2.3 Unsupervised Learning - Transformations and Dimensionality Reduction.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import seaborn as sns; sns.set()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Unsupervised Learning\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Many instances of unsupervised learning, such as dimensionality reduction, manifold learning and feature extraction, find a new representation of the input data without any additional input.\n",
    "\n",
    "<img src=\"figures/unsupervised_workflow.svg\" width=\"100%\">\n",
    "\n",
    "The most simple example of this, which can barely be called learning, is rescaling the data to have zero mean and unit variance. This is a helpful preprocessing step for many machine learning models.\n",
    "\n",
    "Applying such a preprocessing has a very similar interface to the supervised learning algorithms we saw so far.\n",
    "Let's load the iris dataset and rescale it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(150, 4)\n"
     ]
    }
   ],
   "source": [
    "from sklearn.datasets import load_iris\n",
    "\n",
    "iris = load_iris()\n",
    "X, y = iris.data, iris.target\n",
    "print(X.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The iris dataset is not \"centered\" that is it has non-zero mean and the standard deviation is different for each component:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean : [ 5.84333333  3.054       3.75866667  1.19866667] \n",
      "standard deviation : [ 0.82530129  0.43214658  1.75852918  0.76061262] \n"
     ]
    }
   ],
   "source": [
    "print(\"mean : %s \" % X.mean(axis=0))\n",
    "print(\"standard deviation : %s \" % X.std(axis=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To use a preprocessing method, we first import the estimator, here StandardScaler and instantiate it:\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import StandardScaler\n",
    "scaler = StandardScaler()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As with the classification and regression algorithms, we call ``fit`` to learn the model from the data. As this is an unsupervised model, we only pass ``X``, not ``y``. This simply estimates mean and standard deviation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "StandardScaler(copy=True, with_mean=True, with_std=True)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scaler.fit(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can rescale our data by applying the ``transform`` (not ``predict``) method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X_scaled = scaler.transform(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "``X_scaled`` has the same number of samples and features, but the mean was subtracted and all features were scaled to have unit standard deviation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(150, 4)\n"
     ]
    }
   ],
   "source": [
    "print(X_scaled.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean : [ -1.69031455e-15  -1.63702385e-15  -1.48251781e-15  -1.62314606e-15] \n",
      "standard deviation : [ 1.  1.  1.  1.] \n"
     ]
    }
   ],
   "source": [
    "print(\"mean : %s \" % X_scaled.mean(axis=0))\n",
    "print(\"standard deviation : %s \" % X_scaled.std(axis=0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Principal Component Analysis\n",
    "============================"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An unsupervised transformation that is somewhat more interesting is Principle Component Analysis (PCA).\n",
    "It is a technique to reduce the dimensionality of the data, by creating a linear projection.\n",
    "That is, we find new features to represent the data that are a linear combination of the old data (i.e. we rotate it).\n",
    "\n",
    "The way PCA finds these new directions is by looking for the directions of maximum variance.\n",
    "Usually only few components that explain most of the variance in the data are kept. To illustrate how a rotation might look like, we first show it on two dimensional data and keep both principal components.\n",
    "\n",
    "We create a Gaussian blob that is rotated:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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xyjyxyG5dqttNL1oOnhwNaYcQFhHLkLNSPrlE9h85LT/5xQfS2NQ67rQlkTsPGzEZDdB4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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f4582561c90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(1)\n",
    "X = np.dot(np.random.random(size=(2, 2)), np.random.normal(size=(2, 200))).T\n",
    "plt.plot(X[:, 0], X[:, 1], 'o')\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.75871884  0.01838551]\n",
      "[[ 0.94446029  0.32862557]\n",
      " [ 0.32862557 -0.94446029]]\n"
     ]
    }
   ],
   "source": [
    "from sklearn.decomposition import PCA\n",
    "pca = PCA(n_components=2)\n",
    "pca.fit(X)\n",
    "print(pca.explained_variance_)\n",
    "print(pca.components_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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qKytQ5bEDggBAgN1mQVRRIYl6EEMAIAi4/pq5ab9r8ntVum1YuqAG8+q98Pk8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\n/weSLYflzvid5QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f4579677b10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(X[:, 0], X[:, 1], 'o', alpha=0.5)\n",
    "for length, vector in zip(pca.explained_variance_, pca.components_):\n",
    "    v = vector * 3 * np.sqrt(length)\n",
    "    plt.plot([0, v[0]], [0, v[1]], '-k', lw=3)\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(200, 2)\n",
      "(200, 1)\n"
     ]
    }
   ],
   "source": [
    "clf = PCA(0.95) # keep 95% of variance\n",
    "X_trans = clf.fit_transform(X)\n",
    "print(X.shape)\n",
    "print(X_trans.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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tlb/l9ZztssDP5i1oqoyerkjpSEHAH6Ld+Xcx/N3faTh5cvpYQTGP7JUkwK3R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\nwQqGIUq1nGV5erXz0JCKLVs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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f4579509d10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_new = clf.inverse_transform(X_trans)\n",
    "plt.plot(X[:, 0], X[:, 1], 'o', alpha=0.2)\n",
    "plt.plot(X_new[:, 0], X_new[:, 1], 'ob', alpha=0.8)\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Now try the above example with ``sklearn.decomposition.RandomizedPCA``.\n",
    "\n",
    "You can re-use the plotting snippets from above.\n",
    "Are the results similar to those from standard PCA?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "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": "2.7.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}