supervised_learning.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Supervised learning I\n",
    "\n",
    "### Warning: Since github doesn't display interactive plots, make sure you are viewing it locally or using nbviewer.\n",
    "\n",
    "\n",
    "In this session we will dive into the basic scikit-learn API. After briefly introducing scikit-learn's *Estimator* object, we'll cover **supervised learning**, including *classification* and *regression* problems. By the end of the course, you should be able to build a classifier capable of discriminating different classes of faces:\n",
    "\n",
    "![](https://raw.githubusercontent.com/fabianp/mash_2016_sklearn_intro/master/images/lfw.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "# some imports\n",
    "%pylab inline\n",
    "plt.rcParams['figure.figsize'] = (1 * 10.0, 1 * 9.0)\n",
    "plt.rcParams['font.size'] = 30\n",
    "\n",
    "# numpy\n",
    "import numpy as np\n",
    "\n",
    "# pandas\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Scikit-learn Estimator Object\n",
    "\n",
    "Every model is exposed in scikit-learn via an ''Estimator'' object. For instance a linear regression is implemented as so:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```LinearRegression``` is implements a linear regression model. That is, we assume that our data is generated as\n",
    "\n",
    "$$\n",
    "y_i = x_i^T w + b + \\varepsilon_i \\text{ for $i=1\\ldots n$}\n",
    "$$\n",
    "\n",
    "where $y_i, x_i$ are given data, $\\varepsilon_i$ is random noise and $w, b$ are the parameters to estimate.\n",
    "\n",
    "**Estimator parameters**: All the parameters of an estimator can be set when it is instantiated, and have suitable default values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=True)\n"
     ]
    }
   ],
   "source": [
    "lin_reg = LinearRegression(normalize=True)\n",
    "print(lin_reg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Estimated Model parameters**: When data is *fit* with an estimator, parameters are estimated from the data at hand. But first, lets create some data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = np.arange(10)\n",
    "y = 2 * x + 1 + np.random.randn(x.size)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3 4 5 6 7 8 9]\n",
      "[  0.69826828   2.50177105   5.06193971   8.36080477   9.57888385\n",
      "  11.06570416  12.69750692  16.25128518  17.84256961  20.20245944]\n"
     ]
    }
   ],
   "source": [
    "print(x)\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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cDuzblf27dw5dFjAwoQ1ghhw5tpCDh49n8fSZJMnCycUcPHw8SQQ3uMi5PAowQw4dPfGl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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f78a129ddd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the data and embed in the notebook!\n",
    "plt.scatter(x, y)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0]\n",
      " [1]\n",
      " [2]\n",
      " [3]\n",
      " [4]\n",
      " [5]\n",
      " [6]\n",
      " [7]\n",
      " [8]\n",
      " [9]]\n",
      "[  0.69826828   2.50177105   5.06193971   8.36080477   9.57888385\n",
      "  11.06570416  12.69750692  16.25128518  17.84256961  20.20245944]\n"
     ]
    }
   ],
   "source": [
    "# The input data for sklearn is 2D: (samples == 10 x features == 1)\n",
    "X = x[:, np.newaxis]\n",
    "print(X)\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=True)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# fit the model on our data\n",
    "lin_reg.fit(X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Now the model is estimated. Hurray!.\n",
    "\n",
    "All the estimated parameters are attributes of the estimator object ending by an underscore."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 2.14161797]\n",
