AD-A120 7329 ROBUST PRO.JECTION PURSSUIT ESTIMATOR FOR OISPERSION 1/IMATRICES AND PRINCIPAL COMPONENTS(U) HARVAORD UNIVCAMBRIDGE MASS OEPT OF STATISTICS 0 LI IT AL. NOV II

UNCLASSIFIED PJ J-t1t NOCOI4-79-C- S12 gJn ., .

EllE lll

111.2 5 14 .4

MICROCOPY RESOLUTION TEST CHARTNATIONAL GURfAU OF STAiDS 1I63 - ,

G.~*~ "4 W 4..".

Deprten ofn Isatetms wwwini, wit

W"AMfo "WQqM lo MA 3t teir Sepri thieereity

mim.Nb 11411 fer iescio" 0"i WAXIAe OISIMAhei a rdot otionse of aple, the

optios of the ipe"eim ui is toupoteft.e theeaime

"W" oeturimeo N. , bolow Ig mild peejittos to hewo .

Oppoo dweleube '--et~e wasy -, oathewmeefly oveivertee, Mud.

wttbia" *' llp-i unrluiag desety foolyip. - do"tatell tfioly

.qeiteeimt. Vwrtheroe at elliptic demettiom. toay we inottet .2

*ibis Worbo weiitatse in pert by potioel toies orutios tomut Wtittw. rfebuethm - their bnmfm Pect oft be ft bigh as in2.eMC-79-05!s sod offm tea opo awe nol beec utt WWO1l-?p-C-0OSIIup so Pfoww It apipeIsa I"lacso,

-- - ~ to othe enatimtora oc dipros Mftricee.

4f oeeapaat .f satition

tbiearetty . OCT 2 8 1982_______

7tdqe. ., .,I has been appved

ki , ,I llai-e ndsale, its

-LO 2-rrO 6 033

-" IM

1_

@wvs a 41i -mm@lrmmVtwedta it--.,

1.~~u 1ktomoio i C P1.. is df . p4o w usiO a d i rn t te ha

ANO~~~~~i basic. elmet in mutveIt dat eftusls asott. tb. 1k distlku

esi.te of thes 5.tiie at Very sesiiv to mv!.e In rommut le' 1. de Sir !si r fo) So n

yours.ps severa rous alurnaive be" bassiml -mm studied fee IBMlm

Ill, (1. 141, [@1, 191. 1101. 1141. 1191. an 114111. IM pffaMOusnse LP:a. '

to be that m n is coolly aatisfactocy. 1w ustrIS elmnt usthudst !3 *1 iI8

net orthuqs.ly equivarst. do nt emqurenteto , 1 -auoapsition~2 !

definite Matrix. MAt we Very difficult to *tod* thuoretioully. %&,ie . .. ..

afiey qivariant t-ais- fsvrines, Bettis bmy quite Poor

lug pPR. together .ith 121.* -poe sed discuses * sn twa of@1

a r~siu varmt on') placudut low 121. I1, 161. Ill. *

111W loss 110). p. 200 a"d pp. 203-204).

"w PP pborodere duals with high dimensional data. It 0"rdu 1 10v or Ben temni"

fdlums~niu ptojectlo..a ,hirb maxiizes. luinaizs s obtectise function

caled Ptxtstn

En 0

~~'-.I r,- --

. .-- ~-~ ____. . . -woe oa *1s admebf "malat o aproJ

.S-24r-)~ -mi

SMUT~ aummml -b Is. hftAM m saw ahtd rsaaei- Sir!') . -*weaa 4

61.1)lv uieim m 10,S S 11pid el, isrbtmi

%IF)- i amarmll - amt ft Am is weaed o o

41.3 domita MA view bI.bo ~ Mbem bI.a A..ial.

"DIm . o wemm egt

coo, lcaton f atow Lumu 4. 11 baml) so " lu Guiam

a 1 4 1 4t ha e c t e s a m m I I I M o u e . .1 3 w M i n m m a e m m p

mea d oamamam"saa.

_ _ _ _ _ _ _ _ _ -. m

A 1 19V

2. tev Sd? has a-ilakares past Scal n wofeat.tw ofi scal Wv Sat~4)i stitec tl.Ie~

beft eisy!. MA dsirahie. hoes Imporant. Seticateg, thUs seetie Mie- O IF 4,4w.Pup

teasse their seek coutimmlty ona bresakdw peat.

