Asymptotic study of canonical correlation analysis: from matrix and analytic approach to operator and tensor approach

S a is ics & Ope a ions Resea ch T ansac ions SORT 27 (2) July-Decembe 2003, 165-174 S a is ics & Ope a ions Resea ch T ansac ions Asymp o ic s udy o canonical co ela ion analysis: om ma ix and analy ic app oach o ope a o and enso app oach Jeanne Fine∗ Uni e si ´e Paul Saba ie , F ance Abs ac Asymp o ic s udy o canonical co ela ion analysis gi es he oppo uni y o p esen he di e en s eps o an asymp o ic s udy and o show he in e es o an ope a o and enso app oach o mul idimensional asymp o ic s a is ics a he han he classical, ma ix and analy ic app oach. Using he las app oach, Ande son (1999) assumes he andom ec o s o ha e a no mal dis ibu ion and he non ze o canonical co ela ion coe icien s o be dis inc . The new app oach we use, Fine (2000), is coo dina e- ee, dis ibu ion- ee and pe mi s o ha e no es ic ion on he canonical co ela ion coe icien s mul iplici y o de . O cou se, when ec o s ha e a no mal dis ibu ion and when he non ze o canonical co ela ion coe icien s a e dis inc , i is possible o ind again Ande son’s esul s bu we di e ge on wo o hem. In his me hodological p esen a ion, we insis on he analysis ame (Dauxois and Pousse, 1976), he sampling model (Dauxois, Fine and Pousse, 1979) and he di e en ma hema ical ools (Fine, 1987, Dauxois, Romain and Viguie , 1994) which pe mi o sol e p oblems encoun e ed in his ype o s udy, and e en o ob ain asymp o ic beha io o he analyses andom elemen s such as p incipal componen s and canonical a iables.) MSC: 62E20, 62H20, 62H25, 47N30 Keywo ds: mul i a ia e analysis, canonical co ela ion analysis, asymp o ic s udy, ope a o , coo dina e- ee, dis ibu ion- ee 1 Classical app oach 1.1 Popula ion canonical co ela ion analysis Le Xand Ybe wo andom ec o s, pand qdimensional espec i ely (p≤q) de ined on a same p obabili y space (Ω,A,P), cen e ed and admi ing o de 4 momen s. We ∗Add ess o co espondence: Labo a oi e de S a is ique e P obabili ´ es, Uni e si ´ e Paul Saba ie , 118, ou e de Na bonne, 31062 Toulouse cedex, F ance. e-mail: [email p o ec ed]. Recei ed: Oc obe 2003 Accep ed: Decembe 2003 166 Asymp o ic s udy o canonical co ela ion analysis: om ma ix and analy ic... assume he ma ix co a iance VXo X o be non-singula and we deno e by HX he ec o space o eal- alued andom a iables ( . . .) linea combina ions o Xcomponen s. We in oduce simila ly VYand HYand we deno e by VXY he c oss co a iance he componen s o Xwi h he ones o Y. The aim o canonical co ela ion analysis (CCA) o (X,Y) is o measu e he ela ionships be ween Xand Y. CCA may be de ined as he sea ch o 1and g1, . . . o HXand HYwi h uni a iance and maximal co ela ion ρ1 hen, i e a i ely, o j=2, ..., , ( ≤p), as he sea ch o jand gj, . . . o HXand HYwi h uni a iance, unco ela ed wi h he ( k)k<jand (gk)k<jand wi h maximal co ela ion ρj. The . . . j and gja e called j h canonical a iables and he eal ρjo [0,1] is called j h canonical co ela ion coe icien . Le RX=V−1 2 XVXY V−1 YVYXV−1 2 Xand he same o RYpe mu ing Xand Y oles. I is easy o e i y ha RXand RYha e he same non-ze o eigen alues deno ed by (ρ2 j)j=1,..., when w i en in a dec easing o de . We se : λj=ρ2 j, o j=1, ..., ,and, excep in he pa icula case =p=q, we se λj=ρj=0 o j> . Fo j> we de ine jand gjas . . . o HXand HY espec i ely wi h uni a iance and unco ela ed wi h he ( k)k<jand (gk)k<j. CCA o (X,Y) is hen : ((ρj)j=1,..., +1,( j)j=1,...,p,(gj)j=1,...,q).