PCA_week4.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Week 4 Assessment: Principal Component Analysis (PCA)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Learning Objective\n",
    "\n",
    "In this notebook, we will implement PCA. We will implement the two versions of PCA as described in the lectures, which handles the when the dataset size exceeds the dataset dimensionality, as well as the case when we have the dimensionality greater than the size of the dataset.\n",
    "\n",
    "We will break down the task of implementing PCA into small components and combine them in the end to produce the final algorithm. We will apply PCA to the MNIST dataset and observe how the reconstruction changes as we change the number of principal components used.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# PACKAGE: DO NOT EDIT\n",
    "import numpy as np\n",
    "import timeit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/lib/python3.6/site-packages/matplotlib/__init__.py:1401: UserWarning:  This call to matplotlib.use() has no effect\n",
      "because the backend has already been chosen;\n",
      "matplotlib.use() must be called *before* pylab, matplotlib.pyplot,\n",
      "or matplotlib.backends is imported for the first time.\n",
      "\n",
      "  warnings.warn(_use_error_msg)\n"
     ]
    }
   ],
   "source": [
    "# PACKAGE: DO NOT EDIT\n",
    "import matplotlib as mpl\n",
    "mpl.use('Agg')\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('fivethirtyeight')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's plot a digit from the dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcf4f0f1400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from ipywidgets import interact\n",
    "\n",
    "from sklearn.datasets import fetch_mldata\n",
    "MNIST = fetch_mldata('MNIST original', data_home='./MNIST')\n",
    "%matplotlib inline\n",
    "\n",
    "plt.figure(figsize=(4,4))\n",
    "plt.imshow(MNIST.data[0].reshape(28,28), cmap='gray');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we implement PCA, we will need to do some data preprocessing. In this assessment, some of them \n",
    "will be implemented by you, others we will take care of. However, when you are working on real world problems, you will need to do all these steps by yourself!\n",
    "\n",
    "The preprocessing steps we will do are\n",
    "1. Convert unsigned interger 8 (uint8) encoding of pixels to a floating point number between 0-1.\n",
    "2. Subtract from each image the mean $\\mu$.\n",
    "3. Scale each dimension of each image by $\\frac{1}{\\sigma}$ where $\\sigma$ is the standard deviation of this dimension across the whole dataset.\n",
    "\n",
    "The steps above ensure that our images will have zero mean and one variance. These preprocessing\n",
    "steps are also known as [Data Normalization or Feature Scaling](https://en.wikipedia.org/wiki/Feature_scaling)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. PCA\n",
    "\n",
    "Now we will implement PCA. Before we do that, let's pause for a moment and\n",
    "think about the steps for performing PCA. Assume that we are performing PCA on\n",
    "some dataset $\\boldsymbol X$ for $M$ principal components. \n",
    "We then need to perform the following steps, which we break into parts:\n",
    "\n",
    "1. Data normalization (`normalize`).\n",
    "2. Find eigenvalues and corresponding eigenvectors for the covariance matrix $\\boldsymbol S$.\n",
    "   Sort by the largest eigenvalues and the corresponding eigenvectors (`eig`).\n",
    "\n",
    "After these steps, we can then compute the projection and reconstruction of the data onto the spaced spanned by the top $M$ eigenvectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# GRADED FUNCTION: DO NOT EDIT THIS LINE\n",
    "\n",
    "# ===YOU SHOULD EDIT THIS FUNCTION===\n",
    "def normalize(X):\n",
    "    \"\"\"Normalize the given dataset X\n",
