pca_calc.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Principal Component Analysis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "# np.random.seed(6)\n",
    "import math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[14  5]\n",
      " [16  5]\n",
      " [ 6  0]\n",
      " [10  1]\n",
      " [ 8  2]\n",
      " [15  6]\n",
      " [15  7]\n",
      " [ 8  2]\n",
      " [ 8  1]\n",
      " [15  4]]\n",
      "[1 1 0 0 0 1 1 0 0 1]\n"
     ]
    }
   ],
   "source": [
    "nX=[[14, 5], [16, 5], [6, 0], [10, 1], [8, 2], [15, 6], [15, 7], [8, 2], [8, 1], [15, 4]]  #Lab demo\n",
    "# nX=[[12, 6], [16, 8], [5, 1], [13, 4], [10, 4], [14, 7], [15, 9], [10, 4], [9, 3], [15, 6]]  #  good data\n",
    "X=np.array(nX)\n",
    "y=np.array([1, 1, 0, 0, 0, 1, 1, 0, 0, 1])\n",
    "print(X)\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "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(X[:,0],X[:,1],marker='o',c=y)\n",
    "\n",
    "plt.grid() \n",
    "plt.xlim(0,20)\n",
    "plt.ylim(0,10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(10, 2)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Subtracting mean"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[11.5  3.3]\n"
     ]
    }
   ],
   "source": [
    "avg=X.mean(axis=0)\n",
    "print(avg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2.5  1.7]\n",
      " [ 4.5  1.7]\n",
      " [-5.5 -3.3]\n",
      " [-1.5 -2.3]\n",
      " [-3.5 -1.3]\n",
      " [ 3.5  2.7]\n",
      " [ 3.5  3.7]\n",
      " [-3.5 -1.3]\n",
      " [-3.5 -2.3]\n",
      " [ 3.5  0.7]]\n",
      "[1 1 0 0 0 1 1 0 0 1]\n"
     ]
    }
   ],
   "source": [
    "trainset=X-avg\n",
    "print(trainset)\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "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(trainset[:,0],trainset[:,1],marker='o',c=y)\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(10, 2)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trainset.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Calculating Covariance Matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation*}\n",
    "covM=\n",
    "\\begin{pmatrix}\n",
    "cov(x,x) & cov(x,y)\\\\\n",
    "cov(y,x) & cov(y,y) \n",
    "\\end{pmatrix}\n",
    "\\end{equation*}\n",
    "\n",
    "\\begin{equation*}\n",
    "cov(x,y)= \\frac{1}{n-1}  \\sum_{i=1}^{n}(x_{i}-\\bar{x})(y_{i}-\\bar{y})\n",
    "\\end{equation*}\n",
    "\n",
    "<!---\n",
    "\\begin{equation*}\n",
    "X^\\top Y   =  \\sum_{i=1}^{n}x_{i}y_{i}\n",
    "\\end{equation*}\n",
    "--->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance Matrix: \n",
      "[[14.72222222  8.38888889]\n",
      " [ 8.38888889  5.78888889]]\n"
     ]
    }
   ],
   "source": [
    "#using dot product\n",
    "covM=np.dot(trainset.T, trainset)\n",
    "covM=covM/( len(trainset) -1 )\n",
    "\n",
    "print('Covariance Matrix: ')\n",
    "print(covM)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Covariance Matrix using Numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[14.72222222,  8.38888889],\n",
       "       [ 8.38888889,  5.78888889]])"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.cov(trainset.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Finding Eigenvalue and Eigenvector\n",
    "\n",
    "\n",
    "\\begin{equation*}\n",
    "Ax=\\lambda x\\\\\n",
    "where   \\;x:=eigenvector  \\;and \\; \\lambda:=corresponding  \\;eigenvalue\n",
    "\\end{equation*}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "e,EV=np.linalg.eigh(covM)\t#eigenvalues & eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.75163174 19.75947937]\n"
     ]
    }
   ],
   "source": [
    "print(e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "e=e[::-1] #reverse.  to bring the larger value first.\n",
    "EV=EV[::-1] #reverse."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eigenvalue:  19.75947937160304 eigenvector:  [-0.85731596 -0.51479058]\n",
      "eigenvalue:  0.7516317395080718 eigenvector:  [ 0.51479058 -0.85731596]\n"
     ]
    }
   ],
   "source": [
    "print('eigenvalue: ',e[0], 'eigenvector: ',EV[0])\n",
    "print('eigenvalue: ',e[1], 'eigenvector: ',EV[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PC1 contribution:  96.335489894056  PC2 contribution:  3.6645101059440126\n"
     ]
    }
   ],
   "source": [
    "fc=(e[0]/(e[0]+e[1]))*100\n",
    "sc=(e[1]/(e[0]+e[1]))*100\n",
    "print('PC1 contribution: ',fc,' PC2 contribution: ',sc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting Eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "ax = plt.axes()\n",
    "ax.scatter(trainset[:,0],trainset[:,1],marker='o',c=y) #drawing the points.\n",
    "\n",
    "s=4  #arraw scale.\n",
    "ax.arrow(0,0,  -EV[0][0]*s,  -EV[0][1]*s, head_width=0.5, head_length=0.7, fc='green', ec='black')\n",
    "ax.arrow(0,0,   -EV[1][0]*s,   -EV[1][1]*s, head_width=0.5, head_length=0.7, fc='blue', ec='black')\n",
    "\n",
    "plt.grid()\n",
    "plt.title('Plotting Principal Components',fontsize=10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Projecting data on new dimension"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-3.01843389e+00 -1.70460691e-01]\n",
      " [-4.73306582e+00  8.59120466e-01]\n",
      " [ 6.41404671e+00 -2.20550373e-03]\n",
      " [ 2.46999227e+00  1.19964085e+00]\n",
      " [ 3.66983362e+00 -6.87256273e-01]\n",
      " [-4.39054043e+00 -5.12986075e-01]\n",
      " [-4.90533101e+00 -1.37030204e+00]\n",
      " [ 3.66983362e+00 -6.87256273e-01]\n",
