FRST 531 Applied multivariate statistics

Transcription

1 FRS 5 ppled multvarate tattc a varable a tool developed Decrptve motl, rather tha feretal a clafcato of method pe of varable:. Dcrete veru cotuou. Nomal -- ame Ordal -- have order Iterval relate to oe aother but ot ecear to zero Rato relate to zero e.g. D: matr Ob. X X X X F F Data ca be: Oe matr Separated to two or more matrce pe of multvarate aal ued deped o:. he charactertc of the data. he OJEIVE of the aal Sce ma varable are volved, a udertadg of matr algebra eeded order to udertad the varou multvarate tattc tool. E:\FRS5\ote\tro_matrce.doc

2 lafcato of ultvarate ethod Oe group of varable: Prcple compoet aal, factor aal, cluter aal, ordato, multdmeoal calg wo group of varable X ad Y:. he Y group ha oe varable ad the X group cotuou, or med cotuou ad dcrete wth all cla varable repreeted a a group of dumm varable, ad the Y varable : otuou multple lear regreo Dcrete 0 ad logtc regreo Dcrete more tha categore multple dcrmat aal. he Y group ha more tha oe varable, all cotuou he X group dcrete varable repreeted a dumm varable multvarate aal of varace he X group ha a mture of cotuou ad dcrete varable repreeted a dumm varable, but there o teracto betwee cotuou ad dcrete varable multvarate aal of covarace. he Y group ha more tha oe varable, that are med cotou ad dcrete repreeted b dumm varable ad the X group ha a mture of cotuou ad dcrete varable repreeted a dumm varable caocal correlato aal, correpodece aal, redudac aal Decrpto of a Selecto of ultvarate ale ultple ear Regreo-- mple lear regreo but more tha depedet varable ued to predct the depedet varable ultple Dcrmat al D -- a techque for calbratg a procedure to claf dvdual obervato to oe of a et of group; volve dervg a lear combato of two or more depedet varable that wll dcrmate bet betwee the a pror defed group; acheved b mamzg the betwee- group varace relatve to the wth-group varace; ca alo be ued to tet the cetrod of each group for dfferece; volve dcrmat core ad weght. E:\FRS5\ote\tro_matrce.doc

3 ogt Regreo -- alo Probt regreo; clae; dea to predct the probablt of beg a partcular cla; ue amum kelhood Etmato ad traformato to obta a oluto. Prcple ompoet al P -- a procedure for retatg the formato a partcular et of obervato o oe et of varable, term of a alterate et of varable; dmeo reducg method; ue ege vector ad ege value to obta a oluto; objectve to take the orgal varable that are correlated, ad fd dce that are ucorrelated; dea to decrbe the varato the data et o that ol a few dce accout for mot of the varablt, therefore reducg the umber of varable; bet reult whe orgal varable are hghtl correlated; ca be prelmar to other aale. aocal orrelato al -- multaeou correlato betwee two group of varable; weght are foud whch mamze the betwee group correlato Factor al -- a techque for aalzg the teral tructure of a et of varable, order to decrbe the lkage amog a et of oberved varable term of uobervable uderlg cotruct, called factor; the P oluto ca be ued a the frt tage; a reduced et of prcple compoet from the P oluto the mapulated through rotato matr traformato o that the reult a mple tructure that ea to terpret luter al -- a collecto of procedure for groupg ette obervato or varable that are mlar to oe aother; group are NO KNOWN a pror ; uuall followed b D to get a profle o each defed group ultdmeoal Scalg -- procedrue for covertg put o a gle meaure of dmlart or mlart amog object to geometrc repreetato of thoe object multple dmeo; tr to mplf to or dmeo ultple al of Varace NOV -- have two or more outcome meaured o a epermet depedet varable e.g., eedlg heght, dameter, ad root/hoot legth; lke caocal correlato, but the depedet varable are dcrete cla varable factor level the epermet; ma alo clude etg of tem, ad teracto ultple al of ovarace NOV -- a NOV, but we wh to cotrol for a covarate e.g., tartg eedlg heght; depedet varable are the covarate, ad dcrete cla varable E:\FRS5\ote\tro_matrce.doc

4 Revew of atr lgebra atrce are value elemet grouped to row ad colum. I. pe of matrce:. Square -- umber of row equal the umber of colum. e.g. X matr Vector row called a row vector; colum called a colum vector e.g. colum vector X matr e.g. row vector X matr. 5. [. 5 ]. Smmetrc matr quare matr wth row ad colum the ame. e.g. X matr Null all zero or ut all oe matr e.g. ull matr e.g. ut matr Sg vector elemet are ad ol. E:\FRS5\ote\tro_matrce.doc

5 . Idett matrce quare matr wth all off-dagoal elemet are zero; all dagoal elemet are. E.g. I : Scalar matr ha a gle elemet.. atrce are equal f all elemet are equal 9. Sgular matrce occur whe a row [or colum] of a matr ca be obtaed b a lear combato of the elemet other row [or colum] of the matr Eample: h alo called lear depedece. NOE: he determat wll be zero ad the matr wll ot be of full rak ee later eplaato for th. II. atr operato. ddto of matrce matrce mut be the ame ze order to add them ame umber of colum ad of row. ddto mpl reult from addg correpodg elemet. 7. ultplcato b a calar multpl each elemet b the calar E:\FRS5\ote\tro_matrce.doc

