7 - K-Means Clustering & PCA.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# K-Means Clustering:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.io import loadmat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Finding closest centroids:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def findClosestCentroids(X, centroids):\n",
    "    m = X.shape[0]\n",
    "    k = centroids.shape[0]\n",
    "    idx = np.zeros(m)\n",
    "    dist = np.zeros(k)\n",
    "    \n",
    "    for i in range(m):\n",
    "        for j in range(k):\n",
    "            dist[j] = np.sqrt(np.sum((X[i,:] - centroids[j,:]) ** 2))\n",
    "        idx[i] = np.argmin(dist)\n",
    "    \n",
    "    return idx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 2., 1.])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# test the implementation of findClosestCentroids\n",
    "data = loadmat('machine-learning-ex7/ex7/ex7data2.mat')\n",
    "X = data['X']\n",
    "initialCentroids = np.array([[3,3], [6,2], [8,5]])\n",
    "\n",
    "idx = findClosestCentroids(X, initialCentroids)\n",
    "idx[:3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Computing centroid means:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def computeCentroids(X, idx, k):\n",
    "    n = X.shape[1]\n",
    "    centroids = np.zeros((k,n))\n",
    "    \n",
    "    for i in range(k):\n",
    "        centroids[i, :] = np.mean(X[np.where(idx == i), :], axis=1)\n",
    "    \n",
    "    return centroids"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2.42830111, 3.15792418],\n",
       "       [5.81350331, 2.63365645],\n",
       "       [7.11938687, 3.6166844 ]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "computeCentroids(X, idx, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### K-means on example dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def runKmeans(X, centroids, max_iters):\n",
    "    k = len(centroids)\n",
    "    \n",
    "    # run kmeans\n",
    "    for i in range(max_iters):\n",
    "        idx = findClosestCentroids(X, centroids)\n",
    "        \n",
    "        centroids = computeCentroids(X, idx, k)\n",
    "    \n",
    "    return idx, centroids"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "max_iters = 10\n",
    "\n",
    "idx, centroids = runKmeans(X, initialCentroids, max_iters)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f21cb0f35f8>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f21cd1249e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cluster1 = X[np.where(idx == 0)[0],:]\n",
    "cluster2 = X[np.where(idx == 1)[0],:]\n",
    "cluster3 = X[np.where(idx == 2)[0],:]\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(12,8))\n",
    "ax.scatter(cluster1[:,0], cluster1[:,1], s=50, c='m')\n",
    "ax.scatter(cluster2[:,0], cluster2[:,1], s=50, c='b')\n",
    "ax.scatter(cluster3[:,0], cluster3[:,1], s=50, c='g')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Random initialization:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def initCentroids(X, k):\n",
    "    m = X.shape[0]\n",
    "    \n",
    "    randidx = np.random.randint(0, m, k)\n",
    "    \n",
    "    return X[randidx, :]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3.17109619, 5.5946452 ],\n",
       "       [3.30063655, 1.28107588],\n",
       "       [2.37198785, 1.42789607]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "initCentroids(X, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Image compression with K-means:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image(filename='machine-learning-ex7/ex7/bird_small.png')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'A': array([[[219, 180, 103],\n",
       "         [230, 185, 116],\n",
       "         [226, 186, 110],\n",
       "         ...,\n",
       "         [ 14,  15,  13],\n",
       "         [ 13,  15,  12],\n",
       "         [ 12,  14,  12]],\n",
       " \n",
       "        [[230, 193, 119],\n",
       "         [224, 192, 120],\n",
       "         [226, 192, 124],\n",
       "         ...,\n",
       "         [ 16,  16,  13],\n",
       "         [ 14,  15,  10],\n",
       "         [ 11,  14,   9]],\n",
       " \n",
       "        [[228, 191, 123],\n",
       "         [228, 191, 121],\n",
       "         [220, 185, 118],\n",