    "    Args:\n",
    "        X: ndarray, dataset\n",
    "    \n",
    "    Returns:\n",
    "        (Xbar, mean, std): ndarray, Xbar is the normalized dataset\n",
    "        with mean 0 and standard deviation 1; mean and std are the \n",
    "        mean and standard deviation respectively.\n",
    "    \n",
    "    Note:\n",
    "        You will encounter dimensions where the standard deviation is\n",
    "        zero, for those when you do normalization the normalized data\n",
    "        will be NaN. Handle this by setting using `std = 1` for those \n",
    "        dimensions when doing normalization.\n",
    "    \"\"\"\n",
    "    mu = np.mean(X, axis=0) # EDIT THIS\n",
    "    std = np.std(X, axis=0)\n",
    "    std_filled = std.copy()\n",
    "    std_filled[std==0] = 1.\n",
    "    Xbar = (X-mu)/std_filled                 # EDIT THIS\n",
    "    return Xbar, mu, std\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 3.  2.  1.]\n",
      "[[ 0.  0.  1.]\n",
      " [ 0.  1.  0.]\n",
      " [ 1.  0.  0.]]\n",
      "(array([ 3.,  2.,  1.]), array([[ 0.,  0.,  1.],\n",
      "       [ 0.,  1.,  0.],\n",
      "       [ 1.,  0.,  0.]]))\n"
     ]
    }
   ],
   "source": [
    "w, v = np.linalg.eig(np.diag((1, 2, 3)))\n",
    "argsort = np.argsort(-w)\n",
    "print(w[argsort])\n",
    "print(v[argsort])\n",
    "print(eig(np.diag((1, 2, 3))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# GRADED FUNCTION: DO NOT EDIT THIS LINE\n",
    "\n",
    "# ===YOU SHOULD EDIT THIS FUNCTION===\n",
    "def eig(S):\n",
    "    \"\"\"Compute the eigenvalues and corresponding eigenvectors \n",
    "        for the covariance matrix S.\n",
    "    Args:\n",
    "        S: ndarray, covariance matrix\n",
    "    \n",
    "    Returns:\n",
    "        (eigvals, eigvecs): ndarray, the eigenvalues and eigenvectors\n",
    "\n",
    "    Note:\n",
    "        the eigenvals and eigenvecs SHOULD BE sorted in descending\n",
    "        order of the eigen values\n",
    "        \n",
    "        Hint: take a look at np.argsort for how to sort in numpy.\n",
    "    \"\"\"\n",
    "    eigenvalue, eigenvector = np.linalg.eig(S)\n",
    "    argsort = np.argsort(eigenvalue)[::-1]\n",
    "    eigenvector = eigenvector[:, argsort]\n",
    "    eigenvalue = eigenvalue[argsort]\n",
    "    return (eigenvalue, eigenvector) # EDIT THIS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# GRADED FUNCTION: DO NOT EDIT THIS LINE\n",
    "\n",
    "# ===YOU SHOULD EDIT THIS FUNCTION===\n",
    "def projection_matrix(B):\n",
    "    \"\"\"Compute the projection matrix onto the space spanned by `B`\n",
    "    Args:\n",
    "        B: ndarray of dimension (D, M), the basis for the subspace\n",
    "    \n",
    "    Returns:\n",
    "        P: the projection matrix\n",
    "    \"\"\"\n",
    "    P = B@np.linalg.inv(B.T@B)@B.T # EDIT THIS\n",
    "    return P"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, with the help of the functions you have implemented above, let's implement PCA! __When you implement `PCA`, do take advantage of the functions that you have implemented above.__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# GRADED FUNCTION: DO NOT EDIT THIS LINE\n",
    "\n",
    "# ===YOU SHOULD EDIT THIS FUNCTION===\n",
    "def PCA(X, num_components):\n",
    "    \"\"\"\n",
    "    Args:\n",
    "        X: ndarray of size (N, D), where D is the dimension of the data,\n",
    "           and N is the number of datapoints\n",
    "        num_components: the number of principal components to use.\n",
    "    Returns:\n",
    "        X_reconstruct: ndarray of the reconstruction\n",
    "        of X from the first `num_components` principal components.\n",
    "    \"\"\"\n",
    "    # Compute the data covariance matrix S\n",
    "    S = np.cov(X, rowvar = False, bias=True)\n",
    "\n",
