08a_pca.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "simplified-gentleman",
   "metadata": {},
   "source": [
    "Covariance matrix:\n",
    "\n",
    "$$C = \\frac{1}{m-1} \\sum_{i=1}^m (X_i - \\bar{X})(X_i - \\bar{X})^T$$\n",
    "\n",
    "where $X$ is an $m \\times n$ data matrix ($m$ instances, $n$ features) and $X_i$ is the feature vector for instance $i$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "grateful-christmas",
   "metadata": {},
   "source": [
    "Largest eigenvalue of $C$ indicates first principal component, second largest is second, etc."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dental-supervision",
   "metadata": {},
   "source": [
    "Relation to SVD:\n",
    "\n",
    "$$M = U \\Sigma V^T$$\n",
    "\n",
    "Calculate the covariance matrix of M:\n",
    "\n",
    "$$C = M^T M = V \\Sigma U^T U \\Sigma V^T = V \\Sigma^2 V^T$$\n",
    "\n",
    "$\\Sigma^2$ is diagonal, so $V$ is the rotation matrix that diagonalizes the covariance matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "sapphire-catering",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "united-methodology",
   "metadata": {},
   "outputs": [],
   "source": [
    "X = random.multivariate_normal([0.0, 0.0],[[1.0,0.9],[0.9,1.0]], size=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "innovative-collect",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb8fa3394e0>]"
      ]
     },
     "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": [
    "axis([-5,5,-5,5])\n",
    "plot(X[:,0], X[:,1], 'bo')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "affiliated-programming",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.18137595, 1.01358417],\n",
       "       [1.01358417, 1.057253  ]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dot(X.T-mean(X.T),X-mean(X))*(1.0/(100.0-1.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "spread-juvenile",
   "metadata": {},
   "outputs": [],
   "source": [
    "C = cov(X,rowvar=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "multiple-celtic",
   "metadata": {},
   "outputs": [],
   "source": [
    "w, v = linalg.eig(C)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "motivated-possible",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2.13479522, 0.10206195])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "solved-working",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.72837562, -0.68517804],\n",
       "       [ 0.68517804,  0.72837562]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "distributed-mixer",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.72837562, 0.68517804])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[:,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "stainless-chassis",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.68517804,  0.72837562])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[:,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "governing-better",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb8cd011518>]"
      ]
     },
     "execution_count": 17,
     "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": [
    "axis([-5,5,-5,5])\n",
    "plot(X[:,0], X[:,1], 'bo')\n",
    "plot([0.0,v[:,0][0]],[0.0,v[:,0][1]],'k-')\n",
    "plot([0.0,v[:,1][0]],[0.0,v[:,1][1]],'k-')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "specialized-forth",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.decomposition import PCA\n",
    "\n",
    "pca = PCA(n_components = 2)\n",
    "X2D = pca.fit_transform(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "adapted-thriller",
   "metadata": {},
   "outputs": [],
   "source": [
    "c1 = pca.components_.T[:,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "radio-accreditation",
   "metadata": {},
   "outputs": [],
   "source": [
    "c2 = pca.components_.T[:,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "funded-carroll",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb8c3ece7b8>]"
      ]
     },
     "execution_count": 25,
     "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": [
    "axis([-5,5,-5,5])\n",
    "plot(X[:,0], X[:,1], 'bo')\n",
    "plot([0.0,c1[0]],[0.0,c1[1]],'k-')\n",
    "plot([0.0,c2[0]],[0.0,c2[1]],'k-')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "sound-december",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dot(c1,c2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "headed-whole",
   "metadata": {},
   "outputs": [],
   "source": [
    "Xproj = dot(X, v[:,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "varying-silence",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb8c3bf4fd0>]"
      ]
     },
     "execution_count": 47,
     "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": [
    "axis([-5,5,-5,5])\n",
    "plot(X[:,0], X[:,1], 'bo')\n",
    "plot(Xproj, zeros((len(Xproj))),'ro')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "handled-teacher",
   "metadata": {},
   "outputs": [],
   "source": [
    "V = array([v[:,0],v[:,1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "secondary-theology",
   "metadata": {},
   "outputs": [],
   "source": [
    "Xrot = zeros((100,2))\n",
    "for m in range(100):\n",
    "    xnew = dot(V,X[m,:])\n",
    "    Xrot[m,:] = xnew"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "violent-fusion",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb8c3b2ada0>]"
      ]
     },
     "execution_count": 49,
     "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": [
    "axis([-5,5,-5,5])\n",
    "plot(Xrot[:,0],Xrot[:,1],'bo')\n",
    "plot(Xproj, zeros((len(Xproj))),'ro')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "noticed-seafood",
   "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}