(1) CCA o (X,Y) depends only on HXand HY, which a e also gene a ed by componen s o X0:=V−1 2 XXand Y0:=V−1 2 YY espec i ely. We show ha , i (uj)j=1,...,pand ( j)j=1,...,q deno e uni eigen ec o s bases o RXand RYassocia ed wi h (λj)j=1,...,pand (λj)j=1,...,q espec i ely, we can ob ain canonical a iables jand gjas linea combina ions o X0 and Y0componen s, using ujand jcomponen s as coe icien s, ha is, by se ing: j=huj,X0ipe gj=h j,Y0iq, whe e h., .ipand h., .iqdeno e Rpand Rqusual scala p oduc s. Decomposi ion (1) is no unique because each canonical a iable associa ed wi h a simple eigen alue may be eplaced by i s opposi e and he se o canonical a iables associa ed wi h a mul iple eigen alue may be eplaced by an o he se acco ding o he choice o RXand RYeigen ec o s associa ed wi h his eigen alue. 1.2 Sample canonical co ela ion analysis Le (Xl,Yl)l=1,...,nbe a n-sample i.i.d. as (X,Y). We index by n he elemen s de ined p e iously and calcula ed on he sample : µn X, µn Y,Vn X,Vn Y,Vn XY ,Rn X,Rn Y. Le (λn j)j=1,...,pbe he dec easing sequence o he peigen alues o Rn X(and o he p la ges eigen alues o Rn Y, he o he ones, i q>p, being null), (un j, n j)j=1,...,pa sequence o associa ed uni eigen ec o s o Rn Xand o Rn Yand ( n j,gn j)j=1,...,p he canonical a iables sequence, ec o s o Rn, ob ained by : Jeanne Fine 167 ∀l∈ {1, ..., n}( n j)l=hun j,(Vn X)−1 2(Xl−µn X)ipand (gn j)l=h n j,(Vn Y)−1 2(Yl−µn Y)iq. I he case a ises (q>p), we le λn p+1=0, we comple e ( n j)j=1,...,pwi h Rn Y eigen ec o s in o de o ob ain an o hono mal basis ( n j)j=1,...,qo Rqand we de ine he canonical a iables associa ed. A las , o all jin {1, ..., p+1}le ρn j=qλn j. Sample CCA o (X,Y) is hen: ((ρn j)j=1,...,p+1,( n j)j=1,...,p,(gn j)j=1,...,q).(2) 1.3 Asymp o ic s udy Asymp o ic s udy o CCA consis s in es ablishing a.s. con e gence o he canonical elemen s sequences o he sample CCA (2) o he co esponding canonical elemen s o he popula ion CCA (1) and in es ablishing con e gence in dis ibu ion o he s anda dized canonical elemen s sequences. Di icul ies a e nume ous : canonical a iables a e es ima ed (“p edic ed”) by Rn ec o s, he space dimension inc easing wi h sample size. Then he use is o es ic asymp o ic s udy o Rn Xand Rn Yeigen ec o s : (un j)j=1,...,pand ( n j)j=1,...,q espec i ely, called canonical ec o s and also o he Rpand Rq ec o s de ined by: xn j=(Vn X)−1 2un j and yn j=(Vn Y)−1 2 n j espec i ely, called canonical ac o s; hese ec o s pe mi o ob ain di ec ly canonical a iables by: ∀l∈ {1,...,n}( n j)l=hxn j,Xl−µn Xipe (gn j)l=hyn j,Yl−µn Yiq. Mul iple eigen alues case is di icul o p ocess because he eigen ec o s associa ed wi h a e no uniquely de ined. Then he use is o es ic asymp o ic s udy o he case whe e all eigen alues a e simple. Uniqueness is hen e i ied by choosing sys ema ically he uni ec o (be ween he wo ones) which has he i s non null coo dina e in espec wi h he canonical basis posi i e. As o all mul idimensional analyses, co a iance ma ices o sample andom ma ices Vn X,Vn XY ,Rn X, ... a e “supe -ma ices” ( ha is, ma ices o ma ices). Tools such as he “ ec” ope a o which ans o ms ma ix in o ec o , ha e been in oduced in o de o handle heses supe -ma ices; he di