       "         ...,\n",
       "         [ 14,  16,  13],\n",
       "         [ 13,  13,  11],\n",
       "         [ 11,  15,  10]],\n",
       " \n",
       "        ...,\n",
       " \n",
       "        [[ 15,  18,  16],\n",
       "         [ 18,  21,  18],\n",
       "         [ 18,  19,  16],\n",
       "         ...,\n",
       "         [ 81,  45,  45],\n",
       "         [ 70,  43,  35],\n",
       "         [ 72,  51,  43]],\n",
       " \n",
       "        [[ 16,  17,  17],\n",
       "         [ 17,  18,  19],\n",
       "         [ 20,  19,  20],\n",
       "         ...,\n",
       "         [ 80,  38,  40],\n",
       "         [ 68,  39,  40],\n",
       "         [ 59,  43,  42]],\n",
       " \n",
       "        [[ 15,  19,  19],\n",
       "         [ 20,  20,  18],\n",
       "         [ 18,  19,  17],\n",
       "         ...,\n",
       "         [ 65,  43,  39],\n",
       "         [ 58,  37,  38],\n",
       "         [ 52,  39,  34]]], dtype=uint8),\n",
       " '__globals__': [],\n",
       " '__header__': b'MATLAB 5.0 MAT-file, Platform: GLNXA64, Created on: Tue Jun  5 04:06:24 2012',\n",
       " '__version__': '1.0'}"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "imageData = loadmat('machine-learning-ex7/ex7/bird_small.mat')\n",
    "imageData"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(128, 128, 3)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = imageData['A']\n",
    "A.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(16384, 3)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Divide by 255 so that all values are in the range 0 - 1\n",
    "A = A / 255\n",
    "\n",
    "# reshape the array\n",
    "X = np.reshape(A, (A.shape[0] * A.shape[1], A.shape[2]))\n",
    "X.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(16384, 3)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Reduce the no. of colors to 16\n",
    "k = 16\n",
    "\n",
    "# randomly initialize the centroids\n",
    "initialCentroids = initCentroids(X, k)\n",
    "\n",
    "# run the kmeans algorithm\n",
    "idx, centroids = runKmeans(X, initialCentroids, 10)\n",
    "\n",
    "# get the centroids(colors) for every pixel\n",
    "idx = findClosestCentroids(X, centroids)\n",
    "\n",
    "# map each pixel to it's centroid values\n",
    "X_comp = centroids[idx.astype(int), :]\n",
    "X_comp.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(128, 128, 3)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# reshape to original dimensions\n",
    "X_comp = np.reshape(X_comp, A.shape)\n",
    "X_comp.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f21cb056f98>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f21cd11cbe0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(X_comp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Principal component analysis:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example Dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'X': array([[3.38156267, 3.38911268],\n",
       "        [4.52787538, 5.8541781 ],\n",
       "        [2.65568187, 4.41199472],\n",
       "        [2.76523467, 3.71541365],\n",
       "        [2.84656011, 4.17550645],\n",
       "        [3.89067196, 6.48838087],\n",
       "        [3.47580524, 3.63284876],\n",
       "        [5.91129845, 6.68076853],\n",
       "        [3.92889397, 5.09844661],\n",
       "        [4.56183537, 5.62329929],\n",
       "        [4.57407171, 5.39765069],\n",
       "        [4.37173356, 5.46116549],\n",
       "        [4.19169388, 4.95469359],\n",
       "        [5.24408518, 4.66148767],\n",
       "        [2.8358402 , 3.76801716],\n",
       "        [5.63526969, 6.31211438],\n",
       "        [4.68632968, 5.6652411 ],\n",
       "        [2.85051337, 4.62645627],\n",
       "        [5.1101573 , 7.36319662],\n",
       "        [5.18256377, 4.64650909],\n",
       "        [5.70732809, 6.68103995],\n",
       "        [3.57968458, 4.80278074],\n",
       "        [5.63937773, 6.12043594],\n",
       "        [4.26346851, 4.68942896],\n",
       "        [2.53651693, 3.88449078],\n",
       "        [3.22382902, 4.94255585],\n",
       "        [4.92948801, 5.95501971],\n",
       "        [5.79295774, 5.10839305],\n",
       "        [2.81684824, 4.81895769],\n",
       "        [3.88882414, 5.10036564],\n",
