{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Sveučilište u Zagrebu<br>\n",
    "Fakultet elektrotehnike i računarstva\n",
    "\n",
    "# Strojno učenje\n",
    "\n",
    "<a href=\"http://www.fer.unizg.hr/predmet/su\">http://www.fer.unizg.hr/predmet/su</a>\n",
    "\n",
    "Ak. god. 2015./2016.\n",
    "\n",
    "# Bilježnica 6: Linearni diskriminativni modeli\n",
    "\n",
    "(c) 2015 Jan Šnajder\n",
    "\n",
    "<i>Verzija: 0.1 (2015-11-11)</i>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "import scipy as sp\n",
    "import scipy.stats as stats\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "%pylab inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sadržaj:\n",
    "\n",
    "* Uvod\n",
    "\n",
    "* Poopćeni linearni model\n",
    "\n",
    "* Geometrija linearnog modela\n",
    "\n",
    "* Višeklasna klasifikacija\n",
    "\n",
    "* Klasifikacija regresijom\n",
    "\n",
    "* Gradijentni spust\n",
    "\n",
    "* Perceptron\n",
    "\n",
    "* Sažetak"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Uvod\n",
    "\n",
    "* **Diskriminativni modeli:** Za razliku od generativnih modela, koji modeliraju\n",
    "$\n",
    "P(\\mathcal{C}_j|\\mathbf{x})\\ \\propto\\ P(\\mathbf{x}|\\mathcal{C}_j) P(\\mathcal{C}_j)\n",
    "$, \n",
    "modeliramo:\n",
    "  * izravno aposteriornu vjerojatnost klase, $P(\\mathcal{C}_j|\\mathbf{x})$, ili\n",
    "  * izravno *diskriminacijsku* (klasifikacijsku) funkciju $h(\\mathbf{x})$\n",
    "  \n",
    " \n",
    "* **Linearan modeli:** Granica je linearna (hiperravnina):\n",
    " \n",
    "$$\n",
    "h(\\mathbf{x}) = \\mathbf{w}^\\intercal\\tilde{\\mathbf{x}}\n",
    "$$\n",
    "\n",
    "\n",
    "* Model je linearan u značajkama $\\mathbf{x}$ $\\Rightarrow$ to daje linearnu granica u <u>ulaznom\n",
    "prostoru</u>\n",
    "\n",
    "\n",
    "* Granica je hiperravnina za koju $h(\\mathbf{x})=0$ ili $h(\\mathbf{x})=0.5$ (ovisno o modelu)\n",
    "\n",
    "\n",
    "* Granicu zovemo **diskriminativna funkcija** (engl. *discriminative functions*) ili **granica odluke**, **decizijska granica** (engl. *decision boundary*)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def h(x, w): return sp.dot(x, w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def plot_decision_boundary(h, boundary=0, margins=None):\n",
    "    x = linspace(-10, 10)\n",
    "    y = linspace(-10, 10)\n",
    "    X1, X2 = np.meshgrid(x, y)\n",
    "    XX = sp.dstack((sp.ones((50, 50)), X1, X2))\n",
    "    plt.contour(X1, X2, h(XX), linecolor='red', levels=[boundary])\n",
    "    if margins!=None:\n",
    "        CS = plt.contour(X1, X2, h(XX), colors=['gray', 'gray'], levels=[margins[0],margins[1]])\n",
    "        plt.clabel(CS, fontsize=9, inline=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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T/nA9fPgcvqqqihweeUcmj4LU1tamvr4+zZs3T1OnTtW8efM+MeEHlcnhOzvD5/DLly8P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\nJA8ACcYkDwAJ9l+vox0Q9OuykwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe4193b6810>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w = [-2, 1, -0.5]\n",
    "plot_decision_boundary(lambda x : h(x, w), margins=(-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ddRuMowBQuHId1wAAIsQgDwAFjEEeAAoYgzwAFDAGeQAoYAzyAFDAGOQBoIAx\nyANAAfv/3+qHHk2J/QsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe4194cac90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w = [2, -1, 0.5]\n",
    "plot_decision_boundary(lambda x : h(x, w), margins=(-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe418c44610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w = [2, -1.5, 0.3]\n",
    "plot_decision_boundary(lambda x : h(x, w), margins=(-1,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Poopćeni linearan model\n",
    "\n",
    "* **Aktivacijska funkcija:** nelinearna funkcija \n",
    "$$\n",
    "f:\\mathbb{R}\\to[0,1]\n",
    "$$ \n",
    "ili\n",
    "$$\n",
    "f:\\mathbb{R}\\to[-1,1]\n",
    "$$\n",
    "\n",
    "\n",
    "* Poopćeni linearan model (engl. *generalized linear model*, GLM):\n",
    "\n",
    "$$\n",
    "    h(\\mathbf{x}) = \\color{red}{f\\big(}\\mathbf{w}^\\intercal\\tilde{\\mathbf{x}}\\color{red}{\\big)}\n",
    "$$\n",
    "\n",
    "$\\Rightarrow$ **Linearna granica** u ulaznom prostoru (premda je $f$ nelinearna)\n",
    "\n",
    "$\\Rightarrow$ Model je **nelinearan u parametrima** (jer je $f$ nelinearna)\n",
    "\n",
    "* Tipične funkcije preslikavanja:\n",
    "  * Funkcija identiteta (tj. kao da nema preslikavanja)\n",
    "  * Sigmoidalna (logistička) funkcija\n",
    "  * Funkcija skoka (step-funkcija)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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fax3bV0q6QaN/eNvks5LukHRuqwMztFvSuu0jtr9r+wu2Xz/twCJnobxK2y/62eLxHZb0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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe4191b2610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = sp.linspace(-5,5, 100)\n",
    "plt.plot(xs, xs);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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kh11reb4NsKrS41LguMonmFkbQnHvBXQF1HZLVBs3hkI+Z07sJCKZUVtHXpeifB9wU2rc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\ni4jkOBVyEZEcp0IuIpLjVMhFRHLc/wGQ5zMW//7bEwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe4190a41d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def sigm(x): return 1 / (1 + sp.exp(-x))\n",
    "\n",
    "plt.plot(xs, sigm(xs));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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JX4yIR20Ppr2eVXC+pKsk7YqIB21/QNJ7Jd3ctvHERMQ1bc/ZvkHSp5rtHmxOWH5fRPzP\nxBbYU9v7s/0qjb4CH7ItjcYaD9veHBFfnOASeyl9/iTJ9q9q9K3yz05kQavvaUkbF9zfqNFRfRq2\nL5D0SUkfi4i7p72eFfRaSdttb5P0HZK+x/ZfRsT1U17XSjmh0YTgweb+XRqFfqxZGt3cLekNkmT7\nMklra4p8SUQ8HhHrIuLSiLhUo0/SVTVFvovtrRp9m7wjIv5v2utZIS9c8Gd7rUYX/O2b8ppWjEdH\nHbdJeiIiPjDt9aykiHhfRGxs/r29VdI/JYq8IuIZScebVkrS1ZI+17b9LP1HabdLut32Y5K+KSnN\nJ2WMjCOBP5e0VtLB5ruWf42I35jukvqJiNO2z17wt0bSbcku+HudpF+R9FnbjzaP7Y6Iv5/imlZL\nxn9z75L08eYg5N8l7WzbkAumACC5WRrdAABWAaEHgOQIPQAkR+gBIDlCDwDJEXoASI7QA0ByhB4A\nkvt/W3maWweKfe0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe4190e4cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xs, sign(xs));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Odabir funkcije $f$ nema utjecaja na linearnost granice, budući da će, očigledno, funkcija $f$ za iste ulazne vrijednosti $\\mathbf{w}^\\intercal\\mathbf{x}$ davati iste vrijednosti $f(\\mathbf{w}^\\intercal\\mathbf{x})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe418d85190>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w = [-4, 2, -1]\n",
    "plot_decision_boundary(lambda x : h(x, w), margins=(-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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MaK2VlJQU/PTTT/D39yflsO7jM3nGDIxGo0F4eDimTJmCZcuWkbKqqhoRENAxH566T2t2\ndjYKCgrg7e0NExMTUhbrPi7yjBkQKSXi4+MBAG5ubqQsjUZi9+5I/OEPDtiwgdaHv3//PlJTU+Hn\n5wcLC9ovC9YzXOQZMyBZWVkoKSnB9u3byas4fv75JbS1qfHRR+tIOe3t7YiJicHmzZvJWwuynuOe\nPGMG4tGjR0hNTcW+ffvIG3Gnpt7FX/96DZmZ9PnwFy5cgJWVFXkpBaYdPpNnzABIKXHu3Dn87ne/\nIy/TW1HRAD+/SJw8qcCECbQpjnfv3kVBQQE2b95MymHa4yLPmAG4efMm6uvrsXLlSlKOStWxT+sf\n/7iEvC5NS0sLoqOj4eHhQd55immPizxjA1xzczPOnz+PLVu2kDe9PngwBUOHDsEHH6wljysrKwsz\nZszA9Om0Tb0ZDffkGRvAujbbWLhwISZMmEDKio+/jVOn8pGTs588Hx4AVq1aBbVaTc5hNFzkGRug\npJSIjY2FhYUFnJycSFkVFQ0IDIzGN9/sgKWlbqY4CiF4+eB+gNs1jA1Qly5dQk1NDby8vEiLfKnV\nGuzeHYk33liKtWt7vogZ7/DWv3GRZ2wAKigoQFZWFnx8fGBqakrK+tOfOrZd/uCD7q9L8/DhQ9y9\nexcAeBXJfo4/SzE2wFRUVCAuLg67d+/GiBG0JX8zMkrwj39kITt7P4yNu3fOV1BQgMzMTFhYWODq\n1atYvHgxJk6cSF4EjekHF3nGBhC1Wo2oqCi4ubnBxsaGlFVV1QgfnwgcO+YJW9vu/bLQaDQoLi6G\ns7MzbG1tUVBQgIKCAjx+/Jg3/+inuF3D2ABy+fJljB49mrxMr1qtga9vBP7wB3u4ufVsiuPQoUOR\nn58PAJg7dy5WrVqFmzdv4urVq6QxMf3gIs/YAFFXV4crV67g1VdfJffBDx1Kg7GxUY/WpZFSwsjI\nCOvWrYNGo0FiYiJ++uknWFpawt3dHc3NzdBoNKRxMd3jdg1jA4CUEnFxcVizZg150+v4+Ns4fjwX\nOTkHut2Hr6ysRGFhISwsLGBjY4NFixahrKwMcXFxmDNnDn788UdMnDiRvCga0z0u8owNACkpKVCp\nVFi+fDkpp6ysHoGB0ThzZke314dvbW1FTEwMVqxYgZqaGrS2tsLIyAjTp0+HtbU1iouLMX/+fN7p\nqZ/iIs9YP5ebm4ubN29i3759pDNllaqjD//OO8uxZk3358M3NjZiwoQJWLhwITQaDUpLS1FRUYGK\nigrY29tj4sSJWo+J6R9/tmKsHyspKUFycjJ8fHzIm2188kk6zMyG4N13V/foecOGDUN9fT1yc3Nh\nZGSEyZMnY/Lkyfj+++9RVVVFGhPTPy7yjPVTDx8+xJkzZ6BQKGBpaUnKSk29i3/9KwcnTyq6vS7N\nkydPoFarYWpqCldXV2RnZyMmJgYAYGtrCwsLCzQ3N5PGxfRP9NUtyUIIybdDM/Z8LS0t+Oqrr7Bi\nxQosWbKElFVb2wR7+/+Lo0c94eIyrVvPaW9vh1KpxIgRIyClxOTJkzF37lycO3cODx8+xNixY1Ff\nXw9vb2/S2FjPCSEgpez29CruyTPWD50/fx52dnbkAt8xHz4S/v4Lu13gu45vZWWFZcuWoby8HGlp\naaipqcHmzZtRWVmJ4cOHk3efYr2D2zWM9TP37t3DnTt34OzsTM46ciQdKpUGR46s79Hzxo4dC1tb\nW5ibm2P69Omws7NDU1MT7t27BxsbG4wYMQImJibk8TH94zN5xvoRlUqF2NhYbNq0CWZmZqSsxMRi\nfPXVdWRn7+/xPq2jRo1Cbm4u2tra0NbWhqqqKixYsADl5eWYPHkyL0o2gHCRZ6wfuXjxIiwtLTF7\n9mxSTllZPfbujUJo6DZYWw/v8fPnzp0LKSWqqqpgamoKd3d31NTUoKKiggv8AMNFnrF+oqioCNnZ\n2di/fz8pR6XSwNs7HO+8swyOjnbdfp6UEu3t7T8vXfzKK69gzpw5MDIywuPHj3HlyhV4eXmRxsZ6\nH8+uYawfqKmpwYkTJ7Br1y5MmjSJlPXBBxeQnV2Jc+f8uj1dMjs7GzU1NaisrMSSJUuwYMECAB2F\nXwiBJ0+eoLm5GWPHjiWNjdH1dHYNX3hlrI81NTUhNDQUrq6u5AKfnHwHx4/nISio+/PhGxoacP36\ndaxevRqurq7Iy8tDfHz8zwX+xx9/hLGxMRf4AYqLPGN9qGsj7vnz5/989qyt6upG7N0bhaAgr26v\nSwN0zIkfPnw4zMzMMH78ePj5+UGlUuHEiROoq6uDRqMh7z7F+g4Xecb6iJQSMTExGDFiBNav79kU\nx2dpNBL+/lEIDFwEJ6epPXrumDFjYGNjg9TUVFRWVsLIyAhbtmzBpEmT0N7eTr4IzPoWF3nG+khe\nXp5ONuIGgD//+SKam9tx+LCjVs9fvnw5Ro8e/fPesWq1GsXFxWhububZNAMcz65hrA80NTUhOTkZ\nfn5+5JuKkpKK8fe/ZyIrq+fz4bsMHToU9vb2uH//Pm7cuIHS0lLMmjULU6f27FMB6394dg1jfUCp\nVGLYsGFwdXUl5ZSX12PJkn8hOHgrNmyYoqPRdSxOxhtz9088u4axfu7OnTsoKSkh9+E75sNH4M03\nl+q0wAPgAm9AuMgz1ovq6+sRHR2NzZs3k2esfPhhCiwsTPD++2t0NDpmiLjIM9ZL2traEBoaiqVL\nl2LGjBmkrPPni3Dq1I0erQ//a7Kzs/Hw4UNSBuu/uMgz1guklIiMjIS1tTVWrVpFyqqsbEBAQDRO\nndoKS0vablH5+fnIyMjAkCE8B8NQcZFnrBckJyejpaUF7u7upCmJarUGu3crceDA4h6tS/M8JSUl\nSEhIgK+vL0aOHEnKYv0XF3nG9CwnJweFhYXYuXMnjI2NSVmff34JarUGBw+uJeW0tbVBqVTCy8sL\nVlZWpCzWv/FnNMb0qLa2FhcuXEBgYCCGDRtGykpLu4e//vUasrL2w9iYdn6WlJQEOzs78rUB1v/x\nmTxjeiKlRGxsLNatW4dx48aRsiorG+DrG4GTJxWYMIHWWikqKsKtW7fg5uZGymEDAxd5xvQkJycH\narWavE9r13z4AwcWw9m5+/u0Pk9zczNiYmLg6enJe7QOEnor8kIINyFEoRDithDiP/R1HMb6o8bG\nRqSkpGDLli0wMqL9M/voo1SYmRnjww9pfXgpJeLi4jB37lxermAQ0UuRF0IYA/gCgBuAuQB8hBBz\n9HEsxvobjUaDqKgoODg4kC9qnjt3G0FBeTh1aiu5D3/jxg3U1NTAycmJlMMGFn2dyS8DUCSlvCel\nbAdwGoCnno7FWL8hpUR8fDyEEORlC8rL6/H730cjJGRbj9aHf56SkhIkJiZi+/bt5AXR2MCiryI/\nHkDpU4/LOr/HmEG7du0aSkpKsH37dlKbRq3WwM8vEm++uRRr104mjenBgwc4c+YMtm7dytMlByF9\nTaHs1vKShw8f/vlrR0dHODo66mk4jOnf7du3cenSJezbtw9mZmakrE8/zYCxsRF5XZqmpiYEBwfD\nyckJ06bRLtqyvpGWloa0tDStn6+XpYaFECsAHJZSunU+fg+ARkr5n0/9DC81zAxGdXU1goKC4O3t\njYkTJ5Ky0tPvwds7Ajk5+2FjM0LrHCklwsLCMHbsWDg7O5PGxPqP/rLUcBaAGUIIOyGEKYBdAGL0\ndCzG+lR7ezvCwsLg5uZGLvA1NU3YvVuJY8c8SQUeAG7evIm6ujrytQE2sOmlyEspVQDeAnAeQAGA\nMCnlD/o4FmN9LT09Hba2tpg/fz4pR63WwMcnAgEBC+HmNp2UVV9fj/j4eHh5efHiY4Oc3v7rSynj\nAcTrK5+x/qC6uhrXr1/HH//4R3LW4cNpEAJa79PapWuD8KVLl2L8eJ7vMNjxr3jGtKTRaHD27Fls\n2LABw4cPJ2UlJBTh2LFcZGfT16XJzMzEkydPsGYNbybCeFkDxrR26dIlGBsbw8HBgZRTVlaPgIAo\nhIRsg5UV7ZfFgwcPkJaWBoVCQV7xkhkGLvKMaeHmzZvIzs7G9u3bSevDq1Qdffh33llOng/f1NSE\nkJAQODs7kxdEY4aDizxjPVReXo5z587Bx8cHI0bQZsB8/HEahg4dgnffXU3KaW9vR2hoKObNmwd7\ne3tSFjMs3JNnrAceP36MsLAweHh4wNrampR14cIdfP31deTkHCDt06rRaBAZGYmxY8fydEn2C3wm\nz1g3tba2IjQ0FCtWrMCsWbNIWdXVjfD3j8KJE16wtqb14b/77js0NzfDw8OD1DpihomLPGPdFBsb\nC1tbW6xcuZKUo1Zr4O8fBX//BeT14R8+fIiLFy/C09OTL7Sy5+Iiz1g33Lp1C+Xl5di0aRP5bPmT\nTzLQ2qrCJ59sIOVoNBoolUqsXbsWY8aMIWUxw8U9ecZ+Q1tbG86dOwcPDw/yMr2JicX4179ykJ29\nH0OG0M6xLl26BBMTEyxfvpyUwwwbn8kz9htSU1NhZ2dH3k2pvLwee/dGITh4K7kPX1lZie+++w6e\nnp7ch2cvxEWesRcoKSlBfn4+XFxcSDnt7Wrs2hWOt99eBkdHO2JWO5RKJdzc3DBq1ChSFjN8XOQZ\n+xV1dXU4c+YMFAoFhg0bRsr66KNUjBhhRp4PL6WEUqmEjY0N5s2bR8pigwP35Bl7jidPniA0NBTr\n168nb7aRmFiMkydvkOfDd2Qlorm5Gbt37+Y2DesWPpNn7BlqtRpnzpzBjBkzsHjxYlJWVVUjAgKi\nEBSkIO/T+t1336GoqAi7du3i5YNZt3GRZ+wpUkrExcXBxMSEvJuSRiOxZ48Sf/iDAzZsmELKKisr\nw8WLF+Hr6wtzc3NSFhtcuMgz9pSsrCxUVFRg27ZtpI24AeDzzy+hpUWFjz5aR8pRqVSIiYmBm5sb\nXnrpJVIWG3z4Mx9jnR4/fozU1FT8/ve/h6mpKSnrwoU7+OtfryEz8zXyfPiUlBSMGzcOr7zyCimH\nDU58Js9Yp/j4eCxbtoy8TG9FRQN271bi1CkFJkwYScq6d+8e8vPz4e7uzhdamVa4yDMG4IcffkBd\nXR1Wr6ZNcVSpNPD2DscbbyyBkxPt5qnW1lZERUVhy5Yt5CmcbPDiIs8GvaamJsTHx8Pd3Z08a+XD\nD1MwbJgJPvhgLXlcCQkJmDZtGmbOnEnOYoMXF3k2qKlUKpw+fRqLFi3C5Mm0nZni428jODgfJ08q\nyPPhr127htLSUri6upJyGOMizwYtKSWio6MxatQo8mYbFRUN+P3vYxAcvBWWlrT58D/++CMuXrwI\nPz8/8gVgxrjIs0ErPT0djx49Ii/ypVZrsHt3JP74xyXkfVrLy8sRExMDb29vni7JdIKLPBuU8vPz\nkZubC29vb/LywX/+80VICXzwwRpSzpMnTxAWFoYtW7Zg/PjxpCzGuvA8eTboVFZWIiEhAf7+/hg+\nnLbkb0ZGCf7+9yxkZ++HsTHtnCkxMRGzZs3C7NmzSTmMPY3P5NmgotFoEBMTAxcXF1hZWZGyqqoa\n4eMTgePHPWFrO4KU9eOPP+LevXvYuHEjKYexZ3GRZ4PK1atXYW5ujgULFpByuubDv/aaA1xdp5Oy\nmpqaEBsbCy8vL5iZmZGyGHsWF3k2aPz000+4dOmSTu4ePXQoFSYmxjh4kDYfXkqJ2NhYLFiwgDyF\nk7Hn4Z48GxQ0Gg1iY2OxcuVK8qbX8fG3ceJEHnJyDpD78Lm5uXj06BG2bdtGymHs1/CZPBsUkpOT\noVKp8Lvf/Y6UU1ZWj8DAaISEbCOvD19RUYHk5GQoFApeH57pDRd5ZvBycnLw448/YufOnTA2NtY6\nR6XSwM8vEm+/vYw8H/6nn37C6dOnsWXLFvIFYMZehIs8M2h3795FSkoKfH19yYt8ffJJOkxMjMj7\ntD558gTBwcFYtWoVT5dkesefEZnBevDgASIiIrBt2zaMHTuWlJWaehdffpmD69dpfXgpJSIiIjBt\n2jQsX76cNCbGuoPP5JlBamlpQWhoKDZs2IApU2hb79XWNmHPHiWOH/eEtTXt5qm8vDw0NzfDxcWF\nlMNYd3GRZwbpwoULmDx5MhwcHEg5arUGPj4R8PdfSJ4P//jxYyQlJcHDw4O8tSBj3cXvNGZwSktL\nUVhYSN4Hl5nSAAASZElEQVSIGwA+/jgdGo3EkSO0VSqllIiKisKKFStgbW1NHhdj3cU9eWZQ1Go1\nYmNj4erqCnNzc1LW+fNF+Prr68jO3k/ep/Xq1atQq9VYtWoVKYexnuIzeWZQvv32W4wcOZK86XVZ\nWT0CAqIRErKV3IevqanBpUuX4OXlxW0a1uv4HccMxq1bt5CZmUletkCl6ujDv/32MqxbZ0caU0tL\nC8LDw+Hk5ES+05YxbXCRZwahuroa0dHR2LVrF0aNGkXKOnw4DcOGmZDnw6vVaoSFhWHKlCmwt7cn\nZTGmLe7JswGvsbERoaGh2LRpEyZMmEDKSk6+g2PHcpGTs5+0T6uUEjExMRg6dChcXV3JC6Ixpi0+\nk2cDWnt7O06fPg17e3vMmzePlFVd3Yi9e6MQFOQFKytaH/7SpUuoq6vD1q1buQ/P+hS/+9iA1bUR\n95gxY7B2LW3JX41GYu/eKAQGLoKT01RSVm1tLa5evYodO3aQtxZkjIqLPBuwvv/+e9TW1sLDw4Pc\nDvn00ww0N7fj8GFHUo5arYZSqcT69evJ1wYY0wXuybMB6cmTJ0hMTMSuXbvIy/QmJRXjn//M0sl8\n+IyMDFhYWGDx4sWkHMZ0hc/k2YCUlJSEOXPmkC+0lpfXw98/CsHBW2FjQ9untaysDNnZ2Tr5ZMGY\nrnCRZwNOWVkZiouL4eTkRMrp2Kc1Am+9tRTr19MWMWtra4NSqcSrr76KESNovywY0yUhpdR9qBCH\nAfwBQG3nt96TUiY88zNSH8dmhk+j0eDRo0fk5YPfey8Z169X4dw5P/J0yejoaEgpoVAoSGNi7LcI\nISCl7PYbVl89eQngL1LKv+gpnw1iRkZG5AKfmFiMkydv4Pr1A6QCDwDp6emora3F3r17STmM6YM+\n2zXclGRk+vi0V1XViICAKJw8qYClJW2f1tzcXOTl5cHHxwempqY6GiFjuqPPIv+2ECJPCPG1EGK0\nHo/DDMzjx49RVlYGKaXOL2Cq1Rrs3h2J115zIPfhS0pKkJycDD8/PwwfTrt5ijF90bpdI4RIAvC8\nhbE/APAPAEc6H38C4L8A7Hv2Bw8fPvzz146OjnB0dNR2OMxAlJaWIikpCS+99BJSU1OxbNkyTJw4\nkbw/a5fPP7+E9nYNDh5cR8ppb29HdHQ0PDw8MG7cOJ2MjbHnSUtLQ1pamtbP18uF1387gBB2AM5K\nKec/832+8Mp+ISkpCZMmTcKsWbNQWFiIwsJC2NraYtmyZeTs1NS78PWNRGbma5gwYSQpKz4+Hs3N\nzdi2bRt5XIz1RE8vvOqlXSOEsHnqoQJAvj6OwwyLlBLm5ua4ffs2AGD27NlYs2YNbt26hfT0dFJ2\nZWUD/PwiERTkRS7wd+7cQWFhIV599VVSDmO9QV89+f8UQtwQQuQBWAfgf+rpOMxAdPXfV65cCWNj\nYyQkJPw8TVKhUKC1tRVqtVqr7K758AcOLIaz8zTSOJ88efJzm4a68xRjvUHv7ZpfPTC3a1inuro6\n3Lx5E2ZmZpgwYQKMjIxQXl6OoqIizJo1C/fv34eZmRnc3Ny0yn///QvIyqpAfLwfjI1p5zWRkZEY\nOnQon8WzPtNf5skz1i0ajQZRUVFYuHAhHj58iNbWVhgZGWHq1KmwsbHBzZs3MWnSJK033YiPv42T\nJ28gO3s/ucDn5uaioqICBw4cIOUw1pu4yLM+1dDQAEtLSyxZsgRSSpSWlqKiogIVFRVYsmQJxo8f\nr3V2eXk9AgOjcebMDrz8Mm0+/N27d5GUlISAgABePpgNKLx2DeszXRdaW1tbkZOTAyEEJk2aBDs7\nOxQWFqKsrEzr7I758Eq8+eZSrFkzmTTOmpoahIeHY8eOHbC0tCRlMdbbuMizPiOEgKmpKdauXYvs\n7GzExMQAAKytrTFq1Cg0NjZqnf2nP12EEMD7768hjbGxsREhISFwc3ODnZ0dKYuxvsAXXlmvk1Ii\nLy8P8+fPh7Gx8c/fi4mJwePHj2FlZYXq6mr4+/trlZ+efg/e3hHIzt4PW1vaipDh4eEYNWoUnJ2d\nSTmM6QpfeGX93pkzZ/DDDz+gqqoKq1atwogRIyCEgKenJ8rKyjB8+HCYmZlplV1V1Qg/v0gcPepB\nLvAFBQWorKyEp6cnKYexvsTtGtarWlpaMGXKFLz//vswNTXF2bNncffuXQAdZ/MjR47E6NGjtZqD\nrlZr4OsbgcDARdi0aQZpnA0NDTh37hwUCgVfaGUDGrdrWK9raWnB0KFDAQD5+fnIycnBokWLUFtb\nC2tra8ybN0+r3A8/TMHVq2U4f343abqklBIhISGwsbHBhg0btM5hTB+4XcP6NSklhg4d+vMdrvPn\nz8eMGTPw5ZdfYuTIkdi4caNWufHxt3H8eK5O5sNnZ2ejqakJ69bRFjFjrD/gdg3rFVJKtLa2/rx0\ncOfZCKSUMDExwUsvvYQtW7ZolV1W1jEfPiRkG6ysaEv+1tbWIjU1FQqF4ueLwowNZNyuYXqXk5OD\nyspKVFdXw97e/rl3rz5+/BijRo3qcbZKpcH69SewadN08nTJhoYGHD16FI6Ojli4cCEpizF96Rer\nUDLWpbGxEdevX8fq1auxadMm3Lx5E+fOnYNGowEA3Lp1C62trVoVeAA4ciQdQ4cOwbvvriaNs62t\nDaGhobC3t+cCzwwKF3mmVyqVChYWFjAxMYGNjQ38/PwgpcTx48dRV1cHtVqt9XTJlJS7+OqrHJw8\nqSBvxB0eHg5ra2usWUP7NMBYf8NFnunV6NGjMWHCBGRkZKC8vBxCCGzevBlTp05FW1sbZs+erVVu\nTU0T9uxR4vhxL1hb0/rwmZmZaGlpwebNm3W+3SBjfY2LPNO7JUuWYMyYMSgsLERmZiY0Gg2KiorQ\n1NSkVVFVqzXYs0cJf/8FcHGhrQ//6NEjpKenY8uWLXyhlRkkvvDKekVbWxtKS0tx48YNAMCYMWO0\nnqJ46FAqMjLuIylpD4YM0f48RaPR4Pjx45gzZw5WrlypdQ5jvYnnybN+ydTUFNOmTcO0adPw5MkT\nrXdVSkwsxldfXUd29n5SgQeAy5cvw9jYGCtWrCDlMNafcbuG9TptC3xZWT38/ZUIDt5K7sNXVVXh\nypUr8PLy4j48M2hc5NmA0N6uhrd3ON55ZzkcHe1IWW1tbVAqlXBxcdF66iZjAwUXeaY3Dx48wIMH\nD3SSdehQGoYPNyXPh9doNIiIiICtrS0WLFigk7Ex1p9xkWd60dzcjNDQUJSXl5OzkpKKERSUh6Ag\n+nz4+Ph4qFQquLu7c5uGDQpc5JnOqVQqfPPNN5gzZw757tGqqkbs3RuFoCAFeZ/Wy5cv4/79+9ix\nYwdPl2SDBhd5plNSSsTGxsLc3BxOTk6kLI1Gwt9fiX377LFhwxRSVkVFBa5cuQJfX9+flzlmbDDg\nIs90KicnB9XV1VAoFOR2yGefXURLiwqHDjmSclQqFWJiYuDs7MwXWtmgw/Pkmc40NDQgJSUF/v7+\nMDU1JWVduHAHf/tbJrKy6PPh09LSMHr0aL7QygYlPpNnOnP+/Hk4ODjAysqKlFNR0YA9e5Q4dWor\neZ/W+/fvIy8vD1u2bOELrWxQ4iLPdOL27duoqKjA2rVrSTkqlQbe3uF4442l5D58a2srlEol3N3d\nYWFBu2jL2EDFRZ6RNTQ0IDY2Fu7u7uRNrw8eTMGwYSbkDUCklEhISMCUKVMwa9YsUhZjAxkXeUbS\n3t6O06dPw8HBAVOnTiVlJSQU4dSpfPL68ABw9epVlJeXw9XVlZTD2EDHRZ5pTUqJqKgojBs3jtym\nqahoQGBgNE6dUsDSktZaKSgowJUrV+Dn56f1hiSMGQou8kxrqampaGhoIF/U7Fof/vXXF2PdOjvS\nmEpLSxEXFwcfHx+eLskYuMgzLeXl5SE/Px+7du3CkCG0mbiffXYJGo3Ehx/SPg20trYiPDwcXl5e\nsLGxIWUxZih4njzrsaqqKiQmJiIgIIA8ayU9/R6++OIasrP3w9iYds6RlJSEadOmYcaMGaQcxgwJ\nn8mzHtFoNIiNjYWTkxMsLS1JWVVVjfD1jURQkALjx48kZd2+fRtFRUVwcXEh5TBmaLjIsx7JzMyE\nsbEx7O3tSTld8+H373cg79Pa3NyMs2fPwsvLi9elYewZXORZt9XX1yM9PV0ny/QeOpQKU1Njch++\na0G0efPmwc7OjpTFmCHinjzrFo1Gg7Nnz2LZsmXkNk18/G0EBd3QSR/+xo0bqKurw9atW0k5jBkq\nPpNn3ZKYmAi1Wo01a2h3opaX1yMwMBohIVvJ68OXlZUhMTERCoWCPMOHMUPFRZ79pszMTBQXF2Pn\nzp2kzTbUag38/CLx1lvLsGbNZNKYHj58iLCwMHh6esLa2pqUxZgh4yLPXqi4uBgZGRnw8fEhX9T8\n9NMMGBsb4b33aPu0Njc3Izg4GOvWrcPMmTNJWYwZOv6My35VbW0tIiMjsWvXLowZM4aUlZ5+D//8\nZzZycmh9+K6lFGbPno0lS5aQxsTYYMBn8uy52traEBoaChcXF0yaNImUVVPThN27lTh2zBM2NrT1\n4b///ns8fvwYGzZsIOUwNlhwkWfPlZKSgkmTJpE34larNfDxiYC//wK4uU0nZdXX1yMhIQFeXl68\nETdj3cRFnv1CRUUFvv/+e53cPXr4cBqklDhyZD0pp6tNs3z5cl6XhrEe4J48+zdd8+GdnZ0xbNgw\nUlZCQhGOHs0l9+EB4Nq1a2hvb8fq1bSLtowNNnwmz/7Nt99+C3Nzc/Km12Vl9QgIiEJIyFZYWQ0n\nZdXW1iI9PR1eXl4wMuK3LGM9ofW/GCHEDiHETSGEWgjh8MzfvSeEuC2EKBRC8IpRA0RhYSEyMzPh\n6elJWrZAperow/+P/7GcvD58c3MzwsLCsHHjRowdO5aUxdhgRDktygegAJDx9DeFEHMB7AIwF4Ab\ngL8LIfj0S8/S0tJIz6+srMTZs2fh7e1N3mzj44/TYG4+BP/xH7TWikqlQlhYGGbNmgUHB4fffoIO\nUV9P9u/49ew7WhdfKWWhlPLWc/7KE0ColLJdSnkPQBGAZdoeh3UP5R9RfX09Tp8+DXd3d9ja2pLG\nceHCHRw9mkvep7XrQuuIESOwceNG0pi0wUVJt/j17Dv6OMO2BVD21OMyAOP1cBymA21tbTh9+jSW\nLl2KOXPmkLJqaprg7x+FEye8yH34b7/9FvX19fDy8iKveMnYYPbC2TVCiCQAz1sY5H0p5dkeHEf2\naFSs1+Tl5eHll1/GqlWryFnHjl1HQMBCbNw4lZSjVqtx8+ZNnWwtyNhgJ6Sk1V8hRCqA/yWlzOl8\n/C4ASCk/73ycAOCQlPK7Z57HhZ8xxrQgpez2x1tdnSY9fcAYACFCiL+go00zA8C1Z5/Qk0EyxhjT\nDmUKpUIIUQpgBYA4IUQ8AEgpCwB8A6AAQDyANyT14wJjjDGtkNs1jDHG+q9en7/ON1HpjxDisBCi\nTAhxvfOPW1+PaaARQrh1vv9uCyH+o6/HM9AJIe4JIW50vh9/0bZlLyaEOCqEqBZC5D/1vTFCiCQh\nxC0hRKIQYvSLMvriJiW+iUp/JIC/SCntO/8k9PWABhIhhDGAL9Dx/psLwEcIQZtXyiQAx873I98v\n03PH0PF+fNq7AJKklDMBXOh8/Kt6vYjyTVR6xxe0tbcMQJGU8p6Ush3AaXS8LxkNvye1JKW8CODR\nM9/2AHCi8+sTALxelNGfzpT5JirdeFsIkSeE+Pq3PsaxXxgPoPSpx/wepJMAkoUQWUKI1/p6MAbC\nSkpZ3fl1NQCrF/2wXu404Zuo9OcFr+0HAP4B4Ejn408A/BeAfb00NEPA7zfdWyWlrBRCWAJIEkIU\ndp6dMh2QUsrfuudIL0VeSumsxdPKAUx86vGEzu+xp3T3tRVCfAWgJ79Q2S/fgxPx758uWQ9JKSs7\n/7dWCKFER0uMizxNtRDCWkpZJYSwAVDzoh/u63bNszdReQshTIUQU/ArN1GxX9f5H7yLAh0XuVn3\nZQGYIYSwE0KYomMiQEwfj2nAEkIME0KM6PzaAoAL+D2pCzEA9nZ+vRdA1It+uNcXBhFCKAD8HwDj\n0HET1XUp5SYpZYEQousmKhX4Jipt/KcQYhE62g53ARzo4/EMKFJKlRDiLQDnARgD+FpK+UMfD2sg\nswKg7FxgbgiAYCllYt8OaWARQoQCWAdgXOfNpx8B+BzAN0KIfQDuAdj5wgyuo4wxZrj6ul3DGGNM\nj7jIM8aYAeMizxhjBoyLPGOMGTAu8owxZsC4yDPGmAHjIs8YYwaMizxjjBmw/we21wUEn+qZTQAA\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe418e91d10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_decision_boundary(lambda x : sigm(sp.dot(x, w)), boundary=0.5, margins=(0.1,0.9))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Kao i kod regresije, možemo koristiti preslikavanje\n",
    "    $\\boldsymbol{\\phi}:\\mathbb{R}^n\\to\\mathbb{R}^m$ iz ulaznog\n",
    "prostora u prostor značajki:\n",
    "\n",
    "$$\n",
    "    h(\\mathbf{x}) = f\\big(\\mathbf{w}^\\intercal\\mathbf{\\phi}(\\mathbf{x})\\big)\n",
    "$$\n",
    "\n",
    "$\\Rightarrow$ Linearna granica u prostoru značajki\n",
    "\n",
    "$\\Rightarrow$ **Nelinearna granica** u ulaznom\n",
    "prostoru\n",
    "\n",
    "$\\Rightarrow$ Model je **nelinearan u parametrima** (jer je $f$ nelinearna)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Primjer:\n",
    "\n",
    "* Preslikavanje iz dvodimenzijskog ulaznog prostora u 5-dimenzijski prostor značajki\n",
    "\n",
    "$$\n",
    "\\boldsymbol{\\phi}(\\mathbf{x}) = (1,x_1,x_2,x_1 x_2, x_1^2, x_2^2)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def h2(x, w): \n",
    "    x2 = sp.dstack((x, x[:,:,1]*x[:,:,2], x[:,:,1]**2, x[:,:,2]**2))\n",
    "    return sp.dot(x2, w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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0dXVJf7y4JQl5MWO0t7eTn59Pfn4+RqORhIQE7rnnHvz9/ef8TdSJkr8PMV4S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\nmIO8mVkd+w9EqXHNNyOumwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe418fa30d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w = [-0.05 , -0.15 , -0.5 ,  0.15  , -0.08  ,  0.05]\n",
    "plot_decision_boundary(lambda x : h2(x, w), margins=[-1, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Geometrija linearnog modela"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* [Skica]\n",
    "\n",
    "\n",
    "* Za točke $\\mathbf{x}_1$ i $\\mathbf{x}_2$ na hiperravnini:\n",
    "$$\n",
    "h(\\mathbf{x}_1)=h(\\mathbf{x}_2)=0\\quad \\Rightarrow\\quad \\mathbf{w}^\\intercal(\\mathbf{x}_1-\\mathbf{x}_2) = 0\n",
    "$$\n",
    "$\\Rightarrow$ $\\mathbf{w}$ je normala hiperravnine\n",
    "\n",
    "\n",
    "* **NB:** U iduće dvije točke $\\mathbf{w}$ ne uključuje $w_0$\n",
    "\n",
    "\n",
    "* Za točku $\\mathbf{x}$ na hiperravnini:\n",
    "$$\n",
    "\\mathbf{w}^\\intercal\\mathbf{x} + w_0 = 0 \\quad\\Rightarrow\\quad\n",
    "\\frac{\\mathbf{w}^\\intercal\\mathbf{x}}{\\|\\mathbf{w}\\|}=-\\frac{w_0}{\\|\\mathbf{w}\\|}\n",
    "$$\n",
    "$\\Rightarrow$ udaljenost ravnine od ishodišta je $-w_0/\\|\\mathbf{w}\\|$\n",
    "\n",
    "\n",
    "* Za točku $\\mathbf{x}$ izvan hiperravnine:\n",
    "$$\n",
    "\\begin{align*}\n",
    "\\mathbf{x} &= \\mathbf{x}_{\\bot} + d\\frac{\\mathbf{w}}{\\|\\mathbf{w}\\|}\\\\\n",
    "\\mathbf{w}^\\intercal\\mathbf{x} + w_0 &= \\mathbf{w}^\\intercal\\mathbf{x}_{\\bot} + w_0 + d \\frac{\\mathbf{w}^\\intercal\\mathbf{w}}{\\|\\mathbf{w}\\|}\\\\\n",
    "h(\\mathbf{x}) &= d \\|\\mathbf{w}\\|\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "$\\Rightarrow$ udaljenost točke $\\mathbf{x}$ od ravnine je $d=h(\\mathbf{x})/\\|\\mathbf{w}\\|$\n",
    "  * $h(\\mathbf{x}) > 0\\ \\Rightarrow\\ $ $\\mathbf{x}$ je na strani hiperravnine u smjeru normale $\\mathbf{w}$\n",
    "  * $h(\\mathbf{x}) < 0\\ \\Rightarrow\\ $ $\\mathbf{x}$ je na suprotnoj strani hiperravnine\n",
    "  * $h(\\mathbf{x}) = 0\\ \\Rightarrow\\ $ $\\mathbf{x}$ je na hiperravnini"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXQAAAD7CAYAAAB68m/qAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAHEtJREFUeJzt3WtsVOeZB/D/YwwGO+ZmGBPYFtshGFyTlBWbABbtkAZs\nLklA1DXVpkEiYKRV0khxSQKVGqv9sM1K9MNqtRWoN1fpYrAR2GQDLhBmCW0J8UUNjSFAxEBsfMEQ\nrhmPjf3uB884Y8945tieM+cy/59kxZ4z4/N6NHry8p9nniNKKRARkfUlGL0AIiKKDhZ0IiKbYEEn\nIrIJFnQiIptgQScisgkWdCIim0g06sQiwn5JIqIRUEpJqNsN3aErpUb89fbbb4/q8fH2xeeLzxef\nL/N8jeb5CoeRCxGRTbCgExHZhGULutPpNHoJlsLna3j4fA0Pn6/h0ev5kkiZTMRfIPI7AGsAtCul\nFvhumwpgH4DZANwAfqCUuj3ocWq05yYiijciAqXjm6K/B1Aw6La3ABxTSs0FcML3MxER6WjUBV0p\n9SGALwfd/DyAMt/3ZQDWjfY8REQUnl4ZerpSqs33fRuAdJ3OQ0REPrq/KeoLyhmWE9GwKKVw/vz5\nIXuvr169jdra60M+/vLly+jq6tJreaak1ydF20RkhlKqVUQeBdAe6k6lpaX93zudTr5TTkT9amtr\nUVdXh8cffxyJiQNLVVdXD4qKKlFYmINFi2YGPbapqQkHDx7Eli1bMG7cuFgtWRculwsul0vTfUfd\n5QIAIpIB4HBAl8t/ALiplHpHRN4CMFkp9dagx7DLhYhCamlpwbvvvovNmzcjLS0t6HhJSQ0uXbqF\nqqqNEBnY8OHxeLB7927k5+dj/vz5sVpyzITrchn1Dl1E9gL4LoBpIvIFgJ8B+CWA/SLyMnxti6M9\nDxHFB6/Xi8rKSqxatSpkMa+u/gwHDpxHff22oGKulEJVVRXmzZtny2IeyagLulLqh0Mcena0v5uI\n4otSCocPH0ZmZiZyc3ODjl+9ehtbtx7GoUNFmDp1QtDxM2fO4N69eygsLIzFck3Hsp8UJSL7qaur\nQ0dHBwoKBn+05evcfPv2pViy5BtBx5uamnD69Gl8//vfx5gxY2KxXNNhQSciU2hpacHJkydRWFgY\n9CYoAOzYcRzTp6egpGRJ0DGPx4PKykqsXbsWU6ZMicVyTcmweehERH6RcvOqqgvMzTXgDp2IDKUl\nNy8ufg97924Im5uvWLEiFss1NRZ0IjIUc/PoYUEnIsNoyc0djvC5+XPPPRfXuXkgZuhEZIho5ebz\n5s2L1ZJNjzt0Ioo55ub6YEEnopirra1FR0cH8vPzg44xNx85FnQiiqmWlha4XC4UFhZi7NixQce1\n9JszNw+NGToRxYzX60VFRQUKCgqYm+uAO3Qiigl/bp6VlYUFCxYEHWduPnos6EQUE1py8zfeYG4+\nGizoRKQ7Lbm5w5GC11/nnJbRYIZORLqKlJtzvnn0cIdORLrRkptv3Xp4yNz8o48+Ym4+DCzoRKQb\n/5yWkfSbNzc348MPP2RuPgws6ESki8A5Lew3jw1m6EQUdf45LaPJzbOzs9lvPkzcoRNRVAXOaRlJ\nbs5+85FjQSeiqIqUm2/ceIBzWnTCgk5EUaMlN582LTlsvzlz85Fjhk5EUREpN6+quoDKyvOory9G\nQgLntOiBO3QiGjUtuXlx8XsoL9+AtLTkoOPsN48OFnQiGrXRzDdvbm5mbh4lLOhENCqR5rTs3HkC\n06dzTkssMEMnohGLdF3Q6urPUFHRGDY3Z7959HCHTkQjouW6oFu3Hh4yN2e/efSxoBPRiPj7zQsK\nCoKO+fvNOd88tljQiWjYAvvNExODk1v2mxuDGToRDUuk3JzXBTUOd+hEpFmk3Nzt5nVBjcSCTkSa\nRc7NeV1QI7GgE5EmWnJz9psbixk6EUWkpd+8svI8Ghp4XVAjcYdORGFpyc39/ebMzY3Fgk5EYTE3\ntw4WdCIaEnNza2GGTkQhRavfnLl57HCHTkRB2G9uTSzoRBTEP9+cubm1sKAT0QCB882Zm1uLrhm6\niLgB3AXQA6BbKfWUnucjotHxer2oqKhgv7lF6b1DVwCcSqmFLObGqKmpwcqVG7By5QbU1NQYvRwy\nMX9unpWVNWRuvmVLNfvNTSwWkYtEvgvpoaamBuvXb8KxY8/j2LHnsX79JhZ1GpK23DyPubmJxWKH\nflxEakVkq87nokF27doDj+cdAJsAbILH8w527dpj9LLIhLTm5iUlzM3NTO8+9DylVIuITAdwTEQu\nKKU+9B8sLS3tv6PT6YTT6dR5OUQ0WKTcnP3mxnK5XHC5XJruK0opfVfjP5HI2wDuK6V2+X5WsTp3\nvPJHLn27dGDChDdx8GAZ8vPzDV4ZmYVSCgcOHMD48eOxdu3aoONu9208/fRvcOhQUcio5W9/+xv+\n8Y9/YPPmzYxaYkREoJQKGWXrFrmISLKIpPq+TwGwEsA5vc5HwfLz83HwYBlWrKjGihXVLOYUhP3m\n9qLbDl1EMgEc9P2YCOBPSql/DzjOHTqRgVpaWvDuu+9i8+bNIaOWkpIaXLx4C9XVG4OiFo/Hg927\ndyM/P59RS4yF26HrlqErpa4A+LZev5+IRo795vbET4oSxRmt/eac02I9LOhEcUZrv/nSpczNrYYF\nnSiOaOk3dzjC95s/99xz7Dc3Kc5DJ4oTWvrNKyvPo76+OGxuPm/evFgtmYaJO3SiOKAlNy8ufg/l\n5RuQlpYcdJy5uTWwoBPFAX9uHupzCP7cfPt29ptbHQs6kc0F5uZjx44NOq5lvjlzc2tghk5kY/7c\nvKCgIGy/eX19MRISmJtbHXfoRDYVmJsvWLAg6LjbfRtbtx5mbm4jLOhENqUlNx9qTktzczP+8pe/\nMDe3GBZ0IhvSkps7HOFz8zVr1jA3txhm6EQ2o3VOS7h+8+zsbM5psSDu0IlsJFK/+dWr4XPzjz76\niLm5hbGgE9lIpDktRUXhc3P2m1sbCzqRTWid0xIuN+d1Qa2NGTqRDXi9XlRWVkac0xJuvnl2djb7\nzS2OO3Qii/Pn5pmZmUPm5v45LZxvbm8s6EQWV1dXN+LcnHNa7IUFncjCWlpacPLkyVHl5pzTYh/M\n0IksKlrXBWVubh/coRNZ0HD6zUPl5uw3tycWdCILYr85hcKCTmQxkfrNd+48wX7zOMUMnchCtOTm\nFRWN7DePU9yhE1nEaHNz9pvbHws6kUWMJjdnv3l8YEEnsgAtc1oiXReUubn9MUMnMjn/nBYt1wUN\n12/O+eb2xx06kYkFzmkJdV1QzjenQCzoRCYW6bqgRUWV2L6d/ebUhwWdyKQiXRfU329eUsLcnPow\nQycyIW25eSPq69lvTl/jDp3IZPy5eUZGRtjcfO9e9pvTQCzoRCajZb75ULk5+83jGws6kYkEzjcP\nlZv755uHy8053zx+MUMnMolI1wWtrv4MBw6cD5ubc755fOMOncgEtFwXlLk5RcKCTmQCnNNC0cCC\nTmSwaMw3Z25OADN0IkNFo9+cuTn5cYdOZBCtc1qYm5NWLOhEBmG/OUWbbgVdRApE5IKIXBKRN/U6\nD5EVBfabDzXfnP3mNFy6ZOgiMgbAfwF4FkAzgI9FpFopdV6P8xFZSaR+86qqC+w3pxHRa4f+FIDL\nSim3UqobQDmAF3Q6F5FlROo3d7tvo7j4PebmNCJ6FfRZAL4I+LnJdxtRXIvUb75xI/vNaeT0altU\nWu5UWlra/73T6YTT6dRpOUTG8/ebb968OWxuzn5zCuRyueByuTTdV5TSVHuHRUQWAyhVShX4ft4B\noFcp9U7AfZQe5yYyI6/Xiz179sDpdIZsUayquoDXXjuK+vptQVGLUgr79u3D5MmTQ+7sKb6ICJRS\nEuqYXpFLLYDHRSRDRMYBKAJQrdO5iEwtUr85c3OKFl0KulLqIYBXANQAaASwjx0uFK+05ObsN6do\n0O2j/0qpIwCO6PX7iaxAa27OfnOKBs5yIdKJ1+tFRUUF+80pZvjRfyId+HPzrKysIeebMzenaGNB\nJ9JBvMw3r6mpwcqVG7By5QbU1NQYvZy4x8iFKMripd+8pqYG69dvgsfT1418+vQmHDxYhvz8fINX\nFr9Y0ImiKJ7mtOzatcdXzDcBADyevttY0I3DyIUoSrRcF5S5OemJBZ0oSiL3mx+wRW7uV1JSjAkT\n3gRQBqAMEya8iZKSYqOXFddY0ImiINJ1QXfsOI5p05Itn5sHys/Px8GDZVixohorVlQzPzcBXWa5\naDoxZ7mQTXi9XuzevRvPPPNMyKiFc1oomoyY5UIUF9hvTmbCgk40ClquC2qn3JzMjQWdaIS0XhfU\nTrk5mRv70IlGIJ76zck6uEMnGib2m5NZsaATDZOWOS2cb05GYEEnGgYt/eacb05GYYZOpJGW+eaV\nlefR0MDcnIzBHTqRBlr7zcvLw+fmzz77bCyWS3GKBZ1IA39uHuqj7f7c/Cc/WRIxNw8V0xBFCws6\nUQSBufnYsWODju/YcRzTp6egpGRp0DF/br527Vrm5qQ7bheIwtDSb15ZeR719cVISBg6N58/f36s\nlkxxjDt0oiFo7TcvL9+AtLTkoOPsN6dYY0EnGoKWfnMtuTn7zSlWWNCJQtDSb87cnMyGGTrRIFr6\nzf1zWpibk5lwh04UgPPNycpY0IkCRKvfnLk5GYEFncgnGv3mnNNCRmKGToSvc/OCgoJR9ZtzTgsZ\niTt0inuBufmCBQuCjrvdt7F162H2m5PpsaBT3IuUm2/cyPnmZA0s6BTXOjs7cfr0ac5pIVsQpZQx\nJxZRRp2bKFBXVxfGjRsXdHtV1QW89tpR1NdvC2pRVEph3759mDx5cshPkhLpRUSglJJQx7hDp7gQ\nbvMQqpj7c3P2m5OVsKCTrfX09ADAgCsIRfqXIXNzsiq2LZKtnTx5Evfu3UNmZiYmTZqEzMzMoMvD\nDcZ+c7IqFnSyrSNHjiAhIQELFy5Ea2srbt26hTt37iA3N3fIKwdpndPCfnMyI0YuZFtTpkzB/Pnz\nkZGRgSeeeAKzZs1CR0cHbt++HfL+bjfntJC1saCTbaWmpuL48eNob29HcnIysrOz8cgjj6Curg69\nvb0D7svcnOyABZ1sxx+v5OTkYMmSJdi/fz/q6uoAACkpKejq6kJCwsCX/o4dx+FwpKCkZEnQ72Nu\nTlbBPnSylZqaGty+fRuTJ0/G/fv3sXjxYqSmpqK6uhpTp07FjRs3sHr1akyfPr3/MVVVF/DjHx9F\nfX1x0Ef72W9OZhOuD51vipJtXL9+HW1tbXjppZfQ2dmJ5uZmnDp1Ck8++SRefPFFPHjwAImJiUhK\nSup/jH+++aFDRWHntBQWFsbyTyEaEV0iFxEpFZEmEWnwfXFrQ7pzOBxwOBxoa2tDUlISHnvsMSxf\nvhzXrl2Dx+NBSkrKgGLel5sfwBtvhM7Nm5ubmZuTpeiVoSsAv1JKLfR9HdXpPET9EhMTkZaWhrq6\nOty9exdKKcyYMQPd3d1wu91B99+58wSmTUvG668zNyd70DNyCf/pDaIoOXv2LEQEXV1dyMvLw9mz\nZ3H06FHMmTMHCQkJ6OjoQF5e3oDHVFd/hsrKRtTXbwv6oJG/3zw7O5v95mQpena5vCoifxeR34rI\nZB3PQ3GssbERjY2NcDgcaG9vR1lZGebPn49nnnkGSik8ePAAeXl5mDp1av9jrl7lnBaypxHv0EXk\nGIAZIQ79FMCvAfzc9/MvAOwC8PLgO5aWlvZ/73Q64XQ6R7ocikO9vb24efMmcnNzMXv2bMyePRsf\nf/wx/vjHP2LVqlVYtGhR0GMi5eb+fvMtW7YwNydTcLlccLlcmu6re9uiiGQAOKyUWjDodrYt0qhd\nv34df/3rX7Fo0SJkZGQAANxuN65cuYLly5cH3b+kpAYXL95CdfXGoKjF4/Fg9+7dKCgoYNRCphXz\ntkUReVQp1eL7cT2Ac3qch+KbUgrp6enIyMjAtWvX0NnZiTlz5mDixIn44osvguac+68L2tAwdG7O\nOS1kZXq9KfqOiHwbfd0uVwBs0+k8FMdEBGPGjEFOTg7Onz+P1tZW1NbWIjExEU888cSAYu6f03Lo\nUFHY3Jz95mRl/KQoWZJSyv9Pz/7dtsfjwYMHD9DV1YWZM2f237erqwff+c7vUViYE3IkblNTE/bu\n3YstW7awRZFMj1csIttobm4G0Pei7u3t7f+v/7Zp06YNKObA1/PN2W9OdseCTpZx6NAhHDhwAB98\n8AEAICEhoX+H3tnZiffffx+dnZ0DHuOfb15Wto65OdkeCzpZQnt7Ox4+fIgf/ehH8Hq9qKysxM2b\nNyEiEBGMHz8eq1evxvjx4/sfw/nmFG+YoZNl3LlzB5MmTYLX60VDQwMuXbqEvLw83LhxAxkZGUhP\nT++/L3NzsitOWyRL88cqEydOBAAkJSVh8eLFmD17NioqKpCcnIynn356wGOYm1M8YuRCpufPvv3/\n/eSTT/Dw4UMkJycjNTUV69atG3D/vjktzM0p/rCgk6V8+umnOHfuHBITE/HgwQN873vfw7Rp0/qP\nu919c1rKy5mbU/xh5EKWoZSCx+PB6tWrASCoPdF/XVDOaaF4xTdFyVL8eXpvb2/QdUE5p4XiAd8U\nJdvwF+rBxdzfbx5uvjlzc7I7Zuhkev655kNhvzlRHxZ0Mr3a2locOHAg5DGtuTmvC0rxgAWdTK2l\npQUulwtr1qwJeVxLv/natWvZb05xgRk6mZbX60VFRQVWrVqFtLS0oOPV1Z9pys3nz58fqyUTGYo7\ndDIlpRQOHz6MrKws5ObmBh3395szNyf6Ggs6mVJtbS06OjpQUFAQdIy5OVFoLOhkOv7cvLCwEImJ\nwakgc3Oi0Jihk6lEys391wWtry9mbk40CHfoZBpac/Py8g1IS0sOOs7cnOIdCzqZhrbcPI+5OdEQ\nWNDJFJibE40eM3QynD83LygoGLLf3J+bJyQwNycaCnfoZKjA3HzBggVBx5mbE2nHgk6G8ufm+fn5\nQcfYb040PCzoZJjA3Hzs2LFBx3fsOA6Hg9cFJdKKGToZIlr95pxvTvQ17tAp5thvTqQPFnSKOfab\nE+mDBZ1iajT95gDQ09ODF154gbk5UQjM0ClmhpObD+4395s9e7beyySyLBZ0igktuXlx8Xs4dKgo\nZG5ORJExcqGY0Npvnps7GZ9//jkaGxsB9P2PgIi0YUEn3WnpN/fn5nv37sX169dx9uxZ1NTUoLm5\nGb29vSzsRBowciFdac3NGxq2obW1FTNnzsSyZcuwbNkynDlzBnV1dejp6WF2TqQBd+ikm0i5+dWr\nfbl5eXnfdUHHjx+Ppqam/rhl8eLFmDNnDo4cOYI7d+7EevlElsOCTrqJ1G9eVFSJ7duX4qmnZgIA\npkyZgjVr1uDy5cs4c+YMAOBb3/oWZs2aha+++iqmayeyIkYupAt/br558+aI/eYffHAC9+7dQ0ZG\nBiZOnIiFCxfi4sWLKCsrQ1ZWFlpbW/Hoo48a8FcQWYsY9WaTiCi+0WVPXq8Xe/bsgdPpDDkSt7r6\nM7z66hHU1xfj7Nn/Q0JCArKzs9Ha2orOzk5MmjQJubm5uHTpEsaNG4f09HSkpqYa8JcQmY+IQCkV\n8oMa3KFTVPlz88zMzJDF/OrVvjkt/n7zKVOmYObMmfjmN78Jh8OBa9euoampCXfu3EFOTo4BfwGR\ndTFDp6iqq6sL229eVDRwvnlqaiqOHz+O9vZ2JCcnY968eXjkkUdQW1uLnp6eWC+fyNJY0ClqWlpa\ncPLkySH7zXfuPNGfm7e2tuLWrVvIycnBkiVLsH//ftTV1QEAUlJS0N3dzeFbRMM04gxdRAoBlAKY\nB+BflFL1Acd2ANgMoAfAj5VSfw7xeGboNhIpNz98+DO88soRNDRsw8cfn8Lt27cxefJk3L9/H4sX\nL0Zqaiqqq6sxdepU3LhxA6tXr8b06dMN+EuIzE2vDP0cgPUAdg86WQ6AIgA5AGYBOC4ic5VSvaM4\nF5mYltx8y5a+3Lyz80u0tbXhpZdeQmdnJ5qbm3Hq1Ck8+eSTePHFF/HgwQMkJiYiKSnJgL+EyNpG\nHLkopS4opS6GOPQCgL1KqW6llBvAZQBPjfQ8ZH7+3Dxcv7k/N3c4HEhPT0dbWxuSkpLw2GOPYfny\n5bh27Ro8Hg9SUlJYzIlGSI8MfSaApoCfm9C3UycbCszNQ/Wb79x5ov+6oEopiAhSU1PR0NCAu3fv\nQimFGTNmoLu7G263O/Z/AJGNhC3oInJMRM6F+HpumOdhWG5Tly9fHnJOy1dfdaOx8Qb+8Id1/dcF\nHTNmDJYuXYqkpCTU1NSgoaEBDQ0N6OjoQHp6eqyXT2QrYTN0pdRILtrYDCDw2mH/5LstSGlpaf/3\nTqcTTqdzBKcjIy1btmzIY8nJY/H++/+K3t6+t09EBL29vUhISMDy5cvx+eef48svv4TH40FeXh6m\nTp0aq2UTWYbL5YLL5dJ031F/UlRETgL4iVKqzvdzDoD/QV9uPgvAcQBzBre0sMslPty8eROffvop\n5s6dixkzZgDou4wcWxKJRkaXLhcRWQ/gPwFMA/C/ItKglFqllGoUkf0AGgE8BPBvrNzxq7y8HA6H\nA729vbh79y7mzp3bX8ybm5tx//59ZGdnG7xKInvgLBfSTW9vLz755BNkZmbiypUruHnzJiZOnIhF\nixahu7sbt27dwqRJkzBhwgSjl0pkGeF26CzoFBNerxdutxvt7e1QSuHSpUtYvnw5srKyjF4akaWE\nK+j86D/FRFJSErKzs5GTk4OLFy9i1qxZLOZEUWbZgq71XV/qY5bnq6enBxMnTgz5ISQzMcvzZRV8\nvoZHr+eLBT1OmOX5cjgcWLdundHLiMgsz5dV8PkaHhZ0so1x48YZvQQiW2JBJyKyCUO7XAw5MRGR\nxZmubZGIiKKLkQsRkU2woBMR2YSlCrqIFIrIpyLSIyL/POjYDhG5JCIXRGSlUWs0KxEpFZEmEWnw\nfZm7EdwgIlLgew1dEpE3jV6PFYiIW0Q+8b2uzhq9HrMRkd+JSJuInAu4bapvPPlFEfmziEyOxrks\nVdDx9WXvTgXeOOiydwUA/ltErPa36U0B+JVSaqHv66jRCzIbERkD4L/Q9xrKAfBDEZlv7KosQQFw\n+l5XvDpZsN+j7zUV6C0Ax5RScwGc8P08apYqerzs3aiFfGec+j0F4LJSyq2U6gZQjr7XFkXG19YQ\nlFIfAvhy0M3PAyjzfV8GICqftrNUQQ+Dl73T5lUR+buI/DZa/8SzmVkAvgj4ma8jbRT6LgZfKyJb\njV6MRaQrpdp837cBiMrlukY8D10vInIMwIwQh3YqpQ4P41fFXT9mmOfupwB+DeDnvp9/AWAXgJdj\ntDSriLvXTJTkKaVaRGQ6gGMicsG3KyUNlFIqWp/LMV1B1/uyd3am9bkTkd8AGM7/HOPF4NfRNzDw\nX34UglKqxfffGyJyEH3RFQt6eG0iMkMp1SoijwJoj8YvtXLkEpjZVQPYKCLjRCQTwOMA+G57AN+L\nxm89+t5gpoFqATwuIhkiMg59b7RXG7wmUxORZBFJ9X2fAmAl+NrSohrAJt/3mwAcisYvNd0OPRxe\n9m5U3hGRb6MvVrgCYJvB6zEdpdRDEXkFQA2AMQB+q5Q6b/CyzC4dwEERAfrqyZ+UUn82dknmIiJ7\nAXwXwDQR+QLAzwD8EsB+EXkZgBvAD6JyLtY9IiJ7sHLkQkREAVjQiYhsggWdiMgmWNCJiGyCBZ2I\nyCZY0ImIbIIFnYjIJljQiYhs4v8B/BMQciQdyV4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe4193e6950>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w = sp.array([-4, 2, -1])\n",
    "X = sp.array([[1, 5, -1],\n",
    "              [1, -5, 5]])\n",
    "\n",
    "plot_decision_boundary(lambda x : h(x, w), margins=(-1, 1))\n",
    "plt.scatter(X[:,1],X[:,2]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(X[0], w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-19"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(X[1], w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.2360679774997898"
      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.linalg.norm(w[1:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def distance(x,w): return sp.dot(x, w) / sp.linalg.norm(w[1:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.1304951684997055"
      ]
     },
     "execution_count": 196,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance(X[0], w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-8.4970583144992009"
      ]
     },
     "execution_count": 180,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance(X[1], w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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eXI/Rmx84wN7cCqM3LywsZG8eQwx0+gqjNy8oKGBvboHRm//0pyvYm1tg7s1zcnKcXo6v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\nMgDWdk2QaQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe4195fb290>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w2 = w/10.0\n",
    "\n",
    "plot_decision_boundary(lambda x : h(x, w2), margins=(-1,1))\n",
    "plt.scatter(X[:,1],X[:,2]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.69999999999999996"
      ]
     },
     "execution_count": 182,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(X[0], w2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.8999999999999999"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(X[1], w2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.223606797749979"
      ]
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.linalg.norm(w2[1:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.1304951684997051"
      ]
     },
     "execution_count": 186,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance(X[0], w2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-8.4970583144991991"
      ]
     },
     "execution_count": 187,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance(X[1], w2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Višeklasna klasifikacija ($K>2$)\n",
    "\n",
    "\n",
    "* Shema **jedan-naspram-jedan** (engl. *one-vs-one*, **OVO**)   \n",
    "  * $K\\choose 2$ binarnih klasifikatora, razdjeljuju sve parove klasa\n",
    "  * Model:\n",
    "$$\n",
    "h(\\mathbf{x})=\\mathrm{argmax}_i\\sum_{i\\neq j}\\mathrm{sgn}\\big(h_{ij}(\\mathbf{x})\\big)\n",
    "$$\n",
    "gdje $h_{ji}(\\mathbf{x})=- h_{ij}(\\mathbf{x})$\n",
    "\n",
    "\n",
    "* [Skica: OVO]\n",
    "\n",
    "\n",
    "* Shema **jedan-naspram-ostali** (engl. *one-vs-rest*, **OVR**, *one-vs-all*) \n",
    "  * $K$ binarnih klasifikatora s pouzdanošću, po jedan za svaku klasu\n",
    "  * Model:\n",
    "$$\n",
    "h(\\mathbf{x}) = \\mathrm{argmax}_j\\ h_j(\\mathbf{x})\n",
    "$$\n",
    "\n",
    "\n",
    "* [Skica: OVR]\n",
    "\n",
    "\n",
    "* Prednost OVR nad OVO je da imamo manje modela, međutim OVR lako rezultira neuravnoteženim brojem primjera kroz klase"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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fbrfdEMm92HIh6kJv9k9nDZESzZIMXUQyRaReRA6LSL5Zj0PUW1lZw5GTE1+eHgy2oaho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\nCQ50IiKX4EAnInKJ/wf/OcEjXRx6uwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe41907a190>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "h1 = lambda x: h(x, [0, 2, 1])\n",
    "plot_decision_boundary(h1)\n",
    "h2 = lambda x: h(x, [-0.2, 0.7, -0.8])\n",
    "plot_decision_boundary(h2)\n",
    "h3 = lambda x: h(x, [-1.5, 0.1, 0.5])\n",
    "plot_decision_boundary(h3)\n",
    "plt.scatter(X[:,1],X[:,2]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9 4.1 -1.5\n"
     ]
    }
   ],
   "source": [
    "print h1(X[0]), h2(X[0]), h3(X[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-5 -7.7 0.5\n"
     ]
    }
   ],
   "source": [
    "print h1(X[1]), h2(X[1]), h3(X[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def ovr(x): return sp.argmax([h1(x), h2(x), h3(x)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 199,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ovr(X[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 200,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ovr(X[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = linspace(-10, 10)\n",
    "y = linspace(-10, 10)\n",
    "X1, X2 = np.meshgrid(x, y)\n",
    "XX = sp.dstack((sp.ones((50, 50)), X1, X2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n, m, _ = shape(XX)\n",
    "YY = sp.zeros((n,m))\n",
    "for i in range(0,n):\n",
    "    for j in range(0,m):\n",
    "        YY[i,j] = ovr(XX[i, j])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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BSV8e9+QiTX4c2ycl/aWkd6k3ieq7EbEcEc/b3ppE9RMxiWoW52zfqd6ww8uSHqp5f1ol\nIn5i+48kfVXSTZJWI+KFmnerzfZLesy21MuaL0TEhXp3qV1sr0m6V9K7+pNP/0TSI5K+ZPuMpGuS\nPjz2Z5CjAJBX3cM1AICCCHkASIyQB4DECHkASIyQB4DECHkASIyQB4DECHkASOz/Ad5IofkbFOIE\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe418fdd050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.contourf(X1, X2, YY);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Klasifikacija regresijom"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Podsjetnik\n",
    "\n",
    "* [Skica]\n",
    "\n",
    "\n",
    "* Funkcija pogreške (empirijsko očekivanje kvadratnog gubitka):\n",
    "$$\n",
    "E(\\mathbf{w}|\\mathcal{D})=\\frac{1}{2}\n",
    "\\sum_{i=1}^N\\big(\\mathbf{w}^\\intercal\\boldsymbol{\\phi}(\\mathbf{x}^{(i)}) - y^{(i)}\\big)^2 = \n",
    "\\frac{1}{2}\n",
    "(\\boldsymbol\\Phi\\mathbf{w} - \\mathbf{y})^\\intercal\n",
    "(\\boldsymbol\\Phi\\mathbf{w} - \\mathbf{y})\n",
    "$$\n",
    "\n",
    "\n",
    "* Minimizator pogreške:\n",
    "$$\n",
    "    \\mathbf{w} = (\\boldsymbol\\Phi^\\intercal\\boldsymbol\\Phi)^{-1}\\boldsymbol\\Phi^\\intercal\\mathbf{y}\n",
    "    = \\color{red}{\\boldsymbol\\Phi^{+}}\\mathbf{y}\n",
    "$$\n",
    "\n",
    "\n",
    "* **Q:** Kako iskoristiti regresiju za klasifikaciju?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Binarna klasifikacija $(K=2)$\n",
    "\n",
    "* Ideja: regresijska funkcija $h(\\mathbf{x})$ koja za primjere iz klase $\\mathcal{C}_1$ daje $h(\\mathbf{x})=1$, a za primjere iz klase $\\mathcal{C}_2$ daje $h(\\mathbf{x})=0$ \n",
    "\n",
    "\n",
    "* Primjer $\\mathbf{x}$ klasificiriamo u klasu $\\mathcal{C}_j$ za koju je $h(\\mathbf{x})$ najveći\n",
    "\n",
    "\n",
    "* Granica između $\\mathcal{C}_1$ i $\\mathcal{C}_2$ je na $h(\\mathbf{x})=0.5$\n",
    "\n",
    "\n",
    "* Primjer za $n=1$\n",
    "  * $h(x) = w0 + w1 x$\n",
    "  * [Skica]\n",
    "\n",
    "\n",
    "* Primjer za $n=2$\n",
    "  * $h(x) = w0 + w1 x_1 + w2 * x_2$\n",
    "  * [Skica]\n",
    "  \n",
    "  \n",
    "* Alternativno, model možemo trenirati tako da za primjere iz klase $\\mathcal{C_2}$ bude ciljana vrijednost bude $y=-1$\n",
    "  * Diskriminativna funkcija onda je $h(\\mathbf{x})=0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Višeklasna klasifikacija  $(K>2)$\n",
    "\n",
    "* Shema jedan-naspram-ostali (OVR)\n",
    "\n",
    "\n",
    "* Treniramo po jedan model $h_j$ za svaku klasu $\\mathcal{C}_j$\n",
    "\n",
    "$$\n",
    "h_j(\\mathbf{x}) = \\mathbf{w}_j^\\intercal\\boldsymbol{\\phi}(\\mathbf{x})\n",
    "$$\n",
    "\n",
    "* Primjeri za učenje $\\mathcal{D}=\\{\\mathbf{x}^{(i)},\\color{red}{\\mathbf{y}^{(i)}}\\}_{i=1}^N$:\n",
    "\n",
    "\\begin{align*}\n",
    "\\boldsymbol\\Phi = \n",
    "\\begin{pmatrix}\n",
    "    \\boldsymbol{\\phi}(\\mathbf{x}^{(1)})^\\intercal \\\\\n",
    "    \\boldsymbol{\\phi}(\\mathbf{x}^{(2)})^\\intercal \\\\\n",
    "\\vdots\\\\\n",
    "    \\boldsymbol{\\phi}(\\mathbf{x}^{(N)})^\\intercal \\\\\n",
    "\\end{pmatrix}_{N\\times (m+1)}\n",
    "&\n",
    "\\qquad\n",
    "    \\color{red}{\\mathbf{y}_j} = \n",
    "\\begin{pmatrix}\n",
    "    y_j^{(1)}\\\\\n",
    "    y_j^{(2)}\\\\\n",
    "    \\vdots\\\\\n",
    "    y_j^{(N)}\n",
    "\\end{pmatrix}_{N\\times 1}\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "* [Primjer] \n",
    "\n",
    "\n",
    "* Rješenje koje minimizira kvadratnu pogrešku:\n",
    "\\begin{equation*}\n",
    "\\mathbf{w}_j = (\\boldsymbol\\Phi^\\intercal\\boldsymbol\\Phi)^{-1}\\boldsymbol\\Phi^\\intercal\\mathbf{y}_j = \\boldsymbol\\Phi^{+}\\mathbf{y}_j\n",
    "\\end{equation*}\n",
    "\n",
    "\n",
    "* Model (OVR):\n",
    "$$\n",
    "h(\\mathbf{x}) = \\mathrm{argmax}_j\\ \\mathbf{w}_j\\boldsymbol\\phi(\\mathbf{x})\n",
    "$$\n",
    "\n",
    "* Primjer ($K=3$)\n",
    "\n",
    "![Tri ravnine](images/ravnine3.png)\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Prednosti i nedostatci klasifikacije regresijom\n",
    "\n",
    "* Prednosti:\n",
    "  * Rješenje u zatvorenoj formi\n",
    "  * Jednostavan postupak\n",
    "\n",
    "\n",
    "* Nedostatci:\n",
    "   * Izlazi modela nemaju vjerojatnosnu intepretaciju ($h(\\mathbf{x})$ nije ograničena na interval $[0,1]$)\n",
    "   * Nerobusnost: osjetljivost na vrijednosti koje odskaču (kažnjavanje \"pretočno''\n",
    "        klasificiranih primjera)\n",
    "   * Zbog toga, u nekim slučajevima: pogrešna klasifikacija unatoč tome što su primjeri linearno odvojivi\n",
    "\n",
    "\n",
    "* [Primjer]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Gradijentni spust"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Perceptron"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sažetak\n",
    "\n",
    "* **Diskriminativni linearni modeli** modeliraju granicu između klasa, ali ne i način generiranja primjera\n",
    "\n",
    "\n",
    "* **Poopćeni linearan model** je linearan model s aktivacijskom funkcijom\n",
    "\n",
    "\n",
    "* **Klasifikacija regresijom** je jednostavan postupak, ali nije robusan\n",
    "\n",
    "\n",
    "* **Konveksna optimizacija** važna je za strojno učenje jer je funkcija gubitka (a time i funkcija pogreške) tipično konveksna\n",
    "\n",
    "\n",
    "* Ako minimizacija pogreške nema rješenje u zatvorenoj formi, a možemo izračunati gradijent, onda možemo primijeniti **gradijentni spust**\n",
    "\n",
    "\n",
    "* **Perceptron** je linearan klasifikacijski model koji gradijentnim spustom mimimizira aproksimaciju broja pogrešnih klasifikacija\n",
    "\n",
    "\n",
    "* Perceptron **ne konvergira** za linearno nedvojive probleme, dok za linearno odvojive rješenje ovisi o inicijalizaciji i redoslijedu primjera\n",
    "\n",
    "\n",
    "* Niti regresija niti perceptron ne daju probabilistički izlaz"
   ]
  }
 ],
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