Lot Ftc) he owe-diaeaaiccal distrihetion IMn. as to-.wmetj rn s m o tedo

1wr scale of F. decotei by SCY, Is defined by m liiit equationVg PPG

art.j~) -l . WiN~ issieW IWeeMIJ. fsa t tha Quiet a,. 83 10 ' sV

usuall itt) is aeae. sm.) af .Nriw Ppommons P0taiI i-f.L .

Alsn.) -j Wam 12.11

so Ivin is a cars paist goAtomE.. PON) Is a"e difteihfim eaction at [ (

RAPM .1. ssumethatSit) s or se aeseiae ow t > 0 outxte,fl exsts Nor at teast a a d:. assVSm am If a en . grow caw it Naton %bt

acAtf."s It, 4[ -- M. I' Ii~ -~ 1 1.) (1 ( ) e4t * x iIIE

%to, to s--utcal Inrasn in-- F. - - .

(so ~ ~ itur #~t to *&a -wi a m~ttesxaT ncoi tI

MONSOON

I(.p I hi" -.ltC - -. jQ---0-. 0 .C 2J.t

I re -e

* ) E. 0'I)3Fij .

- x0 oi -i (s) - -) £uac A.'@ u 1 vj , 4 x 1.4)

Im s. (A(*. i . 4a -C al e tmistp .s atmos set ?ha-

hi i - aE ( ) - t 1 . - l ) *i*.o es x-o a IF.4 %

YCM MMN 2.1. t sm0 tht CMt s ti 0,ttt..~.V*) - 1ta.1.li(<

?ts 0. ada , w that 0~ CAN e vao s bu . bome of* tbhe f seeh iewtiusein of hic. o v

e1 1i 1 ) 1..12 mmeaiR Ifmt ra &AN e (s.. .r S l.seineiI).

qV ee is t e t l e s e i t o O. , ii i t a s f eh s

is ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ muhsm I * s t tafe h etpoiye i ?~m e~e e

T a@5 Si~t)

F1*

sty.) - no- t I ~

saw we.-ie mn' -8:2ftmo f&.OtI h ca"N 9lt<nt(o

t 2 Ito4 It( k")2 rft)4 It2 ars)-f2 126 c e u

Ito0-

With.f 11 > 0 A -o I2ir

-~~~~~m 4.i~ lot~ <-ain~ of outk ).t slot 3 1

witht2n do)2 .~k h2l do.1

ItlIa ItIAls."I.

0121 ia is the Manue b l gifteihutida fuetica. Cbly. attoGlnnto

a - h, me uixit) is news boun . cinuous MA iucresio n t 0

as it mtleffien s osuymitiom In t 2.1. vuatbnuga. sm tao. i.V 4 S6.0

111111023 sueta i).reI-via) I i- tow No fit Itm hwiuayd ftom 42.0)-0l S0. A" (2) Is auossow

oinS tm illi Ma ss Ce 10.P 10. 114 0U~ oG OS A tOt

0 0 t Xii) to gam an tonses es t 1'0. "am 1 2.1

4 Formy 9 O snb. Pat

.4 l10 - xi"~ - x"S).

k aria) l-r~o1)1(8 0.41

.. # AS

1 -W-- ; W .f" - "0I

A . - .--

10I A S& i i i W Pitt i VO&IS I U. iM is Ate. IiiipC~ 0i06 i i.f- <, , ,-

* . ... .. ... . .. . . . ...... - ...... ... ... +- 4

CA 64 F .- c MASI£AI

I t cI c

tb""

-12- -U-

5100 .01) 'for-, I- 1.1i 110 16.?)(0 ft 0)5. 0 hlhe NOW I"1A oimt iist. 10). Ib. 3) VM* 0o Was ~. 1A disrear

It . (1-t1o*C6, - 4l-€)x40) * qx- a0 o

A is. • lity meow. patt. a pia s at s I* ia is pa i" 4.tW.4t i(1€x(|4cr.3( (2.14

.qpmimI00t to

Ai eoIt ieas to lJst ih ow .M "a

€ e i me h()| In.r l.

>cst a if0 0

A i i i M O SO

tlem

,- metohm m hTp. 10*4 0 s • -,. dte 0gt iot 0). a thifi0 psialnel ml 400 4101 p. 110) 1101 00 Ol mi 1 dis000a.i

tea., if i41) ' 0 for my l .