    "    # Next find eigenvalues and corresponding eigenvectors for S by implementing eig().\n",
    "    eig_vals, eig_vecs = eig(S)\n",
    "    \n",
    "    # Reconstruct the images from the lowerdimensional representation\n",
    "    # To do this, we first need to find the projection_matrix (which you implemented earlier)\n",
    "    # which projects our input data onto the vector space spanned by the eigenvectors\n",
    "    P = projection_matrix((eig_vecs[:, :num_components])) # projection matrix\n",
    "    \n",
    "    # Then for each data point x_i in the dataset X \n",
    "    #   we can project the original x_i onto the eigenbasis.\n",
    "    X_reconstruct = (P@X.T).T\n",
    "    return X_reconstruct\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "## Some preprocessing of the data\n",
    "NUM_DATAPOINTS = 1000\n",
    "X = (MNIST.data.reshape(-1, 28 * 28)[:NUM_DATAPOINTS]) / 255.\n",
    "Xbar, mu, std = normalize(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The greater number of of principal components we use, the smaller will our reconstruction\n",
    "error be. Now, let's answer the following question: \n",
    "\n",
    "\n",
    "> How many principal components do we need\n",
    "> in order to reach a Mean Squared Error (MSE) of less than $100$ for our dataset?\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def mse(predict, actual):\n",
    "    return np.square(predict - actual).sum(axis=1).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "loss = []\n",
    "reconstructions = []\n",
    "for num_component in range(1, 100):\n",
    "    reconst = PCA(Xbar, num_component)\n",
    "    error = mse(reconst, Xbar)\n",
    "    reconstructions.append(reconst)\n",
    "    # print('n = {:d}, reconstruction_error = {:f}'.format(num_component, error))\n",
    "    loss.append((num_component, error))\n",
    "\n",
    "reconstructions = np.asarray(reconstructions)\n",
    "reconstructions = reconstructions * std + mu # \"unnormalize\" the reconstructed image\n",
    "loss = np.asarray(loss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[   1.00000000+0.j,  445.17648843+0.j],\n",
       "       [   2.00000000+0.j,  403.94197923+0.j],\n",
       "       [   3.00000000+0.j,  377.59243065+0.j],\n",
       "       [   4.00000000+0.j,  353.30012434+0.j],\n",
       "       [   5.00000000+0.j,  335.44289452+0.j],\n",
       "       [   6.00000000+0.j,  320.51589546+0.j],\n",
       "       [   7.00000000+0.j,  307.28420723+0.j],\n",
       "       [   8.00000000+0.j,  294.95357442+0.j],\n",
       "       [   9.00000000+0.j,  283.97661502+0.j],\n",
       "       [  10.00000000+0.j,  274.01335119+0.j],\n",
       "       [  11.00000000+0.j,  264.53517783+0.j],\n",
       "       [  12.00000000+0.j,  255.41151940+0.j],\n",
       "       [  13.00000000+0.j,  246.97777313+0.j],\n",
       "       [  14.00000000+0.j,  238.61248858+0.j],\n",
       "       [  15.00000000+0.j,  230.82394538+0.j],\n",
       "       [  16.00000000+0.j,  223.74508667+0.j],\n",
       "       [  17.00000000+0.j,  217.25288549+0.j],\n",
       "       [  18.00000000+0.j,  211.10980526+0.j],\n",
       "       [  19.00000000+0.j,  205.19126619+0.j],\n",
       "       [  20.00000000+0.j,  199.45937602+0.j],\n",
       "       [  21.00000000+0.j,  193.83064587+0.j],\n",
       "       [  22.00000000+0.j,  188.72992977+0.j],\n",
       "       [  23.00000000+0.j,  183.68020736+0.j],\n",
       "       [  24.00000000+0.j,  178.71644680+0.j],\n",
       "       [  25.00000000+0.j,  173.83889867+0.j],\n",
       "       [  26.00000000+0.j,  169.24008665+0.j],\n",
       "       [  27.00000000+0.j,  164.70992918+0.j],\n",
       "       [  28.00000000+0.j,  160.39904031+0.j],\n",
       "       [  29.00000000+0.j,  156.24217637+0.j],\n",
       "       [  30.00000000+0.j,  152.31470003+0.j],\n",
       "       [  31.00000000+0.j,  148.58551212+0.j],\n",
       "       [  32.00000000+0.j,  144.96751302+0.j],\n",
