{\\phi}_i\\boldsymbol{\\phi}_i^{\\top}\\right]\\mathbf{w} +\\text{const}.\\end{align}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Multivariate Derivatives Reminder\n",
    "\n",
    "-   We will need some multivariate calculus.\n",
    "    $$\\frac{\\text{d}\\mathbf{a}^{\\top}\\mathbf{w}}{\\text{d}\\mathbf{w}}=\\mathbf{a}$$\n",
    "    and\n",
    "    $$\\frac{\\text{d}\\mathbf{w}^{\\top}\\mathbf{A}\\mathbf{w}}{\\text{d}\\mathbf{w}}=\\left(\\mathbf{A}+\\mathbf{A}^{\\top}\\right)\\mathbf{w}$$\n",
    "    or if $\\mathbf{A}$ is symmetric (*i.e.*\n",
    "    $\\mathbf{A}=\\mathbf{A}^{\\top}$)\n",
    "    $$\\frac{\\text{d}\\mathbf{w}^{\\top}\\mathbf{A}\\mathbf{w}}{\\text{d}\\mathbf{w}}=2\\mathbf{A}\\mathbf{w}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Differentiate\n",
    "\n",
    "Differentiating with respect to the vector $\\mathbf{w}$ we obtain\n",
    "$$\\frac{\\text{d} E\\left(\\mathbf{w},\\sigma^2 \\right)}{\\text{d} \\mathbf{w}}=-\\frac{1}{\\sigma^2} \\sum _{i=1}^{n}\\boldsymbol{\\phi}_iy_i+\\frac{1}{\\sigma^2} \\left[\\sum _{i=1}^{n}\\boldsymbol{\\phi}_i\\boldsymbol{\\phi}_i^{\\top}\\right]\\mathbf{w}$$\n",
    "Leading to\n",
    "$$\\mathbf{w}^{*}=\\left[\\sum _{i=1}^{n}\\boldsymbol{\\phi}_i\\boldsymbol{\\phi}_i^{\\top}\\right]^{-1}\\sum _{i=1}^{n}\\boldsymbol{\\phi}_iy_i,$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Matrix Notation\n",
    "\n",
    "Rewrite in matrix notation:\n",
    "$$\\sum _{i=1}^{n}\\boldsymbol{\\phi}_i\\boldsymbol{\\phi}_i^\\top = \\boldsymbol{\\Phi}^\\top \\boldsymbol{\\Phi}$$\n",
    "$$\\sum _{i=1}^{n}\\boldsymbol{\\phi}_iy_i = \\boldsymbol{\\Phi}^\\top \\mathbf{y}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Update Equations\n",
    "\n",
    "-   Update for $\\mathbf{w}^{*}$.\n",
    "    $$\\mathbf{w}^{*} = \\left(\\boldsymbol{\\Phi}^\\top \\boldsymbol{\\Phi}\\right)^{-1} \\boldsymbol{\\Phi}^\\top \\mathbf{y}$$\n",
    "\n",
    "-   The equation for $\\left.\\sigma^2\\right.^{*}$ may also be found\n",
    "    $$\\left.\\sigma^2\\right.^{{*}}=\\frac{\\sum _{i=1}^{n}\\left(y_i-\\left.\\mathbf{w}^{*}\\right.^{\\top}\\boldsymbol{\\phi}_i\\right)^{2}}{n}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Avoid Direct Inverse\n",
    "\n",
    "- E.g. Solve for $\\mathbf{w}$\n",
    "  $$\\left(\\boldsymbol{\\Phi}^\\top \\boldsymbol{\\Phi}\\right)\\mathbf{w} = \\boldsymbol{\\Phi}^\\top \\mathbf{y}$$\n",
    "  \n",
    "- See `np.linalg.solve`\n",
    "\n",
    "- In practice use $\\mathbf{Q}\\mathbf{R}$ decomposition (see lab class notes)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Polynomial Fits to Olympic Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlEAAAFQCAYAAACfyCvmAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ntPEdXIkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a4625c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "basis = mlai.polynomial\n",
    "\n",
    "data = pods.datasets.olympic_marathon_men()\n",
    "f, ax = plt.subplots(1, 2, figsize=(10,5))\n",
    "x = data['X']\n",
    "y = data['Y']\n",
    "\n",
    "data_limits = [1892, 2020]\n",
    "max_basis = 7\n",
    "\n",
    "ll = np.array([np.nan]*(max_basis))\n",
    "sum_squares = np.array([np.nan]*(max_basis))\n",
    "\n",
    "for num_basis in range(1,max_basis+1):\n",
    "    \n",
    "    model= mlai.LM(x, y, basis, num_basis=num_basis, data_limits=data_limits)\n",
    "    model.fit()\n",
    "    sum_squares[num_basis-1] = model.objective() \n",
    "    ll[num_basis-1] = model.log_likelihood()\n",
    "    mlai.plot_marathon_fit(model=model, data_limits=data_limits, \n",
    "                           objective=sum_squares, objective_ylim=[0,8],\n",
    "                           fig=f, ax=ax)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg height=\"360pt\" version=\"1.1\" viewBox=\"0 0 720 360\" width=\"720pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <defs>\n",
       "  <style type=\"text/css\">\n",
       "*{stroke-linecap:butt;stroke-linejoin:round;}\n",
       "  </style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\" M0 360 L720 360 L720 0 L0 0 z \" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\" M90 315 L343.636 315 L343.636 36 L90 36 z \" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_1\">\n",
       "    <defs>\n",
       "     <path d=\" M0 3 C0.795609 3 1.55874 2.6839 2.12132 2.12132 C2.6839 1.55874 3 0.795609 3 0 C3 -0.795609 2.6839 -1.55874 2.12132 -2.12132 C1.55874 -2.6839 0.795609 -3 0 -3 C-0.795609 -3 -1.55874 -2.6839 -2.12132 -2.12132 C-2.6839 -1.55874 -3 -0.795609 -3 0 C-3 0.795609 -2.6839 1.55874 -2.12132 2.12132 C-1.55874 2.6839 -0.795609 3 0 3 z \" id=\"m231f9b0f49\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "    </defs>\n",
       "    <g clip-path=\"url(#p4bbcf401cb)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"118.987012987\" xlink:href=\"#m231f9b0f49\" y=\"131.7125\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"126.233766234\" xlink:href=\"#m231f9b0f49\" y=\"132.280178838\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"133.480519481\" xlink:href=\"#m231f9b0f49\" y=\"61.84625\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"140.727272727\" xlink:href=\"#m231f9b0f49\" y=\"161.11488328\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"147.974025974\" xlink:href=\"#m231f9b0f49\" y=\"184.491542289\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"162.467532468\" xlink:href=\"#m231f9b0f49\" y=\"215.535321637\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"169.714285714\" xlink:href=\"#m231f9b0f49\" y=\"191.817336177\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"176.961038961\" xlink:href=\"#m231f9b0f49\" y=\"210.390152862\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"184.207792208\" xlink:href=\"#m231f9b0f49\" y=\"213.365623889\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"191.454545455\" xlink:href=\"#m231f9b0f49\" y