Lecture 3: A brief background to multivariate statistics

Transcription

1 Lecture 3: A brief bckgroud to multivrite sttistics Uivrite versus multivrite sttistics The mteril of multivrite lysis Displyig multivrite dt The uses of multivrite sttistics A refresher of mtrix lgebr Displyig multivrite dt L3. Multivrite versus uivrite sttistics I uivrite sttisticl lysis, we re cocered with lyzig vritio i sigle rdom vrible. I multivrite sttisticl lysis, we re cocered with lyzig vritio i severl rdom vribles which my or my ot be relted (correlted). L3. The mteril of multivrite lysis Multivrite dt cosists of set of mesuremets (usully relted) of P vribles X, X,, X P o smple uits. The vribles X j my be rtio, ordil, or omil. Smple Vrible Vrible Vrible P X X X P X X X P X X X P L3.3

2 Exmple : Bumpus sprrow dt 5 morphologicl mesuremets (i mm) of 49 sprrows recovered from storm i 898. Bird Legth Alr Extet Hed Legth Humerus legth Keel legth L3.4 Exmple : Biodiversity of SE Otrio wetlds Species richess (umber of species) of 5 differet tx i 57 wetlds i southester Otrio. Wetld Birds Amphibis Reptiles Mmmls Plts L3.5 The mteril of multivrite lysis I some pplictios, the mesured vribles comprise both depedet (X) d idepedet (Y) vribles. Smple Idepedet vrible Idepedet vrible Depedet vrible Depedet vrible X X Y Y X X Y Y X X Y Y L3.6

3 Exmple : Pgi frequecies i Clifori Euphydrs edith coloies i reltio to evirometl fctors. Coloy PgiI Pgi Aul precip. (i.) Altitude (ft) Aul mx. temp. SS SB GL..9 5,5 8 L3.7 Exmple : Aurs i SE Otrio wetlds i reltio to surroudig forest cover d rod desities Wetld LF GTF MF Rod desity ( km) Forest cover ( km) L3.8 Multivrite LS estimtors The vector of smple mes, vrices d covrices is estimte of the true ( popultio ) mes, vrices d covrices. As such, ifereces to the ltter bsed o the former ssume rdom smplig. x i xij x ( x, x,, x p) si ( xij xi ) C c ( )( ) ik xij xi xkj xk x,c µ, σ Popultio Smple L3.9

4 The smple covrice mtrix The smple covrice mtrix is squre mtrix whose digol elemets give the smple vrices for ech mesured vrible (s i ), d whose off-digol elemets re the smple covrices betwee pirs of vribles (c ik ). si cik s c C cm ( x ij ( x c c ij s m x ) i x )( x i kj x ) c m cm sm k L3. A review of mtrix lgebr A mtrix of size m x is rry of umbers (either rel or complex) with m rows d colums. Mtrices with oe colum re colum vectors, mtrices with oe row re row vectors. A m m m c c c r ( r, r,, r ) c m L3. Specil mtrices A zero mtrix hs ll elemets equl to zero. t A digol mtrix T is squre mtrix (m ) T with ll elemets equl to zero except the mi digol. A idetity mtrix I is digol mtrix with ll digol terms equl to I zero. t t L3.

5 Mtrix opertios The trspose of mtrix A (A T ) is obtied by iterchgig rows d colums. The trspose of row vector is colum vector, d the trspose of colum vector is row vector. A m m m m T m A m c c T c c ( c, c,, c ) c m L3.3 The trce of mtrix The trce of mtrix A, deoted tr(a), is the sum of the digol elemets. The trce is defied oly for squre mtrices. A m i m m tr ( A) ii L3.4 Mtrix dditio d subtrctio Two mtrices A d B re coformble for dditio if they re of the sme size (sme umbers of rows d colums). The resultig mtrix A + B (A - B) is obtied by ddig (subtrctig) idividul mtrix elemets. b b A, B b b + b + b A + B, + b + b b b A B b b L3.5

6 Mtrix multiplictio by sclr The multiplictio of mtrix A by sclr k ivolves multiplyig ech elemet of A by k. A k, ka k k k L3.6 Mtrix multiplictio Two mtrices A (m x ) d B ( x p) re coformble for multiplictio (A B) if the umber of colums i A equls the umber of rows i B. A B d B A re both defied oly whe both A d B re squre, but eve whe true, i geerl A B B A. A B m j j b b m b b mj j A A B b p p jbjp mjbjp, B 3, B A L3.7 The iverse of mtrix A, deoted A -, is the mtrix solvig the mtrix equtio A A I where I is the idetity mtrix. Oly squre mtrices re ivertible, d some mtrices cot be iverted ( sigulr mtrices) Mtrix iversio A A A, A / 3 / 3 / 3 / 3 I / 3 / 3 / 3 / 3 L3.8

7 The covrice mtrix A multivrite smple is described by covrice mtrix, whose digol elemets give the smple vrices for ech mesured vrible (s i ), d whose offdigol elemets re the smple covrices betwee pirs of vribles (c ik ). si cik s c C cm ( x ij ( x c c ij s m x ) i x )( x i kj x ) c m cm sm k L3.9 Clcultig the smple covrice mtrix X d x x x 4 L O L O L 4 3 O NM QP NM QP x, NM P 7 4 3Q L NM O QP L N M O Q P X T d 3 SSCP SSCP X X L SS CP T 3 d d N M, C CP SS 3 8 L3. 3 O QP The determit of mtrix: X mtrices The determit of mtrix A, deoted det(a) or A, is uique umber ssocited with every squre mtrix. L A A N M O Q P, L C C N M O 4 Q P 3 I multivrite sttistics, the determit of the smple covrice mtrix C plys crucil role i hypothesis testig. L3.