      "0.788838447486\n"
     ]
    }
   ],
   "source": [
    "# underscore at the end indicates a fit parameter\n",
    "print(lin_reg.coef_)\n",
    "print(lin_reg.intercept_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each class can have different estimated parameters and they are named differently.At some point you will probably forget their names: fortunately, ipython has the tab-complete feature that makes introspection easy. Try it out!\n",
    "\n",
    "Also, scikit-learn comes with excellent documentation: both [reference documentation](http://scikit-learn.org/stable/modules/classes.html) and [examples](http://scikit-learn.org/stable/auto_examples/index.html).\n",
    "\n",
    "Coming back to our recently trained model, the great thing now is that we can use this model to \"predict\". We will see why this is great in the next section."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  0.78883845,   2.93045641,   5.07207438,   7.21369235,\n",
       "         9.35531031,  11.49692828,  13.63854625,  15.78016422,\n",
       "        17.92178218,  20.06340015])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg.predict(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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LZ2dn+Pj44IsvvkB4eDgaNmyoVUySJOHDDz9EXFwclixZgt69e6Nu3bqwsrKC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\nU9IQ59zWSM4HAAAQbYK4EUGSFHo79GxJV0p6R951bsWS1kn6RF4tVT/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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f789940ae80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x, y, label='original data') \n",
    "plt.plot(x, lin_reg.predict(X), label='estimated model')\n",
    "plt.legend()\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Supervised Learning: Regression and Classification\n",
    "\n",
    "What we have just seen is a simple instance of supervised learning. We will now see some more challenging scenarios. But first, what is supervised learning anyway?\n",
    "\n",
    "In **Supervised Learning**, we have a dataset consisting of both features and labels.\n",
    "The task is to construct an estimator which is able to predict the label of an object\n",
    "given the set of features. Some examples are:\n",
    "\n",
    "- given a multicolor image of an object through a telescope, determine\n",
    "  whether that object is a star, a quasar, or a galaxy.\n",
    "- given a photograph of a person, identify the person in the photo.\n",
    "- given a list of movies a person has watched and their personal rating\n",
    "  of the movie, recommend a list of movies they would like\n",
    "  (So-called *recommender systems*: a famous example is the [Netflix Prize](http://en.wikipedia.org/wiki/Netflix_prize)).\n",
    "  \n",
    "This is in oposition to **Upsupervised Learning**, where we are only given features but no labels, and the task is to find some structure in the data, either by grouping observations (clustering) or by finding a low-dimensional representation of the data (dimensionality reduction, manifold learning). Examples of such are:\n",
    "\n",
    " - given a collection of images, decide how many people are in the collection.\n",
    "\n",
    "\n",
    "Supervised learning is further broken down into two categories, **classification** and **regression**.\n",
    "In classification, the label is discrete, while in regression, the label is continuous. For example,\n",
    "in astronomy, the task of determining whether an object is a star, a galaxy, or a quasar is a\n",
    "classification problem: the label is from three distinct categories. On the other hand, we might\n",
    "wish to estimate the age of an object based on such observations: this would be a regression problem,\n",
    "because the label (age) is a continuous quantity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Regression Example\n",