      " [ 4.18462420e+00  1.70059690e-01]\n",
      " [-3.36095928e+00  1.20164585e+00]]\n"
     ]
    }
   ],
   "source": [
    "pc=[]\n",
    "for x in trainset:\n",
    "    pc1=np.dot(EV[0], x)\n",
    "    pc2=np.dot(EV[1], x)\n",
    "    pc.append([pc1,pc2])\n",
    "pc=np.array(pc)\n",
    "print(pc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "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(pc[:,0],pc[:,1]  ,marker='o',c=y)\n",
    "plt.title('Data projected on principal components')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Dimentionality Reduction: Taking only first principal component( which holds around 96% information for the given data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-3.01843389 -4.73306582  6.41404671  2.46999227  3.66983362 -4.39054043\n",
      " -4.90533101  3.66983362  4.1846242  -3.36095928]\n"
     ]
    }
   ],
   "source": [
    "print(pc[:,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "#Plotting the 1 dimentional data.\n",
    "ar=pc[:,0].copy()\n",
    "val=0\n",
    "plt.scatter(ar, np.zeros_like(ar) + val , marker='x' ,c=y )\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reconstructing Data\n",
    "\n",
    "\\begin{equation*}\n",
    "\\boxed{\\text{PCA reconstruction} = \\text{Mean} + \\text{Projection on PC1} \\cdot \\text{Eigenvector1}   + \\text{Projection on PC2} \\cdot \\text{Eigenvector2}  +\\text{...}  }\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first pc contains maximum info and the 2nd pc contains much less inf. Its possible to construct back the data.\n",
    "If we use all the principal components we can get the same original data. If we use first few pc we can get most likely to the origin data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[14.08775156  4.85386133]\n",
      "[15.55773288  5.73653769]\n",
      "[ 6.00113537e+00 -1.89081355e-03]\n",
      "[9.38243619 2.02847125]\n",
      "[8.35379305 1.41080423]\n",
      "[15.2640804   5.56020885]\n",
      "[15.70541858  5.82521819]\n",
      "[8.35379305 1.41080423]\n",
      "[7.91245487 1.14579489]\n",
      "[14.38140404  5.03019017]\n"
     ]
    }
   ],
   "source": [
    "#Reconstructing data.\n",
    "# np.dot(pc, EV)\n",
    "# print(pc[:,0].shape)\n",
    "\n",
    "fpc1=pc[:,0]\n",
    "fpc2=pc[:,1]\n",
    "rows=[]\n",
    "for p,p2 in zip(fpc1, fpc2):\n",
    "    contribution_1=p*EV[0]\n",
    "    contribution_2=p2*EV[1]\n",
    "#     row=avg+contribution_1+contribution_2\n",
    "    row=avg+contribution_1\n",
    "    rows.append(row)\n",
    "    print(row)\n",
    "rows=np.array(rows)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "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(rows[:,0],rows[:,1]  ,marker='o',c=y)\n",
    "plt.title('Data projected on principal components')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Helper"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 3],\n",
       "       [5, 4],\n",
       "       [3, 8]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([[1, 3], [5, 4], [3, 8]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4., 1.],\n",
       "       [1., 7.]])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.cov(A, rowvar=False )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculateCovariance(X):\n",
    "    meanX = np.mean(X, axis = 0)\n",
    "    lenX = X.shape[0]-1\n",
    "    X = X - meanX\n",
    "    covariance = X.T.dot(X)/lenX\n",
    "    return covariance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[4. 1.]\n",
      " [1. 7.]]\n"
     ]
    }
   ],
   "source": [
    "covM=calculateCovariance(A)\n",
    "print(covM)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3.69722436 7.30277564]\n",
      "[[-0.95709203  0.28978415]\n",
      " [ 0.28978415  0.95709203]]\n"
     ]
    }
   ],
   "source": [
    "e,EV=np.linalg.eigh(covM)\t#eigenvalues & eigenvectors\n",
    "print(e)\n",
    "print(EV)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "ex=[ [ 10, 5, 3],\n",
    "  [15, 4, 2],\n",
    "  [8, 8, 1] ]\n",
    "ex=np.array(ex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[10,  5,  3],\n",
       "       [15,  4,  2],\n",
       "       [ 8,  8,  1]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-1.         -0.66666667  1.        ]\n",
      " [ 4.         -1.66666667  0.        ]\n",
      " [-3.          2.33333333 -1.        ]]\n"
     ]
    }
   ],
   "source": [
    "avg=ex.mean(axis=0)\n",
    "ex=ex-avg\n",
    "print(ex)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[13.         -6.5         1.        ]\n",
      " [-6.5         4.33333333 -1.5       ]\n",
      " [ 1.         -1.5         1.        ]]\n"
     ]
    }
   ],
   "source": [
    "covm=np.cov(ex.T)\n",
    "print(covm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "e,EV=np.linalg.eigh(covm)\t#eigenvalues & eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3.25757251e-16 1.69263756e+00 1.66406958e+01]\n",
      "[[ 0.27196415  0.40151787 -0.87453925]\n",
      " [ 0.65271395  0.59081674  0.47423631]\n",
      " [ 0.70710678 -0.69979925 -0.10139534]]\n"
     ]
    }
   ],
   "source": [
    "print(e)\n",
    "print(EV)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9076743149959756"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e[2]/sum(e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.25757251e-16\n"
     ]
    }
   ],
   "source": [
    "print(3.25757251e-16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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