=\"218.390863846\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"213.194805195\" xlink:href=\"#m231f9b0f49\" y=\"206.180412371\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"220.441558442\" xlink:href=\"#m231f9b0f49\" y=\"232.202926887\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"227.688311688\" xlink:href=\"#m231f9b0f49\" y=\"227.912371134\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"234.935064935\" xlink:href=\"#m231f9b0f49\" y=\"249.357803057\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"242.181818182\" xlink:href=\"#m231f9b0f49\" y=\"256.153631947\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"249.428571429\" xlink:href=\"#m231f9b0f49\" y=\"237.977544733\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"256.675324675\" xlink:href=\"#m231f9b0f49\" y=\"255.867105107\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"263.922077922\" xlink:href=\"#m231f9b0f49\" y=\"261.156831378\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"271.168831169\" xlink:href=\"#m231f9b0f49\" y=\"258.658905084\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"278.415584416\" xlink:href=\"#m231f9b0f49\" y=\"262.405794525\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"285.662337662\" xlink:href=\"#m231f9b0f49\" y=\"259.7976656\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"292.909090909\" xlink:href=\"#m231f9b0f49\" y=\"253.516115654\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"300.155844156\" xlink:href=\"#m231f9b0f49\" y=\"255.242623534\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"307.402597403\" xlink:href=\"#m231f9b0f49\" y=\"260.569084015\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"314.649350649\" xlink:href=\"#m231f9b0f49\" y=\"258.952778765\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"321.896103896\" xlink:href=\"#m231f9b0f49\" y=\"268.613876052\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"329.142857143\" xlink:href=\"#m231f9b0f49\" y=\"265.344531343\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_2\">\n",
       "    <path clip-path=\"url(#p4bbcf401cb)\" d=\" M111.74 151.782 L113.538 142.035 L115.336 134.189 L117.133 128.062 L118.931 123.487 L120.728 120.304 L122.526 118.366 L124.324 117.532 L126.121 117.674 L127.919 118.67 L129.717 120.407 L131.514 122.781 L133.312 125.695 L135.11 129.06 L136.907 132.794 L140.503 141.074 L145.895 154.583 L151.288 168.203 L154.884 176.849 L158.479 184.905 L162.074 192.231 L165.67 198.739 L167.467 201.672 L169.265 204.388 L171.063 206.888 L172.86 209.174 L174.658 211.252 L176.455 213.128 L178.253 214.809 L180.051 216.304 L181.848 217.623 L183.646 218.778 L185.444 219.781 L187.241 220.643 L190.837 222 L194.432 222.957 L198.027 223.624 L203.42 224.319 L210.611 225.195 L214.206 225.807 L217.801 226.628 L221.397 227.71 L224.992 229.09 L228.587 230.787 L232.182 232.806 L235.778 235.132 L239.373 237.737 L242.968 240.575 L248.361 245.142 L259.147 254.429 L262.742 257.245 L266.338 259.765 L269.933 261.902 L271.731 262.8 L273.528 263.574 L275.326 264.214 L277.124 264.715 L278.921 265.07 L280.719 265.276 L282.516 265.33 L284.314 265.231 L286.112 264.983 L287.909 264.587 L289.707 264.051 L291.505 263.383 L295.1 261.704 L298.695 259.68 L305.886 255.428 L307.683 254.527 L309.481 253.781 L311.279 253.244 L313.076 252.974 L314.874 253.038 L316.672 253.506 L318.469 254.458 L320.267 255.979 L322.065 258.16 L323.862 261.103 L325.66 264.914 L327.458 269.709 L329.255 275.613 L331.053 282.759 L332.85 291.29 L334.648 301.358 L336.446 313.125 L338.243 326.763 L340.041 342.455 L341.892 361 L341.892 361\" style=\"fill:none;stroke:#0000ff;stroke-linecap:square;stroke-width:2;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\" M343.636 315 L343.636 36\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\" M90 315 L90 36\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\" M90 36 L343.636 36\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\" M90 315 L343.636 315\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 -4\" id=\"mc7db9fdffb\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"90.0\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_4\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L0 4\" id=\"m5a7d422ac3\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"90.0\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 1880 -->\n",
       "      <defs>\n",
       "       <path d=\" M12.4062 8.29688 L28.5156 8.29688 L28.5156 63.9219 L10.9844 60.4062 L10.9844 69.3906 L28.4219 72.9062 L38.2812 72.9062 L38.2812 8.29688 L54.3906 8.29688 L54.3906 0 L12.4062 0 z \" id=\"BitstreamVeraSans-Roman-31\"/>\n",
       "       <path d=\" M31.7812 34.625 Q24.75 34.625 20.7188 30.8594 Q16.7031 27.0938 16.7031 20.5156 Q16.7031 13.9219 20.7188 10.1562 Q24.75 6.39062 31.7812 6.39062 Q38.8125 6.39062 42.8594 10.1719 Q46.9219 13.9688 46.9219 20.5156 Q46.9219 27.0938 42.8906 30.8594 Q38.875 34.625 31.7812 34.625 M21.9219 38.8125 Q15.5781 40.375 12.0312 44.7188 Q8.5 49.0781 8.5 55.3281 Q8.5 64.0625 14.7188 69.1406 Q20.9531 74.2188 31.7812 74.2188 Q42.6719 74.2188 48.875 69.1406 Q55.0781 64.0625 55.0781 55.3281 Q55.0781 49.0781 51.5312 44.7188 Q48 40.375 41.7031 38.8125 Q48.8281 37.1562 52.7969 32.3125 Q56.7812 27.4844 56.7812 20.5156 Q56.7812 9.90625 50.3125 4.23438 Q43.8438 -1.42188 31.7812 -1.42188 Q19.7344 -1.42188 13.25 4.23438 Q6.78125 9.90625 6.78125 20.5156 Q6.78125 27.4844 10.7812 32.3125 Q14.7969 37.1562 21.9219 38.8125 M18.3125 54.3906 Q18.3125 48.7344 21.8438 45.5625 Q25.3906 42.3906 31.7812 42.3906 Q38.1406 42.3906 41.7188 45.5625 Q45.3125 48.7344 45.3125 54.3906 Q45.3125 60.0625 41.7188 63.2344 Q38.1406 66.4062 31.7812 66.4062 Q25.3906 66.4062 21.8438 63.2344 Q18.3125 60.0625 18.3125 54.3906\" id=\"BitstreamVeraSans-Roman-38\"/>\n",
