algorithms Consistency and asymptoticnormalityof somesubspace without observed inputst for systems w. Schener'z D. Baucr.M. Deistl€r.
$!\,,,1 dli1a
ütMtin E\P'4'hn
tu a .tü' al ntb:Fe ote
hn| \+r'h ^hhd. th. edt. ir a nßt eep, n
Ju nr! EqPtat
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o be p{romed ls* e,3,abvrt, 1934!D$a'c' a|..|9115). On theoder hud ihs ndi5ri.rl prcrüriesoI rhes llx.imore. i9*3: va. o\rßche Lnd DeMoor.1994 v{h{scn. Lcc+Pdfndl. 1995:D.isderer rl. rqqsl The rdradrse oI rub5pa( ddhodi .omprrcd ro
(1995) consisttrcy hrs ben provedsinularionludi.s cc in Pdündi(1995)a.d Brße1i|.1199r) indi!.r. rhd rherelarive e6ciencyof $mc $bsl,c n.rbodi compai€d ro rhemaximum likdihood6rinxrs. s .lose h*i obraircd.In vib*B d il. 0991)thc a5tnrioric
rlloirhms mrl:c us oI rhc$rudu( of rhecrl,arioi p.obr if. eg Akäikc 193J)h rddirbn ro dasicrl rerljzarion df cdus in
b*n derned.Thse lios haYcb.rn fu wahlb€rgind JaDsonilq9{) and lrisoi
ll99l) ln
i1995)Our coitibürion lro nor (1993)) r a fuihü nep roqardsrn ullrrnll sandinsorrhisPrehl.mri rhisplp
uidü
,. c, D) desribedb€lo\ is dcn\cd. 1,1. !\lflhcdlcin$n]hünü5FFoilimL!{s
rbedas ol subsplerleonrhtuundn.onsidülriotri\ pEstrredIn rhelhidsdion5oncd ri.is lrd irt dü.rd xs s.ll a \o'ncprclimi0rylads. ' rrcr fqsh,!
PrdchPratMAr;!
Thlr lbc rlt fDrureof rhcnois lnd th. pß ofrh. prceslrri,/areumo'cklcJ...r'va isr|t onholoml prq*rioiof yi o orhc$iesFnnc'l br (h0.L.nDtrß ol l';. (Hsc prorstioi hI b be uiddsrNd i_i thc . o N r o f r b ef r i r b . {5 p N s p r n i r ; , :r e r . i = r . . . r l ctrdovcd\irh rhc iinn prcducr(d.r)-Ed, vhcß
rhproric n.nllny i'nnldrlon! ot nD ßsülr3i.dj\cu$cd
sid.d. *h.r '[. iipuß !E unob*d.d {nir. iois.
rtu !)ism apaenrdion ( I ) Ercq deompoinionof rh.
rr;rrh.projeri.n
. I li inro t$o ßrk ri\cr ! brsii in rhe {d. spre. uJ p6nioo. r, = .v vi - v, € t k I nar sq!.ß. rh.D d.frßs rhc slrcn ftlrdas rir [.idrin! shcc(rt.rd.tr.r.srh.1ninriooaltu !r.dournur. k,td dd'.ks kimcnsio.d -ro ddn s birenoi$ { i1h \flrDe.qßrh rhcjdctrull.n de riotriLdüe ir € R" '.4€ R"-c€ * ' üd D€ s '
ir ,n=I ,^ d: I $h$ Iil0)=, rd d(,r:cr ,r.,>0 I' \ir ,'rprp( riE*y{.n NiLlbrßumed ro b€nininal. nrblc {i. ;-,,llr < I f,,ün andnrictlyminimum-plrNllc. )^,,t1 BD t.t n hdlds.Ld
l ' , : , :t r i , l , .
. ' J . , t t ' m d t , - , - t r l - , . r 1: .
