{ "metadata": { "name": "", "signature": "sha256:6a98804493fcad42b048eda0581d497b08c7669deb8c016ec8600e1ff7fc631b" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Principal Component Analyses" ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "A tutorial by Mickey Atwal (atwal@cshl.edu), with added geometrical insight in dot-product based rotation by Daan van Es (d.m.van.es@vu.nl)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Principal Component Analyses (PCA) is a useful mathematical technique that can reveal the structure of high-dimensional data by looking for linear combinations of variables that best explains the variance in the data. \n", "\n", "At its heart, PCA transforms data into a new coordinate system such that the variables are no longer correlated. This is useful for dimensional reduction. The output of PCA is a set of eigenvectors, which are the new axes, and a set of eigenvalues, which tell you how much variance the new axes explain. \n", "\n", "We will explore the idea behind PCA using a toy example in Python." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from __future__ import division\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "1. Toy example" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can get some intuition for this procedure by cooking up a simple two-dimensional dataset, where we set most of the variation to lie along one direction. Let's generate random data from a two-dimensional gaussian distribution, rotated around the origin so that x and y variables are now correlated." ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Generate Monte Carlo data" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's first draw random samples from a two-dimensional Gaussian (normal) distribution centered around the origin, where the x and y variables are independent and uncorrelated,\n", "$$\n", "\\begin{array}\n", "pp(x,y)=p(x)p(y)\n", "&=&\\displaystyle \\frac{1}{\\sqrt{2 \\pi \\sigma_x^2}} \\exp{ \\left( - \\frac{x^2}{2 \\sigma_x^2} \\right)} \\frac{1}{\\sqrt{2 \\pi \\sigma_y^2}} \\exp{ \\left( - \\frac{y^2}{2 \\sigma_y^2} \\right)} \\\\\n", "&=&\\displaystyle \\frac{1}{2 \\pi \\sigma_x \\sigma_y} \\exp{\\left( - \\frac{1}{2} \\left[ \\frac{x^2}{\\sigma_x^2} + \\frac{y^2}{\\sigma_y^2} \\right] \\right)}\n", "\\end{array}\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "N=100 # number of datapoints\n", "sx=7 # standard deviation in the x direction\n", "sy=2 # standard deviation in the y direction\n", "\n", "x=np.random.normal(0,sx,N) # random samples from gaussian in x-direction\n", "y=np.random.normal(0,sy,N) # random samples from gaussian in y-direction\n", "\n", "axs=max(abs(x))*1.2 # axes plotting range\n", "t=np.linspace(-axs,axs,1000) # range of mesh points for contour plot\n", "cx,cy=np.meshgrid(t,t) # mesh array\n", "z=(1/(2*np.pi*sx*sy))*np.exp(-((cx*cx)/(2*sx*sx))-((cy*cy)/(2*sy*sy))) # actual 2d gaussian\n", "\n", "plt.figure(figsize=(10,5))\n", "plt.xlim([-axs,axs])\n", "plt.ylim([-axs,axs])\n", "cm = plt.cm.RdBu # coloring scheme for contour plots\n", "\n", "plt.subplot(1,2,1)\n", "plt.contour(t,t,z,np.linspace(0.0000001,1/(2*np.pi*sx*sy),20),cmap=cm)\n", "plt.grid()\n", "plt.title('contour map of 2d gaussian distribution\\n $\\sigma_x=$ %d, $\\sigma_y=$ %d' % (sx,sy),fontsize=12)\n", "plt.xlabel('x',fontsize=12)\n", "plt.ylabel('y',fontsize=12)\n", "\n", "plt.subplot(1,2,2)\n", "plt.contour(t,t,z,np.linspace(0.0000001,1/(2*np.pi*sx*sy),20),cmap=cm)\n", "plt.plot(x,y,'bo',markersize=5)\n", "plt.grid()\n", "plt.title('%d random samples from the 2d gaussian\\n distribution' % N, fontsize=12)\n", "plt.xlabel('x',fontsize=12)\n", "plt.ylabel('y',fontsize=12)\n", "\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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GTVGUlENj0BRFSVU0Bk1RFEVRFCVFUQMN9ZWnKm6WDdwtn5tlu9S4uS3dLBu4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\n+Mdpf+MAJ1/dQ+ure3nplZdZuah2Mn/Vb91IQXXlrP/ebPOXL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"text": [ "" ] } ], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's now rotate the data by $\\theta$ degrees anticlockwise. This can be done my multiplying the (x,y) data by a rotation matrix M,\n", "$$\n", "\\left(\n", "\\begin{array}{c}\n", "x' \\\\\n", "y'\n", "\\end{array}\n", "\\right)\n", "=\n", "\\left(\n", "\\begin{array}{cc}\n", "\\cos (\\theta) & -\\sin (\\theta) \\\\\n", "\\sin (\\theta) & \\cos (\\theta)\n", "\\end{array}\n", "\\right)\n", "\\left(\n", " \\begin{array}{c}\n", " x \\\\\n", " y\n", " \\end{array}\n", " \\right)\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "degrees=30 # theta degrees\n", "theta=2*np.pi*degrees/360 # convert degrees to radians\n", "\n", "M=np.array([[np.cos(theta),-np.sin(theta)],[np.sin(theta),np.cos(theta)]]) #rotation matrix represented as 2d array\n", "print 'rotation matrix, M=',\n", "print M\n", "xyp=np.dot(M,np.vstack([x,y])).T # matrix multiplication\n", "\n", "xd=xyp[:,0] # new x-coordinates\n", "yd=xyp[:,1] # new y-coordinates\n", "\n", "plt.figure(figsize=(5,5))\n", "plt.plot(xd,yd,'bo',markersize=5)\n", "plt.xlim([-axs,axs])\n", "plt.ylim([-axs,axs])\n", "plt.grid()\n", "plt.title('%d random samples from 2d gaussian distribution \\n rotated by $\\Theta$= %d degrees anticlockwise' % (N,degrees))\n", "plt.xlabel('x')\n", "plt.ylabel('y')\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "rotation matrix, M= [[ 0.8660254 -0.5 ]\n", " [ 0.5 0.8660254]]\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that the x and y variables are now correlated. We can quantify this by calculating the **covariance** matrix and the **correlation** matrix:\n", " $$\n", " \\rm{covariance}=\n", " \\left(\n", " \\begin{array}{cc}\n", " \\left< xx \\right> & \\left< xy \\right> \\\\\n", " \\left< yx \\right> & \\left< yy \\right>\n", " \\end{array}\n", " \\right)\n", " \\hspace{1in}\n", " correlation=\n", " \\left(\n", " \\begin{array}{cc}\n", " 1 & \\frac{\\left< xy \\right>}{\\sqrt{\\left< xx \\right> \\left< yy \\right>}} \\\\\n", " \\frac{\\left< yx \\right>}{\\sqrt{\\left< yy \\right> \\left< xx \\right>}} & 1\n", " \\end{array}\n", " \\right)\n", " $$\n", " where the brackets denotes averaging (conventional physics notation), i.e.,\n", " $$\n", " \\left< xy \\right>=\\frac{1}{N}\\sum_{i}^{N} x_i y_i\n", " $$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "np.cov(xyp.T) # covariance" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 4, "text": [ "array([[ 33.78440758, 18.68697036],\n", " [ 18.68697036, 16.95711535]])" ] } ], "prompt_number": 4 }, { "cell_type": "code", "collapsed": false, "input": [ "np.corrcoef(xyp.T) # correlation" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 5, "text": [ "array([[ 1. , 0.78073699],\n", " [ 0.78073699, 1. ]])" ] } ], "prompt_number": 5 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "PCA on the data" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's now find the new axes (eigenvectors) in which the covariance matrix $\\matrix{C}$ is diagonal. Recall that the eigenvalues w and eigenvectors $\\vec{v}$ satisfy the equation $\\matrix{C} \\vec{v}=w \\vec{v}$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "w,v=np.linalg.eig(np.cov(xyp.T)) #finds the eigenvalues and eigenvectors of the covariance matrix" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "print 'The eigenvalues are %.2f and %.2f' % (w[0],w[1]) #eigenvalues" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The eigenvalues are 45.86 and 4.88\n" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the **eigenvalues are equal to the variances of the data along the eigenvectors**. The square root of the eigenvalues are fairly close to the standard deviations of the original unrotated