* 0 0 ui£)x l

OlX4b~) X €* bshe psa at0alil*

fr ae

I~e..

i € • [21- o' (*)rO O - 0 .lb.li0000• 0 S a

n- ag the hadIm m ieo eimhi 8y mm Is myl sins llt) , lb. ho~ima p.ist of 00UJ-

• [r aithoba e • prole51im lndie. fog pe'tacipsi eq n aetlei. 1101

- .l +." -. + + '

++::, " -- - . .. . ... " " -. -+ 4 ++? , #'+ :. ..... . . + ... .. . . ... +

t ti

-1iaosdobr4-ivrmm roete.mx ~r~p - se iI- i

Asito et of e inecti I that of S pa.riow ousa.. pr a .ai Aot ~V3 Sh~ II~W33 -lW4%7~)1

&Lmeion Poroe .atitot s1 that,.. mist moibastsa fiee at 0

Rcallia that at joal a e toot do It a ettasti wo at IRA I 22

to dvotne ayoesiA of ahw 1 astialso aoit hti a

toly .s sbat. Iow dimpetif this 5simAi (Ig -ha .a 2. pI ~n4.3m 2.3 ttU t

aStl tr. doe al. orhooni gias ertiat. I~e.. la 10 be my

ge ri . it i- r aI omd a koio M- I29 that tme oaaisg tm. wm. rless

5m 1 (r2 8 1irI l"S S V)

it I (r. pvheoa ayI ait mI OthS9Magri, 1.& 2.

1y- PSFl~p I S.2

atsMtISS "~a 0 md aI m met me asmset t isae

IFI

"at"

aha

repectively. ma that mept o

Z..FRI -Wi Win)5

DD raem lhaors 3. 1. it talimv that 46as gtimier two ~. ~g INIA gta 0190 "'-1 f*611 il3

%). .. sitrl -A ..

Aa Wi~t Isst~a Imlu ... ta 133. Ps et

useWi iaie4 beh~.a aqaltariy 411tstki *.*,A tistus .stitMsete.are od~dorth""I V"Vefiet. JM ID clasimOtimho be al V se w1331. Pown game 2.1. it I*s traightfesusr that

papp" that rta~V3 helusge to a p-disawlirn elliptic pnftiilty

II desity really. i.e.. * 16.", bas a Aftalt, fix'") Vol3.

ItemV - East V13f fu 3 *63) I P

alast V is a Fueaua - p unstsat I*t% Is sohorteafit uito ad we Get t ~b .t vae oft. bbh %Qtf* t

loa 0010is900a.8te. of -Oa3. i.e., 19to lowlso gusrality, us amo. dot .63

To oo an b Oimi sawwaiicevs su1ts~ an elliptis desity really. aslks

rally Vontric proabwlity Gswity ~u t tal 1... ~4u3 ltt 6 'a .to I

qis * .....s.a 1 iSol ns ... An, 631 41' 4'iijjJ 3.91

be ike usqisal Awaatty. e~a ~a ) for amy unit metal a.

Iir

I~ .- .. -. .

x W,

Im. t ao aapil "lied the .lwdo-wptim ito iVs~ of S. ) I-t Emilyt la. tha ve s Mial epa m at . e" h iti

a-..a- b b fi do. .-..am -..-.- of-vs b7-Ow. s tuant ofatm 0Carctc Corrspodin too~ 161. a 4uml2.4f1iapaql

Undr o ofifttrnrematee W. so mtim. aith diatrbuton ee-

to 2ae~ AM sepm.

lbarfaga. single algebra gives V "agjaanfy set byeih a ophoweal dirttibatlea. l''xl ad

al do MI, belong In MV olilpel distributionu Caly. mmareffe. as m

6 z11V)-glI'' Vn - atios" ft=9801. ise .otia ffiaily eqaireimt.

21 on" . ME the Undrin~g Asimtiona of %. Is a MM~ of we ellipic

-a ~ Emily tboa - $0.) to aayoptatialTv aftITe OiFAue.

looni a.,-ator.aom tm Ease a oslipti mtihetiaer(.T1

MIND ama"~m to' aaame 4.4.

ea .les arse, 3.2. it f es "hat

.

0 Ir

-20- -n1-

4. CoMMet-Mr st Ellpti te prhelit, bust, Posh,"MI

ubm the data cmfrom -m elliptic probability diatrthetims. for *~ *3 £

whirlh beoaoars an up istaitie intstprstatimn - the sksof the

osArlyisq *llipos. it Is possible to Ihe that the robst wF-eatiste

give consistent etimstes. the ides of the ~ oo impile. It cisa lmo£ a *m s- . .

the continuity propertY of the Pcoilftioe lake BI-) (diced 15 ra asupto 13 m ee

section 3 (or H-tiontes) with, a rmyaeti.s agst t I"Ittigg tugs

ino operation is high dimso and for any muderll Be a -ooriesre AV !IC£* *O oo ~ oP o

is quite omplicated. To awns with this problem, w istr se s isses amwe seod only shoo that doetom etreme am Of this Imessity esowr"q

first. thee start with eas pecial affidarlying peeioosb rtsos to ediews to aft* oh 0 h. *slv by ssmtias £43 . (13 me £23 . it follow

general consistency ret.5that.

