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## Applied Multivariate Statistical Analysis

by: Jordane Kemmer

9

0

14

# Applied Multivariate Statistical Analysis ST 731

Marketplace > North Carolina State University > Statistics > ST 731 > Applied Multivariate Statistical Analysis
Jordane Kemmer
NCS
GPA 3.79

Staff

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COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
14
WORDS
KARMA
25 ?

## Popular in Statistics

This 14 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 731 at North Carolina State University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/223967/st-731-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15
Scaling in PCA 0 Recall PCA is the solution to a data compression problem where error is quantified by total error variance 0 Question is total variance appropriate Variables in different units must be scaled 0 Variables in the same units but with very different variances are usually scaled o Simplest scaling divide each variable by its standard devia tion gt covariances are correlations o In other words use eigen structure of correlation matrix R not covariance matrix 2 PCA fOI some Special Cases 0 Diagonal matrix if 01 1 O O 2 Z O 0272 0 then the principal components are just the original variables Compound symmetry if 02 p02 pa2 2 2 pa2 02 p02 p02 p02 a2 then if p gt O A1 1 p 1p and e1 p 12111 Akzl p kgt1 e2e3ep are an arbitrary basis for the rest of RP If p lt O the order is reversed but note that p must satisfy 1p 1p2 O i p2 1p 1 o Time series 1 order autoregression a2 02 gbp1a2 2 02 a2 gbp2a2 gbp1a2 gbp2a2 a2 o No closed form but for large p the eigen vectors are like sines and cosines sample PCA o Essentially the eigen analysis of S or R S k S k ka and 37k Xdev ka where 1 1 XoleV X g11 X I E11 gtX Recall 1 l S mXGeVXdev 1 xd 6V 6V Singular value decomposition 1 xdev W where U and V have orthonormal columns and D is diagonal but may not be square The diagonal entries of D are the square roots of the largest p eigenvalues of both n 11X X S and n 11X X dev dev dev The columns of V are the eigenvectors of XaeVXdeV deV39 0 Also XdevV y17y27quot397yp Vn so the singular value decomposition of n 1 12XdeV pro vides all the details of the sample principal components the coefficients V the values UD 0 Similarly if X is Xdev with its columns normalized sum of squares 1 then R X X and the singular value decomposition of X gives the PCA of R Example 5 stocks DU PONT E I DE NEM NYSEDD a former Dow In dustrials stock HONEYWELL INTL INC NYSEHON a former DOW Industrials stock EXXON MOBIL CP NYSEXOM a DOW Industrials stock CHEVRON CORP NYSECVX a DowIndustrials stock DOW CHEMICAL NYSEDOW former Dow stock 0 SAS proc princomp program and output 0 R code and graphs for an updated set of stocks DD HON and XOM plus MSFT Microsoft and WMT Walmart stocksPCAcor prcompstocks scale TRUE printstocksPCAcor plotstocksPCAcor biplotstocksPCAcor stocksPCAcov prcompstocks printstocksPCAcov plotstocksPCAcov biplotstocksPCAcov 10 Variances 20 15 10 05 00 stocksPCAcor 11 PC2 01 00 01 02 02 Bipjgt for standaddized stock geturns 73 02 01 00 01 02 12 Variances 15 10 stocksPCAcov 13 PC2 02 01 00 01 02 03 Bi lgt for gyostandardzlized sthck retuEBs 73 112 03 02 01 00 01 02 PC1 14

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