6 . Subtracto matrce mut be the ame ze. ultpl each elemet the ecod matr b, the add the lke elemet of the two matrce.. rapoe a matr wtch the colum ad the row. he trapoe of matr labelled a or a ultplcato the two matrce to be multpled mut have the ame teror dmeo 5 0 [ X ] [ X ] [ X ] e.g. for the elemet row ad colum of the ew matr, multpl row oe wth colum : X X X 5. Determat of the matr Relate to the ze of the matr atr mut be quare gular matr ha a zero determat For [ X ] matrce, the determat eal calculated a: a b d e a e b d Where deote the determat of. 7. Dvo Qute dffcult wth matrce mut be a quare matr E:\FRS5\ote\tro_matrce.doc

7 called atr Ivero proce mlar to that for gle value. o dvde b a value, ou ca mpl multpl b the vere of that value e.g. to dvde 5 b, we ca ue 5 X /. Note that tme the vere of equal. For matrce, to dvde b, we ue X -, where - the vere of. What -? ut atf: I X - [ X ] [ X ] [ X ] ethod ued: ofactor Pvotal approach a other he determat of the matr eeded, order to ue ma of thee method. If the determat zero, matr vero ot poble. Ug the cofactor approach: of For the cofactor approach, the cofactor matr [X] ad the determat calar are eeded. he elemet of the cofactor matr are foud b: cj j j where j the determat of a ubmatr for the th row ad jth colum. lo, oce the cofactor matr foud, the determat of the matr ca be calculated a: a c j j j h ca be ummed over the elemet of a row or a colum. E:\FRS5\ote\tro_matrce.doc

8 Eample: Ug Row of matr : ce the determat of a X matr ca be calculated a ad bc: What the cofactor of? h alo called the adjot of. of of he vere of the foud b: E:\FRS5\ote\tro_matrce.doc

9 It ca be verfed that X - I What about a larger matr? 7 Epadg over Row : 7 herefore, the determat of each X ubmatr mut frt be determed.. Rak the order of the larget quare ubmatr wth determat ot equal to zero. Full Rak occur whe the rak equal to the dmeo of the matr, meag that the determat of the full matr ot equal to zero. atrce that are ot full rak have lear depedece ad the matr caot be verted. 9. race of the matr the um of all of the dagoal elemet. E:\FRS5\ote\tro_matrce.doc

10 III. Rule for atr Operato λ λ λ IV. Properte of Smmetrc atrce:. f mmetrc.. If - X, the kew mmetrc.. ad wll reult mmetrc matrce f mmetrc.. If quare, the alo mmetrc. 5. If mmetrc, the a calar tme wll alo reult a mmetrc matr.. he um of a umber of mmetrc matrce mmetrc. 7. he product of two mmetrc matrce IS NO ecear mmetrc. E:\FRS5\ote\tro_matrce.doc

11 ommo Data atrce Y q q q q O Y X X q for a gle depedet varable for q depedet varable p p p p O X X p for p depedet varable Sum of quare ad cro product for X a pxp matr produced b: SSX X X. lo called the ucorrected um of quare matr. SSX p p p p p p p O he um of quare ad cro product for Y matr a q X q matr produced b: SSYY Y, ad the ame a the SSX matr ecept that the elemet are baed o um for Y. E:\FRS5\ote\tro_matrce.doc

12 he corrected um of quare ad cro product for X matr S; called the SSP matr SS are produced b ubtractg average value from each value pror to multplg. For eample, the elemet the frt row ad frt colum of S : where the mea of the frt colum of value frt varable of the X matr. h repeated ug all of the X varable, reultg um of quare for each of the dagoal elemet of S. he off-dagoal elemet are the corrected cro product. For eample, for row ad colum the ame value occur for row ad colum ce the matr mmetrc: ug matr otato, S ca be calculated a S X X X X ce X reult um of each X varable. lteratvel, ever elemet the X matr ca be adjuted b ubtractg the mea for that varable. For varable, th would be: d he ew matr would the be the X value adjuted for the mea matr Xd, ad S ca be calculated a: S Xd Xd o obta the covarace matr for the X varable baed o the ample data ; called OV SS, all of the elemet of S would be dvded b -. he matrce S ad would be calculated the ame maer. he reultg covarace matrce would have varace o the dagoal ad covarace o the off-dagoal. E:\FRS5\ote\tro_matrce.doc

13 p p p p p p p O p X p Smple correlato value rage from to ad the varable mut be at leat terval cale amog varable ca be calculated a how for ad : var var, cov r For the X varable, the correlato matr called ORR SS calculated b D D R p X p where D a dagoal matr ad the elemet alog the dagoal are the vere of the tadard devato for the X varable. he dagoal elemet of the correlato matr are all equal to ce the varable perfectl ad potvel related to telf. h mea that the trace of the correlato matr wll equal the umber of varable. lteratvel, ever elemet the X matr ca be ormalzed b ubtractg the mea for that varable, ad dvdg b the tadard devato. For varable, th would be: d E:\FRS5\ote\tro_matrce.doc

14 he ew matr would the be the X value adjuted for the mea ad caled ug the tadard devato matr X, ad R ca be calculated a: R X X h mea that the correlato of the orgal data the ame a the covarace of the tadardzed data. he correlato matr for the Y varable ca be calculated a mlar maer. E:\FRS5\ote\tro_matrce.doc