    "\n",
    "One of the simplest regression problems is fitting a line to data, which we saw above.\n",
    "Scikit-learn also contains more sophisticated regression algorithms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AmGUzdfclAECrhDIAgAYIZQAADRDKAAAaIJQBADRAKAMAaECptQ52gAvnKXtF\nrfXlq3zf1UmuTpItW7ZcefDgwYH6kSRnzpzJpk2bBj4Oo6VOk0OtJoM6TQZ1mhwr1Wr37t3X11p3\nDrvdfiePHZpa67VJrk2SnTt31l27dg18zKNHj2YYx2G01GlyqNVkUKfJoE6TY1y1cvkSAKABQhkA\nQAOEMgCABghlAAANEMoAABoglAEANKCvKTFKKY9I8qMXbX5cz/pTSykXH/Nttdbja+kcAMCs6Hee\nsocn+fllfv6U7tLrE0mEMgCAZbh8CQDQgL7OlNVajyYpo+kKAMDscqYMAKABQhkAQAOEMgCABghl\nAAAN6HdKDIC7OHz8VA4cOZnT8wvZunku+/Zsz94d28bdLYCJIpQBAzl8/FT2HzqRhbPnkiSn5hey\n/9CJJBHMAPrg8iUwkANHTt4ZyBYtnD2XA0dOjqlHAJNJKAMGcnp+oa/tAFyaUAYMZOvmub62A3Bp\nQhkwkH17tmdu44YLts1t3JB9e7aPqUcAk8lAf2Agi4P53X0JMBihDBjY3h3bhDCAAbl8CQDQAKEM\nAKABQhkAQAOEMgCABghlAAANEMoAABoglAEANEAoAwBogFAGANAAoQwAoAFCGQBAA4QyAIAGCGUA\nAA0QygAAGiCUAQA0QCgDAGiAUAYA0AChDACgAaXWOu4+pJTyhSQ3DuFQD0zyxSEch9FSp8mhVpNB\nnSaDOk2OlWr18Frrg4bdaBOhbFhKKcdqrTvH3Q+Wp06TQ60mgzpNBnWaHOOqlcuXAAANEMoAABow\nbaHs2nF3gFVRp8mhVpNBnSaDOk2OsdRqqsaUAQBMqmk7UwYAMJGEMgCABow9lJWO55ZS3l5K+Wwp\n5WullP9VSrmulPKCUsplI2jzXqWUnyml/Gkp5fOllK+WUm4spbyllLJn2O1Ng/WsUynlYaWUF5ZS\n3lxK+Ugp5SullNtKKV8opby3lPJLpZSHDau9aTOO36lL9OF+pZTPlVJqz3L5qNudJOOqUyllQynl\nOaWUg6WUT5RSznR/xz5eSnlnKeVlpZRHjaLtSTSmf6O+u5TyX7o1OdP9/vtiKeV9pZR/U0r5+8Nu\nc5J1/5t+bCnlqlLKq0op7y+l3Nrz3fPyEbU7/CxRax3bkuR+Sa5LUpdZrk/ysCG2uSPJJ1do881J\n7j7Oz6alZT3rlORwkjtWaKsm+WqSfzHuz6a1ZRy/U0v043WXaPfycX8+rSzjqlP3++9Dq/j9+tVx\nf0YtLOvSOF+cAAALMklEQVRdp3QmLP3jVdTna0leNu7Pp5UlydtW+LxePoI2R5IlRv4X81JKKXdP\n8vtJntLd9Jl07nb4RJKHJvlnSb4xyeOTvLOU8m211q8M2ObDk7wzyZbupg8keVM6s/ZekeTqJA9I\n8oPpBIMfGaS9aTCGOj02SemufzCdL6iPJflKt71nJ3lyknskOVBKuUet9ZUDtDc1xvE7tUQ/npbk\nqnR+h25Lcs9htzHJxlWnUsqT0vn+u0930weT/Pckn+6+/vpuu88ctK1psN516p5xe2eSxQlLv5rk\njUk+nOTv0qnP9yb535PcPcn/U0o5U2v99bW2OUU2XPT6S0luSjKSM74jzRJjTLY/mQv/0rjfRT+/\nZ5I/7NnnwBDa/L2e4/2nJHe76OcPS+dxT4v7fPe4/wIY97LedUrykSSvTrJ9mX1+qqe9s0m+Ydyf\nUwvLOH6nLtGHr0vyqe7xfy3JDT3tXT7uz6iFZUzffVu6/2DUJAtJfmCZfTck+Xvj/pzGvYzhu+9H\neo71N+k8xudS+/1Yz35fSHLZuD+rcS9Jfi7JNUn+UZJHdLdd1fM5vXzI7Y0sS4zrA7wsyee7nb0j\nyWOW2O/BSc7k/OWqBwzQ5jf3fEA3JrnnEvs9s2e/D477P7ZxLmOq0/1Wud/v9tTpFeP+rMa9jKNW\nSxz/33WP/dkk945Q1kSdkhzsqcNzx/05tL6M6bvvTT01+vEV9j3Ws+8V4/68WlxGFcpGnSXGNdD/\nqUkWH+R5Xa31o5faqdb6+XS+TJLO5ap/OECbz+1Zv7bW+tUl9ntnOqenk2TnjA+oXPc61Vr/bpW7\n/k7P+hVrbW+KjON36gKllCekc3YhSV5Sa715WMeeIutep+6llud0X/5JrfUtaz3WDBnH79ODe9Y/\nvsK+H+tZv9cAbdK/kWaJcYWy7+pZ/8MV9u39+dNH3WbtRNwjPZtm+W7McdRptXr/wZ9bh/ZaN9Za\nlVI2JnlNOt8pv19r/b1hHHcKjaNOz8v57/rXDHCcWTKOOv1tz/pKY6EWf34uFwY0Rm+kWWJcoeyx\nPevXr7DvsSXet2qllLulMyAzSW5P8uejbnNKrGud+tTbxo3r0F7rxl2rn03njOWZJC8e0jGn0Tjq\n9O096+8ppTyglPKKUsqJnukwPtqdSuAbBmhnmoyjTr/fs/6z3TOcd1FKeUHO3wzwxlrrlwZokz6s\nR5YY192Xvb/4N6yw72fT+WtgQ5JHlVJKN4H246HpDEBOklO11ttX2L/3H/lZ/pJa7zqtSvcupef3\nbHrHKNqZMGOrVSnlG5P8fPflz9daP7vWY82AcdRp8R/wLyd5eDrTBzzkon2+qbu8qJSyv9Z6YA3t\nTJNx1Olt6Qwg//507rT861LKf86Fd19+Xzp3X6a770vW0A5rN/IsMa5Qtrln/YvL7Vhrvb2U8pV0\n5ou5LJ3r52dG1V7XTUu8d9asd51W618keXR3/USEsmRMter+5fiadMbTHEvnzlmWtq51KqXcI8l9\nuy/PJXl793ifSvLadMa83C+dsVBPTydY/Eop5bZa63/op60ps+6/T7XWWkp5TpJfSids3TudqRUu\n9qEkv5jkD0b1hy9LGnmWGNfly00960sNkuu10LN+7wlob1o097mVUnYn+b+7L29P8sJa6x2jaGvC\njKtWP57kSen8g3+1Wqxovet0v571+3df/2E6dxO+stb6llrrb9Zan5Hkn/fs+yullIeuob1pMZbf\np1rruSQHkrwinQliL+Xx6QwX+La1tsOajfy/i7E/ZmkM/GUxoUopj07nrsvFM7w/V2t9/xi7NNNK\n51FX13Rf/mqt9fg4+8MlXfwdf3OSH77UHWO11t9K5xJa0pmc9EUj7hsXKaU8PZ3Lpf9vkvenMzh8\nczpnoh+Vznxct6RzCfO6UsrQ7p6mbyPJEuMKZb2ndlcz23fv3XVruc2+t73V3Kk3aHvTYr3rtKRS\nyiOS/FE6syQnnRAw6+Neeo2jVr+Rzl+ONyb5V2s8xqxZ7zpd/J5DtdabLrlnx7U969+5hvamxbr/\nPnUD2TvSudz8u0m+o9b6rlrrl2utt9VaP1FrvSad6Tq+1u3XG0spF48PZHRGniXGFcrme9YfsORe\nuXNQ9+JjQW5P56+EkbV3iX3ml9xr+q13nZY69tcneU86gyyT5DdrrT89rONPiXWtVSnlh3L+cTz/\nZ611aPWecuv9O3Wm+95FH1ph/947DWd5jsZxfPf923