       "       <path d=\" M31.7812 66.4062 Q24.1719 66.4062 20.3281 58.9062 Q16.5 51.4219 16.5 36.375 Q16.5 21.3906 20.3281 13.8906 Q24.1719 6.39062 31.7812 6.39062 Q39.4531 6.39062 43.2812 13.8906 Q47.125 21.3906 47.125 36.375 Q47.125 51.4219 43.2812 58.9062 Q39.4531 66.4062 31.7812 66.4062 M31.7812 74.2188 Q44.0469 74.2188 50.5156 64.5156 Q56.9844 54.8281 56.9844 36.375 Q56.9844 17.9688 50.5156 8.26562 Q44.0469 -1.42188 31.7812 -1.42188 Q19.5312 -1.42188 13.0625 8.26562 Q6.59375 17.9688 6.59375 36.375 Q6.59375 54.8281 13.0625 64.5156 Q19.5312 74.2188 31.7812 74.2188\" id=\"BitstreamVeraSans-Roman-30\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(78.15625 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-38\"/>\n",
       "       <use x=\"127.24609375\" xlink:href=\"#BitstreamVeraSans-Roman-38\"/>\n",
       "       <use x=\"190.869140625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"126.233766234\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"126.233766234\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 1900 -->\n",
       "      <defs>\n",
       "       <path d=\" M10.9844 1.51562 L10.9844 10.5 Q14.7031 8.73438 18.5 7.8125 Q22.3125 6.89062 25.9844 6.89062 Q35.75 6.89062 40.8906 13.4531 Q46.0469 20.0156 46.7812 33.4062 Q43.9531 29.2031 39.5938 26.9531 Q35.25 24.7031 29.9844 24.7031 Q19.0469 24.7031 12.6719 31.3125 Q6.29688 37.9375 6.29688 49.4219 Q6.29688 60.6406 12.9375 67.4219 Q19.5781 74.2188 30.6094 74.2188 Q43.2656 74.2188 49.9219 64.5156 Q56.5938 54.8281 56.5938 36.375 Q56.5938 19.1406 48.4062 8.85938 Q40.2344 -1.42188 26.4219 -1.42188 Q22.7031 -1.42188 18.8906 -0.6875 Q15.0938 0.046875 10.9844 1.51562 M30.6094 32.4219 Q37.25 32.4219 41.125 36.9531 Q45.0156 41.5 45.0156 49.4219 Q45.0156 57.2812 41.125 61.8438 Q37.25 66.4062 30.6094 66.4062 Q23.9688 66.4062 20.0938 61.8438 Q16.2188 57.2812 16.2188 49.4219 Q16.2188 41.5 20.0938 36.9531 Q23.9688 32.4219 30.6094 32.4219\" id=\"BitstreamVeraSans-Roman-39\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(114.390016234 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-39\"/>\n",
       "       <use x=\"127.24609375\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "       <use x=\"190.869140625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"162.467532468\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"162.467532468\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 1920 -->\n",
       "      <defs>\n",
       "       <path d=\" M19.1875 8.29688 L53.6094 8.29688 L53.6094 0 L7.32812 0 L7.32812 8.29688 Q12.9375 14.1094 22.625 23.8906 Q32.3281 33.6875 34.8125 36.5312 Q39.5469 41.8438 41.4219 45.5312 Q43.3125 49.2188 43.3125 52.7812 Q43.3125 58.5938 39.2344 62.25 Q35.1562 65.9219 28.6094 65.9219 Q23.9688 65.9219 18.8125 64.3125 Q13.6719 62.7031 7.8125 59.4219 L7.8125 69.3906 Q13.7656 71.7812 18.9375 73 Q24.125 74.2188 28.4219 74.2188 Q39.75 74.2188 46.4844 68.5469 Q53.2188 62.8906 53.2188 53.4219 Q53.2188 48.9219 51.5312 44.8906 Q49.8594 40.875 45.4062 35.4062 Q44.1875 33.9844 37.6406 27.2188 Q31.1094 20.4531 19.1875 8.29688\" id=\"BitstreamVeraSans-Roman-32\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(150.623782468 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-39\"/>\n",
       "       <use x=\"127.24609375\" xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "       <use x=\"190.869140625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"198.701298701\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"198.701298701\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 1940 -->\n",
       "      <defs>\n",
       "       <path d=\" M37.7969 64.3125 L12.8906 25.3906 L37.7969 25.3906 z  M35.2031 72.9062 L47.6094 72.9062 L47.6094 25.3906 L58.0156 25.3906 L58.0156 17.1875 L47.6094 17.1875 L47.6094 0 L37.7969 0 L37.7969 17.1875 L4.89062 17.1875 L4.89062 26.7031 z \" id=\"BitstreamVeraSans-Roman-34\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(186.857548701 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-39\"/>\n",
       "       <use x=\"127.24609375\" xlink:href=\"#BitstreamVeraSans-Roman-34\"/>\n",
       "       <use x=\"190.869140625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"234.935064935\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"234.935064935\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 1960 -->\n",
       "      <defs>\n",
       "       <path d=\" M33.0156 40.375 Q26.375 40.375 22.4844 35.8281 Q18.6094 31.2969 18.6094 23.3906 Q18.6094 15.5312 22.4844 10.9531 Q26.375 6.39062 33.0156 6.39062 Q39.6562 6.39062 43.5312 10.9531 Q47.4062 15.5312 47.4062 23.3906 Q47.4062 31.2969 43.5312 35.8281 Q39.6562 40.375 33.0156 40.375 M52.5938 71.2969 L52.5938 62.3125 Q48.875 64.0625 45.0938 64.9844 Q41.3125 65.9219 37.5938 65.9219 Q27.8281 65.9219 22.6719 59.3281 Q17.5312 52.7344 16.7969 39.4062 Q19.6719 43.6562 24.0156 45.9219 Q28.375 48.1875 33.5938 48.1875 Q44.5781 48.1875 50.9531 41.5156 Q57.3281 34.8594 57.3281 23.3906 Q57.3281 12.1562 50.6875 5.35938 Q44.0469 -1.42188 33.0156 -1.42188 Q20.3594 -1.42188 13.6719 8.26562 Q6.98438 17.9688 6.98438 36.375 Q6.98438 53.6562 15.1875 63.9375 Q23.3906 74.2188 37.2031 74.2188 Q40.9219 74.2188 44.7031 73.4844 Q48.4844 72.75 52.5938 71.2969\" id=\"BitstreamVeraSans-Roman-36\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(223.091314935 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-39\"/>\n",
       "       <use x=\"127.24609375\" xlink:href=\"#BitstreamVeraSans-Roman-36\"/>\n",
       "       <use x=\"190.869140625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"271.168831169\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_14\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"271.168831169\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 1980 -->\n",
       "      <g transform=\"translate(259.325081169 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-39\"/>\n",
       "       <use x=\"127.24609375\" xlink:href=\"#BitstreamVeraSans-Roman-38\"/>\n",
       "       <use x=\"190.869140625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_7\">\n",
