PCA from scratch for 2 dimensional dataset.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Step 1: Get Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('original input: ', array([[2.5, 2.4],\n",
      "       [0.5, 0.7],\n",
      "       [2.2, 2.9],\n",
      "       [1.9, 2.2],\n",
      "       [3.1, 3. ],\n",
      "       [2.3, 2.7],\n",
      "       [2. , 1.6],\n",
      "       [1. , 1.1],\n",
      "       [1.5, 1.6],\n",
      "       [1.1, 0.9]]))\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x10bfcf8d0>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10bf606d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "input = np.array([[2.5 ,2.4],[0.5, 0.7],[2.2 ,2.9],[1.9 ,2.2],[3.1 ,3.0],[2.3 ,2.7],[2 ,1.6],[1 ,1.1],[1.5, 1.6],[1.1 ,0.9]])\n",
    "print(\"original input: \", input)\n",
    "plt.title(\"original dataset\")\n",
    "plt.scatter(input[:,0], input[:,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Step 2: Center the datapoints along the origin by subtracting the points by their mean.\n",
    "For PCA to work properly, you have to subtract the mean from each of the data dimensions.\n",
    "The mean subtracted is the average across each dimension. So, all the values\n",
    "have (the mean of the values of all the data points) subtracted, and all the values\n",
    "have subtracted from them. This produces a data set whose mean is zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('mean adjusted input: ', array([[ 0.69,  0.49],\n",
      "       [-1.31, -1.21],\n",
      "       [ 0.39,  0.99],\n",
      "       [ 0.09,  0.29],\n",
      "       [ 1.29,  1.09],\n",
      "       [ 0.49,  0.79],\n",
      "       [ 0.19, -0.31],\n",
      "       [-0.81, -0.81],\n",
      "       [-0.31, -0.31],\n",
      "       [-0.71, -1.01]]))\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x10c0e26d0>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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bT/w9FoHDPtU7LWmn7ftt3y7pcUk9cyVMZlLSgWz5gKRb/sdj+y7bd2TLWyR9RdK7pVW4cWt5fVr3/5uS3ojeu4Gm7X4uOfa9V9L5Eusry6Skb2VX/XxZ0icthzT7S0Tw6NBD0td045jhp5I+kDSVtW+TdKql36OSfqkbn4KfrbruDeznPbpxlc97kl6TdHfWXpN0LFv+c0nTkn6e/TxYdd3r2L9bXh9Jz0namy3fKenfJc1K+m9JD1Rdc4f28+8lnctew59I+v2qa97APr4o6Yqkq9nf5kFJT0t6Ottu3bjq6f3s97RWdc2denCHLwAkiMM+AJAgwh8AEkT4A0CCCH8ASBDhDwAJIvwBIEGEPwAkiPAHgAT9P+Plq2Rkq1FZAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10bf9c490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "adjusted_input = input - np.mean(input, axis=0)\n",
    "print(\"mean adjusted input: \", adjusted_input)\n",
    "plt.scatter(adjusted_input[:,0], adjusted_input[:,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "scrolled": true
   },
   "source": [
    "## Step 3: Find the covariance matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.61655556 0.61544444]\n",
      " [0.61544444 0.71655556]]\n"
     ]
    }
   ],
   "source": [
    "cov = np.cov(adjusted_input[:,0], adjusted_input[:,1])\n",
    "print(cov)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 4: Calculate the eigenvectors and eigenvalues of the covariance matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigen Value\n",
      "[0.0490834  1.28402771]\n",
      "Eigen Vector\n",
      "[[-0.73517866 -0.6778734 ]\n",
      " [ 0.6778734  -0.73517866]]\n"
     ]
    }
   ],
   "source": [
    "from numpy import linalg as LA\n",
    "w, v = LA.eig(cov)\n",
    "print(\"Eigen Value\")\n",
    "print(w)\n",
    "print(\"Eigen Vector\")\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Step 5: PCA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.82797019]\n",
      " [ 1.77758033]\n",
      " [-0.99219749]\n",
      " [-0.27421042]\n",
      " [-1.67580142]\n",
      " [-0.9129491 ]\n",
      " [ 0.09910944]\n",
      " [ 1.14457216]\n",
      " [ 0.43804614]\n",
      " [ 1.22382056]]\n"
     ]
    }
   ],
   "source": [
    "#multiply the original data with the eigen vector\n",
    "eigen_input = np.dot(adjusted_input, np.array(v.T[1]))\n",
    "print(eigen_input.reshape(10,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.82797019]\n",
      " [ 1.77758033]\n",
      " [-0.99219749]\n",
      " [-0.27421042]\n",
      " [-1.67580142]\n",
      " [-0.9129491 ]\n",
      " [ 0.09910944]\n",
      " [ 1.14457216]\n",
      " [ 0.43804614]\n",
      " [ 1.22382056]]\n"
     ]
    }
   ],
   "source": [
    "from sklearn.decomposition import PCA\n",
    "pca = PCA(n_components=1)\n",
    "pca.fit(input)\n",
    "print(pca.transform(input))  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
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