ml-assign3-maasencioh-jejhernandezpa.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Assignment 3: Kernels and SVMs\n",
    "* Miguel Angel Asencio Hurtado - 1026573414\n",
    "* Jefferson Javier Hernández Panqueba - 1019096996"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import math\n",
    "from matplotlib import pyplot as plt\n",
    "from sklearn.cross_validation import train_test_split"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 1\n",
    "Let $x = \\{x_1, . . . , x_n\\}$ be a subset of an input data set $X$. Consider a kernel function $k : X \\times X \\rightarrow \\mathbb{R}$, which induces a feature space $\\Phi(X)$:\n",
    "\n",
    "__(a)__ Deduce an expression (using kernels) that, given a vector $w \\in X$, calculates the norm of the projection of the image of a point $(x,\\phi(x))$, onto the image of the vector $(w,\\phi(w))$:\n",
    "\n",
    "$$P_{\\phi(w)}(\\phi(x)) = \\frac{\\langle \\phi(w),\\phi(x)\\rangle}{\\| \\phi(w)\\|}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__R/__ Given the projection of the image of a point $(x,\\phi(x))$ onto th image of the vector $(w,\\phi(w))$ as:\n",
    "\n",
    "$$P_{\\phi(w)}(\\phi(x)) = \\frac{\\langle \\phi(w),\\phi(x)\\rangle}{\\| \\phi(w)\\|}$$\n",
    "\n",
    "Based in the relation of inner product and norm:\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "\\langle x,x\\rangle &=& \\|x\\|^2 \\\\\n",
    "\\|x\\| &=& \\sqrt{\\langle x,x\\rangle}\n",
    "\\end{eqnarray*}\n",
    "\n",
    "And accepting the kernel function $k : X \\times X \\rightarrow \\mathbb{R}$ as:\n",
    "\n",
    "$$k(x, w) = \\langle \\phi(x),\\phi(w)\\rangle$$\n",
    "\n",
    "So, the projection of the image of the point could be calculated as:\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "P_{\\phi(w)}(\\phi(x)) &=& \\frac{\\langle \\phi(w),\\phi(x)\\rangle}{\\| \\phi(w)\\|} \\\\\n",
    "&=& \\frac{\\langle \\phi(w),\\phi(x)\\rangle}{\\sqrt{\\langle \\phi(w), \\phi(w)\\rangle}}\\\\\n",
    "&=& \\frac{k(w,x)}{\\sqrt{k(w,w)}}\n",
    "\\end{eqnarray*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__(b)__ Deduce an expression (using kernels) to calculate the sample variance of the projections in the feature space of a set of points along a vector $w$:\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "\\sigma_{\\phi (w)}^2(x) &=& \\frac{1}{n} \\sum_{x_i \\in X}(P_{\\phi (w)}(\\phi (x_i))- \\mu)^2 \\\\\n",
    "\\mu &=& \\frac{1}{n} \\sum_{x_i \\in X}P_{\\phi (w)}(\\phi (x_i))\n",
    "\\end{eqnarray*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__R/__ Given the definition of projections before:\n",
    "\n",
    "$$P_{\\phi(w)}(\\phi(x)) = \\frac{k(w,x)}{\\sqrt{k(w,w)}}$$\n",
    "\n",
    "The mean could be calculated as:\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "\\mu &=& \\frac{1}{n} \\sum_{x_i \\in X}P_{\\phi (w)}(\\phi (x_i)) \\\\\n",
    "&=& \\frac{1}{n} \\sum_{x_i \\in X}\\frac{k(w,x_i)}{\\sqrt{k(w,w)}} \\\\\n",
    "&=& \\frac{\\mu_k}{\\sqrt{k(w,w)}} \\\\\n",
    "\\mu_k &=& \\frac{1}{n} \\sum_{x_i \\in X}k(w,x_i)\n",
    "\\end{eqnarray*}\n",
    "\n",
    "And the variance as:\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "\\sigma_{\\phi (w)}^2(x) &=& \\frac{1}{n} \\sum_{x_i \\in X}(P_{\\phi (w)}(\\phi (x_i))- \\mu)^2 \\\\\n",
    "&=& \\frac{1}{n} \\sum_{x_i \\in X}(\\frac{k(w,x_i)}{\\sqrt{k(w,w)}} - \\mu)^2 \\\\\n",
    "&=& \\frac{1}{n} \\sum_{x_i \\in X}(\\frac{k(w,x_i)}{\\sqrt{k(w,w)}} - \\frac{\\mu_k}{\\sqrt{k(w,w)}})^2 \\\\\n",
    "&=& \\frac{1}{n} \\sum_{x_i \\in X}\\frac{(k(w,x_i) - \\mu_k)^2}{k(w,w)}\n",
    "\\end{eqnarray*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__(c)__ Use the previous expression to calculate the variance of the projections of the images of the elements of the following point set in $\\mathbb{R}^2$, $x = \\{(0, 1),(-1, 3),(2, 4),(3, -1),(-1, -2)\\}$ over the images of the vectors $w_1 = (1, 1)$ and $w_2 = (-1, 1)$, in the feature spaces induced by the following kernels:\n",
    "\n",
    "<ol type=\"I\">\n",
    "<li> $k(x, y) = \\langle x,y\\rangle$ </li>\n",
    "<li> $k(x, y) = \\langle x,y\\rangle^2$ </li>\n",
    "<li> $k(x, y) = (\\langle x,y\\rangle + 1)^5$ </li>\n",
    "<li> Gaussian kernel with $\\sigma = 1$ </li>\n",
    "</ol>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__R/__ We define basic structure for the different kernels as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X = [(0,1), (-1,3), (2,4), (3,-1), (-1,-2)]\n",
    "W = [(1,1), (-1,1)]\n",
    "\n",
    "def mean(k, X, w):\n",
    "    m = 0\n",
    "    for x in X:\n",
    "        m += k(w, x)\n",
    "    return float(m) / len(x)\n",
    "\n",
    "def var(k, X, w):\n",
    "    s = 0\n",
    "    m = mean(k, X, w)\n",
    "    for x in X:\n",
    "        s = (k(w,x) - m)**2\n",
    "    return float(s) / (len(x) * k(w,w))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I. We create the kernel $k(x, y) = \\langle x,y\\rangle$ and prints the result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w_1: 12.25\n",
      "w_2: 1.0\n"
     ]
    }
   ],
   "source": [