       "     <g id=\"line2d_15\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"307.402597403\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_16\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"307.402597403\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_7\">\n",
       "      <!-- 2000 -->\n",
       "      <g transform=\"translate(295.376034903 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "       <use x=\"127.24609375\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "       <use x=\"190.869140625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_8\">\n",
       "     <g id=\"line2d_17\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"343.636363636\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_18\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"343.636363636\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 2020 -->\n",
       "      <g transform=\"translate(331.609801136 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "       <use x=\"127.24609375\" xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "       <use x=\"190.869140625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_9\">\n",
       "     <!-- year -->\n",
       "     <defs>\n",
       "      <path d=\" M32.1719 -5.07812 Q28.375 -14.8438 24.75 -17.8125 Q21.1406 -20.7969 15.0938 -20.7969 L7.90625 -20.7969 L7.90625 -13.2812 L13.1875 -13.2812 Q16.8906 -13.2812 18.9375 -11.5156 Q21 -9.76562 23.4844 -3.21875 L25.0938 0.875 L2.98438 54.6875 L12.5 54.6875 L29.5938 11.9219 L46.6875 54.6875 L56.2031 54.6875 z \" id=\"BitstreamVeraSans-Roman-79\"/>\n",
       "      <path d=\" M56.2031 29.5938 L56.2031 25.2031 L14.8906 25.2031 Q15.4844 15.9219 20.4844 11.0625 Q25.4844 6.20312 34.4219 6.20312 Q39.5938 6.20312 44.4531 7.46875 Q49.3125 8.73438 54.1094 11.2812 L54.1094 2.78125 Q49.2656 0.734375 44.1875 -0.34375 Q39.1094 -1.42188 33.8906 -1.42188 Q20.7969 -1.42188 13.1562 6.1875 Q5.51562 13.8125 5.51562 26.8125 Q5.51562 40.2344 12.7656 48.1094 Q20.0156 56 32.3281 56 Q43.3594 56 49.7812 48.8906 Q56.2031 41.7969 56.2031 29.5938 M47.2188 32.2344 Q47.125 39.5938 43.0938 43.9844 Q39.0625 48.3906 32.4219 48.3906 Q24.9062 48.3906 20.3906 44.1406 Q15.875 39.8906 15.1875 32.1719 z \" id=\"BitstreamVeraSans-Roman-65\"/>\n",
       "      <path d=\" M34.2812 27.4844 Q23.3906 27.4844 19.1875 25 Q14.9844 22.5156 14.9844 16.5 Q14.9844 11.7188 18.1406 8.90625 Q21.2969 6.10938 26.7031 6.10938 Q34.1875 6.10938 38.7031 11.4062 Q43.2188 16.7031 43.2188 25.4844 L43.2188 27.4844 z  M52.2031 31.2031 L52.2031 0 L43.2188 0 L43.2188 8.29688 Q40.1406 3.32812 35.5469 0.953125 Q30.9531 -1.42188 24.3125 -1.42188 Q15.9219 -1.42188 10.9531 3.29688 Q6 8.01562 6 15.9219 Q6 25.1406 12.1719 29.8281 Q18.3594 34.5156 30.6094 34.5156 L43.2188 34.5156 L43.2188 35.4062 Q43.2188 41.6094 39.1406 45 Q35.0625 48.3906 27.6875 48.3906 Q23 48.3906 18.5469 47.2656 Q14.1094 46.1406 10.0156 43.8906 L10.0156 52.2031 Q14.9375 54.1094 19.5781 55.0469 Q24.2188 56 28.6094 56 Q40.4844 56 46.3438 49.8438 Q52.2031 43.7031 52.2031 31.2031\" id=\"BitstreamVeraSans-Roman-61\"/>\n",
       "      <path d=\" M41.1094 46.2969 Q39.5938 47.1719 37.8125 47.5781 Q36.0312 48 33.8906 48 Q26.2656 48 22.1875 43.0469 Q18.1094 38.0938 18.1094 28.8125 L18.1094 0 L9.07812 0 L9.07812 54.6875 L18.1094 54.6875 L18.1094 46.1875 Q20.9531 51.1719 25.4844 53.5781 Q30.0312 56 36.5312 56 Q37.4531 56 38.5781 55.875 Q39.7031 55.7656 41.0625 55.5156 z \" id=\"BitstreamVeraSans-Roman-72\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(194.805681818 348.875)scale(0.2 -0.2)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-79\"/>\n",
       "      <use x=\"59.1796875\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"120.703125\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"181.982421875\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_19\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L4 0\" id=\"md7965d1ba0\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"90.0\" xlink:href=\"#md7965d1ba0\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_20\">\n",
       "      <defs>\n",
       "       <path d=\" M0 0 L-4 0\" id=\"md9a1c1a7cd\" style=\"stroke:#000000;stroke-width:0.5;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"343.636363636\" xlink:href=\"#md9a1c1a7cd\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 2.5 -->\n",
       "      <defs>\n",
       "       <path d=\" M10.6875 12.4062 L21 12.4062 L21 0 L10.6875 0 z \" id=\"BitstreamVeraSans-Roman-2e\"/>\n",
       "       <path d=\" M10.7969 72.9062 L49.5156 72.9062 L49.5156 64.5938 L19.8281 64.5938 L19.8281 46.7344 Q21.9688 47.4688 24.1094 47.8281 Q26.2656 48.1875 28.4219 48.1875 Q40.625 48.1875 47.75 41.5 Q54.8906 34.8125 54.8906 23.3906 Q54.8906 11.625 47.5625 5.09375 Q40.2344 -1.42188 26.9062 -1.42188 Q22.3125 -1.42188 17.5469 -0.640625 Q12.7969 0.140625 7.71875 1.70312 L7.71875 11.625 Q12.1094 9.23438 16.7969 8.0625 Q21.4844 6.89062 26.7031 6.89062 Q35.1562 6.89062 40.0781 11.3281 Q45.0156 15.7656 45.0156 23.3906 Q45.0156 31 40.0781 35.4375 Q35.1562 39.8906 26.7031 39.8906 Q22.75 39.8906 18.8125 39.0156 Q14.8906 38.1406 10.7969 36.2812 z \" id=\"BitstreamVeraSans-Roman-35\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(71.703125 317.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "       <use x=\"95.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_21\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"90.0\" xlink:href=\"#md7965d1ba0\" y=\"268.5\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_22\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"343.636363636\" xlink:href=\"#md9a1c1a7cd\" y=\"268.5\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 3.0 -->\n",
       "      <defs>\n",