.... ).1,I b. hrir.dimdsiomrrd'o^ of nrcr.d ourruh. olliN r, = t.r. -rrcl. .... I-IA ßD \C)BD IAD r, d.n.rc\ rhc 6ßr/ bl&k 6\s or a Thcn-of.ouB. .{.llr $d rh. Iollosin-q.tlrlrion: ' t , l .-j t , / , f ; , + [ t.1- 3D 'cf.f,ti-,+[tE;.
nalr nos ($mp. lrl Rcsrr$IiJ otr v;,,ro -adxn.ninrtl/,or.i./;, rf r; md ,,., dende y,.r rnd rhc $mple coyinine bcls(n /,) nnd I., tu)r{I..U 'h6 r:r ^ppro',nr.ir,b. r Err'mJ'nrindd(nmn,* ,hß rnn'.$mJrui ,n'o,b. prdtri 4,.t, ',r r$" nnrlmdnc'oedan6'iDr'.a.or'rl) u$ rh.srimareolrh.n'ri, = i;li,ro6rin!rc C b! rcldioc ! d i; h 'nc nqr ncp $rin!r. _'l by G8ß.n-q i, - , oi i rnd .; 'hc ßid' tr. trr ulk otrhcrißrrcEresioo. rinillt. rh.6rin!r. ofD n rhclapt
co!.nana of 4
Th!imroimarionlepllJis p.don.d6yaiisullr vrluc d&rmposirionor rhe mdrir ri;i'rrt; tl+.(*h.r ti; md t. wcj-lhriu ü!tr= O"t !n
Lt; =rr;r ''ror 'hccc^ xr-!.n'hfrmd |]'; = I for rh! N4srr) sori'hmrnd rl lariltre of l',:r rhus rh( r nön
Drrn{\. a,€P"". 4,Eer"' rn
rtu mr spdd! b tLrF\inrding/, culj & oI tullnik Ju. rD iirire dx
t arc di\ü.r.
LhLsl.e reE{rd r" Nhilb \l lrj
rhed lor ?
v r s L r i r s r o * o r L i , i/,r , i : \
a,r:"= trri;r'ali"l'ttlt,r':
rrr!coiliiuny of 'hc !ngd!r vJüs I
tltt;) I Nor!.rr ,lrcsi',gulr\(trr\. i...llrcol u n n r o l r l E d d . i \ C " r r e ü t r , q k u p b s g r c h r n g ! s . i f mirimum.rhu5rüaidü fttricliorsi orl{cllillrn dc,q(t r h r s nr f t n 5 l o m r ( l ootrr' h cn D n d r - c l 1 r .
. 1r)
sdL dinrcn:ionn G kno\n In Dftdl!! rb! svD n i'l\o n$d b ddqmiDdth. nod.l ordü drlrr bl i" \ . { 1 i o i o f r h c5 r g u l i , r l u c \ x f d u itlds of x üircmtr r'ndiü (s.cF Thcdrcnco lli! $dighifgirkrc*r ssnriil. D'r \hicl, h!!c djilüud Iu.n (hoic$ lcid ro ilsorirhm5. .5rmprdi. rior.ni.r Lrti.Ntr\ c.a prcllrm lki nr ( r0!r i.!.hrJi,rrhj!iu.i ert;=rf, sfr LL;=if;r \hs!ri Jü,n!. Irc r:/. n ramnr ofN.lslD lv orf ln r e i a r s r i e s p m d!s r i i = / , ! t u 1r l r = r i . l : rhd rhc $tnrA
,J 1tr.4.4.D) do tror dep€trdotr of (lr. \qMr. roor 1I:l '.