    "def dot(x,y):\n",
    "    return x[0]*y[0] + x[1]*y[1]\n",
    "\n",
    "print 'w_1:', var(dot, X, W[0])\n",
    "print 'w_2:', var(dot, X, W[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "II. We create the kernel $k(x, y) = \\langle x,y\\rangle^2$ and prints the result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w_1: 40.5\n",
      "w_2: 40.5\n"
     ]
    }
   ],
   "source": [
    "def square(x,y):\n",
    "    return (x[0]*y[0] + x[1]*y[1])**2\n",
    "\n",
    "print 'w_1:', var(square, X, W[0])\n",
    "print 'w_2:', var(square, X, W[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "III. We create the kernel $k(x, y) = (\\langle x,y\\rangle + 1)^5$ and prints the result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w_1: 154971.938786\n",
      "w_2: 5126.87705761\n"
     ]
    }
   ],
   "source": [
    "def pol(x,y):\n",
    "    return (x[0]*y[0] + x[1]*y[1] + 1)**5\n",
    "\n",
    "print 'w_1:', var(pol, X, W[0])\n",
    "print 'w_2:', var(pol, X, W[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "IV. We define the kernel as a Gaussian kernel with $\\sigma = 1$, so we create the kernel:\n",
    "\n",
    "$$k(x, y) = e^{-\\|x-y\\|^2 / \\, 2}$$\n",
    "\n",
    "And prints the result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w_1: 0.0210470345361\n",
      "w_2: 0.0308321049036\n"
     ]
    }
   ],
   "source": [
    "def gaussian(x,y):\n",
    "    return math.e**(((x[0]-y[0])**2 + (x[1]-y[1])**2) / -2)\n",
    "\n",
    "print 'w_1:', var(gaussian, X, W[0])\n",
    "print 'w_2:', var(gaussian, X, W[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 2\n",
    "\n",
    "Load all data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Loading training set\n",
    "a = np.genfromtxt(\"assign3-train.txt\", delimiter=\" \", dtype=[('x_train', 'S9'), ('y_train', float)])\n",
    "a = np.sort(a, order='y_train')\n",
    "a[:] = a[::-1]\n",
    "\n",
    "x_train = np.array([a[i][0] for i in xrange(a.shape[0])])\n",
    "y_train = np.array([a[i][1] for i in xrange(a.shape[0])])\n",
    "\n",
    "# Loading test set\n",
    "a = np.genfromtxt(\"assign3-test.txt\", delimiter=\" \", dtype=[('x_test', 'S9'), ('y_test', float)])\n",
    "a = np.sort(a, order='y_test')\n",
    "a[:] = a[::-1]\n",
    "\n",
    "x_test = np.array([a[i][0] for i in xrange(a.shape[0])])\n",
    "y_test = np.array([a[i][1] for i in xrange(a.shape[0])])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (a)\n",
    "\n",
    "Implement a function that calculates a kernel over fixed-length strings,\n",
    "\n",
    "$$ k: \\Sigma^d \\times \\Sigma^d \\rightarrow \\mathbb{R},$$\n",
    "\n",
    "which counts the number of coincidences between two strings.\n",
    "\n",
    "__R/__ We define the kernel function as:\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "k(s_1 ... s_d,t_1 ... t_d) &=& \\sum_{i=1}^d equal(s_i,t_i) \\\\\n",
    "equal(s,t) &=& \\begin{cases} 1 &\\mbox{if } s = t \\\\ \n",
    "0 & \\mbox{if } s \\neq t \\end{cases} \n",
    "\\end{eqnarray*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "AATTAAAAG GTGTGTAAC\n",
      "Coincidences: 3.0\n"
     ]
    }
   ],
   "source": [
    "def str_kernel(a, b):\n",
    "    result = 0\n",
    "    for i in xrange(9):\n",
    "        if a[i] == b[i]:\n",
    "            result += 1\n",
    "    return result * 1.0\n",
    "\n",
    "# test kernel\n",
    "print x_train[0], x_train[1]\n",
    "print \"Coincidences:\", str_kernel(x_train[0], x_train[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (b)\n",
    "\n",
    "Implement Kernel Ridge Regression (KRR).\n",
    "\n",
    "__R/__ The KRR could be described as:\n",
    "\n",
    "\\begin{equation*}\n",
    "g(x) = y^\\prime (K+\\lambda I_n)^{-1} \\left[ \\begin{matrix}\n",
    "k(x,x_1) \\\\\n",
    "\\vdots \\\\\n",
    "k(x,x_n)\n",
    "\\end{matrix} \\right]\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class KRR:\n",
    "    def __init__(self, X_train, y_train, kernel, regularization=1.0):\n",
    "        self.X_train = X_train\n",
    "        self.train_size = X_train.shape[0]\n",
    "        self.kernel = kernel\n",
    "        size = X_train.shape[0]\n",
    "        K = np.array([[kernel(X_train[i], X_train[j]) for i in xrange(size)] for j in xrange(size)])\n",
    "        self.alpha = np.dot(np.linalg.inv(K + regularization*np.eye(size)), y_train).T\n",
    "\n",
    "    def predict(self, X):\n",
    "        size_predict = X.shape[0]\n",
    "        K = np.array([[self.kernel(X[i], self.X_train[j]) for i in xrange(size_predict)] for j in xrange(self.train_size)])\n",
    "        return np.dot(self.alpha, K).T\n",
    "\n",
    "    # R^2 score for regression\n",
    "    def score(self, X, y):\n",
    "        predicted = self.predict(X)\n",
    "        u = np.square(y - predicted).sum()\n",
    "        v = np.square(y - y.mean()).sum()\n",
    "        return 1 - (u / v)\n",
    "    \n",
    "    # Square error for regression\n",
    "    def sqr_error(self, X, y):\n",
    "        predicted = self.predict(X)\n",
    "        l = len(y)\n",
