{
 "metadata": {
  "name": "",
  "signature": "sha256:45eb4d5f790f900da8ac7932d3e3f0ec6702d57f7c3d54b7475b4f1274aeead3"
 },
 "nbformat": 3,
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "rcParams['figure.figsize'] = 14, 7"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Regularized least squares and cross validation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "While calculating CSD, we often have to deal with noisy measurements. \n",
      "\n",
      "Finding a set of optimal parameters for our model, we have to solve the bias\u2013variance dilemma which is the problem of simultaneously minimizing the bias (how accurate a model is across different training sets) and variance of the model error (how sensitive the model is to small changes in training set). "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The error may be decomposed into:\n",
      "\n",
      "$Err(x) = \\left(E[\\hat{f}(x)]-f(x)\\right)^2 + E\\left[\\hat{f}(x)-E[\\hat{f}(x)]\\right]^2 +\\sigma_e^2$\n",
      "\n",
      "$Err(x) = \\mathrm{Bias}^2 + \\mathrm{Variance} + \\mathrm{Irreducible\\ Error}$"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Example - noisy measurements"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Suppose we took some noisy measurements of function $f(x) = 0.08 x^2 + 1.0$ for $x \\in \\lbrace{ 0,1,2,3,4,5,6,7,8,9 \\rbrace}$ \n",
      "The noise will add variance, but not so much bias. The noise is from normal distribution so $E(noise) = 0$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "\n",
      "X = np.array(range(0,10))\n",
      "Y = 0.08*X**2 + 1.0\n",
      "\n",
      "fig, (ax1, ax2) = plt.subplots(1, 2, sharex=True, sharey=True)\n",
      "ax1.plot(X,Y)\n",
      "ax1.set_title(\"model function\")\n",
      "\n",
      "Y2 = Y + np.random.random_sample(size(X))-0.5\n",
      "ax2.scatter(X,Y2)\n",
      "ax2.plot(X,Y)\n",
      "ax2.set_title(\"noisy measurements\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 2,
       "text": [
        "<matplotlib.text.Text at 0x36c5310>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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vr6CgIIWGhqpEiRJ69tln3Y4nvJ+flUrpPGbMGM2fP1+5cuXS9ddfr9mzZytfvnw5lQ8A\nACDHjRo1Svv379fcuXNNRwF8VoodlSNHjmjmzJnavn27fvnlFyUkJCS5MwEAAOBtzp8/r1mzZql/\n//6mowA+LcVCpXDhwsqTJ4+io6MVHx+v6Ojoay5IAwAA8AYzZ85UhQoV1K5dO7Vo0cJ0HMCnpfro\n14wZM/TMM8+oQIECuv3225Oc7wAAAAAA2SHFQuXgwYNq37691q1bpyJFiui+++5Tly5dkuwnXrVq\nVbfbsgIAck6VKlV04MAB0zEciWsVAJiX3utUio9+bdu2Tc2bN1fx4sWVO3duderUSRs3bkzymoMH\nD7r2+faFX8OHDzeegc/KZ+Wz8lmv9Yu/iLvnS9cqX/rvns/qnb986bP62udN73UqxUKlZs2a2rRp\nky5fvizLsrR69WrVrl07UxcLAAAAAEhNioVKvXr11KtXL9cZD5LYAQMAAABAtsud2guee+45Pffc\nczmRxSOEhYWZjpBj+Kzeic8KeDdf+u+ez+qdfOmzSr73edMj1V2/Un0DPz9l8i0AAFmA+dg9xgYA\nzEvvXJzio18AAAAAYAKFCgAAAADHoVABAAAA4DgUKgAAAAAch0IFAAAAgONQqAAAAABwHAoVAAAA\nAI5DoQIAAADAcShUAAAAADgOhQoAAAAAx6FQAQAAAOA4FCoAAAAAHIdCBQAAAIDjUKgAAAAAcBwK\nFQAAAACOQ6ECAAAAwHEoVAAAAAA4DoUKAAAAAMehUAGALJSYaDoBAADuJSZKlmU6RdpQqABAFho2\nTJo82XQKAACubcYM6bnnTKdIGz/LylxN5efnp0y+BQB4hehoqWJFacMGqXr1nP/5zMfuMTYAYHdT\natWyi5Vbbsn5n5/euZiOCgBkkblzpebNzRQpAACkZuVKqVAh6eabTSdJm9ymAwCAN0hMlCZOtO9S\nAQDgRBMmSE8/Lfn5mU6SNnRUACALeNpdKgCAb9mxQ/r9d6lrV9NJ0o5CBQCywIQJ0jPPeM5dKgCA\nb5k4UXrqKSlPHtNJ0o7F9ACQSTt2SB06SIcOmb0AMB+7x9gA8GXHj0vXXy8dPCgVK2YuB4vpASCH\nTZwoDRjgWXepAAC+4513pAcfNFukZAQdFQDIBKfcpZKYj1PC2ADwVVFRUmiotGmTVKWK2Sx0VAAg\nB3nqXSoAgG/48EOpZUvzRUpG0FEBgAxy0l0qifk4JYwNAF+UmCjVqCHNni21aGE6DR0VAMgxnnyX\nCgDg/ZYvtzv+N91kOknGcOAjAGRAQoK9iH72bNNJAAC4tvHjPeuAx/+iowIAGeDpd6kAAN5t2zbp\n8GGpc2fTSTKOQgUAMmDCBM++SwUA8G6eeMDjf7GYHgDSads2qVMne0tiJ10AmI/dY2wA+JJjx6R6\n9eyDiIsWNZ3mXyymB4Bs5g13qQAA3mvqVKlXL2cVKRlBRwUA0sGpd6kk5uOUMDYAfEVkpL11/pYt\nUuXKptMkld65mF2/ACAdvOUuFQDAO82eLYWFXbtIOXLkiN59d4aio6+oR4/71KxZsxzPlx50VAAg\njf65S7V1q1Spkuk0yTEfu8fYAPAFCQlS9erS/PnSf2uQQ4cOqUGD5oqM7KnExCAFBEzSkiVz1K5d\nuxzLl+VrVH777Tc1aNDA9atIkSKaMmVKpkICgCeaPVtq1cqZRQoAAMuWSaVKJS9SJGny5HcVGfmQ\nEhPflPSioqOn6cUXx+Z4xvRI9dGvGjVqaMeOHZKkxMRElS1bVh07dsz2YADgJAkJ0qRJ9l0qAACc\n6J+t86/l0qVoJSZWv+rflFZ0dHSO5MqodO36tXr1alWpUkXly5fPrjwA4Egp3aUCAMC0LVvsDV/c\n9RN69OisgIA3JX0jaZsCAgaqV68uORkx3dK1mH7RokV64IEHsisLADhWSnepAAAwbeJEaeBAKbeb\nv923bt1ac+ZM1ssvD1VMTIweeuh+vfTSszkbMp3SvJg+NjZWZcuW1d69e1WyZMl/38DPT8OHD3d9\nHRYWprCwsCwPCgCmbNkide0qHTjg/gJgQnh4uMLDw11fjxw5kgXjbnCtAuDN/vhDatBAOnxYKlzY\ndJp/ZfY6leZC5YsvvtB7772nVatWJX0DdlIB4OXuv1+68UZp8GDTSVLGfOweYwPAmw0ZIlmWNH68\n6SQpy7ZzVD766CN17949Q6EAwFMdPSp9+600Y4bpJAAAJBcRYe9KuX276SRZL00dlaioKFWsWFGH\nDx9WYGBg0jfgLhUAL+Ypd6kk5uOUMDYAvNWkSdKPP0off2w6SerSOxdz4CMAuBERYZ+Zsn27VLGi\n6TSpYz52j7EB4I3i46Vq1aRFi+xHlJ0uyw98BABfNWuW1KaNZxQpAADfs3SpFBLiGUVKRjho/xoA\ncI74eGnyZPsuFQAATjRhgvTMM6ZTZB86KgBwDd5+lwoA4Nl+/FE6dUq6917TSbIPhQoAXAMHPAIA\nnOyfAx79/U0nyT4spgeA//jxR6lHD2n/fs+6ADAfu8fYAPAmR45IjRrZ//zPhryOxmJ6AMikiROl\nQYM8q0gBAPiOKVOkvn09q0jJCDoqAHCVI0ekxo2lw4c97wLAfOweYwPAW1y8KFWuLO3cKZUvbzpN\n+tBRAYBMmDJFevhhzytSAAC+4YMPpNtv97wiJSPoqADA/1y8aB/wuGuXZ14AmI/dY2wAeIP4eKlK\nFenTT+3uv6ehowIAGfT++9Idd3hmkQIA8H6ffmofQuyJRUpGcOAjAOjfAx4/+8x0EgAAkrMsafx4\n6aWXTCfJOXRUAED2XarQUN+5SwUA8CwbN0rnz0vt25tOknMoVAD4vH/uUnHAIwDAqSZM8L2t81lM\nD8Dnbdgg9e4t/fabZ18AmI/dY2wAeLKDB6Ubb7S30C9UyHSajGMxPQCkky/epQIAeI4pU6RHHvHs\nIiUj6KgA8GnecpdKYj5OCWMDwFNduGAf8Pjzz1K5cqbTZA4dFQBIB1+9SwUA8AwzZ0p33un5RUpG\n0FEB4LO86S6VxHycEsYGgCeKi7OvU0uXSo0amU6TeXRUACCNfPkuFQDA+ZYssU+i94YiJSM48BGA\nT4qLsx/7+uIL00kAAEjOsuzNXoYNM53EHDoqAHzS4sVS1apSw4amkwAAkNy6ddLFi9Jdd5lOYg6F\nCgCfwwGPAACnmzBBGjxYyuXDf1vn0S8APmfdOunSJd++SwUAcK79++3DiBcsMJ3ELB+u0QD4Ku5S\nAQCcbPJkqX9/qWBB00nMYntiAD5l/36peXP7gEdvuwAwH7vH2ADwFOfP2zt97dkjhYSYTpO12J4Y\nAFLAXSoAgJPNmCF16OB9RUpG0FEB4DO8+S6VxHycEsYGgCeIjZUqVZJWrJDq1zedJuvRUQEAN7hL\nBQBwsk8+kWrW9M4iJSPoqADwCd5+l0piPk4JYwPA6SzLPoF+1Cjv3ZWSjgoAXAN3qQAATvbDD1J0\ntNSunekkzkGhAsDrWZa9JTEHPAIAnIqt85PjwEcAXo+7VAAAJ/v9d2nTJmnRItNJnIWaDYDX4y4V\nAMDJJk2SHn1UCggwncRZWEwPwKv99pvUsqV09KhUoIDpNNmL+dg9xgaAU507J1WtKv36q1SmjOk0\n2YvF9ABwlUmTpMce8/4iBQDgmaZNkzp29P4iJSPoqADwWmfPStWq+cZdKon5OCWMDQAniomRQkOl\nb76Rrr/edJrsR0cFAP5n+nTuUgEAnGvRIrtA8YUiJSPoqADwSjEx9gGPX3/tOxcA5mP3GBsATmNZ\n9tleb7wh3XGH6TQ5g44KAEj64AP7AuArRQoAwLN8+aX9z9tvN5vDyeioAPA6V67YO6h89pl0ww2m\n0+Qc5mP3GBsATmJZUsOG0rBh9iPKviLLOyoXLlxQly5dVKtWLdWuXVubNm3KVEAAyG4zZ0oNGvhW\nkQIA8BxLl0p+ftK995pO4mypdlR69+6tW265RQ8//LDi4+MVFRWlIkWK/PsG3KUC4CCXL9vdlGXL\npEaNTKfJWczH7jE2AJwiMdG+mTZqlNShg+k0OSu9c3HulL558eJFrVu3Th9++KH94ty5kxQpAOA0\nM2ZIjRv7XpECAPAMn38u5ckjtW9vOonzpfjo1+HDh1WyZEk99NBDatiwofr166fo6OicygYA6RId\nLY0dK40YYToJAADJJSba16iRI+1Hv5CyFDsq8fHx2r59u6ZOnaomTZpo0KBBGjt2rF599dUkrxtx\n1d8KwsLCFBYWlh1ZASBF06ZJzZrZLXVfEB4ervDwcNMxPAbXKgCmLVkiBQRId95pOknOyOx1KsU1\nKqdOnVKzZs10+PBhSdL69es1duxYLV++/N834LlfAA4QFSVVqWKf7lu3ruk0ZjAfu8fYADAtIcG+\nPr31ltSunWRZlqKiolSwYEH5+Uh7JUt3/SpTpozKly+v33//XZK0evVq1alTJ3MJASAbvPuu1KKF\n7xYpAABn++QTKTDQPtxx8+bNKl06VMWKlVSJEuW1fv160/EcKdVdv3bt2qVHHnlEsbGxqlKlimbP\nns2uXwAcJTLS7qasWSNdd53pNOYwH7vH2AAwKSHBvj5Nniw1bx6pcuWq6eLFaZLukfSVAgP76I8/\nflPRokVNR81WWbrrlyTVq1dPW7duzVQoAMhO77wjhYX5dpECAHCuRYukoCCpbVtp164DsqwSsosU\nSWqnXLkqaN++fWratKnJmI6TaqECAE526ZI0frzEmnIAgBPFx0uvvmrfVPPzk0qXLq3Y2OOSTkoK\nlnRaMTGHFRwcbDip86R6Mj0AONnbb0tt2ki1a5tOAgBAcgsXSqVLS61b218HBwdr6NAXFBBwgwoV\n6qGAgCZ69tlBqlixotmgDpTqGpVU34DnfgEYEhFhr01Zt06qWdN0GvOYj91jbACYEB8v1aplH0bc\nqlXS723btk179+5VjRo1dOONN5oJmMPSOxdTqADwWKNHS/v2SfPnm07iDMzH7jE2AEyYM8f+xePJ\nNgoVAD7hwgWpWjVpwwapenXTaZyB+dg9xgZATouLk2rUkGbPlm65xXQaZ8jSc1QAwKkmT7ZP9qVI\nAQA40dy5UqVKFCmZQUcFgMe5cEGqWlXatMn+J2zMx+4xNgByUmys3U2ZN88+jBg2OioAvN7EiVKH\nDhQpAABn+vBD+/FkipTMoaMCwKOcP29P/lu3SpUrm07jLMzH7jE2AHJKbKx9nVq0SGrWzHQaZ6Gj\nAsCrTZggdexIkQIAcKZZs+wtiSlSMo+OCgCPce6cvXj+p5+k0FDTaZyH+dg9xgZAToiJsbspixdL\nPnI0SrrQUQHgtd56S+rShSIFAOBM778vXX89RUpWoaMCwCOcOWOfPr9jh1Shguk0zsR87B5jAyC7\nXblib/Ly+edSkyam0zgTHRUAXumtt6SuXSlSAADONHOm1KABRUpWoqMCwPFOn7a7Kbt2SeXLm07j\nXMzH7jE2ALLT5ctSlSrS8uVSw4am0zgXHRUAXmfcOOmBByhSAADONH26dMMNFClZjY4KAEc7dUqq\nXVv65RepbFnTaZyN+dg9xgZAdomOtrspX30l1a9vOo2z0VEB4FXGjZN69qRIAQA407RpUvPmFCnZ\ngY4KAMc6eVKqU0fas0cKDjadxvmYj91jbABkh6gou5vyzTdS3bqm0zgfHRUAXmPsWKl3b4oUAIAz\nvfuu1LIlRUp2oaMCwJGOH7cPzdq7VypTxnQaz8B87B5jAyCrRUba3ZQ1a6TrrjOdxjPQUQHgFcaM\nkR5+mCIFAOBMU6dKrVpRpGQnOioAHOfYMalePWnfPqlUKdNpPAfzsXuMDYCsdOmS3U0JD7d3pkTa\n0FEB4PHGjJEeeYQiBQDgTG+/LbVpQ5GS3eioAHCUP/6QGjSwuyklS5pO41mYj91jbABklYgIqWpV\nae1aqWZN02k8Cx0VAB7ttdek/v0pUgAAzjR5snT77RQpOYGOCgDHOHJEatRI+v13qXhx02k8D/Ox\ne4wNgKxw4YJUrZq0YYNUvbrpNJ6HjgoAj/Xaa9Ljj1OkAACcafJk6a67KFJyCh0VAI5w6JB0ww12\nNyUoyHQaz8R87B5jAyCz/v7b7qZs3mzv+IX0o6MCwCONHi098QRFCgDAmSZOlDp0oEjJSXRUABh3\n4IDUtKl5Xmg/AAAgAElEQVS0f79UrJjpNJ6L+dg9xgZAZpw/bz/utWWLVLmy6TSei44KAI8zerQ0\nYABFCgDAmSZMkDp2pEjJaXRUABj1++/STTfZXZUiRUyn8WzMx+4xNgAy6uxZqUYN6aefpNBQ02k8\nGx0VAB5l1CjpqacoUgAAzjR+vNSlC0WKCXRUABizb5908812N6VwYdNpPB/zsXuMDYCMOHPGPthx\nxw6pQgXTaTwfHRUAHuPVV6VBgyhSAADO9OabUrduFCmm0FEBYMTevVJYmHTwoBQYaDqNd2A+do+x\nAZBep0/b3ZSff5bKlTOdxjvQUQHgEV59VXrmGYoUAIAzjRsn9ehBkWISHRUAOW73bqlNG3ttSqFC\nptN4D+Zj9xgbAOlx6pRUu7Z9vQoJMZ3Ge9BRAeB4I0fa3RSKFACAE73xhtSzJ0WKaWnqqISGhqpw\n4cLy9/dXnjx5tGXLln/fgLtUANLh55+l22+3uykFC5pO412Yj91jbACk1cmTUp060p49UnCw6TTe\nJb1zce60vml4eLiCgoIyHAwAJLub8uyzFCkAAGcaO1bq04cixQnSVKhI4k4UgEzbuVP68Udp3jzT\nSQAASO74cfsa9euvppNASuMaFT8/P7Vp00aNGzfWzJkzszsTAC81YoT0/PNSQIDpJAAAJDdmjNS3\nr1S6tOkkkNLYUdmwYYOCg4N15swZtW3bVjVr1lTLli1d3x8xYoTr92FhYQoLC8vqnAA83Pbt0tat\n0kcfmU7iPcLDwxUeHm46hsfgWgUgJceO2dcouilZJ7PXqXRvTzxy5EgVKlRIzzzzjP0GLFAEkAYd\nOkht20oDBphO4r2Yj91jbACk5okn7LO93njDdBLvleXbE0dHR+vSpUuSpKioKH3zzTe6/vrrM54Q\ngM/ZutXuqPTrZzoJAADJHT0qffyxvdkLnCPVR7/++usvdezYUZIUHx+vHj166Lbbbsv2YAC8g2VJ\nL78svfiilD+/6TQAACT36qtS//5SiRKmk+BqnEwPIFt9+aX03HP2+Sl58phO492Yj91jbAC4s22b\n1L69tG+fVKSI6TTeLVvOUQGAjIiJkQYPlt59lyIFAOA8liU99ZT02msUKU6Upu2JASAjJk60T/fl\naVEAgBMtWCDFxdkHPMJ5ePQLQLY4cUKqW1favFmqUsV0Gt/AfOweYwPgvyIjpZo1pcWLpWbNTKfx\nDemdiylUAGSLXr2ksmXtw7OQM5iP3WNsAPzXSy/ZZ6fMm2c6ie9gjQoA4378UVqzxl6YCACA0xw8\nKM2YYW/0AudijQqALJWYaC9MHDvWPjgLAACnefppacgQKSTEdBKkhI4KgCw1Z469w1ePHqaTAACQ\n3DffSHv2SJ98YjoJUsMaFQBZ5uJFe2Hil19KjRubTuN7mI/dY2wASPYOX/Xq2V3/Dh1Mp/E96Z2L\nefQLQJYZNUq6806KFACAM73zjlS+vH3AI5yPjgqALLFvn9SypbR7t1S6tOk0von52D3GBsDp0/bZ\nXmvXSrVqmU7jm9ieGECOsyy7k9KmjfTMM6bT+C7mY/cYGwD9+tmbvEyYYDqJ72J7YgA5bsUK6fBh\nacAA00kAAEjup5+k5culX381nQTpQaECIFNiYqTBg6W335by5jWdBgCApCxLGjjQXkdZtKjpNEgP\nChUAmTJ5sr3T1x13mE4CAEByH30kXb4sPfTQtb+/Y8cOLVz4ifLly6NHHnlYoaGhOZoP7rFGBUCG\nnTwpXX+9fRJ9tWqm04D52D3GBvBNkZH2wvlFi6Sbbkr+/bVr16pdu86Kjn5C/v6RKlhwgX76ab2q\nVq2a82F9AIvpAeSY3r2lMmWkN94wnQQS83FKGBvANw0dKh05Ii1YcO3vN216mzZv7i3JPqU4V67h\neuSRvzV9+pQcy+hLWEwPIEds3ix9+63022+mkwAAkNyhQ9L06dKuXe5fExkZJSnE9XViYoguXDiW\n/eGQJhz4CCDdEhPtHb7GjLG3egQAwGmeeUZ6+mmpbFn3r3nwwU4KCHhW0g5JaxUQ8LoefLBjTkVE\nKuioAEi3uXOlXLmknj1NJwEAILnVq6Wff7YX0qfkuecGKyYmRjNmdFeePHk1YsQotefYesdgjQqA\ndImIsHf5WrpUuuEG02lwNeZj9xgbwHfExUn160uvvSbde6/pNLhaeudiHv0CkC6jR0u3306RAgBw\npvfek0JCpHvuMZ0EmUVHBUCa/f671Ly5tHu3vdsXnIX52D3GBvANZ85ItWtLP/xg/xPOwvbEALLN\nXXdJYWHSs8+aToJrYT52j7EBfMOjj0oFCkiTJplOgmthe2IA2WLlSmn/funzz00nAQAguR07pC++\nkPbtM50EWYU1KgBSFRsrDR5s36HKm9d0GgAAkrIs6amnpFdflYoWNZ0GWYVCBUCqpkyRqlaV7rzT\ndBIAAJL7+GMpKkrq29d0EmQl1qgASNGpU9J110kbN0rVq5tOg5QwH7vH2ADeKyrK3jb/o4+kFi1M\np0FK2J4YQJZ68UXpoYcoUgAAzjR2rNSyJUWKN2IxPQC3tmyRvv6ahYkAAGc6fFh6911p1y7TSZAd\n6KgAuKbERHth4uuvS4ULm04DAEByQ4bYm72UK2c6CbIDHRUA1zR/vr2LSq9eppMAAJDcmjXS9u32\n9QreiUIFQDKXLkkvvGCfmZKLvisAwGHi46WBA6Xx4+0DHuGd+CsIgGRGj5batpVuvNF0EgAAknvv\nPalMGaljR9NJkJ3YnhhAEvv3S82aSb/8IgUHm06D9GA+do+xAbzH2bNS7drS999LdeqYToP0SO9c\nTKECIIn27e0tHp9/3nQSpBfzsXuMDeA9Hn9cypPHPowYniW9czFrVAC4fPWVvRXxkiWmkwAAfFFC\nQoKioqIUGBgoPz+/ZN/fuVP67DO2zfcVrFEBIEmKjZUGDZImTpTy5TOdBgDgaxYs+EiFCgWpePFg\nValSVwcOHEjyfcuyt80fOVIqVsxQSOQoChUAkqS335YqV5buust0EgCAr9mzZ4/69x+kK1fWKz4+\nUkeO9FW7dl2SvOaTT6SICKlfP0MhkeN49AuA/vpLGjNGWr9eukanHQCAbLV161blynWbpOslSZY1\nUIcPP6/o6GgFBAQoOlp69llpwQLJ399sVuScNHVUEhIS1KBBA7Vv3z678wAw4KWXpN69pZo1TScB\nAPiismXLyrK2S7r8v3+zXfnyFVSB/x2S8sYbUvPmUsuWxiLCgDR1VCZPnqzatWvr0qVL2Z0HQA7b\nulVauZKFiQAAc9q0aaO77mqiFSsaKleuukpI+F5z574vPz8/HTkiTZ1qL6SHb0m1o/Lnn39q5cqV\neuSRR9jaEfAyiYn2wsTXXpOKFDGdBgDgq/z8/LRo0Wx9+eW7mjbtHu3YsV6dO3eSJA0ZYp9CX768\n4ZDIcal2VAYPHqw333xTEREROZEHQA5asECKj5f69DGdBADg6/z8/NSqVask/+7776WffpLmzTMU\nCkal2FFZvny5SpUqpQYNGtBNAbzMpUvSCy/YB2blYv8/AIDDxMfbXf+33pL+t1QFPibFjsrGjRu1\nbNkyrVy5UleuXFFERIR69eqluXPnJnndiBEjXL8PCwtTWFhYdmQFkIVef1269VapWTPTSZBR4eHh\nCg8PNx3DY3CtAjzL9OlSyZJSp06mkyCjMnud8rPS2Cr54Ycf9NZbb+nLL79M+gZ+fnRbAA9z4IB0\n443SL79IISGm0yCrMB+7x9gAnuXcOalWLWnNGun6602nQVZJ71ycrgc+/DhgAfAKTz9tL06kSAEA\nONErr0hdu1Kk+Lo0d1TcvgF3qQCP8vXX0v/9n7R7t5Q/v+k0yErMx+4xNoDn2LVLatvW3jY/KMh0\nGmSlbO2oAPBscXHSoEHShAkUKQAA57EseyvikSMpUkChAviUN96QQkOl9u1NJwEAILkPP5QuXpT6\n9zedBE7Ao1+Aj9i+Xbr9dvufHJrlnZiP3WNsAOc7elRq3NheQF+3ruk0yA48+gUgmStXpJ49pYkT\nKVIAAM6TmGgfPjxkCEUK/kWhAviAoUOl2rWlHj1MJwEAILnJk+11lEOGmE4CJ0nxwEcAni88XPro\nI+nnnyV2GAcAOM3evdJrr0mbN0v+/qbTwEnoqABeLCLCbqXPmCGVKGE6DQAAScXF2Y8mv/66VKWK\n6TRwGhbTA17s4Yftu1MzZ5pOgpzAfOweYwM407Bh0k8/ScuX0/X3Bemdi3n0C/BSX3xhP/a1a5fp\nJAAAJLd5szR9urRzJ0UKro1CBfBCp09Ljz0mffKJFBhoOg0AAElFR0u9eklTp0rBwabTwKl49Avw\nMpYldeokVasmjRtnOg1yEvOxe4wN4CwDBkjnz0sLFphOgpzEo1+Aj5s7VzpwQFq0yHQSAACS+/Zb\naelSezdKICV0VAAv8s+pvt9+K9WvbzoNchrzsXuMDeAMf/9tH+g4a5bUtq3pNMhp6Z2LKVQAL5GY\nKLVpY0/8L75oOg1MYD52j7EBnOHBB6WiRe21KfA9PPoF+KgpU6SYGOm550wnAQAgucWLpa1bpR07\nTCeBp6CjAniBvXulm2+WNm2SqlY1nQamMB+7x9gAZp08aT+SvGyZdOONptPAlPTOxZxMD3i4uDh7\ni8fRoylSAADOY1lSv35S//4UKUgfHv0CPNzo0VLJktKjj5pOAgBAcu+/b3dUPvvMdBJ4Gh79AjzY\nli3S3Xfbp/qGhJhOA9OYj91jbAAzDh2yuyjh4VKdOqbTwDQe/QJ8xD+n+r79NkUKAMB5EhKk3r3t\nnSgpUpARFCqAh3rhBalhQ6lbN9NJAABIbvx4yd9fGjTIdBJ4KtaoAB5o9Wr7WV9O9QUAONHPP0tv\nvmlvR5yL2+LIIP7TATzMhQvSww9LH3wgBQWZTgMAQFIxMVLPntK4cVJoqOk08GQspgc8TK9eUqFC\n0rvvmk4Cp2E+do+xAXLOSy9Je/ZIS5dKfn6m08BJOJke8GKffir9+KO9yxcAAE6zcaM0e7a0axdF\nCjKPQgXwEKdOSU88Yd+hKljQdBoAAJKKjLS7/u++K5UqZToNvAGPfgEewLKkDh2kunWl114znQZO\nxXzsHmMDZL/HH5cuX5bmzDGdBE7Fo1+AF/rgA+nPP+1HvwAAyG779+/X+vXrVbx4cd11113y9/dP\n8fVffWX/2rUrhwLCJ9BRARzu0CHphhvsU32vu850GjgZ87F7jA2QdqtWrVLnzj3l53eH/Px+U4MG\nxfXdd18qd+5r398+d87u+M+fL7VqlcNh4VHSOxdTqAAOlpBgT/odOkhDhphOA6djPnaPsQHSrnTp\nyjp9+gNJrSQlqGDBMM2Y8bgeeOCBa77+/vul4GBp4sQcjQkPxKNfgBeZMMHeNWXwYNNJAAC+4u+/\nT0lq/L+v/BUX11CnTp265ms/+sg+3HH27ByLBx/CgY+AQ/3yi/TGG/aixFQeDQYAIMs0adJSuXOP\nlpQgaZ9y516i5s2bJ3vd8ePSwIHSvHlSgQI5HhM+gEIFcKDYWPtU3zfekCpVMp0GAOBLPv30Q9Wt\nu0m5chVQ/vw3aMqU0WratGmS11iW9PDD0pNPSo0aGQoKr8caFcCBhg61OypffMGBWUg75mP3GBsg\n/WJiYpQ3b175XeNC9N57dsd/wwbJzRp7IBkW0wMebuNGqVMne4vH0qVNp4EnYT52j7EBss7+/VLz\n5tL69VKNGqbTwJOkdy7m0S/AQa4+1ZciBQDgNPHx9nVq2DCKFGQ/OiqAgzzxhBQVJX34oekk8ETM\nx+4xNkDWeP116fvvpa+/lnJxuxvpxPbEgIdatUpasYJTfQEAzrRjhzRpkvTTTxQpyBkUKoADnD8v\nPfKI3UkpWtR0GgAAkrpyxd6NcsIEqXx502ngK3j0C3CA7t2lUqWkyZNNJ4EnYz52j7EBMufZZ6XD\nh6XFi9mNEhmX5Y9+XblyRbfccotiYmIUGxure+65R2PGjMlUSAD/WrRI2rlT2r7ddBIAAJJbu1Za\nuNB+NJkiBTkp1UIlf/78+v777xUQEKD4+Hi1aNFC69evV4sWLXIiH+DV/jnVd/lyTvUFADhPRITU\nu7c0fbpUooTpNPA1aVoKFRAQIEmKjY1VQkKCgoKCsjUU4AssS+rbV3r8calJE9NpAABI7umnpTZt\npLvvNp0EvihNhUpiYqLq16+v0qVLq1WrVqpdu3Z25wK83vTp0rlz9in0AAA4zZdfSt99Zy+gB0xI\n065fuXLl0s6dO3Xx4kXdfvvtCg8PV1hYmOv7I0aMcP0+LCwsyfcAJLd1q/TKK/Zzv3nymE4DTxUe\nHq7w8HDTMTwG1yog7Q4ckPr1sxfPBwaaTgNPldnrVLp3/Ro1apQKFCigIUOG2G/ATipAuhw7JjVr\nJr3zjnTPPabTwJswH7vH2ABpd+GCfZ0aOFB67DHTaeBN0jsXp/ro19mzZ3XhwgVJ0uXLl/Xtt9+q\nQYMGGU8I+LDISKlDB3vyp0gBADhNXJzUtat0220UKTAv1Ue/Tp48qd69eysxMVGJiYnq2bOnWrdu\nnRPZAK+SmCg9+KDUsKH0v4YkAACOYVn2jbTcuaXx402nATjwEcgxzz8vbd4sffONlDev6TTwRszH\n7jE2QOreftve6GXjRqlwYdNp4I2y/MBHAJk3e7b02WfSpk0UKQAA5/nqK+n11ylS4Cx0VIBs9sMP\n9vO+P/wg1axpOg28GfOxe4wN4N7u3dKtt0qffy7ddJPpNPBmWb6YHkDGHTggdesmLVhAkQIAcJ7T\np6X27e2zUihS4DQUKkA2+ftv+yTfESPsU30BAHCSK1ekjh2lHj3szV4Ap+HRLyAbxMVJd94p1akj\nTZpkOg18BfOxe4wNkJRlSb162cXKxx9Lubh1jRzAYnrAMMuSnnrKPnGe7R0BAE40Zoy0b5+9fpIi\nBU5FoQJksbffltats3dO8fc3nQYAgKSWLJGmTbN3ogwIMJ0GcI9Hv4As9NVX0sMPSz/+KIWGmk4D\nX8N87B5jA9i2bZPatbPP9GrQwHQa+Boe/QIM2b1b6t1bWrqUIgUA4Dx//inde680cyZFCjwDTyUC\nWeCf7R0nTpSaNzedBgCApKKipA4dpAED7GIF8AQ8+gVk0pUrUuvW9mFZo0aZTgNfxnzsHmMDX5aY\nKHXuLBUtKs2aJfn5mU4EX5XeuZhCBcgEtneEkzAfu8fYwJe9+KK0YYO0erWUN6/pNPBlrFEBctDr\nr7O9IwDAuebMkT75RNq8mSIFnodCBcigJUuk6dPtyZ/tHQEATrNunfTcc/bNtBIlTKcB0o9CBciA\nbdukxx+3t3cMDjadBgCApA4elO67T5o/X6pVy3QaIGN4WAVIJ7Z3BAA42YUL9k6Uw4ZJt91mOg2Q\ncSymB9IhKkpq2VK6/367nQ44CfOxe4wNfEV8vHTXXVKNGtKUKabTAEmx6xeQTf7Z3rFYMemDD9je\nEc7DfOweYwNf8eST0oED0vLlUm4e8IfDsOsXkE1eekk6f97ehpgiBQDgNFOnSt9/L23cSJEC78B/\nxkAazJlj7/K1aRPbOwIAnGfVKum11+wipUgR02mArMGjX0Aq1q6VunSxt3dk5xQ4GfOxe4wNvNne\nvVJYmPTZZ1KLFqbTAO6ldy5m1y8gBQcPSl27SgsWUKQAAHLehg0bdPPNd6thw1aaNGlqsr/knTkj\n3X23NH48RQq8D49+AW5cuGBP/sOHS23bmk4DAPA1O3fu1G233avo6LckBeu3357V5cuX9eKLz0qS\nYmKkjh2l7t2lnj3NZgWyA49+AdcQHy/deadUsybbO8JzMB+7x9jAEw0Z8oLGj88naeT//s02lSvX\nW8eO7ZFlSX362Nvmf/KJlItnZOAB2PULyAIDB9qT/oQJppMAAHxVnjy5lSvXZSUm/vNvLiv3/7bz\nGjtW2r3bXkdJkQJvRUcF+I+pU6X33mPnFHge5mP3GBt4ogMHDqhhw+aKjBwoywpWQMAoTZkyTMWK\nPaSBA+2dKMuWNZ0SSDsOfAQyYdUq6aGH7CKlUiXTaYD0YT52j7GBp9q3b59ef32iIiKi1KtXZ1Ws\n2FF33GFfrxo1Mp0OSB8KFSCD9uyRWrVie0d4LuZj9xgbeIPjx6WmTaXJk6VOnUynAdKP7YmBDDhz\nRmrfnu0dAQDOFBUldeggPfEERQp8Bx0V+LyYGKl1a+mWW+xTfQFPxXzsHmMDT5aYKN13n1SokDRn\njuTnZzoRkDE8+gWkw+XL0v33S3nzSh9/zM4p8GzMx+4xNvBUCQnS449Lv/4qrV4t5ctnOhGQcTz6\nBaTRxYtSu3ZSQIB98jxFCgDASWJipG7dpEOHpJUrKVLge/irGXzS6dP2wvnataX58+2OCgAAThEZ\nKd19t2RZ0ooVUmCg6URAzqNQgc85etReMH/33dI770j+/qYTAQDwr3Pn7LWTFSvap87TSYGvolCB\nT9m7V2rZUvq//5NefZUFiQAAZ/nzT/s6FRYmzZzJzTT4NgoV+IwtW6Rbb7V39ho40HQaAACS+v13\nu+Pfp4/0xhvcTANymw4A5ITVq6Xu3aVZs+zzUgAAcJLt26W77pJGj5b69jWdBnAGChV4vU8/tbd2\n/PRT6eabTacBACCpH36wz0mZNo3DHIGr8egXvNrMmdKAAdLXX1OkAACcZ9kyu0hZtIgiBfgvOirw\nWm+8Yd+d+uEHqVo102kAAEhq7lzpuefs7YebNDGdBnCeVDsqx44dU6tWrVSnTh1dd911mjJlSk7k\nAjLMsuyJf948af16ihQAgPNMnCi98ooUHk6RArjjZ6Vyjv2pU6d06tQp1a9fX5GRkWrUqJGWLl2q\nWrVq2W/g56dU3gLIMfHx0qOPSnv22Kf4BgWZTgTkHOZj9xgbOIVl2QXKkiXSN99IFSqYTgTknPTO\nxak++lWmTBmVKVNGklSoUCHVqlVLJ06ccBUqgFNcuWLv7BUVZe/yVaiQ6UQAAPwrIcE+x2vbNmnd\nOqlkSdOJAGdL1xqVI0eOaMeOHbrxxhuzKw+QIRER0r33SiVKSF9+ySm+AABniY2VevaUzpyRvvtO\nKlzYdCLA+dJcqERGRqpLly6aPHmyCv3nVvWIESNcvw8LC1NYWFhW5QNSdeaM1K6d1Lix9M47nOIL\n3xEeHq7w8HDTMTwG1yqYEhkpde4sBQTYjyXnz286EZAzMnudSnWNiiTFxcXp7rvvVrt27TRo0KCk\nb8BzvzDojz+k226zLwCjR3OKL3wb87F7jA1MOX/ePsixVi1pxgwpN/utwoeldy5Oddcvy7LUt29f\n1a5dO1mRApi0b5/UsqXUv7/02msUKQAAZzl+3D7D66abpA8+oEgB0ivVjsr69et18803q27duvL7\n398Ex4wZozvuuMN+A+5SwYBt26S775bGjpX69DGdBnAG5mP3GBvktP377Y7/o49Kzz/PzTRASv9c\nnKZHv7LyBwKZ9d130v3326fO33OP6TSAczAfu8fYICft3Cndeac0cqTUr5/pNIBzZPn2xICTfP65\nfXdq8WLplltMpwEAIKl16+x1k+++K3XpYjoN4NlSXaMCOMWsWdITT0irVlGkAACcZ/lyu0hZuJAi\nBcgKdFTgEd56S5o6VfrhB6l6ddNpAABIav58acgQu1i54QbTaQDvQKECR7Ms6cUXpWXLpPXrpXLl\nTCcCACCpKVPsG2rffSfVrm06DeA9KFTgWAkJ0mOPSbt2SWvX2qfOAwDgFJYljRghffSRvTalYkXT\niQDvQqECR4qJkXr0kC5ckNaskQIDTScCAOBfiYnSU09JGzfaHf9SpUwnArwPhQocJzJS6thRKlJE\nWrFCypfPdCIAAP4VG2uf4XX8uPT99/b1CkDWY9cvOMrZs1Lr1lJoqPTxxxQpAABniYqyz/CKirJ3\noaRIAbIPhQoc4+efpZYtpVtvlWbMkPz9TScCAOBfhw9LbdpIpUtLn34qFShgOhHg3ShUYFxsrH16\nb+vW0nPPSWPGSH5+plMBAGBLTLS3yG/SROrUyT7XKzcPzwPZjv+bwajt26WHHrK3Hd6xg+2HAQDO\nsn+/1LevvRPlhg1SjRqmEwG+g44KjIiJkV5+WbrjDumZZ+wDsihSAABOkZAgTZggNWtmd1HWrqVI\nAXIaHRXkuK1b7S5K1ar2GSnBwaYTAQDwr337pIcflvLkkTZtsq9XAHIeHRXkmCtXpOefl+6+Wxo6\nVPr8c4oUAIBzxMdL48ZJLVpIDzxgbz1MkQKYQ0cFOWLjRvvu1PXX27t7lS5tOhEAAP/avdu+TgUG\n2p3/SpVMJwJARwXZKjpaevppqXNnafRoafFiihQAgHPExdnXp1atpEcekVavpkgBnIKOCrLN2rX2\nTilNmki//CKVKGE6EQAA/9q1y14zWaqU9NNPUoUKphMBuBodFWS5yEhpwACpe3fprbekhQspUgAA\nzhEbKw0fLrVta1+vvvqKIgVwIjoqyFLffWe3zlu2tLsoQUGmEwEA8K+ffrK7KBUr2ud3lS1rOhEA\ndyhUkCUiIuxT5VeskKZNk+66y3QiAAD+FRMjjRwpffCBNH681KOH5OeXvvc4efKkfv/9d4WGhqpi\nxYrZExSAC49+IdO+/trezSs+3t41hSIFAOAkmzdLDRrY56Ps2iU9+GD6i5TFiz9V1arX6557hqpm\nzUaaNGlq9oQF4OJnWZaVqTfw81Mm3wIe6sIF+1T5NWukGTOk224znQjwbczH7jE2vunyZWnYMGne\nPGnyZKlr1/QXKJIUGRmpUqXK6/Ll7yQ1kPSHChRopF9+2aQqVapkdWzAa6V3LqajggxZvly67jop\nXz57LQpFCgDASTZskOrXl/74w75OdeuWsSJFkk6cOCF//yDZRYokVVDevNfp0KFDWRUXwDWwRgXp\ncv68NHCgfYDjvHn2vvMAADhFVJQ0dKj0ySfS1KlSp06Zf89y5cpJipD0g6RbJO1RbOwvqlGjRubf\nHOQ0d6cAAA6iSURBVIBbdFSQZkuX2l2UoCD7dHmKFACAk/zwg1SvnnT2rN1FyYoiRZICAgL0+ecf\nqVChLgoMrKn8+Vto+vTJqsCexkC2Yo0KUnX2rL3P/E8/SbNmSS1amE4E4FqYj91jbLxbZKT0/PPS\nF19I770ntW+fXT8nUkePHlXZsmVVtGjR7PkhgBdjjQqy1OLF9o5eZctKO3dSpAAAnGX1avs6FR1t\nd1Gyq0iRpEKFCqlOnToUKUAOYY0KrunUKenJJ6U9e6TPP5eaNjWdCACAf124YJ/ftWqVNH261K6d\n6UQAshodFbhYlvT991L37lLNmlK1avapvRQpAAAnsCxp82ZLDRrsUFDQRc2bt1B9+07VHXfwWB/g\njeioQH/9JX34oTRzplSggNSvn/Tuu1KxYqaTAQBgd08WLLDP7Dp+/KIiItbLsgrqypXrNG7c/apY\nsbD69OllOiaALEZHxUclJkrffCN16WJ3T377zd5ueNcue+E8RQoAwCTLss9C6d1bCg2V1q2TJkyQ\nKlfupri4KpKqS6qr6OiXtWjRl4bTAsgOdFR8zIkT9s5dH3xgFyP9+9tfFy5sOhkAANK5c9LcuXaX\nPyHBvk699ZZUsqT9/aCgQEmHXa/PleuwihcvYiYsgGzF9sQ+ICHBXmw4Y4Z9R6prV/vxrkaNTCcD\nkJWYj91jbJzNsuwzUGbMkFautHfu6tdPatky+WnyO3fuVIsWbXXlyoPy84tVQMCn2rZtnapVq2Ym\nPIA0S+9cTKHixf74w+6czJplby/cr5/UrZtUqJDpZACyA/Oxe4yNM50+Lc2ZI73/vpQvn32devBB\n+2DhlOzfv18ff/yJcuf2V/fu3VWxYsUcyQsgcyhUfFxcnLRihX1XavNm6YEH7Im/bl3TyQBkN+Zj\n9xgb50hMlNassa9Tq1dLHTva16mmTZN3TwB4l/TOxaxR8RKH/r+9+4+pqv7jOP7CRBlSBiWgXJgK\n8vVeRNDZWLYKo0vfucHSWFMzGGB/hG7Vt/VPf/VP/vi6lb/a97vxHYpfm1a6KRrStH1pOCJrZJnR\nSOX2RTGWBTRDQeh8/zhf8EeCkvdyDuc8HxvbvVx27vsc7z5vX/ecz/mcNb+R2rFDSk42B/29e6XI\nSKsrAwDAnCO5fbt5pn/yZHPuyb/+ZT4GgFshqIxhvb3S/v3mhMMTJ6Tnnze/nfL5rK4MAABzjuRH\nH5lnTz75xJwj+f775hxJzp4AuB2CyhjU3GyGk507pbQ08+zJkiVSRITVlQEAYM6RrKgwf6ZONfvU\nrl3MkQQwMgSVMeLKFWnfPjOgfPedeV/5Y8fM1eMBALDawBzJ8nKpoUFavlw6eFDKyLC6MgBjFUHF\nhgxD6ugw552cPSvV15sr8s6fL61ZI+XnSxMmWF0lAMDNurqklhazT33+uVRZKc2caZ49+eAD5kgC\nuHu3DSolJSX68MMPFRsbq5MnT45GTa7Q0yP98MO1MDIw2A/8SOaAP3Omeceu48elGTOsrRkA4B5X\nr0qtrTf2put7VU+P2aNmzDDnRjJHEkCw3fb2xHV1dYqKilJhYeEtgwq3fLw1w5Da228dQlpazNc8\nnmthZGCwH3gcHc1EQwAjw3g8NI7NHxmGuQr8UF+YtbWZ80uu703X96opU+hTAEYmJOuoBAIB5eXl\nEVRu8ttvUiBw48A+MNi3tJinvW8VQmbONEPKeC68AxBEbh6Pb8etx+bKlWt96lZfmt1zz4296fp+\nlZTEZcYAgot1VELorbekxsZrg3xXlzR9+o0D+6JF157fe6/VFQMA3KSiwrwN8EAo+eknM3BcH0Ie\nfvja4+hoqysGgKEFJai88cYbg4+zs7OVnZ0djM3azuTJUm7utVAydao0bpzVVQFwq9raWtXW1lpd\nxpjhhl4VFSU9/rhUXGz2qYQE86wJAFjhbvsUl34BgEMwHg+NYwMA1hvpWMz5AAAAAAC2c9ugsnz5\nci1cuFDNzc1KTEzU9u3bR6MuAABwh3p6elRW9jdNm/YXeb1ZOnLkiNUlAcBdu6NLv4bdAKfTAcAW\nGI+H5vRjU1q6Rrt3n9Xly3+XdFaRkatUX39EGSwLD8BGuPQLAACX2bt3ry5f/qekOZLy1dNTpEOH\nDlldFgDcFYIKAABjXETEJEkXBp+PH39BkyZNCvn7NjU16ZFH/iqPx6tnny1WZ2dnyN8TgHtw6RcA\nOATj8dCcfmwqK/+tsrLX1d29WuHhZ/Tgg//RN98cV0xMTMje8+LFi0pNzVBn5+syjMc1YcIWzZ9/\nVp9+ejRk7wlgbGPBRwAAXKao6Hl5PNNUVXVYMTFJWr26IaQhRZKOHTumvr65MozVkqTe3n+osTFa\nHR0dimYlSQBBQFABAMABcnJylJOTM2rvFxkZKem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       "text": [