u ' ; l . r i ; r . .= i ; ü r i ; l , . i i ; r i l i r l ' = r ' t
, \rnh r coNrDr 1{m. Nhi.h n loNr dilsonll s'd hr nricrl) poiirjrc 'li!-qdfuL.i(t r/ ld - Itrr/'rdüd. r|t n'bd oli iE,11l,'r rof \rhld) i hrs n dnlin! rr olr|n ind dLns.n'bsd or llir). \hcrc ul', is eo do$!d Nnh rh! ro üll.d poid\ri5erlpolo,sl Sinlr rhe lll r ler,irb! and ior r$ufulIü rhr und*nandingoI 'hc \olc' ll'rr llr sc| V 1,, rs $ell ß oi rltr
crroi.e olrhe\.ishtinsman\rt ;. I o1 hi crlli&ll n! Io\ rü üclr trD\fcrlundiorioE rr-l,tr pxnrcurrr EPrc5rddlonnll bcobhi'rd asto eigtfyrltrr'lccomposirün ofj = u,t"rLrl ft$ ! htrns L.:l r. m th! {dc .llr b} rlf .hoie .,:rir;r
r:r"\lui,i;/tn
ci!.n\rlrß of t m dnrind.rhe liscn,(trr\ L, xre rlrcsr. s0ac n uniqudrdslincdi dl!posnic In rhn sirionsiD Ll, kJ b.
m a r ers, l , . i t ' . 4 . r L r o Ir l Ns r i l m prdirlrü rcne\drri.i .r 'hdftr. r
donelromr anaitr 7! on\ ds.si' .i rs. trndcror ß\!nprioir
Nhichi51hcLmnor rlt $hmrs ftr lcnrs)nenrrmric$ cotrspoDdiDs ro rhct!nJrcrtunc brn is doocimplicillr6r dlomDo p6\inrbi b l, r rb ar.tr rh
o r r J , ; / , , r r r r- f d *
consincncr orüc !lsonlhfrs\!ß sh
ü,iltu..ifllre hß | singrlrr yrlo* ofrbis matn harc
md.ir*rimats. HeE ir ir Fa,rcd.r N$i
r;
ti;i,"t; ',rr
xdr ',
Y,= n ;/r, Jr. rrl irr i
\ii ri; ! üe! 'lLrL,r=,, !,tr
rfds ro rlinnr r.
=
. i= ' t"li,,
eniin nre rF lh.orm r) Hcrc
E.d ro inliii$ ThiJ(ß{ilirllvdo n' üc iiir ncp r r.grsioD is p{fornrd. ncgLcdine rorfldion tom rhc hr nN l.D.r i ental limit rhco!_m.oitrger.d or,lü \ 7 k nedrJ, rhß 1 lo$fl botrndon rheio$bu
ihe .fied ot ncct.dins rh. fü pln dos mr showuDtn rhclimirinsdi*iburion.oi rhcorh{ha'd. rhehlcd uou.r ordxu impN! trpp.r boundsfor rh. incEar ot / in orde. ro .nsDft . utritom dnEBcr oarh.
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rh.
Th@M f, rer 0r{ l's ßn4dcd bt a tnp t.tslcl \hü. th {lodri trrr? 4d! (/iri.? tunüiunta.!\r\ J,ül\ thr th\\ 4 ,andttlrt
Th. ftnrar riün ücorcm tor rhc sysen natir ßrin,k.arso npljB 3 cinatümir rhcorem tor orhü quanfiies. vbjcn rE d.nrcd tron rt. sys.n idnr Cmlrr_r z rz r Rr']_ar'":-n" ,ir d 2ro4 B tt.thhs th. e.tu \ e 9: ,, th. nntnrü Q_ 3. c_ Dt.tl, (nt6 th. dt41i\5 4 rLloron r. t^b a,,, .* Di at nrb thepati:orinn n[ ka. M ' ln, d&tib.l in sdiln J dnt itu n dilii.ntkbk al tar. 3., ct,, D,,t,1tLnthetourr
\ lt
ar üt. a.. b.t - 1 a,,.B, .,,.Dxr\ 2.
rtttu z h a müiratiak notrnlt.r llnihnlüt rnntth, -. ,dridrb :cr!!,,,t td,,ut, t, = r,ta,r:.a "rhwan öt Nflht ,türdnö oI ! lnntatd d Ll^ Br. c6 D^l \ hq" E dlaott:..t1ttuti,n, t, th. pa! ntut pttüt dl tn! nrte hT itdkat anturcd:01tu ndnt 'at d4{ttiry ant nna t>
ttu n otul allditiorat vb$tirß ,rr h i ' jxed ktesü
I p!,,utktha Jotbtnt|!üttitil,ß ( r ) r > i r r o s r : k ' 3 t r o D ,>v r , l o ' K t ß d > t . tturc poL ou,oatk ddaximil nodut6. l2l rllt ET)'-o lot \Dk t < a.