T+Tb4jyU8uNRSg1vqBJK/vvrx3OpOksj5G\nniXGNdD/Y0ke0V2/PMtPafDQnH+u1cfXOLDxs0luTeeuiYeWUi5b4a6J3luRZ3kOmPWu012UUram\nE8gu7256bTrjmLjQetfqBd3//VySHaWUHUvsd9+e9ReXUha/mH671vq3l3rDlFvXOnUHj38852/j\n//IKb+n9+X2X3Gv6rWudulcCvqn78i9rradXeMu7k7ywu/6EfttjzUaeJcYVyj6S8xOp7UzyP5bZ\nd2fP+kfW0lit9Y5Syl8luTKd/8/fnOXnnhm4zSmxrnW6WPe0/HuSPLK76c1JfswdR5e03rVafGj8\nQ3L+xouVvLRn/e25cLLMWTGO36m/yPlQdp/ldsyFQWylADfN1rtOW3vWV/NQ897amNF/naxHlhjX\n5ct+ZrntnSF5pZmVB26zlFIu+vmRpfadAeOoU5KklPKgJNcl2d7d9JYkz3N335LGViv6Mo46vbNn\n/coV9u39+SxfJVjvOvUGsa9fxf69Z2CWGyPI8I02SwzrIZ39LFnbw14X4oHkU1+n7vHun85MyYt1\neFuSy8b9ebS8jKtWq+jXDT11vHzcn9O4lzF9921OZ5xTTecMy5LHSmeA+WK9fnHcn9es1CmdAeEL\nPZ/9k1bY/909+/7MuD+vFpd4IPnq1c412Fd2X5Yk/7mU0jufTkop90zyhpw/NfvqusRdQ6WU15dS\nand5+RJt/nmSw92XD0vy6u7El73HeVg6d5QtuuSxZsU46lRKuW+SdyV5XHfTf0vyA3XlmZNn2jhq\nRf/G9N03n84g8qRz+fKN3TYuPtbVSZ7dfXlLLvwunCnrXada60IufMzSG7r/Hl3qWD+X5Du6L7+W\n5K2r+3/FSlrIEuMaU5Z0OvzsJE9JZzK8Py+l/FY6M0w/NMmP5vw4iL9M8q+H0OZPpTPh3pbu8R9b\nSnljOqd/r0hn4OTi3RJvrrWaKX796/TOnL+EcjqdcWTf3TkTvKRba63vGrDdaTCO3yn6N446/XKS\nZ6QzxuUZST5SSrl4Rv9n9Oz/wlrrF4bQ7iRb7zr9XJKnpXOl4JHp1OhNSf4snbNoD0vynCRP7HnP\nK6rHmi3eKPGjF21+XM/6U7t3yfZ6W1373IqjyxJjPr14v3TGDdVlluuTPGyF47w+qzxNmWRHkk+u\n0Oabk9x9nJ9NS8t61mmFNpZabhj3Z9TKMo7fqRWOc0PPcS4f9+fTyjKm774HJfnjFdq8NZ2xm2P/\njFpY1rtO6YS/j63iO+9sOs+ZHftn1MKSZNca/t24aq116u47kiwx1hn9a61/l84EhT+QzqR5p5Pc\nls5dWe9J57lfT6y1/s0Q2zyeToJ+aTozJn8xnVPAn0lntvhn1Fp/qNZ627DanHTjqBNro1aTYUzf\nfV9IZ+LRf5zOpbLPdNv8cpLj6Tyd4ZG11jcMq81Jt951qrV+KJ1/n56XzgPHb0gnKN+ezlmYP0vn\nrOf2WusrlzgM62BUWaJ0Ex8AAGM0i8++BABojlAGANAAoQwAoAFCGQBAA4QyAIAGCGUAAA0QygAA\nGiCUAQA0QCgDAGiAUAYA0AChDACgAUIZAEAD/n/8MD0JK7f9LAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f789931a390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create some simple data\n",
    "import numpy as np\n",
    "np.random.seed(0)\n",
    "X = np.random.random(size=(20, 1))\n",
    "\n",
    "# y = linear model + noise\n",
    "y = 3 * X.ravel() + 2 + np.random.randn(20)\n",
    "\n",
    "plt.scatter(X.ravel(), y)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As above, we can plot a line of best fit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AIzHG1Ju3IYTfB/4BOLYM9w0xxjs7qb0KuApg5MiRr16yZElX+u6WAwcOMGSIr56WE8ek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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f78993bba90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lin_reg = LinearRegression()\n",
    "lin_reg.fit(X, y)\n",
    "\n",
    "X_all = np.linspace(0, 1, 100)[:, np.newaxis]\n",
    "y_predict = lin_reg.predict(X_all)\n",
    "\n",
    "\n",
    "plt.scatter(X.ravel(), y, label='original data')\n",
    "plt.plot(X_all.ravel(), y_predict, label='estimated model', color='orange', lw=4)\n",
    "plt.grid()\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scikit-learn also has some more sophisticated models, which can respond to finer features in the data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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pci+OBaD8re1s13XrVqq9JSIjqb0louvWrazc+VmSr2tobezu1bh1L0Y5Meet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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f78a129df28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Fit a Random Forest\n",
    "from sklearn.ensemble import RandomForestRegressor\n",
    "ran_for = RandomForestRegressor()\n",
    "ran_for.fit(X, y)\n",
    "\n",
    "# Plot the data and the model prediction\n",
    "X_all = np.linspace(0, 1, 100)[:, np.newaxis]\n",
    "y_predict = ran_for.predict(X_all)\n",
    "plt.scatter(X.ravel(), y, label='original data')\n",