       "       <path d=\" M40.5781 39.3125 Q47.6562 37.7969 51.625 33 Q55.6094 28.2188 55.6094 21.1875 Q55.6094 10.4062 48.1875 4.48438 Q40.7656 -1.42188 27.0938 -1.42188 Q22.5156 -1.42188 17.6562 -0.515625 Q12.7969 0.390625 7.625 2.20312 L7.625 11.7188 Q11.7188 9.32812 16.5938 8.10938 Q21.4844 6.89062 26.8125 6.89062 Q36.0781 6.89062 40.9375 10.5469 Q45.7969 14.2031 45.7969 21.1875 Q45.7969 27.6406 41.2812 31.2656 Q36.7656 34.9062 28.7188 34.9062 L20.2188 34.9062 L20.2188 43.0156 L29.1094 43.0156 Q36.375 43.0156 40.2344 45.9219 Q44.0938 48.8281 44.0938 54.2969 Q44.0938 59.9062 40.1094 62.9062 Q36.1406 65.9219 28.7188 65.9219 Q24.6562 65.9219 20.0156 65.0312 Q15.375 64.1562 9.8125 62.3125 L9.8125 71.0938 Q15.4375 72.6562 20.3438 73.4375 Q25.25 74.2188 29.5938 74.2188 Q40.8281 74.2188 47.3594 69.1094 Q53.9062 64.0156 53.9062 55.3281 Q53.9062 49.2656 50.4375 45.0938 Q46.9688 40.9219 40.5781 39.3125\" id=\"BitstreamVeraSans-Roman-33\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(71.5234375 271.259375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-33\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "       <use x=\"95.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_23\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"90.0\" xlink:href=\"#md7965d1ba0\" y=\"222.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_24\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"343.636363636\" xlink:href=\"#md9a1c1a7cd\" y=\"222.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 3.5 -->\n",
       "      <g transform=\"translate(71.7328125 224.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-33\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "       <use x=\"95.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_25\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"90.0\" xlink:href=\"#md7965d1ba0\" y=\"175.5\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_26\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"343.636363636\" xlink:href=\"#md9a1c1a7cd\" y=\"175.5\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 4.0 -->\n",
       "      <g transform=\"translate(71.25 178.259375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-34\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "       <use x=\"95.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_27\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"90.0\" xlink:href=\"#md7965d1ba0\" y=\"129.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_28\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"343.636363636\" xlink:href=\"#md9a1c1a7cd\" y=\"129.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_14\">\n",
       "      <!-- 4.5 -->\n",
       "      <g transform=\"translate(71.459375 131.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-34\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "       <use x=\"95.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_29\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"90.0\" xlink:href=\"#md7965d1ba0\" y=\"82.5\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_30\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"343.636363636\" xlink:href=\"#md9a1c1a7cd\" y=\"82.5\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_15\">\n",
       "      <!-- 5.0 -->\n",
       "      <g transform=\"translate(71.5328125 85.259375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-35\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "       <use x=\"95.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_7\">\n",
       "     <g id=\"line2d_31\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"90.0\" xlink:href=\"#md7965d1ba0\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_32\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"343.636363636\" xlink:href=\"#md9a1c1a7cd\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_16\">\n",
       "      <!-- 5.5 -->\n",
       "      <g transform=\"translate(71.7421875 38.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-35\"/>\n",
       "       <use x=\"63.623046875\" xlink:href=\"#BitstreamVeraSans-Roman-2e\"/>\n",
       "       <use x=\"95.41015625\" xlink:href=\"#BitstreamVeraSans-Roman-35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       "  <g id=\"axes_2\">\n",
       "   <g id=\"patch_7\">\n",
       "    <path d=\" M394.364 315 L648 315 L648 36 L394.364 36 z \" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_33\">\n",
       "    <g clip-path=\"url(#p447f0539f8)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"426.068181818\" xlink:href=\"#m231f9b0f49\" y=\"45.6963141566\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_34\">\n",
       "    <g clip-path=\"url(#p447f0539f8)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"426.068181818\" xlink:href=\"#m231f9b0f49\" y=\"45.6963141566\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"457.772727273\" xlink:href=\"#m231f9b0f49\" y=\"245.948066059\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_35\">\n",
       "    <g clip-path=\"url(#p447f0539f8)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"426.068181818\" xlink:href=\"#m231f9b0f49\" y=\"45.6963141566\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"457.772727273\" xlink:href=\"#m231f9b0f49\" y=\"245.948066059\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"489.477272727\" xlink:href=\"#m231f9b0f49\" y=\"274.875775291\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_36\">\n",
       "    <g clip-path=\"url(#p447f0539f8)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"426.068181818\" xlink:href=\"#m231f9b0f49\" y=\"45.6963141566\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"457.772727273\" xlink:href=\"#m231f9b0f49\" y=\"245.948066059\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"489.477272727\" xlink:href=\"#m231f9b0f49\" y=\"274.875775291\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"521.181818182\" xlink:href=\"#m231f9b0f49\" y=\"277.200026829\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_37\">\n",