icul y comes om he necessi y o ixing he o de o lines and columns elemen s. In o he espec s, we know ha he sequence ( √n(Vn X−VX)) con e ges in dis ibu ion o a cen e ed no mal a iable, he co a iance supe -ma ix o which is known in some special cases, when Xhas a no mal o ellip ical dis ibu ion o example. In he CCA ame wo k, we need o s udy con e gence in dis ibu ion o he sequence (√n(Vn Z−VZ)) wi h Z=(X,Y), hen o s udy con e gence in dis ibu ion o he sequence (√n(Rn Z−RZ)) wi h RZ=(RX,RY) and Rn Z=(Rn X,Rn Y),be o e s udying con e gence 168 Asymp o ic s udy o canonical co ela ion analysis: om ma ix and analy ic... o he Rn Zeigenelemen s sequences and con e gence o he sample canonical elemen s sequences. This asymp o ic s udy is much mo e complex han he one o P incipal Componen Analysis (PCA) because p incipal alues and p incipal ec o s o he X PCA a e eigenelemen s o he Xco a iance ma ix. I is only in 1999 ha Ande son publishes a CCA asymp o ic s udy when (X,Y) has a no mal dis ibu ion and when all non ze o eigen alues a e simple. Canonical ac o s componen s and canonical co ela ion coe icien s o he popula ion CCA a e di e en iable unc ions o VZ. Resul s a e hen ob ained om Taylo expansions. So, his classical app oach may be quali ied as ma icial and analy ic. In o de o simpli y calcula ions, we p opose o change a iables om (X,Y) o (X0,Y0), which is equi alen o changing he basis in Rpand Rq. We hen ha e: VX0=Ip, and RX=RX0=VX0Y0VY0X0, and simila ly o VY0and RY. 2 Ope a o and enso app oach 2.1 In oduc ion Di icul ies p e iously desc ibed a e ensued only om he ac ha ma icial ool is no con enien . Wo king di ec ly on linea ope a o s in Euclidean spaces a oid indices p oblems and can be easily ex ended o an Hilbe ian ame. Mo eo e , ins ead o s udying eigen ec o s associa ed wi h simple eigen alues, i is possible o s udy eigenp ojec o s associa ed wi h mul iple eigen alues. Ea on (1983) also ad ices a “ ec o space app oach” o he mul idimensional s a is ics. Dauxois and Pousse (1976) enla ge he PCA de ini ion o a Rp andom ec o o a Hilbe andom a iable and e en o a Hilbe andom unc ion, ha is, a Hilbe andom a iable depending on a pa ame e in o de o p ocess empo al o spa ial da a. They ede ine each ac o ial analysis (PCA, CCA, Co espondence Analysis, Disc iminan Analysis, ...) in an ope a o ial and s ochas ic ame, ha leads hem o de ine, be ween o he s, nonlinea analyses. The i s asymp o ic s udy in his ame has been ealized by Romain (1979) o a Hilbe andom unc ion PCA (see also Dauxois, Pousse and Romain, 1982), s udy comple ed by A con e (1980) who also s a ed on CCA asymp o ic s udy bu all ools we e no a ailable o con inue he s udy. Dauxois, Romain and Viguie (1994) p opose o use some enso p oduc s and es ablish a dic iona y be ween ma icial and ope a o ial o mula. This wo k pe mi s o compa e common esul s ob ained in bo h ames, bu also o ob ain mo e easily complex esul s; w i ings in espec wi h eigen ec o s basis a e es ablished a e concise o mula ions wi h ope a o s. These new ools pe mi o ealize in Fine (2000) he CCA asymp o ic s udy wi hou es ic ion, ha is, wi hou assump ion on he (X,Y) dis ibu ion, in he gene al case whe e eigen alues may be mul iple and wi hou excluding canonical a iables Jeanne Fine 169 asymp o ic s udy (CCA andom elemen s). The e o e, ou app oach may be quali ied as