       "        [3.34323419, 5.89301345],\n",
       "        [5.87973414, 5.52141664],\n",
       "        [3.10391912, 3.85710242],\n",
       "        [5.33150572, 4.68074235],\n",
       "        [3.37542687, 4.56537852],\n",
       "        [4.77667888, 6.25435039],\n",
       "        [2.6757463 , 3.73096988],\n",
       "        [5.50027665, 5.67948113],\n",
       "        [1.79709714, 3.24753885],\n",
       "        [4.3225147 , 5.11110472],\n",
       "        [4.42100445, 6.02563978],\n",
       "        [3.17929886, 4.43686032],\n",
       "        [3.03354125, 3.97879278],\n",
       "        [4.6093482 , 5.879792  ],\n",
       "        [2.96378859, 3.30024835],\n",
       "        [3.97176248, 5.40773735],\n",
       "        [1.18023321, 2.87869409],\n",
       "        [1.91895045, 5.07107848],\n",
       "        [3.95524687, 4.5053271 ],\n",
       "        [5.11795499, 6.08507386]]),\n",
       " '__globals__': [],\n",
       " '__header__': b'MATLAB 5.0 MAT-file, Platform: PCWIN64, Created on: Mon Nov 14 22:41:44 2011',\n",
       " '__version__': '1.0'}"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = loadmat('machine-learning-ex7/ex7/ex7data1.mat')\n",
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x7f21cb0221d0>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f21cb0f39e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X = data['X']\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(12,8))\n",
    "ax.scatter(X[:,0], X[:,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Implementing PCA:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "def pca(X):\n",
    "    # normalize the features\n",
    "    X = (X - X.mean()) / X.std()\n",
    "    \n",
    "    # compute the covariance matrix\n",
    "    X = np.matrix(X)\n",
    "    cov = (X.T * X) / X.shape[0]\n",
    "    \n",
    "    # perform SVD\n",
    "    U, S, V = np.linalg.svd(cov)\n",
    "    \n",
    "    return U, S, V"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(matrix([[-0.79241747, -0.60997914],\n",
       "         [-0.60997914,  0.79241747]]),\n",
       " array([1.43584536, 0.56415464]),\n",
       " matrix([[-0.79241747, -0.60997914],\n",
       "         [-0.60997914,  0.79241747]]))"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, S, V = pca(X)\n",
    "U, S, V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dimensionality Reduction with PCA:\n",
    "### Projecting the data onto the principal components:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def projectData(X, U, k):\n",
    "    U_reduced = U[:, :k]\n",
    "    return np.dot(X, U_reduced)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[-4.74689738],\n",
       "        [-7.15889408],\n",
       "        [-4.79563345],\n",
       "        [-4.45754509],\n",
       "        [-4.80263579],\n",
       "        [-7.04081342],\n",
       "        [-4.97025076],\n",
       "        [-8.75934561],\n",
       "        [-6.2232703 ],\n",
       "        [-7.04497331],\n",
       "        [-6.91702866],\n",
       "        [-6.79543508],\n",
       "        [-6.3438312 ],\n",
       "        [-6.99891495],\n",
       "        [-4.54558119],\n",
       "        [-8.31574426],\n",
       "        [-7.16920841],\n",
       "        [-5.08083842],\n",
       "        [-8.54077427],\n",
       "        [-6.94102769],\n",
       "        [-8.5978815 ],\n",
       "        [-5.76620067],\n",
       "        [-8.2020797 ],\n",
       "        [-6.23890078],\n",
       "        [-4.37943868],\n",
       "        [-5.56947441],\n",
       "        [-7.53865023],\n",
       "        [-7.70645413],\n",
       "        [-5.17158343],\n",
       "        [-6.19268884],\n",
       "        [-6.24385246],\n",
       "        [-8.02715303],\n",
       "        [-4.81235176],\n",
       "        [-7.07993347],\n",
       "        [-5.45953289],\n",
       "        [-7.60014707],\n",
       "        [-4.39612191],\n",
       "        [-7.82288033],\n",
       "        [-3.40498213],\n",
       "        [-6.54290343],\n",
       "        [-7.17879573],\n",
       "        [-5.22572421],\n",
       "        [-4.83081168],\n",
       "        [-7.23907851],\n",
       "        [-4.36164051],\n",
       "        [-6.44590096],\n",
       "        [-2.69118076],\n",
       "        [-4.61386195],\n",