Laml 4. 1. Ase that on% to

(11It 0I. cac t eet InRF Jo? 4. - 0. 1. 2. .) Stsleo I0,!3-'Few3 FoI.£.!. - .PIrj1

-' Omw I*P,. -00% -a Is-"

(2) 115.0 £03 - .0 el) uniformly is am on. adwy

£3) ror 0. 11 R (s0. 1, 2. ... 1, them exist a ech that * .- ,£10IO £s)

.f£ usei 0.*G£03 seO.I.2

zmm 4.2. une the, eOslties of ien 4.1. If It the soIqw,

43 there also exist p x p orthegoeat matrics le erch that m o of .0,1!1 us the cluoes Of %. the

SI, a, - I,,!laC0%) sl2.0

(hr e a A t£ovI) as the NOWNs of Estrin A). £4:e .. e t

by do exmpio 44 ain som _._ wekewta

40~~-

enr*s.VNOF o AE~ s 9 b.__ _ 91

lo*uo !~ .n N, Nodfn- " Vb

*w tho . *etAgmuptonMA .a

11. -#o%-l - im lien is ihaeto d eastee e m i the ro~ttion em..t h. We the I..i

-e~~~ in -. -mP4e O 42

a1-3 i.- a. 1? lei IN Iel~~ 4to3 a. tt a...ts Lot a

0-1 .0-4 1043

m imas in 4)Nhta

reand eti % (a)k we IN~ - 1 0 3... aet sy .cak.2) imtm f% adt

433 A-. e- Q I01 n-)i (1) for. t- me toi . 0 1. . fie . is Mk. comtlmsmw

3~~~~~i 9A.I~~.~~ Is coolammm in a e g -~ei

423 1 tht @,SE3. . m. 44.t

piF434 *3 3m ~ dethte lF.elte

9' *P C1

I2

-24- -25-

42) - that a.0It' *-4 had at*. From 41) we M.L.i that

(F2) RP, 101 1. - 0.,2,.. 1.... eaklVO y -Oactg IS~n - -*rt a... Wt*

n3 'f Now-ISI1 a ( ) te _ a that (e,t1.Z.... ) hom lifitly ow different elemets. m

S(" *04 a §-)a~m me ! ot . my. I that

o .

p 40

(~ ) - 0 is F ) . a a.@.! t -

___ d*a- C .clthe as tI x * t the0 44

Smwm *,-o M4I4- - o. s a mo the Uollotqsaaa sidelo

them acrigtoucomto, -I..I!t i hsFrom the 0teo-Ctatl themes. I.e..

?7 J t - - asae'the-a. 'd'ks. o. a - -1 - I

"was vu~lformy 1,% 0e ara tONIoa1 dletwihtioma of U . it flosta

. .F %~m ro "a- (a. t --

by the weak caattaotty of aII-). -hars

-~ hik .a~ seem, the first tag th Willot'.hee side of (4.4) ales cumerges to

(3 sarmi fIl I(- i ar.y mtm so

a . a- viI ituiwsiwmtvta6 to strN air 1 - io-

-26 -27

-24-- -2471-0

SIP,~a SalIP . 1 ditt...... it is sr to no A, 171 114 14 P1 w) an Certain

out taefis to Somm 4.2 balm Nor SW.?) ~I MW o . go. too mmIntone

IF a o, $F I* ar f0lo aw.I) si aif%). boom Ilve).a 1fat0 tsee

wknwalredy that the @0dtern of th right-handt sift WM*M w.2.. be do =auto"s itistpteft m F . F(M.

* --Ne the first term. in". St.) is esehip COVASSti W. If is 0a.".." o toFNW.1 y 40aIitaom sScif

weekly~ ~ ~ ~ ~ ~~~~~~~~~~~~e .Fvct OnS- .AtSi oinuu i.o o eiyiwtosmitrofo 42.231 =4 4.911. so ami th. im~arery Watpleofhi meo o5. svitsl ~ bn 55 -V LTYTfIWlutin.o .A I 4 mot by U.$). we asm reasn ce.,mm e, wis

on % 01mi Im 0,1-1 we obtauined. Lot a b aq, am- "Um, dijtr.-

we need only1 SW3.*~~O

d*1V00

F) *0 ** P. & 9,5 - 4!! tNo. e -a

Ibis is straightforward fromn 44.5). a ~E IS. lei - A. A~~!i2 .2. P

W0014LAN7. Pat t "I be amhl f mieimu

a d 44e!.3p,31 )Ma

off:) M ly ",a li. & .2. toStral n -1malormy I a n .