    "        err = sum([(predicted[i] - y[i])**2 for i in xrange(l)])\n",
    "        return err / l"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (c)\n",
    "\n",
    "Training the KRR model with the training set"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Training R² score: 0.987792375055\n",
      "Training square error: 0.591611921351\n"
     ]
    },
    {
     "data": {
      "image/png": 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Gj+e7vQ5n1l+f49Ovm/P997B8edBjd8oU2GMPOHrjs5yd/xhtCzfGrgElK2eT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\nFwUMERGJy/8HR/cPqiLluQQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10470c290>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "classifier = KRR(x_train, y_train, str_kernel)\n",
    "sc = classifier.score(x_train, y_train)\n",
    "sr = classifier.sqr_error(x_train, y_train)\n",
    "\n",
    "print \"Training R² score:\", sc\n",
    "print \"Training square error:\", sr\n",
    "\n",
    "y_pred = classifier.predict(x_train)\n",
    "plt.plot(range(y_pred.shape[0]), y_pred, 'r.')\n",
    "plt.plot(range(y_pred.shape[0]), y_train, 'b-')\n",
    "plt.xlabel(\"String label\")\n",
    "plt.ylabel(\"Prediction\")\n",
    "plt.title(\"Training data\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (d)\n",
    "\n",
    "Prediction with the test set"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Test R² score: 0.980099835919\n",
      "Test square error: 1.02530848376\n"
     ]
    },
    {
     "data": {
      "image/png": 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IGzak8KFkZT7ka2pJiEjhyHAdwurVcNppsGlTaIS0bJmDGIuYuptEpOxt3AjD\nh4eWxZQpYRtWCdTdJCJlr1EjuPXWsBnSoYeGqbNf78kt9aaWhIiUrHHj4Morw2K9vfeG9u2hXbvw\ns317OPhg2HffuKPMn5LobjKzUcAFwMfRoWvc/Yka5yhJiEjK1qwJe1y8+254LF0K77wTps7uuScM\nGQInnxxWeJeyUkkSI4Ev3f3GJOcoSYhIxjZtColiwoSw3mKnnUKyOPnkUDuqKIoO1kMpJYnV7v7H\nJOcoSYhIVm3eDC+/HBLGxIlhptRhh4UNlPr0CfWjirbEeaSUksS5wOfAq8CV7r6qxjlKEiKSO5WV\nLJu7khe/6slL372CF1/djkWLoFu3sN7vggvC2EaxKZokYWbTgD1qeWsEMJMt4xHXAW3c/fwan1eS\nEJHcqWXl9+rVoaUxZQqMGQNHHQWXXx5aG2YURS2odJJEo1wFk4y7H5vKeWZ2FzC5tvdGjRr19fOK\nigoqKiqyEZqISK0rv5s3D5slHX00XHst/PWvcPbZYRzjiitgyMJ3aDI9SiyVlQVRC6qqqoqqqqqM\nrlGI3U1t3H159PwK4GB3P6PGOWpJiEjupLjye9Mm+Mc/4OabYeGLnzB83R+4pNtMdnx2Usm0JAox\nSdwHdAMceBe4yN1X1DhHSUJECsqc6V/w+zPn8MTqPlx8SQMuuwx23TXuqLZWEkkiFUoSIpKSGMYJ\n3nkHfv97ePhhOOssuOoqaNs251+bEpXlEBFJtHhxGICeOjUkjDzYd1+47TZ44w1o0iTMiDr//LDq\nuxgpSYhI6Uqz9Hg27LlnaFEsWRLWYFRWwjc6QCorw0yqgQPDOEgBUpIQkdJVAHtGtG4Nt98OixaF\ncLYSQ0unvjQmISKSB6+9BscfD7Nnh1YGkPfd8TRwLSJSwK69dsuCPDMy3mSpvpQkREQK2IYN0Ls3\nXHJJKO2Rb0oSIiIF7o034Mgjw2ZI+a7/pCmwIiIF7oADwtqJ884rjh3zlCRERPLsqqtg7Vq45ZYk\nJxXI9Fh1N4mIxGDx4lBB9qWXoFOnWk6opRJtptTdJCJSiGppFXTqBL/4Ragku2ZNLZ+JcSFgIrUk\nRERyrY5WwebNcM45Yd/tyZNrzILNwfRYtSRERApRHa2CBg3CvhS9ekHfvrB8ecJnWrUKySTmkuNK\nEiIiuZakPEiDBnDjjTB0aNhL+623YoqxDupuEhEpEHfcAaNGhRXZ3btn//pFs32piIh8U2Ul7Lwz\nHHccjB95dbdOAAAIE0lEQVQP/frFHZG6m0RECsrJJ8O4caF36vHH445G3U0iIgVpxgw46aSwH0XL\nltm5pmY3iYiUiEMPhQEDwsZFcVJLQkSkQL33HvToEYoCtmmT+fVUBVZEpMRcdRWsXh12t8uUkoSI\nSIn57DPo3BleeCH8zITGJERESkzr1qE18fOfx/P9akmIiBS4tWtDQcDx48POdulSS0JEpAQ1bRr2\nx/7pTyHffx8rSYiIFIGzzw7jE1Om5Pd7lSRERIpAw4Zwww1w9dWwaVOSE7O8o52ShIhIkRg0KAxk\n33dfkpMWLw57V0ydGhJGhmJJEmZ2ipnNN7NNZtajxnvXmNkSM1toZv3jiE9EpBCZwe9+B78Y/hkr\n+wxi7XEnsW7FKtavhw0bQgtjc9MdcMjajnZxtSTmAYOB5xMPmllX4DSgKzAAuNXM1NpJoqqqKu4Q\nCobuxRa6F1uU2r3o3RuObTGTvV56hNZPPUTLPXegefMwuN2kCTR8YgorTzyn1r0r0hHLL2B3X+ju\ni2t560TgIXff4O5LgbeAQ/IaXJEptf8BMqF7sYXuxRaleC/u6T6aNezA2l59Wffpf1i/HjZuDC0J\nd2g96d6s7WhXaH+l7wl8kPD6A2CvmGIRESlMSXa6y7acbTpkZtOAPWp56+fuPrkel9KqORGRRNX7\nX+dBrCuuzexZ4Ep3fz16fTWAu/8mev0EMNLdX67xOSUOEZE0FOP2pYkBPwaMNbMbCd1MHYFXan6g\nvv9IERFJT1xTYAeb2TKgNzDFzKYCuPsC4BFgATAV+JGKNImIxKcoC/yJiEh+FNrspm0yswHRQrsl\nZvazuOPJJzO7x8xWmNm8hGOtzWyamS02s6fMLLdTHQqEmbU1s2ejRZlvmNmPo+Nldz/MbHsze9nM\nZpvZAjO7ITpedveimpk1NLNZZjY5el2W98LMlprZ3OhevBIdq9e9KKokYWYNgdGEhXZdgdPNbL94\no8qrewn/9kRXA9PcvRPwz+h1OdgAXOHu+xO6LS+N/lsou/vh7l8BR7p7N+Ag4EgzO5wyvBcJLiN0\nW1d3lZTrvXCgwt27u3v1mrN63YuiShKEhXVvuftSd98AjCMswCsL7j4dWFnj8AnAmOj5GOCkvAYV\nE3f/t7vPjp6vBt4kTHYo1/uxJnraBGhI+O+kLO+FmX0LGAjcxZaJMWV5LyI1J/rU614UW5LYC1iW\n8FqL7WB3d18RPV8B7B5nMHEws3ZAd+BlyvR+mFkDM5tN+Dc/6+7zKdN7AdwE/ATYnHCsXO+FA0+b\n2atmdmF0rF73ohCmwNaHRtmTcHcvtzUkZtYceBS4zN2/NNvyR1M53Q933wx0M7MdgSfN7Mga75fF\nvTCzQcBH7j7LzCpqO6dc7kWkj7svN7NdgWlmtjDxzVTuRbG1JP4FtE143Zaty3iUoxVmtgeAmbUB\nPoo5nrwxs8aEBHG/u0+KDpft/QBw98+BKUBPyvNeHAacYGbvAg8BR5nZ/ZTnvcDdl0c/PwYmErrs\n63Uvii1JvAp0NLN2ZtaEUDH2sZhjittjwNnR87OBSUnOLRkWmgx3Awvc/eaEt8rufpjZLtUzVMys\nKXAsMIsyvBfu/nN3b+vu7YGhwDPufhZleC/MrJmZtYie7wD0J1Tgrte9KLp1EmZ2PHAzYXDubne/\nIeaQ8sbMHgL6AbsQ+hJ/AfydsABxb2ApcKq7Z74dVYGLZu88D8xlSzfkNYQV+mV1P8zsQMIAZIPo\ncb+7/97MWlNm9yKRmfUjlP05oRzvhZm1J7QeIAwtPOjuN9T3XhRdkhARkfwptu4mERHJIyUJERGp\nk5KEiIjUSUlCRETqpCQhIiJ1UpIQEZE6KUlIWTCzEVFJ8TlR2eSDo+OXRwvQ6vrcndmqNGxmq7fx\nfrvEMvApXvOvZnZyZpGJ1K3YajeJ1JuZHQp8D+ju7huixUTbRW9fBtwPrK3lcw3c/cKaxzOQi0VJ\nnqPrigBqSUh52AP4JCovj7t/FhU9+zGwJ/Csmf0Twl/7ZvaHqKLqoWZWZWY9Et77VbS5zwwz2y06\n3sHMZkabu/zKzL5MFoyZNTezp83stegzJyS83cjMHog2Dxpf3coxs55RLK+a2RPVtXeqL5mtGyVS\nk5KElIOngLZmtsjMbjGzvgDu/mfgQ8KmLEdH5zYDZrp7N3d/ka3/Sm8GzIg293keqG5l/Am4yd0P\nYutS9nVZCwx2957AUcAfE97rDNzi7l2BL4AfmVkj4C/Aye7ei7D51PX1vAciaVGSkJLn7v8hVEWt\nBD4GHjazs+s4fROhsmxt1rv7lOj5a0C76HlvYHz0/KEUQmoA3GBmc4BpwJ7VrRJgmbvPiJ4/ABxO\nSBz7E/YFmAWMQPuoSJ5oTELKQrTfwnPAc9Hg8Nls2Z0r0Vded0GzDQnPN5P+/z/DCEUae7j7pqis\n9fbVoSacZ9FrA+a7+2Fpfp9I2tSSkJJnZp3MrGPCoe6E6pcAXwItM/yKmcCQ6PnQFM5vSdgYZ1O0\nOdA+Ce/tbWa9o+dnANOBRcCu1cfNrLGZdc0wZpGUKElIOWgO/NXM5kddPF2AUdF7dwBPVA9ck3ym\nkNd4Xv36cuC/o8HuDsDn2/j8g0AvM5sLnEXYn7vaIuBSM1sA7AjcFg24DwF+G33HLODQOuISySqV\nChfJkJk1dfe10fOhwGnuPjjmsESyQmMSIpnraWajCWMHK4HzYo5HJGvUkhARkTppTEJEROqkJCEi\nInVSkhARkTopSYiISJ2UJEREpE5KEiIiUqf/D8AbMoAl0ve5AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1077f2650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sc_test = classifier.score(x_test, y_test)\n",
