pca.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PCA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Goals:\n",
    "- calcualte the initial variance on toy 2d dataset\n",
    "- hand-code the PCA algorithm to reduce the dataset to 1 dimension\n",
    "- calcualte the variance of a reduced dataset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Import basic libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Create a toy dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5.49819804, 13.23551867],\n",
       "       [ 3.12081823,  6.96278967],\n",
       "       [ 4.85927408, 11.66411915],\n",
       "       [ 4.83168255, 13.34882128],\n",
       "       [-2.11881895, -5.70526661]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 100\n",
    "D = 2\n",
    "\n",
    "# data along x axis is normally distributed with mean data_mu and variance data_sigma\n",
    "data_mu = 2.5\n",
    "data_sigma = 3.0\n",
    "\n",
    "# data along y axis has a true relation to x as y = k * x, \n",
    "# except for the measurement error\n",
    "k = 2.5\n",
    "\n",
    "# error is normally distributed random variable with mean err_mu and variance err_sigma\n",
    "# that is, the error is sum of many different random influences\n",
    "err_mu = 0\n",
    "err_sigma = 1.5\n",
    "\n",
    "# make up the dataset\n",
    "data = np.empty([N, D], dtype=float)\n",
    "data[:, 0] = data_sigma * np.random.randn(N) + data_mu\n",
    "data[:, 1] = err_sigma * np.random.randn(N) + err_mu # errors\n",
    "data[:, 1] = data[:, 0] * k + data[:, 1]\n",
    "\n",
    "# peek into the dataset\n",
    "data[:5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot the toy dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x2563539f588>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(data[:, 0], data[:, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Prepare the data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Zero-center"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x2563544bf60>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "data[:, 0] = data[:, 0] - np.mean(data[:, 0])\n",
    "data[:, 1] = data[:, 1] - np.mean(data[:, 1])\n",
    "plt.scatter(data[:, 0], data[:, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scale"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x256354bea90>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "data[:, 0] = data[:, 0] / np.std(data[:, 0])\n",
    "data[:, 1] = data[:, 1] / np.std(data[:, 1])\n",
    "plt.scatter(data[:, 0], data[:, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate variance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9999999999999999, 0.9999999999999999)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.var(data[:, 0]), np.var(data[:, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[100.        ,  97.83960949],\n",
       "       [ 97.83960949, 100.        ]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# calculate the covariance matrix\n",
    "covariance_matrix = data.T.dot(data)\n",
    "covariance_matrix[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([197.83960949,   2.16039051]), array([[ 0.70710678, -0.70710678],\n",
       "        [ 0.70710678,  0.70710678]]))"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# calculate eigenvectors for the covariance matrix\n",
    "w, v = np.linalg.eig(covariance_matrix)\n",
    "w, v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.70710678,  0.70710678],\n",
       "       [ 0.70710678, -0.70710678]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# get the eigenvector sorted\n",
    "eig = v[np.argsort(w)]\n",
    "eig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# transform the data by calculating the projection on eigenvectors\n",
    "data1 = data.dot(eig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate the new variance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.978396094903176, 0.02160390509682361)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_var = np.var(data1[:, 0]), np.var(data1[:, 1])\n",
    "new_var"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot the new dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x25635950cf8>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "axes = plt.gca()\n",
    "axes.set_ylim([-2.0, 2.0])\n",
    "\n",
    "x = data1[:, np.argmax(new_var)]\n",
    "y = data1[:, np.argmin(new_var)]\n",
    "\n",
    "plt.scatter(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Drop dimension"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x256359aff28>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(data1[:, np.argmax(new_var)], np.zeros(N))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By applying the PCA algorithm, the initial 2d dataset could be reduced to a single dimension dataset, while retaining most of the variance"
   ]
  }
 ],
 "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}