       "    <g clip-path=\"url(#p447f0539f8)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"426.068181818\" xlink:href=\"#m231f9b0f49\" y=\"45.6963141566\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"457.772727273\" xlink:href=\"#m231f9b0f49\" y=\"245.948066059\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"489.477272727\" xlink:href=\"#m231f9b0f49\" y=\"274.875775291\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"521.181818182\" xlink:href=\"#m231f9b0f49\" y=\"277.200026829\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"552.886363636\" xlink:href=\"#m231f9b0f49\" y=\"277.301045206\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_38\">\n",
       "    <g clip-path=\"url(#p447f0539f8)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"426.068181818\" xlink:href=\"#m231f9b0f49\" y=\"45.6963141566\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"457.772727273\" xlink:href=\"#m231f9b0f49\" y=\"245.948066059\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"489.477272727\" xlink:href=\"#m231f9b0f49\" y=\"274.875775291\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"521.181818182\" xlink:href=\"#m231f9b0f49\" y=\"277.200026829\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"552.886363636\" xlink:href=\"#m231f9b0f49\" y=\"277.301045206\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"584.590909091\" xlink:href=\"#m231f9b0f49\" y=\"277.615059681\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_39\">\n",
       "    <g clip-path=\"url(#p447f0539f8)\">\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"426.068181818\" xlink:href=\"#m231f9b0f49\" y=\"45.6963141566\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"457.772727273\" xlink:href=\"#m231f9b0f49\" y=\"245.948066059\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"489.477272727\" xlink:href=\"#m231f9b0f49\" y=\"274.875775291\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"521.181818182\" xlink:href=\"#m231f9b0f49\" y=\"277.200026829\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"552.886363636\" xlink:href=\"#m231f9b0f49\" y=\"277.301045206\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"584.590909091\" xlink:href=\"#m231f9b0f49\" y=\"277.615059681\"/>\n",
       "     <use style=\"fill:#ff0000;stroke:#000000;stroke-width:0.5;\" x=\"616.295454545\" xlink:href=\"#m231f9b0f49\" y=\"284.641882197\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"patch_8\">\n",
       "    <path d=\" M648 315 L648 36\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_9\">\n",
       "    <path d=\" M394.364 315 L394.364 36\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_10\">\n",
       "    <path d=\" M394.364 36 L648 36\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_11\">\n",
       "    <path d=\" M394.364 315 L648 315\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_3\">\n",
       "    <g id=\"xtick_9\">\n",
       "     <g id=\"line2d_40\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_41\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_17\">\n",
       "      <!-- −1 -->\n",
       "      <defs>\n",
       "       <path d=\" M10.5938 35.5 L73.1875 35.5 L73.1875 27.2031 L10.5938 27.2031 z \" id=\"BitstreamVeraSans-Roman-2212\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(387.983948864 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-2212\"/>\n",
       "       <use x=\"83.7890625\" xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_10\">\n",
       "     <g id=\"line2d_42\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"426.068181818\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_43\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"426.068181818\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_18\">\n",
       "      <!-- 0 -->\n",
       "      <g transform=\"translate(423.548650568 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_11\">\n",
       "     <g id=\"line2d_44\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"457.772727273\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_45\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"457.772727273\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_19\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(455.602414773 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_12\">\n",
       "     <g id=\"line2d_46\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"489.477272727\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_47\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"489.477272727\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_20\">\n",
       "      <!-- 2 -->\n",
       "      <g transform=\"translate(487.163210227 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_13\">\n",
       "     <g id=\"line2d_48\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"521.181818182\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_49\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"521.181818182\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_21\">\n",
       "      <!-- 3 -->\n",
       "      <g transform=\"translate(518.782599432 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-33\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_14\">\n",
       "     <g id=\"line2d_50\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"552.886363636\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_51\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"552.886363636\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_22\">\n",
       "      <!-- 4 -->\n",
       "      <g transform=\"translate(550.230113636 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-34\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_15\">\n",
       "     <g id=\"line2d_52\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"584.590909091\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_53\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"584.590909091\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_23\">\n",