       "       [  33.00000000+0.j,  141.46360970+0.j],\n",
       "       [  34.00000000+0.j,  138.04905222+0.j],\n",
       "       [  35.00000000+0.j,  134.80419402+0.j],\n",
       "       [  36.00000000+0.j,  131.63769824+0.j],\n",
       "       [  37.00000000+0.j,  128.51486788+0.j],\n",
       "       [  38.00000000+0.j,  125.44911449+0.j],\n",
       "       [  39.00000000+0.j,  122.44568498+0.j],\n",
       "       [  40.00000000+0.j,  119.61780193+0.j],\n",
       "       [  41.00000000+0.j,  116.86116100+0.j],\n",
       "       [  42.00000000+0.j,  114.12098626+0.j],\n",
       "       [  43.00000000+0.j,  111.49498167+0.j],\n",
       "       [  44.00000000+0.j,  109.04360723+0.j],\n",
       "       [  45.00000000+0.j,  106.67769811+0.j],\n",
       "       [  46.00000000+0.j,  104.41128462+0.j],\n",
       "       [  47.00000000+0.j,  102.28918562+0.j],\n",
       "       [  48.00000000+0.j,  100.20962071+0.j],\n",
       "       [  49.00000000+0.j,   98.18639674+0.j],\n",
       "       [  50.00000000+0.j,   96.20259239+0.j],\n",
       "       [  51.00000000+0.j,   94.23047999+0.j],\n",
       "       [  52.00000000+0.j,   92.32157216+0.j],\n",
       "       [  53.00000000+0.j,   90.45233407+0.j],\n",
       "       [  54.00000000+0.j,   88.71988007+0.j],\n",
       "       [  55.00000000+0.j,   87.01017231+0.j],\n",
       "       [  56.00000000+0.j,   85.35714647+0.j],\n",
       "       [  57.00000000+0.j,   83.73306986+0.j],\n",
       "       [  58.00000000+0.j,   82.13460280+0.j],\n",
       "       [  59.00000000+0.j,   80.55715323+0.j],\n",
       "       [  60.00000000+0.j,   79.04351123+0.j],\n",
       "       [  61.00000000+0.j,   77.59111949+0.j],\n",
       "       [  62.00000000+0.j,   76.17928152+0.j],\n",
       "       [  63.00000000+0.j,   74.83974007+0.j],\n",
       "       [  64.00000000+0.j,   73.53709719+0.j],\n",
       "       [  65.00000000+0.j,   72.26303721+0.j],\n",
       "       [  66.00000000+0.j,   71.01439426+0.j],\n",
       "       [  67.00000000+0.j,   69.78934379+0.j],\n",
       "       [  68.00000000+0.j,   68.60084356+0.j],\n",
       "       [  69.00000000+0.j,   67.43033735+0.j],\n",
       "       [  70.00000000+0.j,   66.29343157+0.j],\n",
       "       [  71.00000000+0.j,   65.18158647+0.j],\n",
       "       [  72.00000000+0.j,   64.09569727+0.j],\n",
       "       [  73.00000000+0.j,   63.02477536+0.j],\n",
       "       [  74.00000000+0.j,   61.96452806+0.j],\n",
       "       [  75.00000000+0.j,   60.92847182+0.j],\n",
       "       [  76.00000000+0.j,   59.92795243+0.j],\n",
       "       [  77.00000000+0.j,   58.93630438+0.j],\n",
       "       [  78.00000000+0.j,   57.95961501+0.j],\n",
       "       [  79.00000000+0.j,   56.99704150+0.j],\n",
       "       [  80.00000000+0.j,   56.06279858+0.j],\n",
       "       [  81.00000000+0.j,   55.15272708+0.j],\n",
       "       [  82.00000000+0.j,   54.25745170+0.j],\n",
       "       [  83.00000000+0.j,   53.37740707+0.j],\n",
       "       [  84.00000000+0.j,   52.51919940+0.j],\n",
       "       [  85.00000000+0.j,   51.66320389+0.j],\n",
       "       [  86.00000000+0.j,   50.85142887+0.j],\n",
       "       [  87.00000000+0.j,   50.04982152+0.j],\n",
       "       [  88.00000000+0.j,   49.25596342+0.j],\n",
       "       [  89.00000000+0.j,   48.48553832+0.j],\n",
       "       [  90.00000000+0.j,   47.72661966+0.j],\n",
       "       [  91.00000000+0.j,   46.98670333+0.j],\n",
       "       [  92.00000000+0.j,   46.26800157+0.j],\n",
       "       [  93.00000000+0.j,   45.55661019+0.j],\n",
       "       [  94.00000000+0.j,   44.85918259+0.j],\n",
       "       [  95.00000000+0.j,   44.18483827+0.j],\n",
       "       [  96.00000000+0.j,   43.52591415+0.j],\n",
       "       [  97.00000000+0.j,   42.89422674+0.j],\n",
       "       [  98.00000000+0.j,   42.27062771+0.j],\n",