        "<matplotlib.figure.Figure at 0x364fb10>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now let's try to fit some functions."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.optimize import curve_fit\n",
      "\n",
      "def poly_7(x, a, b, c, d, e, f, g, h):\n",
      "    return h*x**7+g*x**6 + f*x**5 + e*x**4 + d*x**3 + c*x**2 + b*x + a\n",
      "\n",
      "popt, pcov = curve_fit(poly_7, X, Y2, p0=(10.0, 0.2, 0.1, 0.0,0.0,0.0,0.0,0.0))\n",
      "\n",
      "a = popt[0]\n",
      "b = popt[1]\n",
      "c = popt[2]\n",
      "d = popt[3]\n",
      "e = popt[4] \n",
      "f = popt[5]\n",
      "g = popt[6]\n",
      "h = popt[7]\n",
      "residuals = Y2 - poly_7(X, a,b,c,d,e,f,g,h)\n",
      "fres = np.linalg.norm(residuals)\n",
      "print \"7-degree polynomial error: \",fres\n",
      "\n",
      "fig, (ax1, ax2, ax3) = plt.subplots(1, 3, sharex=True, sharey=True)\n",
      "ax1.scatter(X,Y2)\n",
      "xlin = np.linspace(0,9, 50)\n",
      "ax1.plot(xlin, poly_7(xlin,a,b,c,d,e,f,g,h))\n",
      "ax1.set_title(\"Overfit - high variance\")\n",
      "\n",
      "def poly_1(x, a, b):\n",
      "    return a*x + b\n",
      "\n",
      "popt, pcov = curve_fit(poly_1, X, Y2, p0=(10.0, 0.2))\n",
      "\n",
      "a = popt[0]\n",
      "b = popt[1]\n",
      "residuals = Y2 - poly_1(X, a,b)\n",
      "fres = np.linalg.norm(residuals)\n",
      "print \"linear approximation error: \", fres\n",
      "\n",
      "ax2.scatter(X,Y2)\n",
      "ax2.plot(xlin, poly_1(xlin, a, b))\n",
      "ax2.set_title(\"Underfit - high bias\")\n",
      "\n",
      "def poly_2(x, a, b, c):\n",
      "    return a*x**2 + b*x + c\n",
      "\n",
      "popt, pcov = curve_fit(poly_2, X, Y2, p0=(10.0, 0.2, 0.0))\n",
      "\n",
      "a = popt[0]\n",
      "b = popt[1]\n",
      "c = popt[2]\n",
      "residuals = Y2 - poly_2(X, a,b,c)\n",
      "fres = np.linalg.norm(residuals)\n",
      "print \"error of the correct model: \", fres\n",
      "\n",
      "ax3.scatter(X,Y2)\n",
      "ax3.plot(xlin, poly_2(xlin, a,b,c))\n",
      "ax3.set_title(\"Correct model\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "7-degree polynomial error:  0.161557406365\n",
        "linear approximation error: "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        " 1.48952709068\n",
        "error of the correct model:  0.568171379783\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "<matplotlib.text.Text at 0x3e8d410>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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g4EFttQ8iIhnmzgWcnYH+/WUnISI9WrwY6NdP371hHuckOwCRPfjnHyAoCPDy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\niwYGCFbffvutpHBJ5VdxI1St2g06ccK7n088HunGG6VFi6Q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       "text": [
        "<matplotlib.figure.Figure at 0x3865750>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can see that the higher the polynomial order is, the smallest the error. However if we were to add few more points without recalculating the polynomial coefficients, the incorrect models would fail miserably."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "So how to overcome this problem?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In order to give preference to some particular solution, we are can introduce a regularization parameter $\\lambda$.\n",
      "\n",
      "Now the least squares regularization turns into Tikhonov regularization (Regularized Least Squares)\n",
      "\n",
      "$$\\min_{f \\in H} \\sum_{i = 1}^n\\left(f(X_i) - Y_i\\right)^2 + \\lambda ~ ||{f}||^2_K$$\n",
      "\n",
      "The regularization parameter $\\lambda$ determines how much we distrust our data.\n",
      "\n",
      "Too low $\\lambda$ will leave noise in the measurements and therefore modelled function may not extrapolate well.\n",
      "\n",
      "Too high $\\lambda$ may eventually ignore the data."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "How to choose a correct $\\lambda$? "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To validate our data we can only use our own measurements. So we can split them into a training set and testing set, interpolate the function obtained from processing the training set and find how well we can predict the test samples. This procedure is called **cross validation**.\n",
      "\n",
      "Repeating the procedure for different $\\lambda$, we can find which regularization parameter gives the best results (smallest error). \n",
      "\n",
      "Very high $\\lambda$ suggests that our measurements are too noisy or our model may be invalid.\n",
      "We would expect $\\lambda$ to be somewhere between 0 and 1."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The role of Scikit-learn"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In SciPy (scikit-learn) we have some handful procedures for cross-validation. \n",
      "For example - there are index generators, which enable us to perform:\n",
      "* Leave One Out cross validation\n",
      "* Leave P Out cross validation\n",
      "* K-Fold cross validation\n",
      "* Shuffle split iterated cross validation\n",
      "\n",
      "\n",
      "Some examples:"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Leave One Out"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.cross_validation import LeaveOneOut\n",
      "\n",
      "#for example\n",
      "#X = np.array([[0., 0.], [1., 1.], [-1., -1.], [2., 2.]])\n",
      "Y = np.array([0, 1, 0, 1])\n",
      "\n",
      "loo = LeaveOneOut(len(Y), indices=True)\n",
      "print(loo)\n",
      "print \"train  test\"\n",
      "for train_ind, test_ind in loo:\n",
      "    #do something with the indices and data\n",
      "    print(\"%s %s\" % (train_ind, test_ind))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "sklearn.cross_validation.LeaveOneOut(n=4)\n",
        "train  test\n",
        "[1 2 3] [0]\n",
        "[0 2 3] [1]\n",
        "[0 1 3] [2]\n",
        "[0 1 2] [3]\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Leave P Out"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Leave P Out divides the samples into testing subset of size p and training set of size n-p"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.cross_validation import LeavePOut\n",
      "\n",
      "lpo = LeavePOut(n=len(Y), p=2, indices=True)\n",
      "print(lpo)\n",
      "print \"train test\"\n",
      "for train_ind, test_ind in lpo:\n",
      "    #do something with the indices and data\n",
      "    print(\"%s %s\" % (train_ind, test_ind))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "sklearn.cross_validation.LeavePOut(n=4, p=2)\n",
        "train test\n",
        "[2 3] [0 1]\n",
        "[1 3] [0 2]\n",
        "[1 2] [0 3]\n",
        "[0 3] [1 2]\n",
        "[0 2] [1 3]\n",