    "sr_test = classifier.sqr_error(x_test, y_test)\n",
    "\n",
    "print \"Test R² score:\", sc_test\n",
    "print \"Test square error:\", sr_test\n",
    "\n",
    "y_pred = classifier.predict(x_test)\n",
    "\n",
    "plt.plot(range(y_pred.shape[0]), y_pred, 'r.')\n",
    "plt.plot(range(y_pred.shape[0]), y_test, 'b-')\n",
    "plt.xlabel(\"String label\")\n",
    "plt.ylabel(\"Prediction\")\n",
    "plt.title(\"Test data\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As is possible to see in the __Testing data__ plot, the error is really close to the original traning data error, so the model is consistent for those values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (e)\n",
    "\n",
    "Build a new kernel, $k^\\prime$, composing the kernel $k$ with more complex kernel (polynomial, Gaussian, etc). Repeat steps (b) and (c). For instance, the new kernel may be defined as:\n",
    "\n",
    "$$ k^\\prime (x,y) = (k(x,y)+1)^d $$\n",
    "\n",
    "where $d$ is positive integer exponent.\n",
    "\n",
    "__R/__ For the exercise the kernels will be\n",
    "\n",
    "<ol type=\"I\">\n",
    "<li> $k^\\prime(x, y) = k(x,y)^2$ </li>\n",
    "<li> $k^\\prime(x, y) = (k(x,y) + 1)^3$ </li>\n",
    "<li> $k^\\prime(x, y) = e^{(k(x,x)-2k(x,y)+k(y,y)) / \\, 2}$ </li>\n",
    "</ol>\n",
    "\n",
    "So, for $k^\\prime(x, y) = k(x, y)^2$ the results are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Training R² score: 0.999950895498\n",
      "Training square error: 0.00237972651175\n"
     ]
    },
    {
     "data": {
      "image/png": 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JpXbNvayZuZxkdpJx+m7qfLuUEZ/fSH8mc1jmMRrzUQUpYYhIiR0oeRSWRJJz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10824cd90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Test R² score: 0.933053825591\n",
      "Test square error: 3.44924194076\n"
     ]
    },
    {
     "data": {
      "image/png": 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2+vXVnpML04jnw3xUqm4SkcKybRtceinMmQMzZsC++1b6sZxqkM8StUmIiAC4\nw29/C2PHwhNPhHEWGVwQKV8oSYhIXkt7l9ApU+DKK8NgvXbtoEMHaN8+/OzQAQ47LCzBWiQKIkmY\n2RjgfGB1dOhad3+mwmeUJEQKUMa6hG7cGNa4WLYsbMuXw9//Di+/DPvsEyYkHDw4jPAuYMkkiVxc\n49qBW9391rgDEZHsylhjcpMm0LVr2BJt3x4SxeOPh0kHW7QIyWLw4DB3VA5NOhiXXCxJjAY2uPvv\nqvmMShIiBSjWxuSyMnj99ZAwpk4Nkw4eeST06RO2nj1hl12yHFR6FUp102jgHOBL4E3gSndfV+Ez\nShIiklkrVsArr8Crr4afS5ZA9+6hTuz880PbRp7JmyRhZrOA1pW8NQp4jR3tETcCbdz9vArnK0mI\nSHZt2BBKGjNmwIQJcMwxcPnlobSRJ9VSeZMkasvM2gPT3f3gCsd99OjRX78uKSmhpKQkq7GJSBH7\n6iv405/gjjtCO8YVV4TG70aN4o5sJ6WlpZSWln79+vrrr8//JGFmbdx9ZbR/BXCYu59V4TMqSYhI\n/LZvh7/8BW6/Hd57Lwziu/ji2GekrUpBlCTM7EGgO6GX0zLgQndfVeEzShIiklvmzw8D+J55Bi66\nCC67DFq1ijuqnRREkqgNJQkRyVl//3tIFo88AsOHw1VXQdu2cUcFaBZYEZH47b8/3HMPvPNOaKPo\n3h3OOy+M+s5DKkmIiGTSmjVhapD168MYjBh7QqkkISKSa1q25Ef17mXZM0v4TffJrFtX8ym5RElC\nRCTDFn+4C0M2TuCHC67g6uGfxh1OnShJiIhkWJMm8DY9mbrPJYz99/lhKvM8oSQhIpJh5cuUnjHv\nOhquWQUPPBB3SLWmhmsRkWx65x04+mj429+yPv+TGq5FRHLdt74Vxk6ce26YeTbHKUmIiGTbVVfB\npk1w111xR1IjVTeJiMRh6dIwg+yrr0KnTlm5paqbRETyRadO8ItfwIgRYXnVHKUkISISl//+bzjw\nQDjhBHJ1lJ2ShIhIXOrVC+tS9OoF/frBypVxR/QflCREROJUrx7ceisMHRrW0v7gg7gj2kmDuAMQ\nESl6ZnDddbDnnqFEMWMG9OiR1KVGjgxt4k2ahEF8zZunFppKEiIiuWLkSLjzztBG8eKLSV1i6dJw\n6syZ4XKpUpIQEcklgwfDlClhHo+nn67z6U2ahJ+9esG4camHo3ESIiK5aM4cOPVUeP992G23Wp+2\nbl0oQYyPLynRAAAH8klEQVQb959VTVq+VESkkIwYAe3awY037nQ42XYHJQkRkULy0Udw6KFhUsA2\nbb4+XFKyo8nitNPg0UdrdzmNuBYRKRAjR0LJiP14Yvdz2Hzd9Tu9l+52h+qoJCEikoPKSwstWMPy\nRp3ZbcHL0LkzUH27Q3VUkhARKRDlpYWOvVrS8NqrwjiKSPPmoYop1TEQtaGShIhIDtqptLDLpjAh\n4GOPQe/eSV9TDdciIoVq/Pgwz9OLL4YR2klQdZOISKEaMQLWrAlTdmSRkoSISD6oXx9uuQWuuQa2\nb8/abZUkRETyxcCB0LIlPPhg1m4ZS5Iws9PMbJGZbTezQyu8d62ZvW9m75nZ8XHEJyKSk8zgN78J\nK9qtXRvWyd68GbZsga1bQwmjrAzS2GYbV0liITAIeCnxoJl1A84AugEnAnebmUo71SgtLY07hJyh\nZ7GDnsUOBfcseveG/v1h331DqWK33aBpU2jcGBo1CtVSa9em7XaxfAG7+3vuvrSSt04BHnb3re6+\nHPgAODyrweWZgvsPkAI9ix30LHYoyGcxfnxYFzuxJLFtWyhJuIfkkSa59lf6PsAnCa8/AfaNKRYR\nkaKXsZXpzGwW0LqSt65z9+l1uJQGRIiIxCTWwXRm9gJwpbu/Hb2+BsDdfxW9fgYY7e6vVzhPiUNE\nJAl1HUyXC2tcJwb8FDDZzG4lVDMdCLxR8YS6/pIiIpKcuLrADjKzFUBvYIaZzQRw98XAo8BiYCbw\nI82/ISISn7ycu0lERLIj13o31cjMTowG2r1vZlfHHU82mdl4M1tlZgsTjrU0s1lmttTMnjWzLEwe\nHD8za2tmL0SDMt8xsx9Hx4vueZjZrmb2upnNM7PFZnZLdLzonkU5M6tvZnPNbHr0uiifhZktN7MF\n0bN4IzpWp2eRV0nCzOoDYwkD7boBZ5pZ13ijyqo/En73RNcAs9y9E/B/0etisBW4wt0PIlRbXhL9\nWyi65+Hu/waOdvfuwCHA0WbWlyJ8FgkuI1Rbl1eVFOuzcKDE3Xu4e/mYszo9i7xKEoSBdR+4+3J3\n3wpMIQzAKwruPhuoOJTyZGBCtD8BODWrQcXE3f/p7vOi/Q3Au4TODsX6PDZGu42A+oR/J0X5LMzs\nm8AA4H52dIwpymcRqdjRp07PIt+SxL7AioTXGmwHe7v7qmh/FbB3nMHEwczaAz2A1ynS52Fm9cxs\nHuF3fsHdF1GkzwK4DfgpUJZwrFifhQPPmdmbZnZBdKxOzyIXusDWhVrZq+HuXmxjSMysKfAEcJm7\nf2UJi7EU0/Nw9zKgu5ntDvzVzI6u8H5RPAszGwh85u5zzaykss8Uy7OI9HH3lWbWCphlZu8lvlmb\nZ5FvJYl/AG0TXrdl52k8itEqM2sNYGZtgM9ijidrzKwhIUFMdPdp0eGifR4A7v4lMAPoSXE+iyOB\nk81sGfAwcIyZTaQ4nwXuvjL6uRqYSqiyr9OzyLck8SZwoJm1N7NGhBljn4o5prg9BYyI9kcA06r5\nbMGwUGR4AFjs7rcnvFV0z8PM9izvoWJmjYH+wFyK8Fm4+3Xu3tbdOwBDgefdfThF+CzMrImZNYv2\nvwEcT5iBu07PIu/GSZjZScDthMa5B9z9lphDyhozexg4CtiTUJf4C+BJwgDEdsBy4HR3XxdXjNkS\n9d55CVjAjmrIawkj9IvqeZjZwYQGyHrRNtHdf2tmLSmyZ5HIzI4iTPtzcjE+CzPrQCg9QGhamOTu\nt9T1WeRdkhARkezJt+omERHJIiUJERGpkpKEiIhUSUlCRESqpCQhIiJVUpIQEZEqKUlIUTCzUdGU\n4vOjaZMPi45fHg1Aq+q8P6RrpmEz21DD++0Tp4Gv5TX/ZGaDU4tMpGr5NneTSJ2Z2RHA94Ae7r41\nGky0S/T2ZcBEYFMl59Vz9wsqHk9BJgYleYauKwKoJCHFoTXweTS9PO6+Jpr07MfAPsALZvZ/EP7a\nN7P/jWZUPcLMSs3s0IT3fhkt7jPHzPaKjnc0s9eixV1+aWZfVReMmTU1s+fM7K3onJMT3m5gZg9F\niwc9Vl7KMbOeUSxvmtkz5XPvlF8yXQ9KpCIlCSkGzwJtzWyJmd1lZv0A3P33wKeERVmOjT7bBHjN\n3bu7+yvs/Fd6E2BOtLjPS0B5KeMO4DZ3P4Sdp7KvyiZgkLv3BI4BfpfwXmfgLnfvBqwHfmRmDYA7\ngcHu3ouw+NRNdXwGIklRkpCC5+7/IsyKOhJYDTxiZiOq+Ph2wsyyldni7jOi/beA9tF+b+CxaP/h\nWoRUD7jFzOYDs4B9ykslwAp3nxPtPwT0JSSOgwjrAswFRqF1VCRL1CYhRSFab+FF4MWocXgEO1bn\nSvRvr3pCs60J+2Uk///nbMIkjYe6+/ZoWutdy0NN+JxFrw1Y5O5HJnk/kaSpJCEFz8w6mdmBCYd6\nEGa/BPgK2C3FW7wGDIn2h9bi87sRFsbZHi0OtF/Ce+3MrHe0fxYwG1gCtCo/bmYNzaxbijGL1IqS\nhBSDpsCfzGxRVMXTBRgTvTcOeKa84Zrqewp5hf3y15cDP4kauzsCX9Zw/iSgl5ktAIYT1ucutwS4\nxMwWA7sD90QN7kOAX0f3mAscUUVcImmlqcJFUmRmjd19U7Q/FDjD3QfFHJZIWqhNQiR1Pc1sLKHt\nYC1wbszxiKSNShIiIlIltUmIiEiVlCRERKRKShIiIlIlJQkREamSkoSIiFRJSUJERKr0/7dJXjT7\n0rJEAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1084a0150>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def square_kernel(a, b):\n",
    "    return (str_kernel(a, b)) ** 2\n",
    "\n",
    "classifier = KRR(x_train, y_train, square_kernel)\n",
    "sc = classifier.score(x_train, y_train)\n",
    "sr = classifier.sqr_error(x_train, y_train)\n",
    "\n",
    "print \"Training R² score:\", sc\n",
    "print \"Training square error:\", sr\n",
    "\n",
    "y_pred = classifier.predict(x_train)\n",
    "plt.plot(range(y_pred.shape[0]), y_pred, 'b.')\n",
    "plt.plot(range(y_pred.shape[0]), y_train, 'r-')\n",
    "plt.title(\"Training data\")\n",
    "plt.xlabel(\"String label\")\n",
    "plt.ylabel(\"Prediction\")\n",
    "plt.show()\n",
    "\n",
    "sc_test = classifier.score(x_test, y_test)\n",
    "sr_test = classifier.sqr_error(x_test, y_test)\n",
    "\n",
    "print \"Test R² score:\", sc_test\n",
    "print \"Test square error:\", sr_test\n",
    "\n",
    "y_pred = classifier.predict(x_test)\n",
    "\n",
    "plt.plot(range(y_pred.shape[0]), y_pred, 'b.')\n",
    "plt.plot(range(y_pred.shape[0]), y_test, 'r-')\n",
    "plt.title(\"Test data\")\n",
    "plt.xlabel(\"String label\")\n",
    "plt.ylabel(\"Prediction\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results for $k^\\prime(x, y) = (k(x,y) + 1)^3$ are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Training R² score: 0.999999626008\n",
      "Training square error: 1.81246069907e-05\n"
     ]
    },
    {
     "data": {
      "image/png": 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yJQwRKZa9TV3yfNrzbL5hKP9qPYZt1WuTWi+J75alsCW1EZmjTVVVcUwJQ0RK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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105cf3090>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Test R² score: 0.86813739003\n",
      "Test square error: 6.79390642924\n"
     ]
    },
    {
     "data": {
      "image/png": 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ZNWgAt98eNkM67LAwdTbak1vqrkHSAYiIlIttkNwMLr88lPs46aSwWG+ffaBT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\nIlItJQkREamWkoSIiFTr/wFdgJjG1cb0xAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10886f690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def polynomial_kernel(a, b):\n",
    "    return (str_kernel(a, b) + 1) ** 3\n",
    "\n",
    "classifier = KRR(x_train, y_train, polynomial_kernel)\n",
    "sc = classifier.score(x_train, y_train)\n",
    "sr = classifier.sqr_error(x_train, y_train)\n",
    "\n",
    "print \"Training R² score:\", sc\n",
    "print \"Training square error:\", sr\n",
    "\n",
    "y_pred = classifier.predict(x_train)\n",
    "plt.plot(range(y_pred.shape[0]), y_pred, 'b.')\n",
    "plt.plot(range(y_pred.shape[0]), y_train, 'r-')\n",
    "plt.title(\"Training data\")\n",
    "plt.xlabel(\"String label\")\n",
    "plt.ylabel(\"Prediction\")\n",
    "plt.show()\n",
    "\n",
    "sc_test = classifier.score(x_test, y_test)\n",
    "sr_test = classifier.sqr_error(x_test, y_test)\n",
    "\n",
    "print \"Test R² score:\", sc_test\n",
    "print \"Test square error:\", sr_test\n",
    "\n",
    "y_pred = classifier.predict(x_test)\n",
    "\n",
    "plt.plot(range(y_pred.shape[0]), y_pred, 'b.')\n",
    "plt.plot(range(y_pred.shape[0]), y_test, 'r-')\n",
    "plt.title(\"Test data\")\n",
    "plt.xlabel(\"String label\")\n",