8 Mtrix iversio d the determit: X mtrices If X mtrix A is ivertible, the elemets of its iverse A - re obtied by dividig modified elemets of A by A Hece, if A, the divisio is udefied d the mtrix is oivertible or sigulr. L A A N M O Q P, A A D D L NM O QP 4, D 46, 6 L N M O Q P L 6 O 4 NM QP L3. Multivrite vrice: geometric iterprettio Uivrite vrice is mesure of the volume occupied by smple poits i oe dimesio. Multivrite vrice ivolvig m vribles is the volume occupied by smple poits i m -dimesiol spce. X Lrger vrice X Occupied volume X Smller vrice X L3.3 Multivrite vrice: effects of correltios mog vribles X No correltio Correltios betwee pirs of vribles reduce the volume occupied by smple poits d hece, reduce the multivrite vrice. Occupied volume X Positive correltio X Negtive correltio X L3.4

9 C d the geerlized multivrite vrice L C C N M O Q P 3 4 c o r 5. cos θ, θ 6 The determit of the ss smple covrice mtrix C is geerlized multivrite vrice becuse re of h prllelogrm with sides θ s give by the idividul stdrd devitios d s gle determied by the correltio betwee opposite h vribles equls the si 6 ; h 3. hypoteuse determit of C. Are Bse Height 3, Are C L3.5 The use of determits i multivrite lysis For uivrite smple vrice s, the multivrite log is the determit of the correspodig smple covrice mtrix C, i.e., C d these vrices re ofte used i the clcultio of multivrite test sttistics, e.g., Wilk s Λ. Vritio MS Source Groups SS g / k- Uivrite sigle-clssifictio ANOVA, k groups Test sttistic Error SS e / N-k F MSg/MSe Totl SST/ N- Multivrite sigle-clssifictio ANOVA (MANOVA) Vritio Source Groups C Cg Test sttistic Error Ce Λ Cg / CT Totl CT L3.6 Eigevlues The eigevlues of p X p mtrix A re the p solutios, some of which my be zero, to the equtio A - λi. The trce of mtrix is the 3 A sum of its eigevlues d the determit of mtrix is the product of its eigevlues. 3 λ 3 λ A λi λ 3 λ λ λ λ λ L3.7

10 Suppose v is vector, d L lier trsformtio. If L(v) λv, the v is eigevector of L ssocited with the eigevlue λ. e.g., if L is the reflectio i the lie y mx, the α is the eigevector ssocited with eigevlue, β with -. Note tht α d β re orthogol! Eigevlues d eigevectors I β L(γ ) y mx α L(α) γ L( β ) β L3.8 Eigevlues d eigevectors of C No correltio Eigevectors of the X covrice mtrix C re orthogol directed lie segmets tht sp the vritio i the dt, d the Positive X correspodig (usiged) correltio eigevlues re the legth of these segmets. X ξ so the product of the eigevlues is the volume occupied by the dt, i.e. the determit of the ξ covrice mtrix. X λ ξ ξ λ Negtive correltio ξ ξ L3.9 Displyig multivrite dt I: Drftm s plots (SPLOM) Plot pirs of vribles gist oe other. Advtges: eed oly plottig dimesios, bivrite reltioships mog vribles is cler. Problems: o direct iformtio o reltioships i higher th dimesios, reltioships betwee objects ucler. HUMERUS HEAD ALAR TOTLNGTH TOTLNGTH ALAR HEAD HUMERUS TOTLNGTH ALAR HEAD HUMERUS L3.3 TOTLNGTH ALAR HEAD HUMERUS

11 Displyig multivrite dt II: multiple 3-D 3 D plots Plot 3 vribles gist oe other. Advtges: trivrite reltioships mog vribles is cler. Problems: o direct iformtio o reltioships i higher th 3 dimesios, reltioships betwee objects ucler. L3.3 Displyig multivrite dt III: plottig idex vribles Geerte idex vribles 3 tht combie iformtio from severl mesured vribles, the plot these vribles. Advtges: - D plots mke reltioships mog - vribles cler. - Disdvtges: reltioships mog objects ucler, key iformtio my be lost i dt reductio FACTOR() FACTOR() L3.3 Displyig multivrite dt IV: Ico plots Used to visulize Chiese Golde reltioships mog wolf jckl objects, e.g. differet cie groups. Advtges: All vribles displyed simulteously. X 4 Problems: order of disply X 3 of vribles rbitrry, d impressios my deped o order. Reltioships mog vribles my be ucler. X X Cuo Digo Prehistoric dog Moder dog X5 X 6 L3.33

12 Displyig multivrite dt V: profile plots Represet objects by lies, histogrms or Fourier plots. Advtges: All vribles displyed simulteously. Problems: order of disply of vribles rbitrry, d impressios my deped o order. Reltioships mog vribles my be ucler Degrees L3.34 Fourier Compoets -D plots coditioed by third vrible so tht oe c see whether the reltioship betwee the first two vribles chges s the third vrible chges. LRDWGHT Trellis plots SEX: mle SEX: femle LTOTL L3.35