        "[0 1] [2 3]\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "K-Fold"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "An advantage of K-Fold cross validation is reduction of the number of iterations required to complete the task."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.cross_validation import KFold\n",
      "\n",
      "#for example:\n",
      "#X = np.array([[0., 0.], [1., 1.], [-1., -1.], [2., 2.]])\n",
      "Y = np.array([0, 1, 0, 1])\n",
      "\n",
      "kf = KFold(len(Y), n_folds=2, indices=True)\n",
      "print(kf)\n",
      "print \"train test\"\n",
      "for train_ind, test_ind in kf:\n",
      "    print(\"%s %s\" % (train_ind, test_ind))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "sklearn.cross_validation.KFold(n=4, n_folds=2)\n",
        "train test\n",
        "[2 3] [0 1]\n",
        "[0 1] [2 3]\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Shuffle Split"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sklearn.cross_validation import ShuffleSplit\n",
      "Y = np.array([0, 1, 0, 1])\n",
      "\n",
      "ss = ShuffleSplit(len(Y), n_iter=2, test_size = 0.5, indices=True)\n",
      "print(ss)\n",
      "print \"train test\"\n",
      "for train_ind, test_ind in ss:\n",
      "    print(\"%s %s\" % (train_ind, test_ind))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "ShuffleSplit(4, n_iter=2, test_size=0.5, indices=True, random_state=None)\n",
        "train test\n",
        "[3 2] [1 0]\n",
        "[0 1] [2 3]\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And more.\n",
      "\n",
      "We can recalculate our model with different $\\lambda$ to find out which model works best."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Cross validation for determining model - example"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Using the previous functions we can check how the models extrapolate. This is pure cross validation and curve fitting. No regularization parameter was introduced in this example."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def error_7(x_train, y_train, x_test, y_test):\n",
      "    popt, pcov = curve_fit(poly_7, x_train, y_train, p0=(10.0, 0.2, 0.1, 0.0,0.0,0.0,0.0,0.0))\n",
      "\n",
      "    a = popt[0]\n",
      "    b = popt[1]\n",
      "    c = popt[2]\n",
      "    d = popt[3]\n",
      "    e = popt[4] \n",
      "    f = popt[5]\n",
      "    g = popt[6]\n",
      "    h = popt[7]\n",
      "\n",
      "    y_estimated = poly_7(x_test, a,b,c,d,e,f,g,h) \n",
      "    err = np.linalg.norm(y_test - y_estimated)\n",
      "    return err\n",
      "\n",
      "X = np.array(range(0,10))\n",
      "Y = 0.04*X**2 + 1.0\n",
      "Y2 = Y + np.random.random_sample(size(X))-0.5\n",
      "\n",
      "loo = LeaveOneOut(len(Y), indices=True)\n",
      "error = 0.0\n",
      "for ind_train, ind_test in loo:\n",
      "    error += error_7(X[ind_train], Y2[ind_train], X[ind_test], Y2[ind_test])\n",
      "\n",
      "print \"Total error:\", error"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Total error: 27.2241098604\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we can see how badly this model extrapolates."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def error_1(x_train, y_train, x_test, y_test):\n",
      "    popt, pcov = curve_fit(poly_1, x_train, y_train, p0=(10.0, 0.2))\n",
      "\n",
      "    a = popt[0]\n",
      "    b = popt[1]\n",
      "\n",
      "    y_estimated = poly_1(x_test, a,b) \n",
      "    err = np.linalg.norm(y_test - y_estimated)\n",
      "    return err\n",
      "\n",
      "error = 0.0\n",
      "for ind_train, ind_test in loo:\n",
      "    error += error_1(X[ind_train], Y2[ind_train], X[ind_test], Y2[ind_test])\n",
      "\n",
      "print \"Total error:\", error"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Total error: 3.68014736226\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now this looks better."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def error_2(x_train, y_train, x_test, y_test):\n",
      "    popt, pcov = curve_fit(poly_2, x_train, y_train, p0=(10.0, 0.2,0.0))\n",
      "\n",
      "    a = popt[0]\n",
      "    b = popt[1]\n",
      "    c = popt[2]\n",
      "\n",
      "    y_estimated = poly_2(x_test, a,b,c) \n",
      "    err = np.linalg.norm(y_test - y_estimated)\n",
      "    return err\n",
      "\n",
      "error = 0.0\n",
      "for ind_train, ind_test in loo:\n",
      "    error += error_2(X[ind_train], Y2[ind_train], X[ind_test], Y2[ind_test])\n",
      "\n",
      "print \"Total error:\", error"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Total error: 1.90230633991\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we can see that the 2nd degree polynomial extrapolates best."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is how we manage to minimize the error connected with too high variance."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Visualization of $\\lambda$ parameter influence"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Lets now solve a linear equation $Ax = b$\n",
      "\n",
      "$x = (A+\\lambda I)^{-1} b$ "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = np.array([\n",
      "            [1.0, 0.5, 0.2, 0.0, 0.0],\n",
      "            [0.5, 1.0, 0.5, 0.2, 0.0],\n",
      "            [0.2, 0.5, 1.0, 0.5, 0.2],\n",
      "            [0.0, 0.2, 0.5, 1.0, 0.5],\n",
      "            [0.0, 0.0, 0.2, 0.5, 1.0],\n",
      "          ])\n",
      "b = np.array([[1],[2],[3],[4],[5]])\n",
      "fig, (ax1, ax2) = plt.subplots(1,2)\n",
      "ax1.imshow(A, interpolation=\"none\")\n",
      "ax1.set_title(\"A\")\n",
      "ax2.imshow(b, interpolation=\"none\")\n",
      "ax2.set_title(\"b\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 11,
       "text": [
        "<matplotlib.text.Text at 0x474ab10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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fZwHQm5rGS6Om8dKIeCc077333owXAZyp11vnR44ciaNHj0ZExJtvvhlPPvlk1NfXJx8G\nAMDg1usVzVdffTUWLlwYJ0+ejJMnT8att94a11577UBsAwDoF8ePH4+GhoZ46623orOzM770pS/F\nAw88kPWsstdraF5++eXxhz/8YSC2AAAk4e0as+E3AwEAQ4K3axx4QhMAGBK8XePAO6+fOgcAGKx6\nfbvGiGj+5bufN06IaPz0wG4cDNr2RbTt79uxQhMAGFK6fbvGiGi+IZtNg0njp88M8Pt+1f2xbp0D\nAGXP2zVmwxVNAKDsebvGbAhNAKDsebvGbLh1DgBAEkITAIAkhCYAAEkITQAAkhCaAAAkITQBAEhC\naAIAkITQBAAgCaEJAEASQhMAgCSEJgAASQhNAACSEJoAACQhNAEASEJoAgCQhNAEACAJoQkAQBJC\nEwCAJIQmAABJCE0AAJIQmgAAJCE0AQBIQmgCAJBERb88S7HYL09Tbopb78t6QslaPmdF1hNK1t0n\nsl5Quj7wAf/edOvk8qwXAJzFFU0AAJIQmgAAJCE0AQBIQmgCAJCE0AQAIAmhCQBAEkITAIAkhCYA\nAEkITQAAkhCaAAAk0T+/ghIAoAzc982sF5QXVzQBAEhCaAIAkITQBAAgCaEJAEASQhMAgCSEJgAA\nSQhNAACSEJoAACQhNAEASEJoAgCQhNAEACAJoQkAQBJCEwCAJIQmAABJCE0AAJIQmgAAJCE0AQBI\nQmgCAJCE0AQAIAmhCQBAEn0KzUKhEPX19TF79uzUewAAKBN9Cs01a9ZEbW1t5HK51HsAACgTvYbm\noUOHYtu2bbFkyZIoFosDsQkAgDLQa2guW7YsVq1aFcOGeTknAAB9V9HTg1u3bo1Ro0ZFfX19tLW1\n9XDkex+rOfUBQErtxYj2U583NzdnuATg3Hq8TPnss8/Gli1bYty4cTF//vx46qmnYsGCBec4svE9\nHzX9vRGAc6jJRTSe+hCaQCnqMTRXrFgR+Xw+Dhw4EBs3bozp06dHS0vLQG0DAGAQO68XXvqpcwAA\n+qrH12i+V0NDQzQ0NKTcAgBAGfGj5AAAJCE0AQBIQmgCAJCE0AQAIAmhCQBAEkITABgyCoVC1NfX\nx+zZs7OeMiQITQBgyFizZk3U1tZ6b/ABIjQBgCHh0KFDsW3btliyZEkUi8Ws5wwJQhMAGBKWLVsW\nq1atimHD5M9A6fNvBgIAGKy2bt