\ i,Erir
,o r,-s".ar .o.ü
Drt\2.
thü. z t t ntuiürku trktt küda'n utidhtt rith :sa mun u n'iartz yi. ht,1 ! da,a ,o,u,tn.c rt, tirtibuiot. Hü! lAo,Er,cn,Dt t!1rt^ 1lnpatli.rtdl ttttizdlirn af k .tlnitut tl stdion 1. rh. rymp'o'ic sri!ß /i dqüd! onI (q and on rhcchoiE or$. utishrins frlr.id Hoftrc. üjs ü ior .npnsized iD rh. üpbioDs lor norariood coDr6i. .ß. Norq ih rld asumprionsotr r .ncdy rhc:3m ar rh* gftn in H nan and Dciitcr *hcE rheiiympr.ricnomdtiyornlrimum [r., il933J, Nore rhd rhe lo*d bolnd of rh. incna of rh. tunerion ird.r pd.pci& on ü. tr r ir posiht. ro.sim{r. rhis6ound coisisrnllyis rot_ lo*s:Fr a 0ons)luhrc!rcsionro rn. samptc daa and e$'mrc rheo.derof thß aR mod.t b lf rh. rrucpr6i i' ARMI. rh.n rhe*rim ed.rdcr rt c {i[ tulfill lirr-. -l2rirloEroD{logt= | 4i lR ag HaDornrnd D.ierlc( 1933!ThE orc lÄE (a. r'r^rlIar*n > | sme ro6.a r.eMbr.ddeior
rr plnioh.
rbc rppti.lrioni of rhn @ro ary m of
. Thepolcsof rhesyro d.pc0ddifiq.nriiblyon rh. .ntics in rhemadr r. iI rhccisctrvatu* aE dijincr (s L.n,na 7) Thü ! CLT tor rhe*ri,mls oflhe rydem polesü obraiicd on somc gcnsricsub*r or ti(i) komp.wrhrb.'s !trd J!'$o', t99,11Thc ene ndcm.nt s ru. 16rrhc ßrimlrcr oI rb. srsrm ros . Fnr nr.d tEtremr o rh. rrnlfcr fundion .ratuar.d d @ n qul b lo(.-t=Do + ci.Lr ror ,ro. Thus3 mhr rinir rh.o..n tor rh.6rinar6 ofrh. bandr rudiotr ar hftd lGq*ict poinß is obninrd Thi an be rd, ro onprc diftßor chojs or !r@du6lie ditrernrchoier.r/!trd of ri) 6r . Forgilcnsy$.h frarnß rhcfußfornariorroE!hc. loi coordrds is difsrcnrirbhfor srn.m mfuicd coir*pondingro r r.ßfs tundiddio rbe genen. n.4hborhoodcor*pondins b rh. Echdonpar rncriarion. Trru! I cLT tor ü. Ech.lonFraner.r srimarß .n , 8.Ern D.jlhborhoad hotd! tsine ü. inr.Frjon or $ö scßric Fß ie dil i !.ß]ic rrl Tr'is 6n als be ucd rö .bdpaE difl.Enr pl&durs oir.poDdjns 16rh.i. ßymproric h€harjoui Nok rhlt rhe algoarhmhß bcctrmrd. uniqü.b, rcnndinga{rid d.m.nß in cochlotum or0, ro bc poruv..HoNeveilhc.crurlinpl.m.nhrionotrhcsvD ursonrhmmayü$ r difcß shcdon.rrh. sisnsor rlr sinsulrrv.doß. Thtrs,rlr. syn.n obraf.d by thc ilsoirhm nay b. rrxrcd h r;,,i,.4,_i,) * ddi"""r ibore br a basis trrnsfomdbn .orrcsponding b dr!8(t 1...., 1l) con*hcy rnd !5ympioricno., D iryrorlncdiDlrd orrb..d0.[y imo|mnrd SvD {ill ioll if tbis SvD of r'ß.oninmus ar lI. rud .Fr?m. Th. ßtr16 of cororlory 1 trill botd.if o &pend3 onryon rh. u stu rDmriöi !, .En if rlr adut svD is
