PROCEDURE for PCA

p VARIABLES

h 0.3 Striations

Not ON X DATA MATRIX

Xi ITH row of X

Xj j TH OL of X

EXAMPLE of STUDENTS on various EXAMSXi score

X j Scores on A Particular Exam

SCATTER PLOT VAR I X

X

1r points IN

IpDimensionsAR I

CEN tr.NL X X T I Col MEANS

first PC vector w C IRP HWII I s t

2 w Van l t wz Van 2 t Wp Van p

HAS maximum VARIANCE I E

UAR Lti Van LW Xi IS MAX

Linear COMBINATION of VARIABLES WHICH CAPTURES AS

MUCH As PolfiBCE of VARIATION IN THE DATA

xx

xI

tx t

ProJ with MAX UAR

MAIN STREETtM 21

Vanftif I 2 Hi En l i L

n

IF I _O THEN Van ti L tfn i

T TUm Ltily Van w xi I t w'T

UAL wtf JL 11 44

c T 2ProBeem MAX H X w 11

S.t Hw11 1

Solution X I U 2 VT

MAX Aeitievers For w V

i e first PC IS RIGHT finaucan Vector of X Twith LARGEST LWh VALUE

C X 5 TX 5 EX tX covariance matrix

first PC EIGENVECTOR of C WITH LARGEST EIGENVALUE

SECOND PC WE CRP HWY I 1 First PC

set Um wt X IS MAX

Max 11 7 11

S t 11W11 1 d W L V

solution Va

THizD PC W E IRP Llull L L L first 2Pc'sS t Van wt x IS MAX

se i Vz

a o

NEW VARIABLES PCs

ft y.plt xcv u i

NEW VANIABLES

COL MEANS of XC o Col MEANS OF Z o

umLZ.jfLHZ.jlf rfllujli rj 7 jo

EIGEN AWES ARE VARIANCES of PCs

T.givarht.jh.ee Ig Ht.gl Httpvmhxc.jgx jlxoj.tl uxclk

sum of variances of PCs Sum of VARIANCES of

INDIVIDUAL VARIABLES

TOTAL VARIATION IN DATA

Cov 7 j Zoe L Z'T 7 e

rjre u'The

o If g ye

PC VARIABLES AZE uncorrected

PROCEDURE for PCs

1 COMPUTE COVARIANCE MATRIX C X 7 F

l find EIGENVALUES I EIGENVECTORS

Al Ap Vi Vp

3 DISCARD ANY COMPONENT THAT Accounts for ONLYA SMALL proportion of variation IN THE DATA

log 20 VARIABLES

3 PC s contribute 7 90 of VARIATIONon this BASIS IGNORE 17 COMPONENTS