odf

'1 ~~t "Oral*. s b

-- 4

- inns of $( a) w eag i I. ae h o a l -hato byatl o. t 1In~ t %4 - 1 949 p1 m i i 4

irnam.N 4.1. SOMS that A I 1 As. tbmbm , stean ammal Aa t

-1..11, os pro,*ag win .n I . an bm to 1111ta Mliia 413-44

~i~ni 4 . 1P. a*.*. aqizNrom 4.1. Actually. was moot III 42 ().

41) Out 9 ha ftemma ia 44.6), tim ..atlaodt, af 6A.~ an OStA in

I! an a Ites"twm 4.4 41).

Forced. First of all, as adjust am pmUn to Am 4.2. a ma. 42 nolay. .4 iAa

mntioned halo.. slam. tim .igaofolano.... , a. all 419t6940t. us @A 8 . a5 a.*.Y Ibas only tuo po.Oblal a veo ! and am ofm half ;V hIWor-

01-ll (01 a'M. 101-1. NAPk. @) anmy III. contains only am of 1:11 IS admommny ino lso ILmaim points t a . we chooa. Ot oatai"Iag !I. Lama 4.2 alan maoine a t.wa t a4)~L)

that tha Natemns is aniqa, nar tim classes ofatt tompla., an an ano" to fi w.i Novos ie r 1 ~%IW

Ipeciflcally. an Seek mo that ?Ito its aanta. IN Zn it. lt N and .at obip III - -9 P. to timb Immuime 4.1 ani 4.2.

la 0 0, ow *Io oI

%- IT""l-fl. 1o U U 03

I 4e-I ' -0 - . . . . . . .. a'. > )Ucosim mo" fb . lt i earott s"a t si

92~~~~~. 1 Ioa4306ta

(It 0 to 1

I F *

.- -- - - ~ - -max- -- SAM-- 161 - q P.-PM. 1 1 . ... I.--

a~ 20

(a aP a..%.-v 0 V o@

SON it- lss tk

Pi - I (a- a 3 . .on*

bwt~~~~ thi 1.1ow tw a. -ahi So hat 4e~iamv t. aa uala.s halfsi we eumo Massi ana~

In "ss thtA a

9t It) 0

P1 -. 1 (-dob .a.... owa .s sa..

salts tht iW-

I-01

m

o)M1. of.l-

goo) P- ac. 1-1.3. .p. AUI blto

Mi or jl iof* atope some9 Pt e.. U -1.3 .. so OW N 1.1....3 .su h.

go,

mi.l ?I pll "i 2. kl 44

-- , ~ ~ ~ ~ ~ ~ ~ ~ bm 6 &to %ar.- Aa Afa, .m mm l,( .a... - . ,. .... 0 1*rand A. Sam

.41.

P p"I min.,L.- I ! I u).81~l|4 P.68.8. . !.* gP• • • a

p 1 .1 * Pha...

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my-a------------------------of--------------------ore----..aft 4

W -4 hP. a1 adt. H.ap 01. . or, IMoV.aitg

-~~. onha. f.4.40.m .63

INK .6et owpek i wm 4.1 adin .... uA A .amb

IF.S~ a .. *A&"sawne f d 11.1t a - W I:a

led.-~~~~~ ~ ~ ~ <19Wk"ugW.. po wac2 w& n so s

64ol m fwat-ily IIIUI 1),'insi n s he l as~ a -. so of12 Sn s w 1. mi soM me 39 bag am A. R.6 ) U u si.-

SomiavoN. a4.03 6o11 sm) . 2 t m te40MNRM~

40

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4'4

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Sm VM oua 1 ml b

ft om" bsm " lvo alewa I L asGlssft m om..

ft Clam, 01

wt (D -Cd 0@ . Inu 4 W/2) be the Mole be"*= n L, for Sw ss a4..a.. 4 is S o" ) 4f e s "% 0164..m ptwoms a I M. - .

an 4 .mdbIAta

a 41 1

L 1k

11 !i*) is a -oi me L'. v*** IE o3l

Ib~~~~~~~~~~~m~~- -fsAit -41) ow m w o Da Mf

030 ,14>&Otwdlv.~~ -g"I4t' ef

wr as in t at a , low1) 14.11 Voam pv dtt -02 belm ao"@ t

I -, noa s aa* 46

IV. - .5 aN or.f)) as*'

Vfto m w e"Rf awa prob2m Iso 01""a alw tm bk16 m to

It is.m W. I

on WaL. ~gl Si -62

$aa Mt. em b k . t

biS2. 12 4.4 fat aa

to~~6,- a"m4II bd AI

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ba ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ e - 16 agas pbmme-a a~ ae.S.