    "plt.plot(X_all.ravel(), y_predict, label='estimated model', lw=4, color='orange')\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compare the performance of the different classifiers using the method .score(X, y). This returns a number between [-1, 1] representing the accuracy of the model (higher is better)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.46752494572004183"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg.score(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.90916154001749661"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ran_for.score(X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, the ```score``` function implements an \"R2 score\", which is defined as\n",
    "$$\n",
    "1 - \\frac{\\|y - \\hat{y}\\|^2}{\\|y - \\bar{y}\\|^2}\n",
    "$$\n",
    "where $\\bar{y}$ is the mean of $y$. The best possible score is $1$.\n",
    "\n",
    "**Warning**. Different estimators might implement different scoring functions. Make sure you are not comparing apples and oranges."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Classification\n",
    "\n",
    "Suppose now that instead of predicting a real number you want to predict a categorial variable. For example, yes/no (2 classes), yes/no/maybe (3 classes), etc.\n",
    "\n",
    "We will consider an example dataset with 3 classes. Its a classical example in which the task is to predict from 3 species of iris (a flower) given a set of measurements of its flower. \n",
    "\n",
    "K nearest neighbors (kNN) is one of the simplest learning strategies: given a new, unknown observation, look up in your reference database which ones have the closest features and assign the predominant class.\n",
    "Let's try it out on our iris classification problem:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# the dataset\n",
    "from sklearn import datasets, neighbors\n",
    "iris = datasets.load_iris()\n",
    "X = iris.data\n",
    "y = iris.target"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['versicolor']\n"
     ]
    }
   ],
   "source": [
    "# create the model\n",
    "knn = neighbors.KNeighborsClassifier(n_neighbors=5)\n",
    "\n",
    "# fit the model\n",
    "knn.fit(X, y)\n",
    "\n",
    "# What kind of iris has 3cm x 5cm sepal and 4cm x 2cm petal?\n",
    "# call the \"predict\" method:\n",
    "result = knn.predict([[3, 5, 4, 2],])\n",
    "\n",
    "print(iris.target_names[result])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, classification accuracy can be measured with ```.score()```. In this case, the score is just number of correctly classified samples divided by the total number of samples (i.e., the average number of correctly classified samples)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.96666666666666667"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "knn.score(X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Quiz**: \n",
    "\n",
    "* Predicting the height of a person, it is a classification or a regression problem?\n",
    "\n",
    "* Predicting the grade of a student, is it a classification or a regression problem?\n",
    "\n",
    "* [Ridge()](http://scikit-learn.org/stable/modules/generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge) and a [RidgeClassifier()](http://scikit-learn.org/stable/modules/generated/sklearn.linear_model.RidgeClassifier.html) implement the same underlying model. Why do we have two different classes?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Recap: Scikit-learn's estimator interface\n",
    "\n",
    "Scikit-learn strives to have a uniform interface across all methods,\n",
    "and we'll see examples of these below. Given a scikit-learn *estimator*\n",
    "object named `model`, the following methods are available:\n",
    "\n",
    "- For all **supervised estimators**\n",
    "  + `model.fit()` : fit training data. For supervised learning applications,\n",
    "    this accepts two arguments: the data `X` and the labels `y` (e.g. `model.fit(X, y)`).\n",
    "    For unsupervised learning applications, this accepts only a single argument,\n",