0ao0aNivr6+mhra+v2uO3vudBZExHj3GE/y4FiRHsfjxWaAEDZ\ne/bZZ2PLli2xbdu2OH78ePzjH/+IBQsWREtLyxnHTROWvRqXixj3nq939PAqBNeOAYCyt2LFisjn\n83HgwIHYuHFjTJ8+/azIpP8JTQBgyPFT5wPDrXMAYEhpaGiIhoaGrGcMCa5oAgCQhNAEACAJoQkA\nQBJCEwCAJIQmAABJCE0AAJIQmgAAJCE0AQBIQmgCAJCE0AQAIAmhCQBAEkITAIAkyjA027MeULqK\nB7JeULKKR9qynlCy2tteyXpCyWovZr0AoLQJzSGlPesBpet/27JeULJe2XEw6wklqz3rAQAlrgxD\nEwCAUiA0AQBIIlcsFi/oVUaNjY2xY8eO/toDwPvU0NAQbW1tWc+AQSuXy0VzLusVg09zMaK7nKy4\n0Cf3lxoAAOfi1jkAAEkITQAAkhCaAAAkUVah2draGhMnTowJEybEypUrs55TMhYtWhRVVVVx+eWX\nZz2l5OTz+Zg2bVpMnjw5LrvssnjooYeynlQyjh8/HlOnTo26urqora2Nu+66K+tJJaVQKER9fX3M\nnj076ykAJatsQrNQKMTSpUujtbU19uzZExs2bIi9e/dmPask3HbbbdHa2pr1jJJUWVkZq1evjpdf\nfjmee+65ePjhh/17c8pFF10U27dvjxdffDF2794d27dvj2eeeSbrWSVjzZo1UVtbG7mcH1EF6E7Z\nhOauXbti/PjxUVNTE5WVlTFv3rzYvHlz1rNKwjXXXBMjRozIekZJGj16dNTV1UVExPDhw2PSpElx\n+PDhjFeVjosvvjgiIjo7O6NQKMTIkSMzXlQaDh06FNu2bYslS5Z0+5YeAJRRaHZ0dMTYsWO7vq6u\nro6Ojo4MFzHYtLe3xwsvvBBTp07NekrJOHnyZNTV1UVVVVVMmzYtamtrs55UEpYtWxarVq2KYcPK\n5q9QgCTK5m9Jt6+4EMeOHYu5c+fGmjVrYvjw4VnPKRnDhg2LF198MQ4dOhRPP/20982NiK1bt8ao\nUaOivr7e1UyAXpRNaI4ZMyby+XzX1/l8PqqrqzNcxGBx4sSJuPHGG+OWW26JOXPmZD2nJF1yySVx\nww03xPPPP5/1lMw9++yzsWXLlhg3blzMnz8/nnrqqViwYEHWswBKUtmE5pQpU2L//v3R3t4enZ2d\nsWnTpmhqasp6FiWuWCzG4sWLo7a2Nu68886s55SUI0eOxNGjRyMi4s0334wnn3wy6uvrM16VvRUr\nVkQ+n48DBw7Exo0bY/r06dHS0pL1LICSVDahWVFREWvXro2ZM2dGbW1t3HzzzTFp0qSsZ5WE+fPn\nx1VXXRX79u2LsWPHxqOPPpr1pJKxc+fOWL9+fWzfvj3q6+ujvr7eT+if8uqrr8b06dOjrq4upk6d\nGrNnz45rr70261klx8t2ALqXK3qREQBA5HK5aPb/Hc9bczG6fc162VzRBABI6cAFXpq7kPMH6/cW\nmgAAfdCe4fmD9XsLTQAAkqjIegAAQKn42H/+Z7ePfejw4fjYxz/+vp/7Qs4v6e/9+z90+5AfBgIA\niIjGxsbYsWNH1jMGnYaGhm5/oYfQBAAgCa/RBAAgCaEJAEASQhMAgCSEJgDAObzxxhsxY8aM+PSn\nPx3XXXddHD16tNtjC4VC1NfXx+zZs/t8bj6fj2nTpsXkyZPjsssui4ceeihaW1tj4sSJMWHChFi5\ncuU5v9e3vvWtmDBhQlx55ZXxwgsvnPFYb+f/7Gc/iyuvvDKuuOKK+PznPx+7d+/u87mn/e53v4uK\nior4xS9+0e0xpwlNAIBzePDBB2PGjBmxb9++uPbaa+PBBx/s9tg1a9ZEbW1t5HK5Pp9bWVkZq1ev\njpdffjmee+65WLt2bdxxxx3R2toae/bsiQ0bNsTevXvPOGfbtm3xpz/9Kfbv3x8//vGP4xvf+EbX\nY4VCIZYuXdrj+Z/85Cfj6aefjt27d8f3vve9uOOOO/p87unjvv3tb8cXv/jFbn/t5HsJTQCAc9iy\nZUssXLgwIiIWLlwYTzzxxDmPO3ToUGzbti2WLFnSFV99OXf06NFRV1cXERHDhw+P0aNHR1VVVdTU\n1ERlZWXMmzcvNm/e3O2mqVOnxtGjR+P111+PiIhdu3bF+PHjezz/c5/7XFxyySVd5x86dKjP50ZE\n/OAHP4i5c+fGRz/60T78CQpNAIBzev3116OqqioiIqqqqrqC7t8tW7YsVq1aFcOGDTvvc09rb2+P\nvXv3xuTJk7v+WXV1dXR0dJxxXEdHR4wdO/aMY07H4rke+/fz3+uRRx6J66+/vs/ndnR0xObNm7uu\nop6+etsTvxkIABiyZsyYEa+99tpZ/3z58uVnfJ3L5c4KqxkzZsT+/fvjn//8Z/zxj3+Mf/3rX3Hk\nyJHYsmVLr+e+17Fjx2Lu3LmxaNGieOONN3rd/O+3rE8/d1/C77Tt27fHT37yk9i5c2efz73zzjvj\nwQcfjFwuF8VisU+3zoUmADBkPfnkk90+VlVVFa+99lqMHj06Xn311Rg1atRZ5373u9+Nn/70p3Hs\n2LE4fvx4FAqF+PnPf97ruaedOHEibrzxxrjlllvis5/9bDQ3N3c9ls/no7q6+ozjx4wZE/l8vuvr\nQ4cOxZgxY8752LnOj4jYvXt33H777dHa2hojRozo87m///3vY968eRERceTIkfjVr34VlZWV0dTU\n1O2foVvnAADn0NTUFOvWrYuIiHXr1sWcOXPOOmbFihWRz+fjwIEDsXHjxpg+fXq0tLT06dxisRiL\nFy+O2trauPPOO2PKlCmxf//+aG9vj87Ozti0adNZEdfU1BQtLS0REfHcc8/Fhz/84a5b9H05/+DB\ng/HlL3851q9fH+PHj+/653059y9/+UscOHAgDhw4EHPnzo0f/vCHPUZmhCuaAADn9J3vfCduuumm\neOSRR6KmpiYef/zxiIg4fPhw3H777fHLX/7yrHNO34Luy7k7d+6M9evXxxVXXBH19fUREfG1r30t\nZs6cGYVCIRYvXhyTJk2KH/3oRxER8fWvfz2uv/762LZtW4wfPz4++MEPxqOPPtr1vSsqKmLt2rU9\nnn/ffffF3/72t67XWVZWVsauXbv6dO774XedAwCQhFvnAAAkITQBAEhCaAIAkITQBAAgCaEJAEAS\nQhMAgCSEJgAASfwfwgyIrh0RmG8AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x383a510>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, (ax1, ax2, ax3, ax4) = plt.subplots(1,4)\n",
      "\n",
      "x = dot(inv(A), b)\n",
      "ax1.imshow(x, interpolation=\"none\")\n",
      "ax1.set_title(\"x for lambda=0.0\")\n",
      "ax4.plot(x)\n",
      "\n",
      "lambd = 1.0\n",
      "x = dot(inv(A + lambd*identity(A.shape[0])), b)\n",
      "ax2.imshow(x, interpolation = \"none\")\n",
      "ax2.set_title(\"x for lambda=1.0\")\n",
      "ax4.plot(x)\n",
      "\n",
      "\n",
      "lambd = 20.0\n",
      "x = dot(inv(A + lambd*identity(A.shape[0])), b)\n",
      "ax3.imshow(x, interpolation = \"none\")\n",
      "ax3.set_title(\"x for lambda=20.0\")\n",
      "ax4.plot(x)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "[<matplotlib.lines.Line2D at 0x4d62c90>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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hw4dr0aJFmj17tkJCQhQaGqqFCxeaWTICkC9fMUgiCAAAgADUrVs3FRUVXXKb\nkSNHauTIkV6qCMHIl08UllgaBAAAAHgEQQAAAAAIQr6+NIggAAAAALiZYTAjAAAAAASd3FypcmXp\nqqvMrqR8BAEAAADAzXx9WZBEEAAAAADczteXBUkEAQAAAMDtCAIAAABAEGJpEAAAABCEmBEAAAAA\ngszRo1J+vhQZaXYll0YQAAAAANzo/GyAxWJ2JZdGEAAAAADcyB+WBUkuBIHs7GwlJCSoXbt2at++\nvWbOnOmJuoCAQL8AjqNfAASK9HT/CAIhzr6hSpUqmj59umJjY5Wfn69OnTrp1ltvVVt/+GkBL6Nf\nAMfRLwACRUaGFB9vdhWX5/SMQMOGDRUbGytJCgsLU9u2bXXo0CG3FwYEAvoFcBz9AiBQBOzSoAvt\n3btXmzdvVteuXd1VDxCw6BfAcfQLAH/166/Sjz9KzZubXcnluRwE8vPzdc8992jGjBkKCwtzZ01A\nwKFfAMfRLwD82c6dUqtWUojTC/C9z6USz549q7vvvlsDBgxQ3759y94oJ+m3x2FWqabVlV0Ftl9s\nUr5NkpS53tRK4EGX7ZcPk357HG2V2lm9VJkf2W6T0m1mVwEvuFy/bEhaWvy4sbWlGltbebM8v3DI\ntkuHbFlmlwEELX9ZFiRJFsMwDGfeYBiGBg0apPr162v69OllD2qxSLFODRv0eneTUmdZ5ORfB3zc\n5frFYrFIC/k7d9r99EogcqRfhhX9zYTK/NvcSmPolwqwWPj/Bs55+mmpalVp4kTv7dPVf6dOLw1a\ns2aN3n33Xa1cuVJxcXGKi4vTsmXLnN4xEAzoF8Bx9AuAQOBPMwJOLw3q1q2bioqKPFELEHDoF8Bx\n9AuAQOBPQYA7CwMAAABuUFAg7dkjtW5tdiWOIQgAAAAAbpCVJTVtKlWrZnYljiEIAAAAAG7gT8uC\nJIIAAAAA4Bbp6VJ0tNlVOI4gAAAAALgBMwIAAABAECIIAAAAAEGmsFDauVNq08a7+12W5fr9VggC\nAAAAQAXt2yddeaVUs6b39nnm3Bk98tkjLr+fIAAAAABUkBnLgmZvmK0ODTq4/H6CAAAAAFBB3r5i\n0PHTx/XXb/6qqT2mujwGQQAAAACoIG/PCLy89mX1atmrQjMCIW6sBwAAAAhKGRlSYqJ39pXzS45e\n3/C6Ng/fXKFxmBEAAAAAKsAwvLs0aNLXk/RwzMNqWrtphcYhCAAAgICTnZ2thIQEtWvXTu3bt9fM\nmTPL3G6OFlacAAAgAElEQVT06NFq1aqVYmJitHlzxT5dRfDKyZGqVZPq1/f8vjJ/ztQH2z/QU92f\nqvBYLA0CAAABp0qVKpo+fbpiY2OVn5+vTp066dZbb1XbCxZxp6amKisrS7t27dL69es1YsQIrVu3\nzsSq4a+8eX7AM189oz/d8CfVD6146mBGAAAABJyGDRsqNjZWkhQWFqa2bdvq0KFDJbZJSUnRoEGD\nJEldu3bVsWPHlJub6/Va4f+8tSzou4PfaU32Gj1+/eNuGY8gAAAAAtrevXu1efNmde3atcT3Dx48\nqCZNmhQ/j4yM1IEDB7xdHgKAN2YEDMPQuBXjNDF+okKrhLplTJYGAQCAgJWfn6977rlHM2bMUFhY\nWKnXDcMo8dxisZTaJikpqfix1WqV1Wp1d5nwcxkZ0v/9n2f38cXuL3Twl4MaHDdYNptNNputwmMS\nBAAAQEA6e/as7r77bg0YMEB9+/Yt9XpERISys7OLnx84cEARERGltrswCABl8fTSoCKjSOO/HK+p\nPaYqpFJIqUCanJzs0rgsDQIAAAHHMAwNGTJE0dHRGjNmTJnb9OnTRwsWLJAkrVu3TnXq1FGDBg28\nWSYCwJEj0unTUuPGntvHwu8XqmrlqvpD2z+4dVxmBAAAQMBZs2aN3n33XV177bWKi4uTJE2dOlX7\n9++XJA0fPly9e/dWamqqWrZsqRo1amjevHlmlgw/lZEhtWkjlbGqzC0KCgv0zFfPaN5d88pculYR\nBAEAABBwunXrpqKiostuN2vWLC9Ug0Dm6WVBczbMUZsr2yg+Kt7tYxMEAAAAABd58opBJ86c0JTV\nU/T5gM89Mj7nCAAAAAAu8mQQeGXtK7qtxW2KaRjjkfGZEQAAAABc5KmlQbn5uZr13SxtHLbR/YP/\nDzMCAAAAgAvy86XDh6WoKPePPenrSRp47UBF1fHA4P/DjAAAAADggh07pFatpMqV3Ttu1pEsLfx+\noXaM2uHegS/CjAAAAADggowMzywLenblsxp7/VhdGXql+we/ADMCAAAAgAs8caLwxkMbtWrvKr35\n+zfdO3AZmBEAAAAAXJCe7v4gMP7L8Xou/jnVqFrDvQOXgSAAAAAAuMDdS4OW716ufcf2aUjcEPcN\negkEAQAAAMBJZ85I+/bZTxZ2hyKjSOO/HK8pPaaoSuUq7hn0MggCAAAAgJN27bJfNrRqVfeM98H2\nD1TZUln3RN/jngEdwMnCAAAAgJPceaJwQWGBnv7qab35+zdlsVjcM6gDPBcEbvfYyIHpGrMLgFme\n7/dns0vwO8/db3YFMMs/Fo81uwS/M9fsAoAA5c4g8MbGN9SqXislNE9wz4AOYkYAAAAAcFJ6unTn\nnRUf55czv2jy6sla2n9pxQdzEucIAAAAAE5y14zAq9++qpub36zYhrEVH8xJzAgAAAAATigstJ8s\n3KZNxcb56defNDNtpjb8cYN7CnMSMwIAAACAE/bskcLDpRoVvOfX5K8na0CHAWpet7l7CnMSMwIA\nAACAE9yxLOiHoz/ovf++p4yRGe4pygXMCAAAAABOcEcQeHblsxrddbSuqnGVe4pyAUEAAAAAcEJ6\nuhQd7fr7N+ds1ld7vtITNzzhvqJcQBAAAAAAnFDRGYHxX47Xszc9q7CqYe4rygUEAQAAAMBBhlGx\