       "        [-5.88236227],\n",
       "        [-7.76732508]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z = projectData(X, U, 1)\n",
    "Z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reconstructing an approximation of the data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "def recoverData(Z, U, k):\n",
    "    U_reduced = U[:, :k]\n",
    "    return np.dot(Z, U_reduced.T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((50, 2), matrix([[3.76152442, 2.89550838],\n",
       "         [5.67283275, 4.36677606],\n",
       "         [3.80014373, 2.92523637],\n",
       "         [3.53223661, 2.71900952],\n",
       "         [3.80569251, 2.92950765],\n",
       "         [5.57926356, 4.29474931],\n",
       "         [3.93851354, 3.03174929],\n",
       "         [6.94105849, 5.3430181 ],\n",
       "         [4.93142811, 3.79606507],\n",
       "         [5.58255993, 4.29728676],\n",
       "         [5.48117436, 4.21924319],\n",
       "         [5.38482148, 4.14507365],\n",
       "         [5.02696267, 3.8696047 ],\n",
       "         [5.54606249, 4.26919213],\n",
       "         [3.60199795, 2.77270971],\n",
       "         [6.58954104, 5.07243054],\n",
       "         [5.681006  , 4.37306758],\n",
       "         [4.02614513, 3.09920545],\n",
       "         [6.76785875, 5.20969415],\n",
       "         [5.50019161, 4.2338821 ],\n",
       "         [6.81311151, 5.24452836],\n",
       "         [4.56923815, 3.51726213],\n",
       "         [6.49947125, 5.00309752],\n",
       "         [4.94381398, 3.80559934],\n",
       "         [3.47034372, 2.67136624],\n",
       "         [4.41334883, 3.39726321],\n",
       "         [5.97375815, 4.59841938],\n",
       "         [6.10672889, 4.70077626],\n",
       "         [4.09805306, 3.15455801],\n",
       "         [4.90719483, 3.77741101],\n",
       "         [4.94773778, 3.80861976],\n",
       "         [6.36085631, 4.8963959 ],\n",
       "         [3.81339161, 2.93543419],\n",
       "         [5.61026298, 4.31861173],\n",
       "         [4.32622924, 3.33020118],\n",
       "         [6.02248932, 4.63593118],\n",
       "         [3.48356381, 2.68154267],\n",
       "         [6.19898705, 4.77179382],\n",
       "         [2.69816733, 2.07696807],\n",
       "         [5.18471099, 3.99103461],\n",
       "         [5.68860316, 4.37891565],\n",
       "         [4.14095516, 3.18758276],\n",
       "         [3.82801958, 2.94669436],\n",
       "         [5.73637229, 4.41568689],\n",
       "         [3.45624014, 2.66050973],\n",
       "         [5.10784454, 3.93186513],\n",
       "         [2.13253865, 1.64156413],\n",
       "         [3.65610482, 2.81435955],\n",
       "         [4.66128664, 3.58811828],\n",
       "         [6.1549641 , 4.73790627]]))"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_recovered = recoverData(Z, U, 1)\n",
    "X_recovered.shape, X_recovered"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f21cad337f0>]"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f21cad33550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(12,8))\n",
    "ax.plot(X_recovered[:, 0], X_recovered[:, 1], 'bo')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Face Image Dataset:\n",
    "### PCA on Faces:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5000, 1024)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "faces = loadmat('machine-learning-ex7/ex7/ex7faces.mat')\n",
    "X = faces['X']\n",
    "X.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f21c827fba8>"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f21caba8ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "face = np.reshape(X[4000,:], (32, 32))\n",
    "plt.imshow(face)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dimensionality Reduction:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "U, S, V = pca(X)\n",
    "\n",
    "# reduce to 100 principal components\n",
    "k = 100\n",
    "Z = projectData(X, U, k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f21c5b30160>"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f21c82e8160>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_recovered = recoverData(Z, U, k)\n",
    "face = np.reshape(X_recovered[4000,:], (32, 32))\n",
    "plt.imshow(face)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}