       "       [  99.00000000+0.j,   41.65512189+0.j]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also put these numbers into perspective by plotting them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/lib/python3.6/site-packages/numpy/core/numeric.py:482: ComplexWarning: Casting complex values to real discards the imaginary part\n",
      "  return array(a, dtype, copy=False, order=order)\n"
     ]
    },
    {
     "data": {
      "image/png": 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RmQjUuvtXAlNDjp/irosq9AkmyufzMfuQCk7eVsfSyp5mwY+yJ3B2RVHcZfQnhoEsx2IZ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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcf480766a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(loss[:,0], loss[:,1]);\n",
    "ax.axhline(100, linestyle='--', color='r', linewidth=2)\n",
    "ax.xaxis.set_ticks(np.arange(1, 100, 5));\n",
    "ax.set(xlabel='num_components', ylabel='MSE', title='MSE vs number of principal components');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But _numbers don't tell us everything_! Just what does it mean _qualitatively_ for the loss to decrease from around\n",
    "$450.0$ to less than $100.0$?\n",
    "\n",
    "Let's find out! In the next cell, we draw the original eight as the leftmost image. Then we show the reconstruction of the image on the right, in descending number of principal components used."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/lib/python3.6/site-packages/ipykernel/__main__.py:6: ComplexWarning: Casting complex values to real discards the imaginary part\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcf4c50cba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact(image_idx=(0, 1000))\n",
    "def show_num_components_reconst(image_idx):\n",
    "    fig, ax = plt.subplots(figsize=(20., 20.))\n",
    "    actual = X[image_idx]\n",
    "    x = np.concatenate([actual[np.newaxis, :], reconstructions[:, image_idx]])\n",
    "    ax.imshow(np.hstack(x.reshape(-1, 28, 28)[np.arange(10)].astype(float)),\n",
    "              cmap='gray');\n",
    "    ax.axvline(28, color='orange', linewidth=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also browse throught the reconstructions for other digits. Once again, `interact` becomes handy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/lib/python3.6/site-packages/ipykernel/__main__.py:5: ComplexWarning: Casting complex values to real discards the imaginary part\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcf4c3dbf60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact(i=(0, 10))\n",
    "def show_pca_digits(i=1):\n",
    "    plt.figure(figsize=(4,4))\n",
    "    actual_sample = X[i].reshape(28,28).astype(float)\n",
    "    reconst_sample = (reconst[i, :] * std + mu).reshape(28, 28).astype(float)\n",
    "    plt.imshow(np.hstack([actual_sample, reconst_sample]), cmap='gray')\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. PCA for high-dimensional datasets\n",
    "\n",
    "Sometimes, the dimensionality of our dataset may be larger than the number of data points we\n",
    "have. Then it might be inefficient to perform PCA with the implementation above. Instead,\n",
    "as mentioned in the lectures, we can implement PCA in a more efficient manner, which we\n",
    "call PCA for high-dimensional data (PCA_high_dim)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the __normalized__ data matrix $\\boldsymbol{\\bar{X}}$ of size $N \\times D$ where $D > N$. To do PCA we perform the following steps:\n",
    "\n",
    "* We solve the following eigenvalue/eigenvector equation for the matrix $\\frac{1}{N} \\boldsymbol{\\bar{X}} \\boldsymbol{\\bar{X}}^T$, i.e. we solve for $\\lambda_i$, $\\boldsymbol c_i$ in\n",
    "$$\\frac{1}{N} \\boldsymbol{\\bar{X}} \\boldsymbol{\\bar{X}}^T \\boldsymbol c_i = \\lambda_i \\boldsymbol c_i.$$\n",
    "\n",