    "plt.ylabel(\"Prediction\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results for $k^\\prime(x, y) = e^{(k(x,x)-2k(x,y)+k(y,y)) / \\, 2}$ are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Training R² score: 0.999998707267\n",
      "Training square error: 6.26490779187e-05\n"
     ]
    },
    {
     "data": {
      "image/png": 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H2ie0DzpcKWNKGCJSIgebuuSptKf44fpR/L3tC+ypWZfUBkl8sTaFXalNyJxg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\nxp59N1tz/QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10860ced0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Test R² score: -188.301862093\n",
      "Test square error: 9753.32687732\n"
     ]
    },
    {
     "data": {
      "image/png": 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0og2SJgOkc4M9W3B+GkbSASSB49qIuD5N7tjrARARW4CfA7PpzGvxZmC+pCeA\nZcBfSrqWzrwWRMQz6eMfgZ+QVP3nuhbtEjx+SzLT7gxJo0lm5r2h4DwV7QbgwnT9QpLZi9teOgvz\nt4GHIiI77U3HXQ9Jh5V7zEgaA7ydZEBux12LiLg8IqZFxNHAecAvI+ICOvBaSOqSND5dHwucBjxA\nzmvRNuM8JL2DZI6skcC3I+KqgrPUMJKWAacAh5HUVf498FNgOTAdWAOcExGbi8pjo6S9ie4E7qev\nOnMhycwEHXU9JB1P0vA5Il2ujYgvSJpIh12LLEmnAB+PiPmdeC0kHU1S2oCk6eJfI+KqvNeibYKH\nmZk1TrtUW5mZWQM5eJiZWW4OHmZmlpuDh5mZ5ebgYWZmuTl4mJlZbg4e1tEkXZFO3X5fOj31G9L0\nj6QD6wZ63v+q1czNkp7fz/4Z2en2h3jOayS9a3g5MxtYu8xtZZabpDcB/xk4ISJ600FSB6a7LwWu\nBXZWeN6IiHh///RhqMdgq6jTec0Alzyss00GNqbT+BMRz6WTxV0CTAVul/QLSEoHkv4pnaH2TZJ6\nJL0+s+8f05su3SXpiDT9lZLuTm+684+Stg2WGUnjJN0m6Xfpc+Zndo+SdF16U6cflktFkmanefmt\npJvLcxOVT1mrC2XWn4OHdbJbgWmSHpX0TUlvBYiIrwFPk9ws523psV3A3RHxuoj4v+z7q74LuCu9\n6dKdQLlU8lXgyxHxGva9ZcBAdgJnRsRs4C+BL2b2/TnwzYiYBWwFLpY0Cvg68K6ImENyU7BP57wG\nZlVx8LCOFRHbSWaZXQD8EfiBpAsHOHwPyUy9leyKiJ+n678DZqTrbwR+mK4vG0KWRgBXSboPWAFM\nLZdigKci4q50/TrgZJKAchzJfRlWAVfg+9hYg7jNwzpaer+LO4A70kbpC+m7m1rWCzHwRHC9mfW9\nVP9/9R6SyS1fHxF70unDDypnNXOc0m0BD0bEm6t8PbOqueRhHUvSsZJmZpJOIJlNFGAbcPAwX+Ju\n4Kx0/bwhHH8wyQ2L9qQ3bXpFZt90SW9M198N/Ap4FDi8nC7pAEmzhplnsyFx8LBONg64RtKDaVXR\nq4BF6b7FwM3lBnMG77kU/dbL2x8BPpY2sr8S2LKf5/8rMEfS/cAFJPdfL3sU+LCkh4AJwNVpQ/9Z\nwOfS11gFvGmAfJnVlKdkN6sTSWMiYme6fh5wbkScWXC2zGrCbR5m9TNb0jdI2iY2ARcVnB+zmnHJ\nw8zMcnMtdTXpAAAAKElEQVSbh5mZ5ebgYWZmuTl4mJlZbg4eZmaWm4OHmZnl5uBhZma5/X9absG5\nTKP2hwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1089600d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def gaussian_kernel(a, b):\n",
    "    return math.e**((str_kernel(a,a) - 2*str_kernel(a,b) + str_kernel(b,b))/2)\n",
    "\n",
    "classifier = KRR(x_train, y_train, gaussian_kernel)\n",
    "sc = classifier.score(x_train, y_train)\n",
    "sr = classifier.sqr_error(x_train, y_train)\n",
    "\n",
    "print \"Training R² score:\", sc\n",
    "print \"Training square error:\", sr\n",
    "\n",
    "y_pred = classifier.predict(x_train)\n",
    "plt.plot(range(y_pred.shape[0]), y_pred, 'b.')\n",
    "plt.plot(range(y_pred.shape[0]), y_train, 'r-')\n",
    "plt.title(\"Training data\")\n",
    "plt.xlabel(\"String label\")\n",
    "plt.ylabel(\"Prediction\")\n",
    "plt.show()\n",
    "\n",
    "sc_test = classifier.score(x_test, y_test)\n",
    "sr_test = classifier.sqr_error(x_test, y_test)\n",
    "\n",
    "print \"Test R² score:\", sc_test\n",
    "print \"Test square error:\", sr_test\n",
    "\n",
    "y_pred = classifier.predict(x_test)\n",
    "\n",
    "plt.plot(range(y_pred.shape[0]), y_pred, 'b.')\n",
    "plt.plot(range(y_pred.shape[0]), y_test, 'r-')\n",
    "plt.title(\"Test data\")\n",
    "plt.xlabel(\"String label\")\n",
    "plt.ylabel(\"Prediction\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (a)\n",
    "\n",
    "Getting all data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a = np.genfromtxt(\"train.csv\", delimiter=',')\n",
    "\n",
    "x_train = a[1:, 1:]\n",
    "y_train = a[1:, 0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (b)\n",
    "\n",
    "Choosing classes 8 and 9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "num1 = 8\n",
    "num2 = 9\n",
    "x_sub = [x_train[i] for i in xrange(x_train.shape[0]) if y_train[i] == num1 or y_train[i] == num2]\n",
    "y_sub = [y_train[i] for i in xrange(y_train.shape[0]) if y_train[i] == num1 or y_train[i] == num2]\n",
    "\n",
    "x_sub_train, x_sub_test, y_sub_train, y_sub_test = train_test_split(x_sub, y_sub, test_size=0.4, random_state=42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Performing Cross validation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Training error [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]\n",
      "Test error [85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85, 85]\n",
      "C parameters [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10886fad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn.svm import SVC\n",
    "\n",
    "reg = range(-5, 16)\n",
    "test_error = []\n",
    "train_error = []\n",
    "\n",
    "for i in reg:\n",
    "    val = 2**i\n",
    "    classifier = SVC(C=val, kernel='linear')\n",
    "    classifier.fit(x_sub_train, y_sub_train)\n",
    "    y_pred_train = classifier.predict(x_sub_train)\n",
    "    y_pred_test = classifier.predict(x_sub_test)\n",