. --. %I.-....Poo

ad" ~e act o& y s" loica "Vunwas lm d cod 09Fie

- tis ad to sat im ow we one40 tot~~s li toiwm , in.*w s.watn nt an 4.3oesa

tau".ah I"A)0 IhPelsmtis 423 i ;. *jj.-- ' -' U ) nabo 4Iu

4 1h) wan .11, was to J UsM9 MLM f,% I II

Io II) piot do.13 -4.sms PA . at 3 V ..

Astmot5. to dai: ee' Ma~L C.. fone ass. -y evi .. mab ,o F o

A" As i mies .93. u. w wita ss,43

5st11 too 6 01 3itimo OT . 6,3. sa I . vs,. tavo INS *A Mth is.~ L ~. . an 41A to ts hi s q ld"I39 1fis. a,£ Il8amt m4

supoe hav down 9P-, 4 .a: a..) P.a..0

Asisim toto &"O f3 w m a

7 -- 7-33m sts-

~h'~ls .,3 *.~t 1 3.. u~£*£u£..s*3-~*3. 9.5.. a3.IVa

SlowL 1 .w ookansm r1er

V ..1 Isan of9...a%..a..lhm I its aeamrlw9

afe bann ia s .,U olz ae m1"ftI ewetai".

siu us r e b. aue ,, w nflefrt 9 lens 41.31 wal 2 een iONwaen of AE1 to a dmn ball of th Awl"*I-ini apuesolmom. 4.AIM is anuall

Ne we coa havelb thenus.) maIt., of cnitnc~y.

alfro , itn - 1. .. 0

*AeI F n we) -~ Son on . &a.. 44.21

tie !' . - i ie! . - -4 . a .s - 0. O el is. a m . U e 4b f 1 rP b5-mo ua e I t P.

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1in tb44 mmm . n maldl brtrs obe

Is.4 ere 3n Iee t l gm ej7

walyeiel efn i t

logtax 4.. It. of. +.ub~ vl si br48Ilibln o .O ua " m 11u al--N W0:11

anelipicPnamit dmmt ridy.thm hereef r-ftme" w eaituwsaqmaitts oatm4 -yacovrlacematix " ts ricipl cmpww~ ar snsitet (. a0..1 tedft m Pdait s A NMRGOIN bf Setin. t i snte fie- tha M

in te smee of taotmi .4. almst r-esimi" mrt M UF ottl m*eaneat fco~. a owtus htalbsb-h PP - . - - - -- ted - .~mlpreblt ssbto tela ?--.

estlate A (F) ad P .) y weuniiml beda" ian, teirwe mver ams~mto blb bl~upto w *11r m &8it famay

9*im the. an ofse Theomtmat 6 1eee tn.tises-~

tho tNMIil saw~el wil thew ammov N.M. w- givet as" tz .14ii

doet

v 1de . r I. 1 E. .0 It l ei .9 .

v-It. 1 m a1'se 00411 lie" do m .~b01 eti ttheehiew elf amumeg bf Amomat to1 as aft-4

-iu of PmgI. bm.uheih ife ni.-fB

MWyse tl ~ 41.~ ee *6.39

't* 'td*EV F 3B Et 3 .;L.)

It*

0~~ ~ ~ -Ata As 1. 41 *

Seem. (S.1) 1. elmo tUm I. *Am amsm see.

CUMJAT1 S.1I. It 014.) is mmt. Sbmeet. thwo S(-) 1. ve"Th"a am t US i1.e...&m t do m of Sals 2. we be"

-~10 - Stot -l a" a. a @1131 1.t LUt.

are. Ameoaine to 0ba 3.2 .u .. w MY to do" twat£am-~- -~a,

W Sem. is -w e oMMAN oin IA . Ine.w)U~. law, is climts.