    "* We want to recover original eigenvectors $\\boldsymbol b_i$ of the data covariance matrix $\\boldsymbol S = \\frac{1}{N} \\boldsymbol{\\bar{X}^T} \\boldsymbol{\\bar{X}}$.\n",
    "\n",
    "* Left-multiply the eigenvectors $\\boldsymbol c_i$ by $\\boldsymbol{\\bar{X}}^T$ yields\n",
    "$$\\frac{1}{N} \\boldsymbol{\\bar{X}}^T \\boldsymbol{\\bar{X}} \\boldsymbol{\\bar{X}}^T \\boldsymbol c_i = \\lambda_i \\boldsymbol{\\bar{X}}^T \\boldsymbol c_i$$ and we recover $\\boldsymbol b_i=\\boldsymbol{\\bar{X}}^T \\boldsymbol c_i$ as eigenvector of $\\boldsymbol S$ with the eigenvalue $\\lambda_i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# GRADED FUNCTION: DO NOT EDIT THIS LINE\n",
    "\n",
    "def PCA_high_dim(X, num_components):\n",
    "    \"\"\"Compute PCA for small sample size. \n",
    "    Args:\n",
    "        X: ndarray of size (N, D), where D is the dimension of the data,\n",
    "           and N is the number of data points in the training set. You may assume the input \n",
    "           has been normalized.\n",
    "        num_components: the number of principal components to use.\n",
    "    Returns:\n",
    "        X_reconstruct: (N, D) ndarray. the reconstruction\n",
    "        of X from the first `num_components` principal components.\n",
    "    \"\"\"\n",
    "    N, D = X.shape\n",
    "    M = (X@X.T)/N # EDIT THIS, compute the matrix \\frac{1}{N}XX^T.\n",
    "    eig_vals, eig_vecs = eig(M) # EDIT THIS, compute the eigenvalues. \n",
    "    U = (X.T)@eig_vecs# EDIT THIS. Compute the eigenvectors for the original PCA problem.\n",
    "    norm=np.linalg.norm(U)\n",
    "    U = U/norm\n",
    "    # Similar to what you would do in PCA, compute the projection matrix,\n",
    "    # then perform the projection.\n",
    "    P = projection_matrix(U[:,:num_components])# projection matrix\n",
    "    X_reconstruct = X@P.T # EDIT THIS.\n",
    "    return X_reconstruct"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given the same dataset, `PCA_high_dim` and `PCA` should give the same output. \n",
    "Assuming we have implemented `PCA` correctly, we can then use `PCA` to test the correctness\n",
    "of `PCA_high_dim`.\n",
    "\n",
    "We can use this __invariant__\n",
    "to test our implementation of `PCA_high_dim`, assuming that we have correctly implemented `PCA`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "correct\n"
     ]
    }
   ],
   "source": [
    "np.testing.assert_almost_equal(PCA(Xbar, 2), PCA_high_dim(Xbar, 2))\n",
    "# In fact, you can generate random input dataset to verify your implementation.\n",
    "print('correct')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's compare the running time between `PCA` and `PCA_high_dim`.\n",
    "\n",
    "__Tips__ for running benchmarks or computationally expensive code:\n",
    "\n",
    "When you have some computation that takes up a non-negligible amount of time. Try separating\n",
    "the code that produces output from the code that analyzes the result (e.g. plot the results, comput statistics of the results). In this way, you don't have to recompute when you want to produce more analysis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def time(f, repeat=100):\n",
    "    times = []\n",
    "    for _ in range(repeat):\n",
    "        start = timeit.default_timer()\n",
    "        f()\n",
    "        stop = timeit.default_timer()\n",
    "        times.append(stop-start)\n",
    "    return np.mean(times), np.std(times)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "times_mm0 = []\n",
    "times_mm1 = []\n",
    "\n",
    "for datasetsize in np.arange(4, 784, step=20):\n",
    "    XX = Xbar[:datasetsize]\n",
    "    mu, sigma = time(lambda : XX.T @ XX)\n",
    "    times_mm0.append((datasetsize, mu, sigma))\n",
    "    \n",
    "    mu, sigma = time(lambda : XX @ XX.T)\n",
    "    times_mm1.append((datasetsize, mu, sigma))\n",
    "    \n",
    "times_mm0 = np.asarray(times_mm0)\n",