nIPDlD19q95Hd+mPHP7q3MBcQBAAAAAAHHTwohYZK9eo5/94io0jjVozTlB5TVKVyFfcX5ySCAAAA\nAOCgiswGLEpfJEm6t929bqzIdVw+FAAAAHCQq0HgbOFZPf3V05p9x2xVsvjGZ/G+UQUAAADgB1y9\nYtCbm95U8zrNdcvVt7i/KBcRBAAAAAAHuTIjkF+Qr0lfT9K0W6Z5pigXEQQAAAAAB7kSBP627m+K\nj4pXx0YdPVOUizhHAAAAAHBAXp5UUCA1auT4ew7/elh/W/c3rR+63nOFuYgZAQAAAMAB52cDLBbH\n3zNl9RQ90P4BtajXwnOFuYgZAQAAAMABzi4L2nN0j97Z9o7SH033XFEVwIwAAAAA4ABnrxj0nO05\nPdblMTUIa+C5oiqAIAAAAAA4wJkZga0/btXy3cv1pxv+5NmiKoAgAAAAADjAmSAw4csJerr706pZ\nraZni6oAzhEAAAAALuOXX6Sff5aaNbv8tiv3rNSOvB1afP9izxdWAcwIAACAgDN48GA1aNBAHTp0\nKPN1m82m2rVrKy4uTnFxcZo8ebKXK4S/2bFDat1aqlz50tsZhqFxK8ZpSo8pqlq5qneKcxEzAgAA\nIOAkJibqscce08CBA8vdJj4+XikpKV6sCv7M0WVBH2V8pHNF59SvfT/PF1VBzAgAAICA0717d9Wt\nW/eS2xiG4aVqEAgcuWLQ2cKzevqrpzXtlmmqZPH9w2zfrxAAAMDNLBaL1q5dq5iYGPXu3Vvp6b55\nnXf4DkdmBN7a/Jaa1GqiW6++1TtFVRBLgwAAQNDp2LGjsrOzFRoaqqVLl6pv377KzMwsc9ukpKTi\nx1arVVar1TtFwqekp186CPxa8KuSVyUr5YEUWZy59bALbDabbDZbhcchCAAAgKBTs+Zvl3Ts1auX\nHn30UR05ckT16tUrte2FQQDB6fRpKTtbatmy/G1mrJ+h7s26q3Pjzh6v5+JAmpyc7NI4BAEAABB0\ncnNzFR4eLovForS0NBmGUWYIACRp1y6peXOpajkXAco7madXv31V3w751ruFVRBBAAAABJwHHnhA\nq1atUl5enpo0aaLk5GSdPXtWkjR8+HAtWrRIs2fPVkhIiEJDQ7Vw4UKTK4Yvu9yyoKmrp+q+dvep\nVf1W3ivKDQgCAAAg4Lz//vuXfH3kyJEaOXKkl6qBv8vIKP+KQfuO7dP8rfO1/dHt3i3KDbhqEAAA\nAHAJl7pi0HO25zTyupFqGNbQu0W5AUEAAAAAuITylgZty92mZVnL9OSNT3q/KDcgCAAAAADlOHdO\nysqS2rQp/dpTXz6lp7o9pVrVanm/MDfgHAEAAACgHHv2SA0bSqGhJb+/au8qbT+8XR/d95E5hbmB\n0zMCp0+fVteuXRUbG6vo6GhNmDDBE3UBAYF+ARxDrwDwVWUtCzIMQ+NWjNOkhEmqFlLNnMLcwOkZ\ngerVq2vlypUKDQ3VuXPn1K1bN33zzTfq1q2bJ+oD/Br9AjiGXgHgq8q6YtDiHYt1+txpPdjhQXOK\nchOXzhEI/d/cSEFBgQoLC7kBB3AJ9AvgGHoFgC+6+IpB54rOacKXEzTtlmmqZPHv021dqr6oqEix\nsbFq0KCBEhISFF3ehVUB0C+Ag+gVAL4oPb3kjMC8zfPUuGZj9WzR07yi3MSlk4UrVaqkLVu26Pjx\n4+rZs6dsNpusVmvJjVYn/fa4qVVqdtHrkPbZpP02SVJmhqmVwIMu1y9fJa0pftzc2kTNrU1NqNK3\n7bHt1x5bttllwMMc+d2StNAofmxtL1nbW7xcpe+zfW/I9r3ZVQCBwTCkHTt+mxE4efakklcl6+N+\nH8ti8f//fyyGYRiX36x8kyZN0hVXXKEnn/zt+qkWi0UaX6Fhg07va6TURIsq+NcBH3dxv1gsFj1v\n+Oe1h830nOVleiXAlfe7pehj///F622V/mDQLxVgsfC7OZhlZ0tdukg5Ofbn076Zpo05G/XhvR+a\nW9hFXP136vTSoLy8PB07dkySdOrUKS1fvlxxcXFO7xgIBvQL4Bh6BYAvunBZ0JFTR/TKt69oSo8p\n5hblRk4vDcrJydGgQYNUVFSkoqIiPfTQQ7r55ps9URvg9+gXwDH0CgBfdOGJwn9d/Vfd3fZuta7f\n2tyi3MjpINChQwdt2rTJE7UAAYd+ARxDrwDwRRkZ0rXXSvuP79dbW97S9yMC6wQc/77mEQAAAOAh\n55cGTbRN1IjOI9SoZiOzS3Irl64aBAAAAAS6jAzJ0uB7pW5MVeaoTLPLcTtmBAAAAICLHD4sFRZK\nr259SuN/N161q9c2uyS3Y0YAAAAAuEh6uhR54zfalrvN5y4X6i4EAQAAAOAi6emGfooZp5cSnle1\nkGpml+MRLA0CAAAALvKf3SmqfMUv6t+hv9mleAxBAAAAALjAiTMntLLyeD3ScpoqV6psdjkeQxAA\nAADwc6dOmV1B4Mg6kqXr37xelj236KHre5ldjkcRBAAAAPzYjz9K4eHSffdJu3ebXY1/W/HDCt3w\nxu9Uc/vjCt/wdzVrZjG7JI8iCAAAAPixF16QHnxQiomRunSRnnhCOnLE7Kr8i2EYesE2U30XPKQz\n736oXuHDtW2bVCnAj5S5ahAAAICfysmR5s+Xtm+XGjWShg6VkpKka66Rxo+XRo2SqgXmBW/c5peT\nZ3TbjJHacOg7/aHgW03/PEqNG5tdlXcEeM4BAAAIXNOmSYMG2UOAJDVoIM2eLX39tbRqldS2rbRw\noWQY5tbpiwxDeuvfuQr/cw/tyTmqbwav0b/nBE8IkAgCAAAAfungQemdd6Rx40q/1ratlJIi/fOf\n0ksvSddfL61e7f0afdX69VJc7016ZFMX3dPxVh2a8aG6xoWZXZbXsTQIAADAD02bJiUmSg0blr9N\nQoL03XfSe+9JAwZInTrZ39e6tffq9CU//CBNmCAtz/lAZ28dqXf/b7bua3+P2WWZhhkBAAAAP3Pw\noPSvf0l/+cvlt61UyR4CduyQunaVbrxReuwx6fBhz9fpK44ckf70J+m6LkXKbfeMav7fX7T6j8uD\nOgRIBAEAAAC/89e/SkOG2M8JcNQVV9iXEWVkSBaLffnQCy8E9j0IzpyRXn1VatNGOnbqF3V55Q8q\navK1vhuWptiGsWaXZzqCAAAAgB/Jzpbef1/6859de/9VV0kzZ0pr19rXyrdpI737rlRU5N46zWQY\n0gcf2MPOypXSO5/9oPXtb1CTeuFaMXCFwmuEm12iTyAIAAAA+JG//tV+mdDwCh7Ltm4tffyxfYnR\n3/8uXXed/aDZ333zjXTDDfbZjn/+U3pi5koN+vpGjeg8QnPunKOqlauaXaLPIAgAAAD4if37pX//\nW3rySfeN2a2btG6d/XyDIUOk3//evnzI3+zaJd19t9S/v/3+CWlphjLCXtcDHz2g9+5+TyO7jJTF\nEth3CnYWQQAAAMBPTJ0q/fGP9uU97mSxSP362QNAQoJ0003SiBFSbq579+MJeXnS6NH2WYAuXewn\nRd/3QIFGLh2h1797XWuHrFWP5j3MLtMnEQQAAAD8wL590ocfunc24GLVqklPPCHt3CmFhkrt2kmT\nJ0snT3pun646fdq+/KdtW/vzjAz7ydD5RYd1y4JblJOfo2+HfKur615tbqE+jCAAAAACzuDBg9Wg\nQQN16NCh3G1Gjx6tVq1aKSYmRps3b/Zida6ZMkUaPly68krP76tePemVV+wnE//3v9I110hvvy0V\nFnp+35dTVGQ/r+Gaa+z1rVljP/n5qqukrT9uVZc3u+imZjfpk36fqGa1mmaX69MIAgAAIOAkJiZq\n2bJl5b6empqqrKws7dq1S3PnztWIESO8WJ3z9uyRPvrIfi18b2rRwn5OwgcfSG+8IXXsKC1f7t0a\nLmSz2Zf/zJxpDwMff/zbzdE+Sv9It7xzi6bdPE2Te0xWJQuHuZfDnYUBAEDA6d69u/bu3Vvu6ykp\nKRo0aJAkqWvXrjp27Jhyc3PVwJkL83vRlCn2Nfv165uz/xtusF+N55NPpEcflVq2lF58UbrEhItb\nnV/289//2u+MfN999vMaJKnIKNLzq57XvC3z9PmAz9WxUUfvFBUAiEoAACDoHDx4UE2aNCl+HhkZ\nqQMHDphYUfl++EFavNi+dt9MFov0hz9I27dLvXpJt9xiv4zpoUOe22durj0A3XSTFB9vPxG4X7/f\nQkB+Qb7u/fBeLf9hudKGphECnMSMAAAACEqGYZR4Xt6lJZOSkoofW61WWa1WD1ZV2uTJ9k/h69Xz\n6m7LVbWq/So9Awfa72nQoYP02GP2k5jDwtyzj5MnpenT7V8DB9pPXr745997bK/uWniXOjXqpPf+\n8J6qhVRzz879gM1mk81mq/A4BAEAABB0IiIilJ2dXfz8wIEDioiIKHPbC4OAt2VlSSkp9mvk+5o6\ndexX7Xn0Uenpp+1r9ZOTpcREKcTFI8zCQumdd6Rnn5VuvNF+MnCLFqW3+3rf1+q3qJ/G/268Rncd\nHXT3B7g4kCYnJ7s0DkuDAABA0OnTp48WLFggSVq3bp3q1Knjk+cHTJ5svzlW3bpmV1K+Zs2kd9+1\nB5Z//UuKjZWWLpUumnC5rBUrpE6d7Cclf/CB/STlskLA3I1zde+H92pB3wV6/PrHgy4EuBMzAgAA\nIOA88MADWrVqlfLy8tSkSRMlJyfr7NmzkqThw4erd+/eSk1NVcuWLVWjRg3NmzfP5IpL27VL+uwz\n+6yAP+jcWVq5Uvr0U2nsWPvlR19+2R4MLuX776U//9n+c06bZj8Poaxj+7OFZzX287H6cs+X+ibx\nG7Wq38ozP0gQIQgAAICA8/777192m1mzZnmhEtdNmmRfi1+njtmVOM5ikfr0sZ9M/Oab0u23278m\nT5YiI0tum5MjPfectGSJ9Mwz0iOP2M8/KEveyTzd9+F9Cq0SqnVD1ql29dqe/2GCAEuDAAAAfMzO\nnfblNY8/bnYlrqlSxX61n8xMewCIibGfR3DihJSfLyUlSe3b208Azsy0B57yQsD3P32vLm90UZeI\nLlpy/xJCgBsRBAAAAHzMpEn2EFDbz495a9WyzwZs3SodPGg/ofiaa+zLnjZutJ9sfKkZjyU7lihh\nfoImJUzStFumqXKlyt4rPgiwNAgAAMCH7NghffGF9PrrZlfiPpGR0ttv228IVlh4+fMGDMPQ1NVT\n9Y+N/1Dqg6m6LuI6r9QZbAgCAAAAPmTSJGnMGPun6YHGkTsRnzx7UolLErX32F6tH7pejWs29nxh\nQcpjQWDiNFYdOaNV795KNbsImKKw0stmlwD4jefvdvJ6hICfyciQli+X/vEPsysxR/bxbN218C61\nD2+vVQ+vUvWQ6maXFNA4WgcAAPARzz8vPfGEVLOm2ZV435r9a9T1za7q36G/5vedTwjwApYGAQAA\n+IDt26WvvrLfUCvYvLX5LY1fMV7z+85Xr1a9zC4naBAEAAAAfMDzz0t/+pMUFmZ2Jd5zruicnvzi\nSS3NWqrViat1zZXXmF1SUCEIAAAAmOz77yWbTfrnP82uxHuOnDqifov6qbKlstYNWae6V9Q1u6Sg\nwzkCAAAAJktOlp58MnhmA9IPp6vrm10V0yBGnz34GSHAJMwIAAAAmGjbNmn1avt19oPBZ5mfafCS\nwXrp1pc0KHaQ2eUENYIAAACAiZKTpT//WapRw+xKPOv7n77X69+9riU7lyjlgRRdH3m92SUFPYIA\nAACASbZskdauld55x+xKPOPk2ZP6cPuHmrNxjvYf368hcUO0cdhGNQxraHZpEEEAAADANMnJ0l/+\nIoWGml2Je23/abvmbJyjf/33X7o+8nqN+9043dH6DoVU4tDTl/C3AQAAYILNm6X166X33jO7Evc4\ndfaUPky3f/q/99heDYkbok3DNqlZnWZml4ZyEAQAAABMkJQkjRsnXXGF2ZVUTPrhdM3ZYP/0v0tE\nF/35xj/rztZ38um/H+BvCAAAwMs2bpQ2bJAWLjS7EtecOntKi9IXae6mudp9ZLcGxw3WhmEbFFUn\nyuzS4ASCAAAAgJclJUnjx/vfbED64XTN3ThX7257V9dFXKcnrn9Cd7a+U1UqVzG7NLiAIAAAAOBF\n331nPz/gww/NrsQxp8+d1qL0RZqzcY6yjmRpcOxgfffH79S8bnOzS0MFEQQAAAC8KClJmjBBql7d\n7EouLeNwhv3T//++q06NOmns9WP1+9a/59P/AEIQAAAA8JK0NPudhD/+2OxKynb63Gl9lP6R5m6a\nq8yfM5UYm6i0oWl8+h+gCAIAAABekpQkPfWUVK2a2ZWUtCNvh97Y+IYWbFuguIZxGt1ltPpc04dP\n/wMcQQAAAMAL1q2Ttm+XPvnE7Erszpw7o48yPtLcjXO1I2+HEmMTtX7oel1d92qzS4OXEAQAAAC8\nwFdmA3bm7dQbm97Qgq0LFNMwRqO6jFKfa/qoauWq5hYGryMIAAAAeNi330o7dkiJiebs/8y5M/o4\n42PN3TRX6YfTlRibqLVD1qplvZbmFASfQBAAAADwsIkTpaeflqp6+UP3zJ8z9cbGNzR/63xd2+Ba\njeg8Qn3b9OXTf0giCAAAAHjUmjXSrl3SoEHe2d+Zc2e0eMdizdk4R9sPb9fDMQ9rzeA1alW/lXcK\ngN8gCAAAAHiQt2YDso5kae7GuZq/db7ah7fX8E7D1bdNX1UL8bFLFMFnEAQAAAA8ZPVq6YcfPDsb\nsPvIbj2a+qg252zWoJhBWp24Wq3rt/bcDhEwCAIAAAAeMnGi9MwzUhUPXI7fMAzN2zJP41aM01Pd\nnlLK/Sl8+g+nEAQAAAA8YNUqad8+6aGH3D923sk8Dft0mHYf3a2Vg1aqfXh79+8EAa+S2QUAAAAE\nookTpWefdf9swOdZnyvmHzFqUbeF0oamEQLgMmYEAAAA3GzlSungQWnAAPeNeersKY1bMU6LdyzW\nO//3jno07+G+wRGUCAIAAABuZBi/zQaEuOlIa8uPW9T/4/5qH95eWx/ZqrpX1HXPwAhqLA0CAAAB\nadmyZWrTpo1atWqlF154odTrNptNtWvXVlxcnOLi4jR58mS37Perr6Qff5QefLDiYxUWFeqlNS/p\n1ndu1YRuE