iorconrinuoos d rhcfß transrülundiof r0 (r" rhis {s rheslim!6 of thc srnen
lralr
. r B . < " r . ' r l r 7 : d i x g-( r , , , l r r l )
l l ; r r : . r J r - r I : x i d Ü , c o n r n. sr h c c o r $ p o n d i n s
(it) ,,i,rii) .1t,:i"i; ,r,ir{
Tlt mdri(sa'rnd
To simrlifrrhenoldion.ln'bulfuoronlJrlr.i* ol Lrrjfrdr\ proledm r be considcrcd. ic rh3 !r$ shüe tii = lri) rßibd in . comolcrdr!n!losd$ m fli[.In rh. DrcoI rle rolLoNiosnoLdion \iLL h. i , a d er,l T ) = o 1 l l T l )m . a n sl i n ! , . / ( l l l l I ) : 0 d i . q ( 7 ) = O l l ( T ) )m e a . s\ u p , . , l i T r r / l t < r , ä s f o r ,- e(7):41l(rrr means'hd ror.rül l>0.P d1r)jlr) >rl -o fo'I- 1 rhc
-d iI.,:r,
dcßicJ s
.i-,=tl'lli;r''=t,'alii)'tj,ri"r'
"[]
x =\;Firnd Lhc/rno'D ofr (po$ibrridrjrtrc i\ driöt.d \ynh -Y L =LlJ, dimen:ion:'l)lcdor
I=Eir;,=rl);,
rd
Y, 1.r= E,Eiv; + srrar i' k nil r!c rhc foloNing 3:pressions:
Fü 'rE d sorargd'bmso\idedd jn-'hise,qeri, n nraighdoNirnh $c.'hll rhc$Lifr
( i , .i , ) : 4 " .
Fbr gne indn$lrnd ! rh! !{jhx rhe imple coudails up t) lrg r+ I I ie on i(0).;ilrl . 11l- r, 1).shü. irlrrd.noEsrheeninde
( , , . i i 1 )= E j t t , i a ; ) ' a l t , ' .
r . r : i - , , 1 r ' r l' o f ir r = E i r l J , i >
rhecoh'.,nrtrncaliotr inderrhtr I renrjnlr(Dcisrrrd rl. reesrT olincluded $rltrjrnc c{im ß ttr
noLdtun
(i,-j.:i,) - !, i:lii;r \l I ti),t 'E")Elatt" +t"'ai\l;t \t t"ti;t 's"ti"l
'rrt) 'ilt,'
(4i.6 ) =
* i i r ! . a o . . ! .r i r
trrll=,!/,,r. rrrf 'h! regr*sioßin na 3 cM b.,ü' 'ilor. . ill)
xnd
l.:!.,,i, = r,,r;t
\
7,
'il"
j"=.i,,r-. '\.iI"
,r
rn..de rd oürliNrlE pnirolrhe CLT somcfunhq
i,:(i,,.i,xi,.i) i,=(ii,.,:,)=r(i,,.r)
( i , , .i , ) a l ) i ; ' . r ? )
*lreE the inDoldions t.!(r, a.r,r Nor rhr rM .iLi rnd rhns.lrrndr, mryrr. obriDcJ
', = veqto)....i(/). r- j. z,).
In rbcforlo\ins.rh. xbör. .rpßiiotri *lll b| tunhs nnalyed Thc c{ifrrts or the nx ii= ;,y ,riLl i, = rrr- -rrnnr,,'r^,/r,-üb.
& :lin,,D,=rqilor....
\.= a Jr;r 'ti"i
followirg$rr RccallftoD sedioi l.