    "times_mm1 = np.asarray(times_mm1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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YeHFoM8Z2sPHnb7NZc8jO4A+OVHO2ujlS5GbSl5kUuTXXpsYzvUdCUMepo4ew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\nV5L+H6nXVU7zks5DAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcf4c4aef98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.set(xlabel='size of dataset', ylabel='running time')\n",
    "bar = ax.errorbar(times_mm0[:, 0], times_mm0[:, 1], times_mm0[:, 2], label=\"$X^T X$ (PCA)\", linewidth=2)\n",
    "ax.errorbar(times_mm1[:, 0], times_mm1[:, 1], times_mm1[:, 2], label=\"$X X^T$ (PCA_high_dim)\", linewidth=2)\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We first benchmark the time taken to compute $\\boldsymbol X^T\\boldsymbol X$ and $\\boldsymbol X\\boldsymbol X^T$. Jupyter's magic command `%time` is quite handy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we benchmark `PCA`, `PCA_high_dim`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "times0 = []\n",
    "times1 = []\n",
    "\n",
    "for datasetsize in np.arange(4, 784, step=100):\n",
    "    XX = Xbar[:datasetsize]\n",
    "    npc = 2\n",
    "    mu, sigma = time(lambda: PCA(XX, npc))\n",
    "    times0.append((datasetsize, mu, sigma))\n",
    "    \n",
    "    mu, sigma = time(lambda: PCA_high_dim(XX, npc))\n",
    "    times1.append((datasetsize, mu, sigma))\n",
    "    \n",
    "times0 = np.asarray(times0)\n",
    "times1 = np.asarray(times1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, use the `time` magic command."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 48 ms, sys: 4 ms, total: 52 ms\n",
      "Wall time: 29.7 ms\n",
      "CPU times: user 56 ms, sys: 0 ns, total: 56 ms\n",
      "Wall time: 28.7 ms\n"
     ]
    }
   ],
   "source": [
    "%time Xbar.T @ Xbar\n",
    "%time Xbar @ Xbar.T\n",
    "pass # Put this here, so that our output does not show the result of computing `Xbar @ Xbar.T`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also compare the running time for __PCA__ and __PCA\\_high\\_dim__ directly. Spend some time and think about what this plot means. We mentioned in lectures that `PCA_high_dim` are advantageous when\n",
    "we have dataset size $N$ < data dimension $D$. Although our plot for the two running times does not intersect exactly at $N = D$, it does show the trend."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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J5nxTl7Tn1s4pOzyxhqKAh95BD32CHnoHxVkOeegdaPxcVP866KHAl9sjfasq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\njAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fcf4c20c518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.set(xlabel='number of datapoints', ylabel='run time')\n",
    "ax.errorbar(times0[:, 0], times0[:, 1], times0[:, 2], label=\"PCA\", linewidth=2)\n",
    "ax.errorbar(times1[:, 0], times1[:, 1], times1[:, 2], label=\"PCA_high_dim\", linewidth=2)\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, with the magic command `time`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 2.41 s, sys: 104 ms, total: 2.51 s\n",
      "Wall time: 1.36 s\n",
      "CPU times: user 6.56 s, sys: 212 ms, total: 6.77 s\n",
      "Wall time: 3.53 s\n"
     ]
    }
   ],
   "source": [
    "%time PCA(Xbar, 2)\n",
    "%time PCA_high_dim(Xbar, 2)\n",
    "pass"
   ]
  }
 ],
 "metadata": {
  "coursera": {
   "course_slug": "mathematics-machine-learning-pca",
   "graded_item_id": "CXC11",
   "launcher_item_id": "ub5A7"
  },
  "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": "3.6.0"
  },
  "widgets": {
   "state": {
    "88f3621a7d8d4539925ec6c530d8372a": {
     "views": [
      {
       "cell_index": 24
      }
     ]
    },
    "8c41a0bcbf4243d8b84af0e3a95885d9": {
     "views": [
      {
       "cell_index": 22
      }
     ]
    }
   },
   "version": "1.2.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}