an ope a o and enso app oach. We gi e below he di e en s eps o he CCA asymp o ic analysis, some ools used and some examples o esul s. 2.2 Di e en s eps o he CCA asymp o ic s udy, ools, esul s 1) Popula ion CCA Fi s , he ma e is o de ine CCA o a pai o Euclidean andom a iables (popula ion CCA). Again, we use classical app oach no a ions subs i u ing (Rp,h., .ip) and (Rq,h., .iq o pand qdimensional Euclidean spaces (X,h., .iX) and (Y,h., .iY) espec i ely. Ob iously, we wo k wi hou e e ence o any basis ( ee coo dina e). Le L2(P) he Hilbe space o . . . de ined on (Ω,A,P) and admi ing o de 2 momen s, scala p oduc o which associa es E( g) o ( ,g). The ope a o ΦX om X o L2(P) which associa es hx,XiX o xplays an essen ial ole in he ope a o app oach o mul idimensional s a is ics. In pa icula , Xis a no mal Euclidean andom a iable i , and only i , ∀x∈ X,hx,XiXis a no mal . . .. The expec ed alue o Xis he unique elemen o X(Riesz heo em), deno ed by E(X), e i ying: ∀x∈ X,hx,E(X)iX=E(hx,XiX). Fo all (x,y)∈ X×Y, we deno e by x⊗y he ope a o om X o Ywhich associa es hx0,xiXy o x0; i is an elemen o he Hilbe space σ2(X,Y) o ope a o s om X o Ywi h he scala p oduc : hA,Bi2= (AB∗).Due o he Riesz heo em, we may hen de ine co a iance ope a o s VXo X,VYo Y, and c ossed co a iance ope a o s VXY and VYX o Xand Y:VX=E((X−µX)⊗(X−µX)),... As in he CCA classical app oach (§1.1), Xand Ya e assumed o be cen e ed. The adjoin ope a o Φ∗ Xo ΦXis he ope a o om L2(P) o Xwhich associa es E( X) o and hen we ha e : Φ∗ X◦ΦX=VX,Φ∗ X◦ΦY=VXY , ... I is con enien o ep esen ope a o s ela ionships in he ollowing commu a i e diag am, also called a duali y scheme; he e, each space is iden i ied wi h i s dual space. The HXand HYspaces a e image spaces o ΦXand ΦY espec i ely and he o hogonal p ojec o s o L2(P) on hese subspaces a e: ΠX= ΦX◦V−1 X◦Φ∗ Xand ΠY= ΦY◦V−1 Y◦Φ∗ Y. X Φ∗ X ←− L2(P) Φ∗ Y −→ Y V−1 X↓↑ VX↑I VY↑↓ V−1 Y X−→ ΦXL2(P)←− ΦYY Ope a o s RXand RY, and also CCA o (X,Y), a e de ined as p e iously (symbols ◦ a e dele ed in o de o educe no a ion). 170 Asymp o ic s udy o canonical co ela ion analysis: om ma ix and analy ic... As in he classical app oach, in o de o acili a e calcula ions, we change he scala p oduc on Xso ha he co a iance ope a o o Xis he iden i y o X, and simila ly o Y. We hen ha e: RX=VXY VYX and RY=VYXVXY . 2) Sample model and sample CCA We use a sample model (Dauxois, Fine and Pousse, 1979) es ablishing a link be ween he sample used in Da a Analysis and he i.i.d. sample o S a is ics. A sample (Xl,Yl)l∈N∗ i.i.d. as (X,Y) is buil om an elemen ωo ΩN∗se ing, o all lo N∗(πldeno ing he l h p ojec ion o ΩN∗on o Ω) : Xl=X◦πland Yl=Y◦πl ha is Xl(ω)=X(ωl) and Yl(ω)=Y(ωl). We hen p o ide L2(P) wi h he scala p oduc ( andom scala p oduc as i depends on ω): ∀( ,g)∈L2(P)×L2(P),En( g)=1 n n X l=1 (ωl)g(ωl). Then we ha e: En(X)=µn X,Φn X=h., X−µn XiX,Vn X=1 n n X l=1 (Xl−µn X)⊗(Xl−µn X), . . . This sample model is he clue o dis inguish andomness implied by he model (L2(P) elemen s) om andomness implied by sampling. I pe mi s o ob ain he canonical a iables asymp o ic dis ibu ion. The duali y scheme o sample CCA is he same as he popula ion one a e subs i u ing L2(P) o (L2(P),En) and indexing ope a o s by n. Sample ope a o s Rn Xand Rn Yand sample CCA a e de ined as p e iously. 