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Tutorial 5 Advanced Statistics with Matlab Page 1 of 5 04 22 2004 Tutorial 5 Advanced Statistics with MATLAB Daniela Raicu draicu cs depaul edu School of Computer Science Telecommunications and Information Systems DePaul University Chicago IL 60604 The purpose of this tutorial is to present several advanced statistics techniques using Matlab Statistics toolbox For this tutorial we assume that you know the basics of Matlab covered in Tutorial 1 and the basic statistics functions implemented in Matlab If you need to refresh your Matlab skills then a good starting point is to read and solve the exercises from Tutorial 1 Introduction to Matlab and Tutorial 3 Statistics with matlab The tutorial purpose is to teach you how to use several Matlab built in functions to calculate advanced statistics for different data sets in different applications the tutorial is intended for users running a professional version of MATLAB 6 5 Release 13 Topics discussed in this tutorial include 1 2 3 4 Covariance matrices and Eigenvalues Principal component analysis Canonical Correlation Polynomial fit for a set of points 1 Covariance matrices and Eigenvalues Covariance matrix from raw data cov m X 1 1 2 3 7 8 9 7 1 3 2 1 10 9 11 9 2 2 1 1 3 4 1 2 X 1 7 1 10 2 3 1 8 3 9 2 4 2 9 2 11 1 1 3 7 1 9 1 2 cov X ans 13 6000 11 6000 15 6000 11 8000 11 6000 10 7000 13 6000 9 9000 15 6000 13 6000 19 8667 14 6667 11 8000 9 9000 14 6667 11 3667 Event Sponsor Visual Computing Area Curriculum Quality of Instruction Council QIC grant Tutorial 5 Advanced Statistics with Matlab Page 2 of 5 04 22 2004 cov X 1 ans 13 6000 cov X 2 ans 10 7000 Eigenvalues from raw data eig m eig cov X ans 0 5731 0 1590 1 4018 second largest eigenvalue that captures the second largest variance 53 3994 largest eigenvalue that captures the largest variance in the data 2 Principal Component Analysis components analysis from raw data princomp m takes a data matrix X and returns the principal components in PC the socalled Z scores in SCORES the eigenvalues of the covariance matrix of X in LATENT and Hotelling s T squared statistic for each data point in TSQUARE Principal pc score latent tsquare princomp X pc 0 4961 0 4316 0 6032 0 4514 0 3697 0 6706 0 4179 0 4889 0 6397 0 4003 0 5296 0 3874 0 4561 0 4515 0 4254 0 6381 2 0736 0 0423 0 2454 0 0763 0 3624 1 5843 0 3947 0 7265 1 1806 0 5080 0 3889 0 6155 0 2489 0 2473 0 1243 0 2441 0 6064 0 4786 score 4 7824 7 2433 4 8221 11 2723 5 3608 3 5502 latent Tutorial 5 Advanced Statistics with Matlab Page 3 of 5 04 22 2004 53 3994 1 4018 0 5731 0 1590 tsquare 4 1571 2 2893 3 0077 3 2088 3 2088 4 1284 3 Canonical Correlation Canonical correlation analysis for two types of attributes Canoncorr m A B R U V CANONCORR X Y computes the sample canonical coefficients for the N by D1 and N by D2 data matrices X and Y X and Y must have the same number of observations rows but can have different numbers of variables cols The jth columns of A and B contain the canonical coefficients i e the linear combination of variables making up the jth canonical variable for X and Y respectively Columns of A and B are scaled to make COV U and COV V see below the identity matrix R containing the sample canonical correlations the jth element of R is the correlation between the jth columns of U and V see below the canonical variables computed as U X repmat mean X N 1 A and V Y repmat mean Y N 1 B Example load carbig X Displacement Horsepower Weight Acceleration MPG nans sum isnan X 2 0 A B r U V canoncorr X nans 1 3 X nans 4 5 plot U 1 V 1 xlabel 0 0025 Disp 0 020 HP 0 000025 Wgt ylabel 0 17 Accel 0 092 MPG A A 0 0025 0 0202 0 0000 size X ans 0 0048 0 0409 0 0027 Tutorial 5 Advanced Statistics with Matlab 406 5 B B 0 1666 0 0916 0 3637 0 1078 r r 0 8782 4 0 6328 Polynomial fitting Y 1 1 7 8 1 3 10 9 2 2 3 4 2 3 Page 4 of 5 04 22 2004 Tutorial 5 Advanced Statistics with Matlab 6 Page 5 of 5 04 22 2004 6 plot Y 2 Y 1 Fit polynomial to data polyfit m P S POLYFIT X Y N returns the polynomial coefficients P and a structure S for use with POLYVAL to obtain error estimates on predictions P S polyfit Y 1 Y 2 2 P 0 0309 1 1623 0 6383 S R 3x3 double df 5 normr 1 9199 Evaluate polynomial polyval m Y POLYVAL P X when P is a vector of length N 1 whose elements are the coefficients of a polynomial is the value of the polynomial evaluated at X Y P 1 X N P 2 X N 1 P N X P N 1 If X is a matrix or vector the polynomial is evaluated at all points in X See also POLYVALM for evaluation in a matrix sense Y DELTA POLYVAL P X S uses the optional output structure generated by POLYFIT to generate error estimates Y delta Y new polyval P Y 1 Y new 1 7697 7 2612 1 7697 9 1731 2 8394 3 8473 2 8394 6 5004 plot Y 2 Y 1 Y new 1 Y 1