    "    the data `X` (e.g. `model.fit(X)`).\n",
    "  + `model.predict()` : given a trained model, predict the label of a new set of data.\n",
    "    This method accepts one argument, the new data `X_new` (e.g. `model.predict(X_new)`),\n",
    "    and returns the learned label for each object in the array.\n",
    "  + `model.score()` : for classification or regression problems, all estimators implement\n",
    "    a score method.  Scores are between 0 and 1, with a larger score indicating a better fit.\n",
    "\n",
    "- Some also implement:\n",
    "  + `model.predict_proba()` : For classification problems, some estimators also provide\n",
    "    this method, which returns the probability that a new observation has each categorical label.\n",
    "    In this case, the label with the highest probability is returned by `model.predict()`.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model Validation\n",
    "\n",
    "An important piece of machine learning is **model validation**: that is, determining how well your model will generalize from the training data to future unlabeled data. Let's look at an example using the *nearest neighbor classifier*. This is a very simple classifier: it simply stores all training data, and for any unknown quantity, simply returns the label of the closest training point.\n",
    "\n",
    "With the iris data, it very easily returns the correct prediction for each of the input points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0\n"
     ]
    }
   ],
   "source": [
    "from sklearn.neighbors import KNeighborsClassifier\n",
    "X, y = iris.data, iris.target\n",
    "clf = KNeighborsClassifier(n_neighbors=1)\n",
    "clf.fit(X, y)\n",
    "print(clf.score(X, y))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For each class, all 50 training samples are correctly identified. But this **does not mean that our model is perfect!** In particular, such a model generalizes extremely poorly to new data. We can simulate this by splitting our data into a *training set* and a *testing set*. Scikit-learn contains some convenient routines to do this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.947368421053\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/fabian/anaconda3/lib/python3.6/site-packages/sklearn/cross_validation.py:41: DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the refactored classes and functions are moved. Also note that the interface of the new CV iterators are different from that of this module. This module will be removed in 0.20.\n",
      "  \"This module will be removed in 0.20.\", DeprecationWarning)\n"
     ]
    }
   ],
   "source": [
    "from sklearn.cross_validation import train_test_split\n",
    "Xtrain, Xtest, ytrain, ytest = train_test_split(X, y)\n",
    "clf.fit(Xtrain, ytrain)\n",
    "print(clf.score(Xtest, ytest))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This paints a better picture of the true performance of our classifier: apparently there the classification is not perfect.\n",
    "\n",
    "This is why it's **extremely important** to use a train/test split when evaluating your models.  We'll go into more depth on model evaluation later in this tutorial."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quick Application: Optical Character Recognition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To demonstrate the above principles on a more interesting problem, let's consider OCR (Optical Character Recognition) – that is, recognizing hand-written digits.\n",
    "In the wild, this problem involves both locating and identifying characters in an image. Here we'll take a shortcut and use scikit-learn's set of pre-formatted digits, which is built-in to the library."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Loading and visualizing the digits data\n",
    "\n",