7Tw7oWEALgNMwIAACDgFBYWatSoUVqxYoUiIiJ03XXXqU+fPmrbtm2J7eLj45WSkuK2\n/Z6fDXjuuYrPBuw/vl+DFg9SYVGhvvvjd4qqE+WWGoHzmBEAAAABJy0tTS1btlRUVJSqVKmi+++/\nX0uWLCm1nWEYbt3vihVSXp70wAMVG+f9/76vznM7q2eLnlo5aCUhAB7BjAAAAAg4Bw8eVJMmTYqf\nR0ZGav369SW2sVgsWrt2rWJiYhQREaGXX35Z0dHRLu/zwtmAypVdG+PY6WMalTpKGw5t0NL+S9Wp\ncSeX6wEuhyAAAAACjsViuew2HTt2VHZ2tkJDQ7V06VL17dtXmZmZpbZLSkoqfmy1WmW1Wsscb/ly\n6dgxqV8/12petXeVBi0epDta3aFNwzcptEqoawMh4NlsNtlstgqP41IQKCwsVOfOnRUZGalPP/20\nwkUAgYx+ARxHv8BdIiIilJ2dXfw8OztbkZGRJbapWbNm8eNevXrp0Ucf1ZEjR1SvXr0S210YBMpT\nkdmAgsICPbfyOS3YukBv/P4N3dH6DucGQNC5OJAmJye7NI5L5wjMmDFD0dHRDqVtINjRL4Dj6Be4\nS+fOnbVr1y7t3btXBQUF+ve//60+ffqU2CY3N7f4HIG0tDQZhlEqBDjq88+lEyeke+917n0ZhzPU\n9c2uSj+cri2PbCEEwKucDgIHDhxQamqqhg4d6vYTbIBAQ78AjqNf4E4hISGaNWuWevbsqejoaPXr\n109t27bVnDlzNGfOHEnSokWL1KFDB8XGxmrMmDFauHChS/s6PxswcaLjswGGYei1tNd009s3aUTn\nEVpy/xKF1wh3af+Aq5xeGjR27Fi99NJLOnHihCfqAQIK/QI4jn6Bu/Xq1Uu9evUq8b3hw4cXPx45\ncqRGjhxZ4f0sXSqdPCndc49j2/+Y/6MSlyTq55M/a83gNWpdv3WFawBc4dSMwGeffabw8HDFxcVd\n9tMam4zir73ik52y7L3gz+ijMk5Ogn9ztF9sxm9fe2mVMu01Sv45IfA42i8rjd++9vBvoUx7jJJ/\nTvCsC2cDKjlwVLV4x2LF/iNW1zW+jhAA0zk1I7B27VqlpKQoNTVVp0+f1okTJzRw4EAtWLCg1LZW\nsb7zcqJkUdT/Hrdq3VqfZO0ysxy4maP9YqVVLivKouJekaRVHNwEHEf7JYF+uazmFqn5Bc/pF8/6\nz3+kggLpD3+49Hb5Bfkau2ysvtzzpT7u97FubHKjdwoELsFiuLgQc9WqVXr55ZfLvKqDxWLRRIKA\nU1r17q0Bqf9hXWyAKq9fLBaLJtIqTks23H8TIPiOS/VLEv3itCT6pUIsFku5f36GIV13nfTUU5cO\nAusPrFf/j/vrpmY36W+3/021qtXyULUIVpf6d3opFbqPAFd1ABxHvwCOo1/gDz79VDp3Turbt+zX\nzxWd09TVU/Xad6/ptd6v6Z5oB08iALzE5SAQHx+v+Ph4d9YCBCz6BXAc/QJ/YBhSUlL55wbsPrJb\nAz4ZoJpVa2rTsE2KqBXh9RqBy3HpPgIAAADBbMkSexi4eDbAMAy9tfktXf/P63V/u/u1bMAyQgB8\nVoWWBgEAAASboiL7bEBysnThKra8k3ka9ukw7T66WysHrVT78Pam1Qg4ghkBAAAAJyxebL9x2IU3\nKv4863PF/iNWLeq2UNrQNEIA/AIzAgAAAA46PxswZYp9NuDU2VMav2K8PtnxiRb83wL1aN7D7BIB\nhxEEAAAAHPTxx1K1atKdd0pbftyi/h/3V/vw9tr6yFbVvaKu2eUBTiEIAAAAOKCoyH5ewNS/Funl\nta/oxbUvanrP6erfoT+XvIVfIggAAAA4YNEiqXK9bL2aN1CFhwv13R+/U1SdKLPLAlzGycIAAACX\nUVgoPfHWQu29rZN6tuyplYNWEgLg95gRAAAAuIRjp4/p97NH6ecOG7T64aXqHNHJ7JIAt2BGAAAA\n4BJi/xGr7Rtra2HCJkIAAgozAgAAAJdwX43X9c0PvdWnl9mVAO5FEAAAALiET1/trZkzS95FGAgE\nLA0CAAC4hPr1pVtuMbsKwP2YEQAAALiE5GRmA2CiwkIpN1c6eLD8LxcRBAAAAC6hRw+zK0DAys+/\n9AH+wYPSTz9J9epJERElv6zW3x63b+/S7gkCAAAAl8BsAJxWWGg/gL/cQf7Zs6UP8Fu2lOLjf3ve\nqAYfUWcAAAwySURBVJFUpYpHyiQIAAAAAI769dfLH+Dn5kp165Y+yL/pppLP69QxNWkSBAAAAIAz\nZ6Qff7R/5eTYv8o6yD9zpvQB/tVXS927l/wUv2pVs3+iyyIIAAAAIDAZhnT8eMmD+7Ie//ijfb1+\ngwZSw4b2A/mGDe0H9d26lTzor1s3YNaLEQQAAADgX86dkw4fvvzBfU6OfX19o0a/Hdyff9yhQ8nv\n16snVQquK+sTBAAAAOAbTp507OD+55/tN3i48MC+YUOpTRspIaHkp/o1apj9U/ksggAAAAA8p7BQ\nysuzH8Tn5pZeh3/h44KC3w7sL/wE/8YbS34vPFwK4TC2ovgTBAAAwP+3d3cxbZVhHMD/nSNZJgtC\nAmXColkpcXys7eJSiZmTLLjABlkcRmLUmX1kQZdlXixGr9ToZJpdqLvYTKaJizEmaDbiCgkmazbH\nmMowXLBFllClbOCAdQIGYfTxgtBR6GnPR7tSzv+XNPTjfZ8+nNOnfV84p682s4P7wcH7g3ul6yMj\nM9+Ok5sbfgx+fj6wcWP4gD8jY8kcf58KOBEgIiIiopnB/fCw8qB+7u3h4ZnB/ezA3mq9f72kJPz+\n7Gz+9X6R4l4hIiIiWqqCQfV/uVca3Fut9wf3s49xcL8kcA8SERERpZJgcOZwm7nH3CsN8oeG7g/u\n5w/wi4vDB/vZ2QlbwZYWJ04EiIiIiJJNBAgEwk+mjTTIHxiY+drMVatmBvXzD8spKgq/j4N7ioIT\nASIiIqJEEAFGR5UH9HOv//03sGJF+Am1sz/t9vD7cnJSYtVaWvw4ESAiIiLSYnxceXA//+dDDy0c\n2FutM9+WM/++FSuS/ZuRyXAiQERERBRNbW344P7evch/uXc4Fh6Hn56e7OyJFD3wdZR9kJTtn+zc\nyXx8Bl8yRvon87nj0Z/Mp9fga8ZI/1R+7qWqpaUFTzzxBOx2O44ePRqxzcGDB2G32+FwONDZ2akc\n7MUXgSNHgHPnAL9/ZvXb3l6gvR04cwY4eRJ47z2gvh54/vmZxa9sNsVJgNfrjcNv+GDiMmb8JSqu\nHkmYCKRu/2Q+N5mTL4n9k/nc8ehP5uNLYv9Ufu6laHp6GgcOHEBLSwu6u7vx7bff4tq1a2FtPB4P\nbty4gZ6eHnzxxReor69XDvjCC8AzzwCFhXFZ8CqVBpiMGX+mnggQERERJdIvv/yCgoICPP7440hL\nS0NdXR3Onj0b1qapqQm7du0CALjdbgQCAQwODiYjXaKkSdg5Aqs3bIh4f/rNm1j96KO64yazfyKf\nO6vApjsupTalWgGW7uvdcP+Oq7rjUmqLVi+rDL7mjPRf1M9twnrp7+/HmjVrQrfz8/Nx5cqVmG38\nfj+sVusDy5Mo6SQBNm/eLAB40XjZvHlzInYHLWKsFdYKqcd6Yb2o1djYKHv37g3dPn36tBw4cCCs\nzfbt2+Xnn38O3d6yZYt0dHQsiGWz2ZK+D3nhJdbFZrPpqpWE/EdgMR37RLSYsVaI1GO9kFp5eXno\n6+sL3e7r60N+fn7UNn6/H3l5eQti3bhxI3GJEiUZzxEgIiKiJeXJJ59ET08PfD4fJicn8d1336Gm\npiasTU1NDb7++msAQHt7Ox555BEeFkSmw3UEiIiIaElZvnw5jh8/jq1bt2J6ehp79uzBunXrcPLk\nSQDA/v37UVVVBY/Hg4KCAjz88MP46quvkpw10YNnERFJdhJERERERPRgJfzQoJGREVRUVKCwsBDP\nPfccAoGAYtvp6Wm4XC5UV1dr6t/X14fy8nIUFxejpKQE9fX1hhYRibUIyTfffAOHw4H169fj6aef\nRldXl6b+s3799VcsX74cP/zwg2IbMhez1QtrhYwwUi96auWzzz4zvEgV62XxiusCZCpjer1eZGRk\nwOVyweVy4YMPPogZc/fu3bBarSgtLVVsozXPWDH15Bmpfozmqiam1lwnJibgdrvhdDpRVFSEt99+\n23CeauPq2a5A5M9/3bnqOsVYg8OHD8vRo0dFRKShoUHeeustxbbHjh2Tl156SaqrqzX1v3XrlnR2\ndoqISCAQkLS0NGltbZXJyUlxOBzS3d0d1v7cuXNSWVkpIiLt7e3idrtDj927d09sNpv09vYq9m9r\na5NAICAiIs3NzZr7z7YrLy+Xbdu2SWNjo+I2IXMxU72wVsgoI/WitVZGR0fFbrfLmjVror5mWS+p\nSc32jbZv9cY8f/582Hu4GhcuXJCrV69KSUlJxMe15qkmpp4859dPYWGh4W2qJqaeXMfHx0VEZGpq\nStxut1y8eNFQnmrj6slVJPLnv95cE/4fgbkLduzatQtnzpyJ2M7v98Pj8WDv3r2QOUcrqemfm5sL\np9MJAOju7kZmZiYsFouuRUTULEJSVlaGjIyMUH+/3x96TE1/APj8889RW1uL7OzsKFuPzMZM9cJa\nIaOM1IvWWklPT0dubi6sVqvuRapYL4tXIhYgU7vPROMR2ps2bUJmZqbi43oWSosVU0+e8+tn3bp1\nuHnzpqFc1cTUk+vKlSsBAJOTk5ienkZWVpahPNXG1ZOr0ue/3lwTPhEYHBwMnYVvtVoVk3nzzTfx\nySefYNmy8JTU9p/V2dmJ8fFxuN1uADMLhPT394e1UVpEROmx+f3nOnXqFKqqqqLGjvT8Z8+eDS1n\nbjG4VDktHWaqF9YKGWWkXrTWis/nw7Vr11BcXBy6j/WydKjdvkr7Vm9Mi8WCtrY2OBwOVFVVobu7\n2+ivojlPNYzm6fP50NnZGfqsiUeuSjH15BoMBuF0OmG1WlFeXo6ioqK45Bkrrp5clT7/9eYal28N\nqqiowMDAwIL7P/zww7DbFosl4huTy+XCX3/9hevXr2N8fBxDQ0MoLS1V3X/W2NgYjh07hrKyMqSn\np0fNef4sajauljfO8+fP48svv8SlS5cWxInm0KFDaGhogMVigYhong1SamO9XFLdl7VCRuqloqIC\nPT09GB0dDauXpqammH3nGhsbQ21tLXbv3o2RkZGYObNeUo/afaO0b/XG3LBhA/r6+rBy5Uo0Nzdj\nx44d+OOPP1TlEq881TCS52z9fPrppxE/a/TkGi2mnlyXLVuG33//HXfv3sXWrVvh9Xrx7LPPGs4z\nVlytuf7444/IycmBy+WKuq6KllzjMhFobW1VfMxqtWJgYAC5ubm4desWcnJyFrSprKzE6dOnMTY2\nhomJidBJEDU1Nar6A8DU1BR27tyJ6upqXL9+PXS/1kVE1CxCAgBdXV3Yt28fWlpawv6dpqZ/R0cH\n6urqAABDQ0Nobm5GWlragu84pqWJ9ZKpui9rhYzUS2trK955550F9dLY2Ki5Vl5++WU89dRTePfd\nd0OPsV6WjnguQKYl5qpVq0LXKysr8frrr2NkZCTiISR6f5dYeaqhN8+59bNjx4645BorppFtmpGR\ngW3btuG3334LG7Ab3aZKcbXm2tbWhqamJng8HkxMTOCff/7Bq6++GloPQ1eums9Q0Ojw4cPS0NAg\nIiIfffRR1JO5RES8Xq9s375dU/9gMCivvPKKHDp0SKampmTt2rXS29sr//33X8wTfi5fvhx2IoWa\n/n/++afYbDa5fPnyglzU9J/rtddek++//z7qNiHzMFO9sFbIKCP1orVWRNS9ZlkvqcnovtUbc2Bg\nQILBoIiIXLlyRR577DFV+fb29qo6WVhNnmpi6slzfv3EI1c1MbXmevv2bblz546IiPz777+yadMm\n+emnnwzlqTau3v0vsvDzX2+uCZ8IDA8Py5YtW8Rut0tFRUVoo/T390tVVdWC9l6vN+wsaDX9L168\nKBaLRRwOhzidTlm7dq3k5eWJzWaTI0eOiIjIiRMn5MSJE6G4b7zxhthsNlm/fr10dHSE5eDxeKSw\nsFCx/549eyQrK0ucTqc4nU7ZuHGjpv5z8c2a5jJbvbBWyAgj9aKnVpxOp7z//vsxX7Osl9SkZvtG\n27d6Yh4/flyKi4vF4XBIWVlZxD8wzldXVyerV6+WtLQ0yc/Pl1OnThnOM1ZMPXlGqh+Px2MoVzUx\nteba1dUlLpdLHA6HlJaWyscffywixve9mrh6tuusue9nRnLlgmJERERERCaU8G8NIiIiIiKixYcT\nASIiIiIiE+JEgIiIiIjIhDgRICIiIiIyIU4EiIiIiIhMiBMBIiIiIiIT4kSAiIiIiMiE/gfUksAm\nUyKN1AAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x48a7a90>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "References"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "1. Computer Science and Artificial Intelligence Laboratory - Technical Report\n",
      "    Notes on Regularized Least Squares\n",
      "    Ryan M. Rifkin and Ross A. Lippert\n",
      "2. http://en.wikipedia.org/wiki/Bias-variance_dilemma\n",
      "3. http://scott.fortmann-roe.com/docs/BiasVariance.html\n",
      "4. http://en.wikipedia.org/wiki/Tikhonov_regularization\n",
      "5. Scikit-learn documentation http://scikit-learn.org/"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    }
   ],
   "metadata": {}
  }
 ]
}