    "    error_train = sum([1 if y_pred_train[i] != y_sub_train[i] else 0 for i in xrange(len(y_pred_train))])\n",
    "    error_test = sum([1 if y_pred_test[i] != y_sub_test[i] else 0 for i in xrange(len(y_pred_test))])\n",
    "    train_error.append(error_train)\n",
    "    test_error.append(error_test)\n",
    "\n",
    "print \"Training error\", train_error\n",
    "print \"Test error\", test_error\n",
    "print \"C parameters\", reg\n",
    "plt.plot(reg, train_error, 'r')\n",
    "plt.plot(reg, test_error, 'g')\n",
    "plt.title('Train and test error')\n",
    "plt.xlabel('C values')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The constant results changing the C values, means that the classes are separables and for that the value of C doesn't affect the behaviour of the model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (c)\n",
    "\n",
    "Getting the coefficients of the model, we choose an arbitrary value of C because it doesn't matter to the model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "classifier = SVC(C=0.03125, kernel='linear')\n",
    "classifier.fit(x_sub_train, y_sub_train)\n",
    "\n",
    "# This are the weights of the linear SVM model\n",
    "weights = classifier.coef_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (d)\n",
    "\n",
    "#### I.\n",
    "\n",
    "Arranging the weights with the same shape as the input images."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "weights = weights.reshape((28, 28))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### II.\n",
    "\n",
    "Ploting the weights of the SVM model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ZuR780M7JmLFfSE85BJjwo+uTMTEub8rh1Dg1K44rz0rHvDkvVc7P/D7GZuV6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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108283b90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.pcolor(weights)\n",
    "plt.axis([0, 28, 0, 28])\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### III. \n",
    "\n",
    "Ploting the weights with the diverging colormap"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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v2tZe5FpQPJInflzMzaxE/p752CqTgLkRsTcwr3i9GUl9gCuAccB+wGmS9s1s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\n/H+8Jbs1Xn2TXwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1077f6910>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.pcolor(weights, cmap=\"seismic\")\n",
    "plt.axis([0, 28, 0, 28])\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### IV. \n",
    "\n",
    "Discuss the results.\n",
    "\n",
    "__R/__ As is possible to see, there are several conclussions:\n",
    "\n",
    "* Not all the pixels are important to define the model\n",
    "* The parts where are common parts have lower weight\n",
    "* In this case we are calssifying the 8's and 9's, so the main difference is in the bottom-right of the image, this happens because in this part the 8's close the circle and the 9 remains as a stick."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (e).\n",
    "\n",
    "Upload to kaggle\n",
    "\n",
    "__R/__ We make the submission as __unal-ml__\n",
    "\n",
    "<img src=\"kaggle.jpg\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (a)\n",
    "\n",
    "Building the dataset of spanish and english words."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000 1000\n"
     ]
    }
   ],
   "source": [
    "import csv\n",
    "\n",
    "# This is the size of the data for each language\n",
    "n = 1000\n",
    "\n",
    "english = np.genfromtxt(\"english.txt\", delimiter=\",\", dtype=str)\n",
    "english = english[:, 1]\n",
    "english = [w.lower() for w in english if len(w) >= 4]\n",
    "\n",
    "spanish = []\n",
    "with open('spanish.txt', 'rb') as f:\n",
    "    reader = csv.reader(f)\n",
    "    for row in reader:\n",
    "        if len(row[1]) >= 4:\n",
    "            spanish.append(row[1].lower())\n",
    "\n",
    "english, spanish = english[:n], spanish[:n]\n",
    "print len(english), len(spanish)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (b)\n",
    "\n",
    "Function for the string kernel\n",
    "\n",
    "__R/__ The idea of this kernel is to find the longest contiguous matching subsequence, and returns a percentage of similarity between this values, so it's 1 if is the same string and 0 if there are totally different."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import difflib\n",
    "\n",
    "def string_kernel(s1, s2):\n",
    "    return difflib.SequenceMatcher(None, s1, s2).ratio()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (c) and (d)\n",
    "\n",
    "Train a SVM with precomputed kernel and do cross validation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Preparing the dataset\n",
    "\n",
    "x = []\n",
    "y = []\n",
    "\n",
    "for s in english:\n",
    "    x.append(s)\n",
    "    y.append(0)\n",
    "\n",
    "for s in spanish:\n",
    "    x.append(s)\n",
    "    y.append(1)\n",
    "\n",
    "x = np.array(x)\n",
    "y = np.array(y)\n",
    "\n",