    "We'll use scikit-learn's data access interface and take a look at this data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1797, 8, 8)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn import datasets\n",
    "digits = datasets.load_digits()\n",
    "digits.images.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAkMAAABECAYAAACCozEKAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAQCA0P4FPPH5pUuaYdkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f78973ccc50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from plotly import tools\n",
    "\n",
    "fig, ax = plt.subplots(1, 10)\n",
    "\n",
    "for j in range(10):\n",
    "    ax[j].imshow(digits.images[j], cmap=plt.cm.gray_r)\n",
    "    ax[j].set_xticks(())\n",
    "    ax[j].set_yticks(())\n",
    "# fig['layout'].update(height=250, width=1000, title='digits examples')\n",
    "# iplot(fig, image_height=200, image_width=200)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot a few of these:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here the data is simply each pixel value within an 8x8 grid:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1797, 8, 8)\n",
      "[[  0.   0.   5.  13.   9.   1.   0.   0.]\n",
      " [  0.   0.  13.  15.  10.  15.   5.   0.]\n",
      " [  0.   3.  15.   2.   0.  11.   8.   0.]\n",
      " [  0.   4.  12.   0.   0.   8.   8.   0.]\n",
      " [  0.   5.   8.   0.   0.   9.   8.   0.]\n",
      " [  0.   4.  11.   0.   1.  12.   7.   0.]\n",
      " [  0.   2.  14.   5.  10.  12.   0.   0.]\n",
      " [  0.   0.   6.  13.  10.   0.   0.   0.]]\n"
     ]
    }
   ],
   "source": [
    "# The images themselves\n",
    "print(digits.images.shape)\n",
    "print(digits.images[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1797, 64)\n",
      "[  0.   0.   5.  13.   9.   1.   0.   0.   0.   0.  13.  15.  10.  15.   5.\n",
      "   0.   0.   3.  15.   2.   0.  11.   8.   0.   0.   4.  12.   0.   0.   8.\n",
      "   8.   0.   0.   5.   8.   0.   0.   9.   8.   0.   0.   4.  11.   0.   1.\n",
      "  12.   7.   0.   0.   2.  14.   5.  10.  12.   0.   0.   0.   0.   6.  13.\n",
      "  10.   0.   0.   0.]\n"
     ]
    }
   ],
   "source": [
    "# The data for use in our algorithms\n",
    "print(digits.data.shape)\n",
    "print(digits.data[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 ..., 8 9 8]\n"
     ]
    }
   ],
   "source": [
    "# The target label\n",
    "print(digits.target)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So our data have 1797 samples in 64 dimensions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Classification on Digits\n",
    "\n",
    "Let's try a classification task on the digits. The first thing we'll want to do is split the digits into a training and testing sample:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1347, 64) (450, 64)\n"
     ]
    }
   ],
   "source": [
    "from sklearn.cross_validation import train_test_split\n",
    "Xtrain, Xtest, ytrain, ytest = train_test_split(\n",
    "    digits.data, digits.target, random_state=2)\n",
    "print(Xtrain.shape, Xtest.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's use a simple logistic regression which (despite its confusing name) is a classification algorithm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,\n",
       "          intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,\n",
       "          penalty='l2', random_state=None, solver='liblinear', tol=0.0001,\n",
       "          verbose=0, warm_start=False)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sklearn.linear_model import LogisticRegression\n",
    "clf = LogisticRegression()\n",