0 5 I'IO -1141) I ~lot - .a - 0.1.2.... II.) soimm~e-U sa~lg~1 5

Acallyu. Am the to~S. mmmus A Is 09mrIm1atI nMRSS Oat m. vat I e am fig". airetim m4

701)-r)- 0 *<1i I - fo .'!.I. 1- 0 lo 9pt aa Its "so at 4i"Ant, lAsh 0059 etin. thm

4 ue frof, fee .ample. gn 32 ia 1101 4.55 1161. P. 32. it #slUse do&

now. $(n to 10110mu. S it as .1.101.15 sm IS no". it warme0 ( 1101 101 4 M r(PIVf.0

a- a- a SO" o t lwItSw " qwcwetak

1'-4

1- 1WM1DW j Nd MSCMid.5- . ht1

-17 i I>at4 -4NM

,v& -awo.. t4 i A.ot A1qvlsr tat any A. CisOm s ft cnq 1s lb. ttfi h ONsos y a t to omswto to. too "1 t.", @i.fts d

0 &051 1b. ot G. el . 41~ .0 * ~kb. SOMISISo mithod to s". b. .til.050 , -mps -b - -soi-

Ii-I P). tn 0. "w05 ore aso the am. 4 %QstA to sot Wlabh- *000616 b4%r ho00 uoft ik. A. smalsltIng sothi.

ad s~i~l~l sotodo. f055 .5605 tb1* sectionlb is am 101s - .0 to assuo.

tll MAtSS thss %k is 60WO I* ~m drOtift 1. ... *CW4 ,..oaim t ly5a is . 0. o ) .f cis varlteism. hit

ye -aassimi lsowi ovtlotpai wep It-& a tatotl agaladsq ot-

-~~ t mb- motis" wo. uuw we-owtuaterv do "ot his.

co"Ald" t"15051015 .0"5 fi. If c el. ONOs too MY~ M km5 pnu 0 - ofMi ns"v to otso. mmptAlV. fit

ths 410uiommas to) inAb, thi siloo Would 101 seop hlAWW0

t dateati., Ithawo ~s wst utthmfaLefObaase ls boo.

Iol.It %. Iaf 4*9stitst is.f~l.i. 1..Uhtc 4lwm hwoad it somes Ok. dol"Woo o .10 fst s t IV asto

AtU c 14 g h f s o Ihi flaothas of empeuaon Is 4ois Ok.. y

at*1 toto . ie. 1 VA -. *1 -A. ft thi ca

93. Ole-) - 0 forl*

adl.4,7 maials 0nth. a, l .. ,P Atn wbt ot aism

404 " r--.--si. o I; ft tswtO feosa fs

6.am o ehegediagt the hgebam point, ift) ea in -fq sts.rs ubetter

(1) hreao the two estimetoo for the oeve;iae. matrix fro et- tj. lh) > 0, in thim Me. is impotant. Pre, the disserni. i M. tat

.1.i.9 and mint.12i.9 P'aooee by titer~) end $.FO. respacti-aiy. 0. and (3) of SOCtieft S, W ushef that the IWOODWOO pint of C, (F) sad Rl?)

Coo defines another esitastor, denoted by t.() by the vs..ea of theeA

t all' -1 ( I. I- . m 1- O11

VMae tslP) end QrI) both are oairtmat en" amebly osetieeM at a"nfo ea mo"mto Bomeda F - it-f)? * to.

amer of en elliptic Praoabilitty doesty trade. obviously. en isQ ) 4)) tit mentiosn. e, M .times" that -~ or pri"Sp)o~

Avo *$() Shou~ld alen hae. the Ma "Ji~elos red brakdown point 55thoimn co do a mc better job for variatioe Iedtaig thee the clase-

St *) which. usually bas a low imeakdwn point thm t$(r) has. me aearb. YM. Me meed say that "nI" Rout PIP laesen . t."sa

The sisolation reselts Ibs thet the aw.ae procedure f A ?) grOad", of the canmeicel e. to substitue a I r-t Noseobts distance eaed

en the whole, better performance then either %IFV) or t*4?). give. beaus of its goodrbstess. hoabtter remIt. In diecrinfeetise

M4 olusterte at &I. maltieriaew anelysis matbode.(21 So far.w, how ba mt mtioed aenythiag about correlatitn note-

netion whlch li I-e important in mltivariats dats enalysis If - aent 44) It abash haD related Ga that te 40thilda W" have OWV is

the .esawete of $P?) by ta.. som Immorality haothe stuy at PU-typ o ee

t~)- (F ()). __ 40.1) NO&a thenaet &MDi4 Lomest 4.4 hae" nothiag to do with either

then the robust PIP-estieas for correspondin.g correlation coefficients 40elitic. probabily endt famil the sth* aliey aesiemf the o.to

ed co-rsatton eetrto. deotad by o~fi) sad 34?). sen ha def (ed by IA=m 811 lis Kasa 4.1. 4.2 206 4.41 and the eniuamar of the mseum

resealing. i.e.,* poit* of 341on the -er~ia regicm it Smm 4.2 cely). Manor.