    "x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.33, random_state=42)\n",
    "\n",
    "train_kernel = [[string_kernel(s1, s2) for s1 in x_train] for s2 in x_train]\n",
    "test_kernel = [[string_kernel(s1, s2) for s1 in x_train] for s2 in x_test]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Training error [262, 255, 251, 288, 363, 419, 435, 451, 452, 454, 446, 454, 458, 449, 456, 451, 459, 457, 453, 454, 472]\n",
      "Test error [140, 133, 133, 152, 175, 211, 216, 217, 231, 229, 229, 230, 229, 225, 225, 232, 238, 232, 229, 222, 217]\n",
      "C parameters [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c4d6050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Training the SVM\n",
    "\n",
    "reg = range(-5, 16)\n",
    "test_error = []\n",
    "train_error = []\n",
    "\n",
    "for i in reg:\n",
    "    val = 2**i\n",
    "    classifier = SVC(C=val, kernel='precomputed')\n",
    "    classifier.fit(train_kernel, y_train)\n",
    "    y_pred_train = classifier.predict(train_kernel)\n",
    "    y_pred_test = classifier.predict(test_kernel)\n",
    "    error_train = sum([1 if y_pred_train[i] != y_train[i] else 0 for i in xrange(len(y_pred_train))])\n",
    "    error_test = sum([1 if y_pred_test[i] != y_test[i] else 0 for i in xrange(len(y_pred_test))])\n",
    "    train_error.append(error_train)\n",
    "    test_error.append(error_test)\n",
    "\n",
    "print \"Training error\", train_error\n",
    "print \"Test error\", test_error\n",
    "print \"C parameters\", reg\n",
    "plt.plot(reg, train_error, 'r')\n",
    "plt.plot(reg, test_error, 'g')\n",
    "plt.title('Train and test error')\n",
    "plt.xlabel('C values')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Part (e)\n",
    "\n",
    "#### I.\n",
    "\n",
    "Build the confussion matrix with the C parameter as 0.0625."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x108561cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Train with the best value\n",
    "classifier = SVC(C=2**-4, kernel='precomputed')\n",
    "classifier.fit(train_kernel, y_train)\n",
    "y_pred_test = classifier.predict(test_kernel)\n",
    "\n",
    "# Build confusion matrix\n",
    "from sklearn.metrics import confusion_matrix\n",
    "\n",
    "names = [\"English\", \"Spanish\"]\n",
    "\n",
    "def plot_confusion_matrix(cm, title='Confusion matrix for test set', cmap=plt.cm.Blues):\n",
    "    plt.imshow(cm, interpolation='nearest', cmap=cmap)\n",
    "    plt.title(title)\n",
    "    plt.colorbar()\n",
    "    tick_marks = np.arange(len(names))\n",
    "    plt.xticks(tick_marks, names)\n",
    "    plt.yticks(tick_marks, names)\n",
    "    plt.tight_layout()\n",
    "    plt.ylabel('True label')\n",
    "    plt.xlabel('Predicted label')\n",
    "    \n",
    "conf_mat = confusion_matrix(y_test, y_pred_test)\n",
    "conf_mat = conf_mat.astype('float') / conf_mat.sum(axis=1)[:, np.newaxis]\n",
    "plot_confusion_matrix(conf_mat)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "#### II."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['jack', 'importante', 'quise', 'value', 'imposible', 'duele', 'ancient', 'donde', 'gente', 'sere', 'sientes', 'camp', 'questions', 'plan', 'moral', 'sientate', 'muerte', 'estes', 'irme', 'coche', 'dormir', 'este', 'understood', 'stop', 'diferente', 'ayudarte', 'vere', 'dijiste', 'increible', 'usual', 'quien', 'dije', 'physical', 'conseguir', 'estare', 'linea', 'otras', 'deal', 'corte', 'tren', 'norte', 'reglas', 'grande', 'considerable', 'foto', 'marriage', 'clase', 'siempre', 'conversation', 'places', 'paid', 'llave', 'eres', 'irse', 'ladies', 'llegar', 'miren', 'caliente', 'dire', 'public', 'sentir', 'hotel', 'jesus', 'copy', 'stay', 'realmente', 'supone', 'curious', 'states', 'crimen', 'fall', 'simplemente', 'idea', 'crees', 'mision', 'seguridad', 'understand', 'darle', 'suerte', 'speak', 'creen', 'sale', 'carcel', 'seguir', 'indian', 'quieren', 'corre', 'real', 'bosque', 'tiene', 'came', 'occasion', 'meses', 'dare', 'suficiente', 'hiciste', 'madre', 'natural', 'perder', 'captain', 'probably', 'decirle', 'especially', 'road', 'hacen', 'paragraph', 'information', 'does', 'porque', 'pale', 'entonces', 'interesante', 'horrible', 'several', 'paper', 'iron', 'dicen', 'case', 'condition', 'ustedes', 'seis', 'tonto', 'pain', 'calmate', 'verte', 'dulce', 'esperen', 'ocurre', 'hice', 'problema', 'nombre', 'quedate', 'conocer']\n"
     ]
    }
   ],
   "source": [
    "mistake_words = [x_test[i] for i in xrange(len(y_test)) if y_test[i] != y_pred_test[i]]\n",
    "print mistake_words"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are two cases when there are missclasification:\n",
    "\n",
    "* When the words are really simmilar between spanish and english, and for that is really complex his classification.\n",
    "* Because of the size of the training data is not big enough and a model has his errors, there some outliers words missclassified.\n",
    "\n",
    "This errors could be minimized choosing a better kernel function and training with a biggest training data."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}