    "clf.fit(Xtrain, ytrain)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can check our classification accuracy by comparing the true values of the test set to the predictions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.94666666666666666"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "clf.score(Xtest, ytest)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This single number doesn't tell us **where** we've gone wrong: one nice way to do this is to use the *confusion matrix*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[42  0  0  0  0  0  0  0  0  0]\n",
      " [ 0 45  0  1  0  0  0  0  3  1]\n",
      " [ 0  0 47  0  0  0  0  0  0  0]\n",
      " [ 0  0  0 42  0  2  0  3  1  0]\n",
      " [ 0  2  0  0 36  0  0  0  1  1]\n",
      " [ 0  0  0  0  0 52  0  0  0  0]\n",
      " [ 0  0  0  0  0  0 42  0  1  0]\n",
      " [ 0  0  0  0  0  0  0 48  1  0]\n",
      " [ 0  2  0  0  0  0  0  0 38  0]\n",
      " [ 0  0  0  1  0  1  0  1  2 34]]\n"
     ]
    }
   ],
   "source": [
    "from sklearn.metrics import confusion_matrix\n",
    "ypred = clf.predict(Xtest)\n",
    "print(confusion_matrix(ytest, ypred))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ds8iQJGlua+5v8ZJJx25otPfqdZKIWEC1ogTgAar7pvRkkSFJUqFRSzJqzcsa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\nveIlrJn51xHxM+B/A3sAr6g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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f78964f1710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.matshow(confusion_matrix(ytest, ypred), cmap=plt.cm.gray_r)\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The interesting thing is that even with this simple logistic regression algorithm, many of the mislabeled cases are ones that we ourselves might get wrong!\n",
    "\n",
    "There are many ways to improve this classifier. To go further, we could use a more sophisticated model, use cross validation, or apply other techniques.\n",
    "We'll cover some of these topics later in the tutorial."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Cross-Validation\n",
    "One problem with validation sets is that you \"loose\" some of the data. Above, we've only used 3/4 of the data for the training, and used 1/4 for the validation. \n",
    "\n",
    "Another option is to use 2-fold cross-validation, where we split the sample in half and perform the validation twice. Thus a two-fold cross-validation gives us two estimates of the score for that parameter. Because this is a bit of a pain to do by hand, scikit-learn has a utility routine to help:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.          0.93333333  1.          0.93333333  0.86666667  1.\n",
      "  0.86666667  1.          1.          1.        ]\n"
     ]
    }
   ],
   "source": [
    "from sklearn.cross_validation import cross_val_score\n",
    "cv = cross_val_score(KNeighborsClassifier(1), X, y, cv=10)\n",
    "print(cv)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise\n",
    "\n",
    "Face recognition engine using the dataset [Labeled Faces in the Wild (LFW)](http://scikit-learn.org/stable/modules/generated/sklearn.datasets.fetch_lfw_people.html#sklearn.datasets.fetch_lfw_people).\n",
    "   * plot some of the images\n",
    "   * build a machine learning system that predicts the **name** of the person in the images.\n",
    "   * plot a confusion matrix to see which are the persons that are most often mistaken."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ariel Sharon\n",
      "Colin Powell\n",
      "Donald Rumsfeld\n",
      "George W Bush\n",
      "Gerhard Schroeder\n",
      "Hugo Chavez\n",
      "Tony Blair\n"
     ]
    }
   ],
   "source": [
    "# get the data\n",
    "from sklearn.datasets import fetch_lfw_people\n",
    "lfw_people = fetch_lfw_people(min_faces_per_person=70, resize=0.4)\n",
    "\n",
    "# print the names\n",
    "for name in lfw_people.target_names:\n",
    "    print(name)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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