4 4) if tOn projectie ladme in enarle watseoen. the at &eas the fenctienei

It rtItci1/ for detaulajate a fir"t meelm veeen flihea I ) howed ha oeistent

44.) 21 Me abliy onetinuous. Pentboee. if the nOffile guests Of Md.IYieg

(r4 1) I d1RiANIhtIOa lIMO elgeMoetore hereS) ae el %Siqenly defitned is U

aMemo. tham the remotionese for uers" amie weloe end maimow points

aih siP end ai1W) here) ld h"e all comatetent ad sely setimei.

v e. .

.I _... . ... .

-33-

7. Acknoledments fill Suiss. P.. (190a1). Dsity estimtion ad peoo)o pacooltmethod@ - asipt).

Us would like to thank Mt *J. 3065! fee expo.9 us to tist 1121 p06.!. P.J. (196103) p rojection perm.it Iaorpt).

direction. as wall as for his careful reading of the first draft and 113iii o B4 . L. 41959, 160 teti tatitiol U1pprthoseo. Johs Wiley

helpful cont.. we a1im wish to "Wooas appreciation to David Dnoes Bo.3 Uam Took.

for hi hel in edtn th aucit 1141 Ibso . N.A. (1976). 1 *t -tstoce of mltivras4te locationA"6 scatter. AM6. sta1195.. 4. pp. 51-67.

1151 mostalr. Is. ems u?. a.m. 41977). Dat An.aslysis and Wsqr..sI-.hadion-Uswle. me"a. lease.

Owformuch.8 1161 Schbohsla. R. f1979), 0.1mo6*. Nowertas Ph.D. ghous. midge..tische Maonae. 0.irioh.

III Chen. N. at &1. (1974), Sttistical Nothoft for groupingq orporation.tootshy.. 336, pp. 1-29.

121 Ch-n..Z. :"d Li. G. (1961). achust principal connts and dispersionsetsio to Projectios paroulL (Saasc~riptl.

(31 Dol to. s.J. at .1. (1975). Behest estimationa nd outli.r detectionwith corroltion coefficients, Zismatrik.. 62. pp. 511-S4S.

141 - (19813, Bobcat s.tonton of diaprsios Matrices .06 Prfis-* cip.l coossts. J. nar. Statist. Ass. 76. pp. 394-362.

* ~ 15 II picoha. .". *t -1. (19003. prajctions permeit rweerani. Tech.sap. no. 146, Dept. of Statist..* Stanford Unix.

161 rriedmos. a. 5. and Stwetsi., U. (1960). Projection pursuit olamsifi-cation (uspubllahsd).

171 rriedso., J. ". and Tukay. a. U. (19743, A projection. pursit algorithfor exploratory data analysIs. 170 Yrammi. Cmpters, C-23. pp. @$1-

161 an Rondshs . 0nd Battan J. R. (19723. Mahout estimates,residuel and otlier detection with mltirespons. dote, Blimtrios.26. pp. 81_-124.

(91 Hoho-, :.J. 191. ootcwrac n Statistical DecisionI)or and R.iate Topic*. 11, B.S. Capta mad 0.8. Ptc.. Ofd..

Acadnisr ft.... Ua- York.

1101 Nwb., P.J. (1901. oat Statistics, .J0hn Uley a Son.. New York.

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Robust Projection Pursuit Estimator for Dispersion Technical ReportMatrices and Principal Components S PERFORMING ORG. REPORT NUMBER

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Guoying LiZhongllan Chen N00014-79-C-0512

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20. ABSTRACT (Continue on reveree ede If neceeeary ani Identify by block number)This paper proposes and discusses the ROBUST PROJECTION PURSUIT ESTIMATOR

for dispersion matrices and their principal components. This estimator findsrobust principal components by searching, successively, for directions whichmaximize (minimize) a robust estimate of scale. the estimate of the dispersionmatrix is constructed from the estimated principal components.

These estimators are shown below (under mild conditions) to have a number odesirable properties. They are orthogonally equivariant and, within anyelli tic underlyin densit family, asymptotically affinel e uivariant.DD , AN m 1473 EDITION OF I NOV SSIS OSOLETE UNCLASSIFIED

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Furthermore, at elliptic densities, they are consistent and weaklycontinuous (i.e., qualitatively robust). Finally they have good quantitativerobustness--their breakdown point can be as high as 1/2.

The robust projection pursuit approach is a promising alternative toother estimators of dispersion matrices.

o .

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