3) Con e gence o sample ope a o s sequence Limi heo ems in Euclidean o Hilbe spaces pe mi o ob ain a.s. con e gence and con e gence in dis ibu ion o he co a iance ope a o s sequence wi hou assump ion on he dis ibu ion o (X,Y) excep he exis ence o o de 4 momen s. Fo CCA, we ob ain ( emind we le Z=(X,Y)): Wn Z:=√n(Vn Z−VZ)D −→ WZ∼N(0 ; KZ), whe e KZis he co a iance ope a o o Z⊗Z. In wha conce ns he sample ope a o s Rn Z=(Rn X,Rn Y), elemen s o σ2(Z) (wi h Z=X×Y), a.s. con e gence de i es om he ac ha i is possible o w i e Rn Xand Rn Y as con inuous unc ion o Vn Z. Le Un Z=√n(Rn Z−RZ) (:=(Un X,Un Y)). We w i e Un X= Ψn X(Wn Z) whe e (Ψn X) is a sequence o andom ope a o s om σ2(Z) o σ2(X) a.s. con e ging o ΨX. We hen deduce he con e gence in dis ibu ion Jeanne Fine 171 o (Un X) o UX= ΨX(WZ), cen e ed no mal a iable, co a iance ope a o o which being LX= ΨX◦KZ◦Ψ∗ X, and he same esul o (Un Y) pe mu ing Xand Y oles. The p oposi ion used he e, is easy o p o e om classical esul s in me ic spaces (Billingsley, 1968). We ob ain o example: UX=−1 2(WXRX+RXWX)+WXY VYX +VXY WYX −VXY WYVYX ∼N(0 ; LX) 4) Con e gence o eigenelemen s and CCA elemen s sequences Wha e e may be he “ ac o ial” me hod, which is an analysis o a model ob ained om a spec al (o singula - alue) decomposi ion, all esul s conce ning eigenelemen s (eigen alues, eigenp ojec o s, eigen ec o s associa ed wi h simple eigen alues, ...) a e easily ob ained hanks o pe u ba ion heo y o linea ope a o s (Ka o, 1980). In Fine (1987), his heo y has been adap ed o bounded pe u ba ions ha pe mi s o use i , due o he i e a ed loga i hm law, in he asymp o ic s udy ame. So we ob ain a.s. expansions o eigenelemen s o a symme ic posi i e ope a o s sequence. We may also consul Dossou-Gbe e and Pousse (1991) o limi esul s bu , o he con e gence in dis ibu ion o some CCA elemen s, limi esul s a e no su icien when pe u ba ion expansions pe mi o conclude. Fo example, o he canonical ac o s associa ed o a simple eigen alue λi, we ha e: xi=uibecause VX=IXand xn i=(Vn X)−1 2un iso: √n(xn i−xi)=−(Vn X)−1 2((Vn X)1 2+IX)−1[√n(Vn X−IX)]un i+[√n(un i−ui)]. We know ha ( √n(Vn X−IX)) con e ges in dis ibu ion o WXand ( √n(un i−ui)) o SXiUXxi(wi h SXi=(RX−λiIX)−) bu , hanks o pe u ba ion expansions, i is possible o es ablish: √n(xn i−xi)D −→ 1 2WXxi+SXiUXxi∼N(0 ; LXi) 5) Asymp o ic co a iance ope a o s in he ellip ical case We ha e al eady seen ha he asymp o ic co a iance ope a o o ( √n(Rn X−RX)) is LX= ΨX◦KZ◦Ψ∗ Xwhe e KZis he asymp o ic co a iance ope a o o ( √n(Vn Z−VZ)) and whe e he ope a o ΨX om σ2(Z) o σ2(X) can be w i en explici ly. All he dis ibu ion limi s o eigenelemen s o CCA elemen s sequences a e cen e ed no mal a iables (o unc ion o cen e ed no mal a iables), co a iance ope a o o which being w i en as unc ion o KZin he same way. Now, we may w i e explici ly hese asymp o ic co a iance ope a o s in he case whe e Zhas an ellip ical dis ibu ion wi h mean µZ, co a iance ope a o VZand ku osis κ( eal pa ame e , which, when i is null, leads o a N(µZ,VZ) dis ibu ion). We hen know ha KZis he ope a o om σ2(Z) o i sel which associa es o T: KZ(T)=(1 +κ)VZ(T+T∗)VZ+κhVZ,Ti2VZ. 172 Asymp o ic s udy o canonical co ela ion analysis: om ma ix and analy ic... A his s ep, we need mo e algeb aic ools. The enso p oduc in spaces o ype σ2 is deno ed by ˜ ⊗.Fo example: ∀(A,B)∈σ2(Z)×σ2(Z),∀T∈σ2(Z),A˜ ⊗B(T)=hT,Ai2B. We de ine also p oduc ` ⊗in spaces o ype σ2.Fo example: ∀(A,B)∈σ2(Z)×σ2(Z),∀T∈σ2(Z),A` ⊗B(T)=BT A∗. We de ine he commu a ion ope a o Cwhich associa es o an ope a o Ti s adjoin T∗. A las , we subs i u e he space p oduc Z=X×Y o he Hilbe ian sum Z=X⊕Y; his pe mi s o plunge all he ope a o s in o σ2(Z) in o de o simpli y no a ion. P ojec o PX om σ2(Z) on o σ2(X) becomes in his ame a symme ic ope a o o σ2(Z). The ope a o KZ om σ2(Z) o i sel may be w i en as: KZ=(1 +κ)VZ ` ⊗VZ(I+C)+κVZ˜ ⊗VZ. Le (xi)i=1,...,pbe an o hono mal basis o med by canonical ac o s o X, hen (xi⊗xj)i,j=1,...,pis an o hono mal basis o σ2(X) and ((xi⊗xj)˜ ⊗((xk⊗xl))i,j,k,l=1,...,p is an o hono mal basis o σ2(σ2(X)). A e calcula ions ob ained in a concise way, i is easy o decompose ope a o s in espec wi h his ype o basis. Fo example, we ob ain o he asymp o ic co a iance ope a o o ( √n(Rn X−RX)): LX=(1 +κ)(I+C)[−3 4R2 X ` ⊗IX+R2 X ` ⊗RX+RX ` ⊗IX−5 4RX ` ⊗RX](I+C) and, in espec wi h he basis o canonical ac o s ( emembe ha (λj)j=1,...,pis he dec easing sequence o eigen alues o RX): LX=1 2(1 +κ) p X j=1 p X k=1Ã−3 4λ2 j−3 4λ2 k+λ2 jλk+λjλ2 k+λj+λk−5 2λjλk! (xj⊗xk+xk⊗xj)˜ ⊗(xj⊗xk+xk⊗xj) When (X,Y) has a no mal dis ibu ion and when all eigen alues a e simple, i is possible o edisco e Ande son’s esul s bu we di e ge on wo o hem. 6) Con e gence o CCA andom elemen s sequences As p e iously announced (§2.1.2) he sample model pe mi s o ob ain a.s. con e gence and con e gence in dis ibu ion o canonical a iables sequences. We ha e o example, o he canonical a iable associa ed wi h a simple eigen alue λi: √n( n i− i)) D −→ h1 2WXxi+SXiUXxi,XiX∼N(0 ; MXi) Jeanne Fine 173 wi h, in he pa icula case whe e (X,Y) has an ellip ical dis ibu ion: MXi =1 4(2 +3κ) i⊗ i+(1 +κ)X j,i (1 −λi)(λi+λj−2λiλj)(λi−λj)−2 j⊗ j 7) In e en ial applica ions and conclusion These esul s on CCA asymp o ic s udy pe mi o ackle easily in e en ial applica ions (con idence in e al es ima ion, s a is ical es s, ...) which imply CCA elemen s, pa icula ly he p oximi y measu es buil on canonical co ela ion coe icien s. See Ande son (1999) and Dauxois and Nkie (2002). Fu he aspec s and esul s may be consul ed in Fine (2000). This me hodological p esen a ion shows ha he ope a o app oach pe o ms qui e well in sol ing asymp o ic p oblems in mul i a ia e s a is ics. 3 Re e ences Ande son, T. W. (1999). Asymp o ic Theo y o Canonical Co ela ion Analysis. Jou nal o Mul i a ia e Analysis, 70, 1-29. A con e, A. (1980). ´ E ude asymp o ique de l’analyse en composan es p incipales e de l’analyse ca- nonique. Th` ese de 3` eme cycle, Uni e si ´ e de Pau e des Pays de l’Adou . Billingsley, P. (1968). Con e gence o p obabili y measu es. Wiley, New Yo k. Dauxois, J., Fine, J. and Pousse A. (1979). ´ Echan illonnage en segmen a ion, ´ e ude de la con e gence. S a- is ique e Analyse des Donn´ees, 3, 45-53. Dauxois, J. and Nkie , G. M. (2002). Measu es o Associa ion o Hilbe ian subspaces and some ap- plica ions. Jou nal o Mul i a ia e Analysis, 82, 263-298. Dauxois, J. and Pousse, A. (1976). 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Sp inge -Ve lag, New Yo k. Romain, Y. (1979). ´ E ude Asymp o ique des app oxima ions pa ´echan illonnage de l’analyse en compo- san es p incipales d’une onc ion al´ea oi e. Quelques applica ions. Th` ese 3` eme cycle. Uni e si ´ e Paul Saba ie . Toulouse.