normal distribution." ] }, { "cell_type": "code", "collapsed": false, "input": [ "print 'The eigenvectors are' \n", "print v #eigenvectors " ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "The eigenvectors are\n", "[[ 0.83980583 -0.54288689]\n", " [ 0.54288689 0.83980583]]\n" ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **eigenvectors are linear combinations of the original axes**." ] }, { "cell_type": "code", "collapsed": false, "input": [ "ah=0.1 # size of arrow head\n", "f=1.1 # axes range\n", "\n", "plt.figure(figsize=(5,5))\n", "plt.subplot(111,aspect='equal')\n", "plt.arrow(0,0,v[0,0],v[1,0],color='r',linewidth=2,head_width=ah,head_length=ah)\n", "plt.arrow(0,0,v[0,1],v[1,1],color='r',linewidth=2,head_width=ah,head_length=ah)\n", "plt.text(f*v[0,0],f*v[1,0],r'Eigenvector 1, $\\vec{v_1}$ = %.2f $\\vec{x}$ + %.2f $\\vec{y}$' % (v[0,0],v[1,0]), fontsize=15)\n", "plt.text(f*v[0,1],f*v[1,1],r'Eigenvector 2, $\\vec{v_1}$ = %.2f $\\vec{x}$ + %.2f $\\vec{y}$' % (v[0,1],v[1,1]), fontsize=15)\n", "plt.xlim([-f,f])\n", "plt.ylim([-f,f])\n", "plt.xlabel('x',fontsize=15)\n", "plt.ylabel('y',fontsize=15)\n", "plt.grid()\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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J3FP7NtHP0yFJbx1Old6wmqiHSTJ3wQXwy19G06edFj13rl273LZJJI3i4mJ2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\nQmCC\n", "text": [ "" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **eigenvectors** (also known as **principal components**) correspond to the **orthogonal directions of greatest and least variation**. Let's plot them on the original data with the size of the vectors equal to the square root of the eigenvalues." ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.figure(figsize=(5,5))\n", "plt.plot(xd,yd,'bo',markersize=5,zorder=1,)\n", "plt.axis('equal')\n", "plt.xlim([-axs,axs])\n", "plt.ylim([-axs,axs])\n", "plt.grid()\n", "plt.title('Principal Components (eigenvectors) of random data', fontsize=12)\n", "plt.xlabel('x', fontsize=12)\n", "plt.ylabel('y', fontsize=12)\n", "scaled_eigenvec1 = [v[0,0]*math.sqrt(w[0]),v[1,0]*math.sqrt(w[0])]\n", "scaled_eigenvec2 = [v[0,1]*math.sqrt(w[1]),v[1,1]*math.sqrt(w[1])]\n", "\n", "plt.arrow(0,0,scaled_eigenvec1[0],scaled_eigenvec1[1]),color='r',linewidth=2,head_width=1,head_length=1)\n", "plt.arrow(0,0,scaled_eigenvec2[0],scaled_eigenvec2[1],color='r',linewidth=2,head_width=1,head_length=1)\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "ename": "SyntaxError", "evalue": "invalid syntax (, line 13)", "output_type": "pyerr", "traceback": [ "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m13\u001b[0m\n\u001b[0;31m plt.arrow(0,0,scaled_eigenvec1[0],scaled_eigenvec1[1]),color='r',linewidth=2,head_width=1,head_length=1)\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The variance along each principal component (PC) is given by the eigenvalue $w$ of each principal component. Therefore the amount of **variance explained** by each PC is given by\n", "$$\n", "\\text{Variance explained by PC i}=\\frac{w_i}{\\sum_j w_j}\n", "$$\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "var_exp=100*w/sum(w)\n", "plt.figure(figsize=(10,5))\n", "plt.subplot(1,2,1)\n", "plt.bar(np.arange(len(w)),w,width=0.6,color='r')\n", "plt.ylabel('Eigenvalue',fontsize=12)\n", "plt.xticks(np.arange(len(w))+0.3,np.arange(len(w))+1)\n", "plt.xlabel('Principal Component',fontsize=12)\n", "plt.subplot(1,2,2)\n", "plt.bar(np.arange(len(w)),var_exp,width=0.6,color='g')\n", "plt.ylabel('Variance explained (%)',fontsize=12)\n", "plt.xticks(np.arange(len(w))+0.3,np.arange(len(w))+1)\n", "plt.xlabel('Principal Component',fontsize=12)\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 11 }, { "cell_type": "raw", "metadata": {}, "source": [ "The dot product can geometrically be thought of as the length of the projection of one vector onto