       "      <!-- 5 -->\n",
       "      <g transform=\"translate(582.232315341 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_16\">\n",
       "     <g id=\"line2d_54\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"616.295454545\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_55\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"616.295454545\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_24\">\n",
       "      <!-- 6 -->\n",
       "      <g transform=\"translate(613.778267045 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-36\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_17\">\n",
       "     <g id=\"line2d_56\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#mc7db9fdffb\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_57\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#m5a7d422ac3\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_25\">\n",
       "      <!-- 7 -->\n",
       "      <defs>\n",
       "       <path d=\" M8.20312 72.9062 L55.0781 72.9062 L55.0781 68.7031 L28.6094 0 L18.3125 0 L43.2188 64.5938 L8.20312 64.5938 z \" id=\"BitstreamVeraSans-Roman-37\"/>\n",
       "      </defs>\n",
       "      <g transform=\"translate(645.65625 326.5984375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-37\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_26\">\n",
       "     <!-- polynomial order -->\n",
       "     <defs>\n",
       "      <path id=\"BitstreamVeraSans-Roman-20\"/>\n",
       "      <path d=\" M30.6094 48.3906 Q23.3906 48.3906 19.1875 42.75 Q14.9844 37.1094 14.9844 27.2969 Q14.9844 17.4844 19.1562 11.8438 Q23.3438 6.20312 30.6094 6.20312 Q37.7969 6.20312 41.9844 11.8594 Q46.1875 17.5312 46.1875 27.2969 Q46.1875 37.0156 41.9844 42.7031 Q37.7969 48.3906 30.6094 48.3906 M30.6094 56 Q42.3281 56 49.0156 48.375 Q55.7188 40.7656 55.7188 27.2969 Q55.7188 13.875 49.0156 6.21875 Q42.3281 -1.42188 30.6094 -1.42188 Q18.8438 -1.42188 12.1719 6.21875 Q5.51562 13.875 5.51562 27.2969 Q5.51562 40.7656 12.1719 48.375 Q18.8438 56 30.6094 56\" id=\"BitstreamVeraSans-Roman-6f\"/>\n",
       "      <path d=\" M18.1094 8.20312 L18.1094 -20.7969 L9.07812 -20.7969 L9.07812 54.6875 L18.1094 54.6875 L18.1094 46.3906 Q20.9531 51.2656 25.2656 53.625 Q29.5938 56 35.5938 56 Q45.5625 56 51.7812 48.0938 Q58.0156 40.1875 58.0156 27.2969 Q58.0156 14.4062 51.7812 6.48438 Q45.5625 -1.42188 35.5938 -1.42188 Q29.5938 -1.42188 25.2656 0.953125 Q20.9531 3.32812 18.1094 8.20312 M48.6875 27.2969 Q48.6875 37.2031 44.6094 42.8438 Q40.5312 48.4844 33.4062 48.4844 Q26.2656 48.4844 22.1875 42.8438 Q18.1094 37.2031 18.1094 27.2969 Q18.1094 17.3906 22.1875 11.75 Q26.2656 6.10938 33.4062 6.10938 Q40.5312 6.10938 44.6094 11.75 Q48.6875 17.3906 48.6875 27.2969\" id=\"BitstreamVeraSans-Roman-70\"/>\n",
       "      <path d=\" M45.4062 46.3906 L45.4062 75.9844 L54.3906 75.9844 L54.3906 0 L45.4062 0 L45.4062 8.20312 Q42.5781 3.32812 38.25 0.953125 Q33.9375 -1.42188 27.875 -1.42188 Q17.9688 -1.42188 11.7344 6.48438 Q5.51562 14.4062 5.51562 27.2969 Q5.51562 40.1875 11.7344 48.0938 Q17.9688 56 27.875 56 Q33.9375 56 38.25 53.625 Q42.5781 51.2656 45.4062 46.3906 M14.7969 27.2969 Q14.7969 17.3906 18.875 11.75 Q22.9531 6.10938 30.0781 6.10938 Q37.2031 6.10938 41.2969 11.75 Q45.4062 17.3906 45.4062 27.2969 Q45.4062 37.2031 41.2969 42.8438 Q37.2031 48.4844 30.0781 48.4844 Q22.9531 48.4844 18.875 42.8438 Q14.7969 37.2031 14.7969 27.2969\" id=\"BitstreamVeraSans-Roman-64\"/>\n",
       "      <path d=\" M52 44.1875 Q55.375 50.25 60.0625 53.125 Q64.75 56 71.0938 56 Q79.6406 56 84.2812 50.0156 Q88.9219 44.0469 88.9219 33.0156 L88.9219 0 L79.8906 0 L79.8906 32.7188 Q79.8906 40.5781 77.0938 44.375 Q74.3125 48.1875 68.6094 48.1875 Q61.625 48.1875 57.5625 43.5469 Q53.5156 38.9219 53.5156 30.9062 L53.5156 0 L44.4844 0 L44.4844 32.7188 Q44.4844 40.625 41.7031 44.4062 Q38.9219 48.1875 33.1094 48.1875 Q26.2188 48.1875 22.1562 43.5312 Q18.1094 38.875 18.1094 30.9062 L18.1094 0 L9.07812 0 L9.07812 54.6875 L18.1094 54.6875 L18.1094 46.1875 Q21.1875 51.2188 25.4844 53.6094 Q29.7812 56 35.6875 56 Q41.6562 56 45.8281 52.9688 Q50 49.9531 52 44.1875\" id=\"BitstreamVeraSans-Roman-6d\"/>\n",
       "      <path d=\" M54.8906 33.0156 L54.8906 0 L45.9062 0 L45.9062 32.7188 Q45.9062 40.4844 42.875 44.3281 Q39.8438 48.1875 33.7969 48.1875 Q26.5156 48.1875 22.3125 43.5469 Q18.1094 38.9219 18.1094 30.9062 L18.1094 0 L9.07812 0 L9.07812 54.6875 L18.1094 54.6875 L18.1094 46.1875 Q21.3438 51.125 25.7031 53.5625 Q30.0781 56 35.7969 56 Q45.2188 56 50.0469 50.1719 Q54.8906 44.3438 54.8906 33.0156\" id=\"BitstreamVeraSans-Roman-6e\"/>\n",
       "      <path d=\" M9.42188 75.9844 L18.4062 75.9844 L18.4062 0 L9.42188 0 z \" id=\"BitstreamVeraSans-Roman-6c\"/>\n",
       "      <path d=\" M9.42188 54.6875 L18.4062 54.6875 L18.4062 0 L9.42188 0 z  M9.42188 75.9844 L18.4062 75.9844 L18.4062 64.5938 L9.42188 64.5938 z \" id=\"BitstreamVeraSans-Roman-69\"/>\n",
       "     </defs>\n",
       "     <g transform=\"translate(436.291193182 348.875)scale(0.2 -0.2)\">\n",
       "      <use xlink:href=\"#BitstreamVeraSans-Roman-70\"/>\n",
       "      <use x=\"63.4765625\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"124.658203125\" xlink:href=\"#BitstreamVeraSans-Roman-6c\"/>\n",
       "      <use x=\"152.44140625\" xlink:href=\"#BitstreamVeraSans-Roman-79\"/>\n",
       "      <use x=\"211.62109375\" xlink:href=\"#BitstreamVeraSans-Roman-6e\"/>\n",
       "      <use x=\"275.0\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"336.181640625\" xlink:href=\"#BitstreamVeraSans-Roman-6d\"/>\n",
       "      <use x=\"433.59375\" xlink:href=\"#BitstreamVeraSans-Roman-69\"/>\n",
       "      <use x=\"461.376953125\" xlink:href=\"#BitstreamVeraSans-Roman-61\"/>\n",
       "      <use x=\"522.65625\" xlink:href=\"#BitstreamVeraSans-Roman-6c\"/>\n",
       "      <use x=\"550.439453125\" xlink:href=\"#BitstreamVeraSans-Roman-20\"/>\n",
       "      <use x=\"582.2265625\" xlink:href=\"#BitstreamVeraSans-Roman-6f\"/>\n",
       "      <use x=\"643.408203125\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "      <use x=\"682.771484375\" xlink:href=\"#BitstreamVeraSans-Roman-64\"/>\n",
       "      <use x=\"746.248046875\" xlink:href=\"#BitstreamVeraSans-Roman-65\"/>\n",
       "      <use x=\"807.771484375\" xlink:href=\"#BitstreamVeraSans-Roman-72\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_4\">\n",
       "    <g id=\"ytick_8\">\n",
       "     <g id=\"line2d_58\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#md7965d1ba0\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_59\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#md9a1c1a7cd\" y=\"315.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_27\">\n",
       "      <!-- 0 -->\n",
       "      <g transform=\"translate(385.324573864 317.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_9\">\n",
       "     <g id=\"line2d_60\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#md7965d1ba0\" y=\"280.125\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_61\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#md9a1c1a7cd\" y=\"280.125\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_28\">\n",
       "      <!-- 1 -->\n",
       "      <g transform=\"translate(386.023011364 282.884375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-31\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_10\">\n",
       "     <g id=\"line2d_62\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#md7965d1ba0\" y=\"245.25\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_63\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#md9a1c1a7cd\" y=\"245.25\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_29\">\n",
       "      <!-- 2 -->\n",
       "      <g transform=\"translate(385.735511364 248.009375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-32\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_11\">\n",
       "     <g id=\"line2d_64\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#md7965d1ba0\" y=\"210.375\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_65\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#md9a1c1a7cd\" y=\"210.375\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_30\">\n",
       "      <!-- 3 -->\n",
       "      <g transform=\"translate(385.565198864 213.134375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-33\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_12\">\n",
       "     <g id=\"line2d_66\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#md7965d1ba0\" y=\"175.5\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_67\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#md9a1c1a7cd\" y=\"175.5\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_31\">\n",
       "      <!-- 4 -->\n",
       "      <g transform=\"translate(385.051136364 178.259375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-34\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_13\">\n",
       "     <g id=\"line2d_68\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#md7965d1ba0\" y=\"140.625\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_69\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#md9a1c1a7cd\" y=\"140.625\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_32\">\n",
       "      <!-- 5 -->\n",
       "      <g transform=\"translate(385.646448864 143.384375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-35\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_14\">\n",
       "     <g id=\"line2d_70\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#md7965d1ba0\" y=\"105.75\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_71\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#md9a1c1a7cd\" y=\"105.75\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_33\">\n",
       "      <!-- 6 -->\n",
       "      <g transform=\"translate(385.329261364 108.509375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-36\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_15\">\n",
       "     <g id=\"line2d_72\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#md7965d1ba0\" y=\"70.875\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_73\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#md9a1c1a7cd\" y=\"70.875\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_34\">\n",
       "      <!-- 7 -->\n",
       "      <g transform=\"translate(385.676136364 73.634375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-37\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_16\">\n",
       "     <g id=\"line2d_74\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"394.363636364\" xlink:href=\"#md7965d1ba0\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"line2d_75\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.5;\" x=\"648.0\" xlink:href=\"#md9a1c1a7cd\" y=\"36.0\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_35\">\n",
       "      <!-- 8 -->\n",
       "      <g transform=\"translate(385.363636364 38.759375)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#BitstreamVeraSans-Roman-38\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p447f0539f8\">\n",
       "   <rect height=\"279.0\" width=\"253.636363636\" x=\"394.363636364\" y=\"36.0\"/>\n",
       "  </clipPath>\n",
       "  <clipPath id=\"p4bbcf401cb\">\n",
       "   <rect height=\"279.0\" width=\"253.636363636\" x=\"90.0\" y=\"36.0\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pods.notebook.display_plots('olympic_LM_polynomial{num_basis}.svg', directory='./diagrams', num_basis=(1,7))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Reading\n",
    "\n",
    "- Section 1.4 of @Rogers:book11.\n",
    "- Chapter 1, pg 1-6 of @Bishop:book06.\n",
    "- Chapter 3, Section 3.1 of @Bishop:book06 up to pg 143.\n",
    "\n",
    " "
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}