'r
i":
k'li'ro)
:l[- Lr)'
j l f ) . i f ! .r ' ! . 2 ! r .
H . E . & t , ß d r n n dJ s r , = ' t . ) . , t ' . i, nd r!=lin,__.Y"=,,rr | ''j ).tvt.tl ft !( *ilt b. urd ro dcndrc 0. !!.ror of nickedv.crrtdi(rs 0f fv6a1mdn..s wirhslish(abusofiohrionx*i h. prfl ordr pmof ro.how rh. qisrne of. .h. rcqurcdlimns as has bei nded xltudy Ä" is i nftrio' ot üt snDh ru'o.oqriaoesr,.,.,,tr= 4rir r.rs!J- Th.npopu, koon muir.old: iß enln bt r, - dllr-r Thc!5ri, obhin.d s r ru* bDn 0r = 'rl,i.r. Ilien rhc proof of ü. cLT mr) bi dcomp*d inrorh.Ionosii! foü nepsl
' ,lllrIyrll,,')t- -\ L(.! Z ß Kotü nünullt
lnn)hd!,J \ nh :no nIM
PruDt ln a finr srrp rhc lsnn r r - , , \ o r . r h a ri . r h n . r $
\ rt.,,.)=t,
ir pov.d fo. rh. cN
r Irit,r,r-r.q',....d,,,1
=rtrIq,..... .4 ,t.
I r) A enkal [nn üeo(fr lor rhccounan$ ]ir.i.{r[ - ' d r $iribG k for ?r, , = t/ + 1?) r3e (:)Thc phol rhar .1,'r-rit-o and dd {l'ro) = 0o {heie ,o .oßF)id5 ro rbe panicutrr Eilt i.n oI üe h. s$r.n des-ih.d in S.dion rl) A enrnl linir rhmnD rq zt, i.. ,'tti', -,, !2, trh.rc z n n itad:tr€nornrtt! disrnburdNirh m.!i 4ro and rmme r., fü 11' . ar a $nrbt. n0 [ü T] ' ls* $di1ri 5.1t, l4) Th. proololüedifleie irbiliryof rh. napping, trl rlr poin!r,, is s(riof Jt. rn orh.r wordsn ir pnrqr rhar\rth, + iJ4r- ül'hr = Jrrn' +,( d- )
\ ttdr o.)= \ rülr.r- {tfl" cz.
r , - r , \ / 7 , I r , r ,t . i= l . . . r - |
6- '\ rLI,+r ,l Nos ld /lD€;n b. r (qr.ne or r-..rr\ {nfyjns 0
r o , l > 0 c k r d r . ' h e ' o \ ( , , I r h cn d . , r t ' b; ?tuidcn $nh ^ro\. convdrsc ii rh. /r rn* rj the"rols of rh. ir:! 4) ndrir tJ=i!".1, '. ..). wh.d Lr / = h, Li,.In rrr b) 'hc.otrr n r o r r o $r 5h d ' r . ; L r < . 1 r , r h o d 5\.h { r r h r . o . . nirr-
Ior Lhc co,rmnc! ' h . , ( r o r , n t ' i , rtr r ar'hhhrJ.r,
\ 4rr., e^r= rr\.r.r,r,. ....s..^) +\.ccnol,$L..nr)l
tr a5 ns ,1h bl4k h$ ,heh.u\.j1. Br h. ***!."(. -", / . " " i n d b y ' h e h l o ( rt l J g o t r r{ti r r u r"r,h; o f r 1 . r "lt r rolowi rhrr,ror fix.d h. rh. uriiiR of ,?(!t].i ,i) yx r h c , b r d , - h h d u .r , h c I ? f l r h r r l u m 5 0 f L h r f u r m .\rß md n giE by - li'L . , a4r r1ri(rr 1r. h L : - ; , . t . : , , . , , . r . , - , / r! E r p r r e d 6 , s / _ , - r . Finallr-l.r4n€Rt b. a lqu$!! of \Eroß ru fit. i 4 0