the other multiplied by the length (norm) of the vector that it is projected upon. However, because the eigenvalues are unit vectors (with length one), the dot product of a data point onto an eigenvector is defined only by the length (norm) of the projected vector. If one wants to find the coordinates of a datapoint after rotation among the eigenvectors, we can simply compute it by taking the dot product of that data point with the first eigenvector to get the x coordinate, and the dot product of that data point with the second eigenvector to get the y coordinate. This is illustrated by the next block of code. Change the data point to view other examplars:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "### PICK RANDOM DATA POINT (x & y between -10/10)\n", "data_point = [8,8]\n", "###\n", "\n", "# plot data\n", "plt.figure(figsize=(10,10))\n", "plt.plot(xd,yd,'bo',markersize=4,zorder=1,alpha = 0.1)\n", "plt.plot(data_point[0], data_point[1], 'bo',markersize=5, color='b')\n", "plt.text(data_point[0]+.5,data_point[1],'('+str(data_point[0])+','+str(data_point[1])+')',color='b',fontsize='16')\n", "plt.plot([-100,100],[0,0],'k')\n", "plt.plot([0,0],[-100,100],'k')\n", "plt.axis('equal')\n", "plt.xlim([-axs,axs])\n", "plt.ylim([-axs,axs])\n", "plt.grid()\n", "plt.xlabel('x', fontsize=12)\n", "plt.ylabel('y', fontsize=12)\n", "plt.arrow(0,0,v[0,0],v[1,0],color='r',linewidth=2,head_width=0.8,head_length=0.8)\n", "plt.arrow(0,0,v[0,1],v[1,1],color='r',linewidth=2,head_width=0.8,head_length=0.8)\n", "plt.text(15,12,'comp 1',color='r',fontsize='16')\n", "plt.text(-7,15,'comp 2',color='r',fontsize='16')\n", "z = v*100\n", "plt.plot([-z[0,0],z[0,0]],[-z[1,0],z[1,0]],'r',linestyle = '--')\n", "plt.plot([-z[0,1],z[0,1]],[-z[1,1],z[1,1]],'r',linestyle = '--')\n", "\n", "# compute new position of data point\n", "dot_product = np.dot(np.linalg.inv(v),data_point)\n", "# derive norms of projections of the data point along both eigenvectors\n", "x_comp = v[:,0]*dot_product[0]\n", "y_comp = v[:,1]*dot_product[1]\n", "# plot geometrical representation of this derivation:\n", "plt.plot([data_point[0],x_comp[0]],[data_point[1],x_comp[1]],'k',linestyle = '--')\n", "plt.plot([data_point[0],y_comp[0]],[data_point[1],y_comp[1]],'k',linestyle = '--')\n", "plt.arrow(0,0,x_comp[0],x_comp[1],color='g',linewidth=1,head_width=0.3,head_length=0.3)\n", "plt.arrow(0,0,y_comp[0],y_comp[1],color='g',linewidth=1,head_width=0.3,head_length=0.3)\n", "plt.text(x_comp[0],x_comp[1]+1,str(int(round(dot_product[0]))),color='g',fontsize='16')\n", "plt.text(y_comp[0],y_comp[1]-1,str(int(round(dot_product[1]))),color='g',fontsize='16')\n", "plt.plot(dot_product[0], dot_product[1], 'bo',markersize=5, color='r')\n", "plt.text(int(round(dot_product[0]))+.5,int(round(dot_product[1])),'('+str(int(round(dot_product[0])))+','+str(int(round(dot_product[1])))+')',color='g',fontsize='16')\n", "\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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VUYtFRMKn7+zEEPE479oVk6WOlJSJRNupp8KsWfDJJ25HIiIiofr8czjqKHjuuagfSj1l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\nV+uCrNJ5lmrEtVwPueS5LU/KQgh3Apv08dIpMcZb1uBX9Tl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"text": [ "" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now compute new location for every point and plot:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "a=np.dot(np.linalg.inv(v),xyp.T)\n", "\n", "plt.figure(figsize=(5,5))\n", "plt.axis('equal')\n", "plt.grid()\n", "plt.plot(a[0],a[1],'bo',markersize=5)\n", "plt.xlim([-axs,axs])\n", "plt.ylim([-axs,axs])\n", "plt.xlabel('Principal Component 1',fontsize=15)\n", "plt.ylabel('Principal Component 2',fontsize=15)\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hopefully this should resemble the original unrotated dataset." ] }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }