Robust Information Fusion using Variable-Bandwidth Density Estimation

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1 Robust Iformato Fuso usg Varable-Badwdth Desty Estmato Dor Comacu Real-Tme Vso ad Modelg Departmet Semes Corporate Research 755 College Road East, Prceto, NJ 854, USA Abstract We preset a ew approach to formato fuso based o kerel desty estmato. Employg a desty estmato strategy wth adaptve badwdths, we develop the varablebadwdth mea shft as a effcet techque for mode detecto. Iterestgly eough, the mea shft procedure leads to the defto of a ew fuso estmate as the locato of the hghest mode of a desty fucto whch takes to accout the ucertaty of the estmates to be fused. We show that the ew Varable-Badwdth Desty-based Fuso VBDF) s cosstet ad coservatve the sese defed by Covarace Itersecto, whle beg robust to outlers. At the same tme, the ew framework ca reveal the presece of multple source models the put data. Expermetal comparsos wth the BLUE fuso ad Covarace Itersecto show some of the advatages of our techque. 1 Itroducto The two ma archtectures for formato fuso are represeted by cetralzed ad dstrbuted fuso. Cetralzed fuso volves processg all sesor measuremets at a sgle locato, whle sesor measuremet errors are assumed depedet across sesors ad tme. Classcal estmato techques, such as the exteded) Kalma flter [3], are employed to obta a estmate.e., a mea vector ad the covarace matrx expressg the ucertaty of the mea) from multple sesor data. Dstrbuted fuso [15] volves a collecto of processg odes, whch each ode processes ts sesor measuremets ad commucates the results wth eghborg odes. I addto, each ode performs a specfc fuso task usg state formato from ts eghbors. Dstrbuted archtectures have a mproved relablty ad flexblty, however, the fuso s more dffcult sce the state estmates are geerally correlated. We preset ths paper a fuso soluto that features three mportat propertes: ablty to accommodate multple source models, cosstet ad coservatve approach hadlg cross-correlato, ad robustess to outlers. The fuso estmate s gve by the locato of the mode of a adaptve kerel desty fucto that explots the ucertaty of the estmates to be fused. Although the ew estmate has a dfferet motvato ad s derved based o desty estmato theory, the cross-correlato s hadled a sprt smlar to that of Covarace Itersecto [13]. The paper s orgazed as follows. Secto 2 formally states the formato fuso problem ad dscusses ts curret solutos. Secto 3 revews the multvarate kerel desty estmato wth fxed ad adaptve badwdth. A geeral form of the varable-badwdth mea shft s troduced ad aalyzed Secto 4. The Varable-Badwdth Desty-based Fuso VBDF) estmator s defed Secto 5, whle Secto 6 presets expermets ad comparsos. 2 Related Work We formulate the followg the formato fuso problem ad revew ts solutos accordg to the type of avalable formato. Let ˆx 1 ad ˆx 2 be two estmates that are to be fused together to yeld a optmal estmate ˆx. The error covaraces are defed by P j =E [ x ˆx )x ˆx j ) ] 1) for =1, 2adj =1, 2. For the smplcty of otato deote P 11 P 1 ad P 22 P 2. Whe the cross-correlato ca be gored,.e., P 12 = P 21 =, theblueestmates also called Smple Covex Combato [8] ad s gve by ˆx CC = P CC P 1 ˆx 1 + P 2 ˆx ) 2 2) P CC = P 1 + P ) 2 3) Whe the tal estmates are correlated P 12 = P 21 ) ad the ose correlato ca be measured, the BLUE estmate ˆx BC, P BC ) s derved accordg to Bar-Shalom ad Campo [2] usg Kalma formulato. The most geeral case of BLUE estmato also assumes pror kowledge of the covarace of x [14]. However, sce the cross-correlato s dffcult to evaluate, t s problematcal practce to rgorously apply the Kalma flter. The reaso s that the assumpto of zero cross-correlato may lead to overly cofdet estmates. A coservatve soluto to ths problem s provded by the Covarace Itersecto algorthm [13]. The tersecto s characterzed by the covex combato of the covaraces ˆx CI = P CI ωp 1 ˆx 1 +1 ω)p 2 ˆx ) 2 4)

2 P CI = ωp 1 +1 ω)p ) 2 5) where ω [, 1]. The parameter ω s chose to optmze the trace or determat of P CI. IarelatedworkChog ad Mor [8] exame the performace of Covarace Itersecto ad Che, Arambel ad Mehra [6] aalyze the optmalty of the algorthm. We ote that the Covarace Itersecto geeralzes to the fuso of estmates as ˆx CI = P CI ω P ˆx ) P CI = ω P 6) wth ω = 1. Aga, the weghts ω are chose to mmze the trace or determat of P CI. Whle the Covarace Itersecto represets a elegat soluto to the cross-correlato problem, t assumes a sgle source model ad s ot robust to outlers. I the ext sectos we show that a robust soluto to the formato fuso problem ca be obtaed usg kerel desty estmato. The ew estmate s defed as the locato of the sample mode of a desty fucto costructed usg varable-badwdth kerels. 3 Kerel Desty Estmato We defe the sequel multvarate kerel desty estmato wth fxed ad adaptve badwdth. Some propertes of the two most commo adaptve badwdth estmators, the balloo desty estmator ad the sample pot estmator are revewed. 3.1 Fxed-Badwdth Desty Estmato Gve the d-varate radom sample x, =1... draw depedetly from some ukow desty f, the fxedbadwdth kerel desty estmate of f computed at locato x wth kerel K s defed by [21, p.91] ˆf K x) = K H x x ) 7) wth K H x) = H /2 KH /2 x) 8) The kerel K s take to be a d-varate desty fucto wth Kx)dx = 1, xkx)dx =, ad xx Kx)dx = c K I, where c K s a costat. The symmetrc postve defte d d matrx H s called the badwdth matrx. Its choce cotrols the amout of smoothg preset the estmate. Sce H s held costat across x R d, the estmator 7) s sad to have fxed badwdth. The desty at each pot s estmated by takg the average of detcally scaled ad oreted kerels cetered at each of the data pots. Propertes of ths estmator related to ts asymptotc) mea tegrated squared error MISE), choce of the kerel, ad badwdth selecto are dscussed [18, 21, 19]. 3.2 Adaptve Desty Estmato Itutvely, more smoothg s ecessary where the data s sparse, such as o tals ad valleys, ad less smoothg s ecessary ear peaks. To acheve ths goal, the badwdth H ca be adapted two ways [17]. By selectg a dfferet badwdth H = Hx) foreachestmato pot x, we obta the balloo desty estmator ˆf K,b x) = K Hx) x x ) 9) The estmate of f at x s also computed as the average of detcally scaled ad oreted kerels cetered at each data pot, but the badwdth matrx vares wth x. Loftsgaarde ad Queseberry [16] troduced ths type of estmator as the kth earest eghbor estmator. Due to the varato of H wth x, balloo estmators are usually osy, dscotuous ad fal to tegrate to oe, but ther propertes mprove as dmesoalty creases [2]. By selectg a dfferet badwdth H = Hx ) assumed full rak) for each data pot x oe ca defe the sample pot desty estmator ˆf K,s x) = K Hx ) x x ) 1) Ths tme, the estmate of f at x s the average of dfferetly scaled ad oreted kerels cetered at each data pot. Brema, Mesel ad Purcell [4] troduced the sample pot estmator by adaptg the badwdth as a fucto of the dstace from x to the kth earest data pot. Aother adaptato crtero was suggested by Abramso [1], makg use of a plot estmate of the desty to calbrate the badwdth. Sample pot estmators are themselves destes, beg o-egatve ad tegratg to oe They are of partcular terest sce ther estmato bas decreases comparso to the fxed-badwdth estmators, whle the varace remas the approxmately the same [2, 12]. I the ext secto we wll develop a techque for the detecto of the peaks of sample pot desty estmates. 4 Mode Detecto usg Varable Badwdth The modes or peaks of a desty fucto represet atural measures of data cetral tedecy. Ths secto troduces a teratve procedure, the varable-badwdth mea shft, for mode detecto whe the uderlyg desty s computed through sample pot estmators. We demostrate the covergece of the ew procedure both value ad argumet to a statoary pot of the desty surface.

3 4.1 Desty Gradet Estmato We derve the followg a useful expresso of the desty gradet. By deotg H Hx ) for all =1...,the sample pot estmate 1) s rewrtte as ˆf ) K x) = H /2 K H /2 x x ) 11) where we dropped the subscrpt s. To smplfy otatos we defe the profle of the kerel K as a fucto k : [, ) R such that Kx) =k x 2 ). Defto 11) becomes ˆf K x) = H /2 k D 2 x, x, H ) ) 12) wth D 2 x, x, H ) x x ) H x x ) 13) beg the Mahalaobs dstace from x to x. A quatty of terest s the data-weghted harmoc mea of the badwdth matrces computed at x ) H h x) = ω x)h 14) The weghts H /2 g D 2 x, x, H ) ) ω x) = H 15) /2 g D 2 x, x, H )) satsfy ω x) = 1, whle the fucto gx) = k x) s the opposte of the dervatve of profle k. Observethatf we defe a kerel Gx) =Cg x 2 ), where C s a ormalzato costat, the deomator of 15) s proportoal to the desty estmated at x usg the kerel G ˆf G x) =C H /2 g D 2 x, x, H ) ) 16) We ca compute ow the gradet of the true desty as the gradet of the estmate 12) ˆ f K x) =2 H /2 H x x)g D 2 x, x, H ) ) 17) By multplyg 17) to the left wth H h x) wehavesuccessvely H h x) ˆ f K x) = = 2 H h H /2 g D 2 x, x, H ) ) H /2 H x g D 2 x, x, H ) ) H /2 g D 2 x, x, H )) where we used 14), 15) ad 16). Equato 18) shows that after a proper scalg ad rotato wth H h x), the gradet of the desty at x becomes proportoal to the dfferece betwee a weghted sum of data pots ad x. 4.2 Varable-Badwdth Mea Shft The vector defed by the edg brackets of equato 18) s called the varable-badwdth mea shft m G x) H h x) ω x)h x x 19) It ca be easly see that the mea shft vector s proportoal to the ormalzed gradet of the desty fucto after a proper scalg ad rotato) m G x) = C 2 H hx) ˆ f K x) ˆf G x) 2) If the badwdth matrces are all equal to a fxed matrx H, expresso 19) reduces to the fxed-badwdth mea shft m G,F B x) = x g D 2 x, x, H) ) g x 21) D2 x, x, H)) Furthermore, f the kerel K s multvarate ormal the mea shft vector becomes m G,N x) = x exp 1 2 D2 x, x, H) ) exp 1 2 D2 x, x, H) ) x 22) Hstorcally, Fukuaga ad Hostetler [11] troduced the fxed-badwdth mea shft as a estmator of the ormalzed desty gradet. Later o, Cheg [7] dscussed applcatos to clusterg, Comacu ad Meer [9] studed the covergece of the fxed badwdth procedure ad developed mage segmetato ad flterg applcatos, whle Comacu, Ramesh ad Meer [1] proposed a verso of the varable-badwdth mea shft, based o dagoal badwdth matrces. Expresso 19) of the varable-badwdth procedure s the most geeral, ad, as t wll be see Secto 5, t makes a terestg lk from kerel desty estmato to the formato fuso problem. Before dscussg ths, however, we wll study the propertes of a teratve procedure for mode detecto that reles o expresso 19). 4.3 Iteratve Procedure for Mode Detecto Deote by { y j the sequece of successve locatos }j=1, H /2 g D 2 x, x, H ) ) x = ˆf G x) C/2 [ H h x) ] ω x)h x x 18) y j+1 = H h y j ) ω y j )H x j =1, 2,... 23) where y 1 s the tal locato of the sequece. The sequece s obtaed by computg the mea shft vector

4 at the curret locato ad traslatg by that amout. The correspodg sequece of desty estmates computed wth kerel K the pots 23) s { { } ˆfK j)} ˆfK y j ) 24) j=1,2... j=1,2... Assume that the kerel K has a covex ad mootocally decreasg profle. The followg propertes are the vald for the sequeces { { } y j }j=1,2... ad ˆfK j) s mooto- { Property 1 The sequece ˆfK j)} cally creasg ad coverget. j=1,2... j=1,2.... Proof The sequece s bouded sce s fte, hece t s suffcet to show that ˆf Kj) s strctly mootoc creasg,.e., f y j y j+1 the ˆf K j) < ˆf K j +1),forall j =1, 2... Wthout loss of geeralty we assume y j = ad wrte ˆf K j +1) ˆf K j) = H /2 [ k D 2 y j+1, x, H )) k D 2, x, H ) )] 25) Sce the profle k s covex kx 2 ) kx 1 )+k x 1 )x 2 x 1 ) 26) for all x 1,x 2 [, ), x 1 x 2, ad usg k = g we have kx 2 ) kx 1 ) gx 1 )x 1 x 2 ). 27) From 25) ad 27) t results that ˆf K j +1) ˆf K j) H /2 g D 2, x, H ) )[ D 2, x, H ) D 2 )] y j+1, x, H = 2y j+1 H /2 H x g D 2, x, H ) ) H /2 D 2 ) y j+1,, H g D 2, x, H ) ) = 2y j+1 H /2 H g D 2, x, H ) ) y j+1 H /2 D 2 ) y j+1,, H g D 2, x, H ) ) = H /2 D 2 ) y j+1,, H g D 2, x, H ) ) where we made use of defto 23). However, k s mootocally decreasg, whch mples that gx) = k x) 28) for all x [, ). I addto, at least oe fucto g D 2, x, H ) ) ) should be strctly greater tha zero otherwse the mea shft vector caot be defed). As a result, as log as y j+1 y j =, the last term of 28) s strctly postve,.e., ˆfK { } j +1) ˆf K j) >. Hece, the sequece ˆfK j) s coverget. j=1,2... Lemma 1 The sequece { y j s bouded. }j=1,2... Proof Accordg to Property 1 for ay startg pot y 1,fy j y 1 we have ˆf K j) > ˆf K 1) for all j =2... { However, the level set y ˆf K j) > ˆf } K 1) s bouded because ˆf K s a desty. Thus, the sequece { y j }j=1,2... s bouded. Lemma 2 The orm of the mea shft vector m G y j ) coverges to zero. Proof Let us rewrte equalty 28) for a geeral y j ˆf K j +1) ˆf K j) H /2 D 2 y j+1, y j, H ) g D 2 y j, x, H )) = D 2 y j+1,y j, H c y j ) ) 29) where H c y j )= H /2 H g D 2 y j, x, H )) 3) s a postve defte matrx. Sce ˆf K j +1) ˆf K j) coverges to zero, t results that D 2 y j+1,y j, H c y j ) ) coverges to zero. Furthermore, sce H c y j ) s postve defte, the quatty y j+1 y j = m G y j ) also coverges to zero. Property 2 The sequece { y j s coverget. }j=1,2... Proof Sce the sequece { } y j s bouded j=1,2... Lemma 1) t has at least oe lmt pot, accordg to Bolzao-Weerstrass theorem [5, p.22]. Deote the lmt pot by y. We wll show that for ay ɛ>thereexsts a dex J such that for ay j J the sequece { } y j j J s cotaed a ope set aroud y. Sce m G y j ) coverges to zero Lemma 2) t results that for a gve <r<ɛthere exsts a dex J such that m G y j ) ɛ r 31) for ay j J. Deoteby ˆf max = ˆf K y,r,e) 32) the maxmum value of ˆf K the set r y y <ɛad defe the ope set S = {y y y <ɛ, ˆf K y) > ˆf } max 33)

5 The, f y j S wth j J we have y j+1 S. Ideed, the codto ˆf K y j ) > ˆf max mples that y j y <rad usg 31) we have By recallg ow that we ca wrte I addto y j y + m G y j ) <ɛ 34) y j+1 = y j + m G y j ) 35) y j+1 y = y j + m G y j ) y y j y + m G y j ) <ɛ 36) ˆf K y j+1 ) > ˆf K y j ) > ˆf max 37) Equatos 36) ad 37) mply that y j+1 S, whchmeas that the lmt y s uque. Q.E.D. The propertes from above show that by smply computg expresso 23) teratvely we obta a hll-clmbg procedure that coverges to a statoary pot of the uderlyg desty the desty gradet s zero at the covergece pot). Due to the ormalzato to ˆf G 2) the mea shft s a self-adaptg procedure that moves faster where the data desty s lower ad moves smaller steps whe approachg the peak of the desty. Nevertheless, certa precautos are eeded practce. Frst, a check for the qualty of the covergece pot should be made, ether by computg the Hessa matrx or through a radomzed procedure [9], to determe f the algorthm coverged to a local maxmum of the desty. Secod, sce the uderlyg desty mght dsplay multple local maxma, the mea shft procedure should be ru wth multple talzatos ad the best result.e., hghest mode) be selected. ) H h ˆx m )= ω ˆx m )H 39) Equatos 38) ad 39) defe the VBDF estmator, whch has the followg propertes: The covarace 39) of the fuso estmate s a covex combato of the covaraces of tal estmates. Thus, the ew algorthm s close to the Covarace Itersecto, although ts dervato had a dfferet motvato based o desty estmato theory. The matrx H h ˆx m ) s a cosstet ad coservatve estmate of the true covarace matrx of ˆx m, rrespectve of the actual correlatos betwee tal estmates. The proof s smlar to the cosstecy proof of the Covarace Itersecto. Whle the weghts the Covarace Itersecto algorthm are chose by mmzg the trace or determat of the covarace, our crtero s based o the most probable value of the data. Ths mght be a more approprate crtero, especally whe the data s multmodal,.e., the tal estmates belog to dfferet source models. The presece of multple modes the desty ladscape dcates that the put data has multple sources. Our fuso framework allows a clusterg-lke aalyss of the put data to detfy the uderlyg models. Sce the estmator s based o the data mode, t s robust to outlers,.e., a small percetage of faulted measuremets do ot fluece, or have a weak fluece the fal estmate. To compute the VBDF estmator oe has to determe the locato of the hghest mode of the desty 11). The varable-badwdth mea shft s ru wth multple talzatos to accomplsh ths task. 5 Varable-Badwdth Destybased Fuso Ths secto troduces a ew fuso estmator that follows the mode searchg prcples dscussed above. The Varable-Badwdth Desty-based Fuso VBDF) defes the fuso estmate as the locato of the sample mode of the data,.e., the maxmum of the desty 11). The covarace of the estmate s defed as the matrx H h gve by the varable-badwdth mea shft formulato ad computed at the locato of the mode. Deote by ˆx m the mode locato. Sce m G ˆx m )=, from 19) t results that ˆx m = H h ˆx m ) ω ˆx m )H x 38) 6 Expermets ad Dscussos A frst set of expermets s preseted Fgure 1. The sythetc put data s gve by 8 tal b-varate estmates expressed as locato ad covarace. They are represeted Fgure 1a where each covarace s dsplayed as a ellpse wth 95% cofdece. Observe that the put data has a clearly detfable structure of 5 measuremets ad aother 3 measuremets that ca be cosdered outlers. By rug the varable-badwdth mea shft wth talzato each tal put data locato, we obta the peaks show Fgure 1b, wth the largest peak marked by a ellpse of thck le. The mea shft trajectores are also show. I Fgure 1c we compare our result wth that of the BLUE fuso 2) ad 3)) ad Covarace Itersecto 6).

6 Note that the BLUE estmate produces the most cofdet result, however, the presece of outlers the data has a strog, egatve fluece o ths estmate. At the same tme the BLUE estmate ca be overly cofdet by eglectg the cross-correlato. The Covarace Itersecto s also egatvely flueced by outlers. We optmzed the weghts to mmze the trace of the covarace matrx, however, sce the optmzato regards oly the covarace ad ot the locato, the resultg estmate s rather poor. The best result seems to be produced wth less cofdece) by the VBDF algorthm. Note that the varablebadwdth mea shft also reles o optmzg the weghts. However the crtero s the mode of a desty fucto that takes to accout the data ucertaty. Observe that the algorthm has ot bee flueced by outlers. A secod set of expermets s preseted Fgure 2. Ths tme the data s real, obtaed from a moto estmato task. There are 49 bvarate measuremets ad ther covaraces Fgure 2a). The correspodg varablebadwdth desty fucto s show Fgure 2b. Observe the presece of a large mode the vcty of org. At the same tme, the exstece of other modes dcate the presece of more tha oe moto model. Ths ca be geerated, for example, by multple objects movg depedetly. The trajectores of the mea shft ad fal covaraces for each ru are show Fgure 2c, whle detals of the trajectores aroud the largest mode are gve Fgure 2d. The expermets from above showed a very promsg behavor of the VBDF fuso estmate. More expermets ad aalyss are however ecessary to uderstad uder what codtos ad for what applcatos our ew estmate has a superor ad useful behavor. For example, we recetly foud that the VBDF estmate s better behaved ad more stable whe t s defed as the locato of the most sgfcat mode across the desty scales. We wll detal these fdgs our future publcatos. Ackowledgmets I thak Radu Bala ad Vsvaatha Ramesh from Semes Corporate Research ad Peter Meer from Rutgers Uversty for stmulatg dscussos. Refereces [1] I. Abramso, O badwdth varato kerel estmates - a square root law, The Aals of Statstcs, vol. 1, o. 4, pp , [2] Y. Bar-Shalom ad L. Campo, The effect of the commo process ose o the two-sesor fused track covarace, IEEE Tras. Aero. Elect. Syst., vol. AES-22, o. 22, pp , [3] Y. Bar-Shalom ad T. Fortma, Trackg ad Data Assocato. Academc Press, [4] L. Brema, W. Mesel, ad E. Purcell, Varable kerel estmates of multvarate destes, Techometrcs, vol. 19, o. 2, pp , [5] I. Broshte ad K. Semedyayev, Hadbook of Mathematcs. Sprger, thrd edto, [6] L. Che, P. Arambel, ad R. Mehra, Estmato uder ukow correlato: Covarace tersecto revsted, IEEE Tras. Automatc Cotrol, vol. 47, o. 11, pp , 22. [7] Y. Cheg, Mea shft, mode seekg, ad clusterg, IEEE Tras. Patter Aal. Mache Itell., vol. 17, pp , [8] C. Chog ad S. Mor, Covex combato ad covarace tersecto algorthms dstrbuted fuso, Proc. of 4th Itl. Cof. o Iformato Fuso, Motreal, Caada, 21. [9] D. Comacu ad P. Meer, Mea shft: A robust approach toward feature space aalyss, IEEE Tras. Patter Aal. Mache Itell., vol. 24, o. 5, pp , 22. [1] D. Comacu, V. Ramesh, ad P. Meer, The varable badwdth mea shft ad data-drve scale selecto, Proc. 8th Itl. Cof. o Computer Vso, Vacouver, Caada, volume I, July 21, pp [11] K. Fukuaga ad L. D. Hostetler, The estmato of the gradet of a desty fucto, wth applcatos patter recogto, IEEE Tras. Iformato Theory, vol. 21, pp. 32 4, [12] P. Hall, T. Hu, ad J. Marro, Improved varable wdow kerel estmates of probablty destes, The Aals of Statstcs, vol. 23, o. 1, pp. 1 1, [13] S. Juler ad J. Uhlma, A o-dverget estmato algorthm the presece of ukow correlatos, Proc. Amerca Cotrol Cof., Alberqueque, NM, [14] X. L, Y. Zhu, ad C. Ha, Ufed optmal lear estmato fuso - part : Ufed models ad fuso rules, Proc. of 3rd Itl. Cof. o Iformato Fuso, Pars, Frace, 2. [15] M. Lggs, C. Chog, I. Kadar, M. Alford, V. Vacolla, ad S. Thomopoulos, Dstrbuted fuso archtectures ad algorthms for target trackg, Proceedgs of IEEE: Specal Issue o Data Fuso, vol. 85, o. 1, pp , [16] D. Loftsgaarde ad C. Queseberry, A oparametrc estmate of a multvarate fucto, The Aals of a Mathematcal Statstcs, vol. 36, pp , [17] S. Sa, Multvarate locally adaptve desty estmato, Computatoal Statstcs ad Data Aalyss, vol. 39, pp , 22. [18] B. W. Slverma, Desty Estmato for Statstcs ad Data Aalyss. Chapma & Hall, [19] J. Smooff, Smoothg Methods Statstcs. Sprger- Verlag, [2] G. R. Terrell ad D. W. Scott, Varable desty estmato, The Aals of Statstcs, vol. 2, pp , [21] M. P. Wad ad M. Joes, Kerel Smoothg. Chapma & Hall, 1995.

7 a) a) b) b) c) c) Fgure 1: Robust Iformato Fuso. a) Iput data represeted as ellpses wth 95% cofdece. b) Trajectores of the varable-badwdth mea shft ad ellpses correspodg to the fal covarace matrces. Thck le s used for the ellpse correspodg to the largest mode. c) Fuso results overlayed o the put data. Ellpses are represeted wth squares for the BLUE estmate, damods for the Covarace Itersecto, ad thck le for the VBDF estmate d) Fgure 2: Fuso of Moto Data. a) Iput data represeted as ellpses wth 95% cofdece. b) Desty estmate correspodg to formula 11) c) Trajectores of the varable-badwdth mea shft ad ellpses correspodg to the fal covarace matrces. Thck le s used for the ellpse correspodg to the largest mode. d) Detal of the trajectores correspodg the largest mode the data.e., the locato of the VBDF estmate).