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1 6 Supplementar Materals 61 Proof of Theorem 31 Proof Let m Xt z 1:T : l m Xt X,z 1:t Wethenhave mxt z1:t ˆm HX Xt z 1:T mxt z1:t m HX Xt z 1:T + mxt z 1:T HX We consder each of the two terms n equaton (16 Let m Xt z 1:T : m Xt z 1:T m Xt z 1:T Forthefrstterm, m Xt z 1:T Snce ξ t HX m X t z 1:T,j1 m Xt X,z 1:t j ξ t ( X, X j ξ t ( X,dP Xt+1 z 1:T ( + ξ t (, dp Xt+1 z 1:T (dp Xt+1 z 1:T ( m Xt+1 z 1:T m Xt+1 z 1:T,ξ t mxt+1 z 1:T ξ t HX <, the frst term decas wth O p (l αt+1 For the second term, we have mxt z 1:T (m Xt X,z 1:t ˆm Xt X,z 1:t m Xt X,z 1:t,j1 j ξ t ( X, X j ˆm Xt+1 z 1:T ˆm Xt+1 z 1:T, ξ t ˆmXt+1 z 1:T ξ t HX Snce ˆm Xt+1 z 1:T HX m Xt+1 z 1:T HX <, the second term decas wth O p (l βt These results lead to the statement mxt z 1:T HX O p (l αt, where α t mn{α t+1,β t } (16 6 Epermental Settng & Vdeo: Trackng a Sngle Object (Eperment 1 State Space Model Settng: The target s state at tme t s descrbed b t ( t, t, ẋ t, ẏ t wth the object s poston ( t, t and the veloct (ẋ t, ẏ t n cartesan coordnates R The dscretzed dnamcs s epressed wth
![Yu Nshama, Amr Hossen Afsharnejad, Shunsuke Naruse Fgure 4: A supplementar vdeo Ths anmaton vsualzes the sequental update of kernel means of the nkbflter [1] and the nkb-er (Algorthm 1 for a test](/docs-images/96/127569252/images/2-0.jpg "sequence z 1:4 n the clutter problem The upper three fgures show the sequental update of kernel means m Xt z 1:t (t 1 : 4 of the nkb-flter The lower three fgures show the estmated ng kernel means m")
2 Yu Nshama, Amr Hossen Afsharnejad, Shunsuke Naruse Fgure 4: A supplementar vdeo Ths anmaton vsualzes the sequental update of kernel means of the nkbflter [1] and the nkb-er (Algorthm 1 for a test sequence z 1:4 n the clutter problem The upper three fgures show the sequental update of kernel means m Xt z 1:t (t 1 : 4 of the nkb-flter The lower three fgures show the estmated ng kernel means m Xt z 1:4 (t 1:4ofthenKB-er Foreach,the left fgure shows the kernel mean projected to state, the mddle fgure shows the kernel mean projected to state, and the rght fgure both Each fgure vsualzes the followng (Left four fgures The black dot vertcal lne shows the true target s state (, The magenta dot vertcal lne shows the (cluttered observaton (, ỹ The kernel mean weghts are shown wth left vertcal as The postve (negatve weght values are vsualzed wth blue (red bars, respectvel The can curve shows the estmated kernel mean (estmated RKHS functon m P ( as a functon of ( wth rght vertcal as The blue dot n the top of the mountan shows the result of the mode estmaton for the target s state (, wth the objectve functon value From the two mddle fgures, t can be that the flterng estmaton s bmodal for uncertant, but ng estmaton correctl dentfes the state b usng the future measurements z 11:4, so that the blue dot s on the black dot vertcal lne a tme-nvarant lnear equaton: t+1 A t + q t, A : 1 t 1 t 1 1 where q t s dscrete Gaussan whte process nose havng moments E[q t ], E[q t q t ] : t 3 /3 t / t 3 /3 t / t / t t / t, (17 wth q> The measurement process for the target s a mture model: p(z t t (1 ρn(z t H t,r+ρ 1 S, (18 q
3 num sample 96 (tran1, test1 flter num sample 96 (tran1, test flter flter flter 1 tme tme num sample 119 (tran, test1 flter num sample 119 (tran, test flter flter flter 1 1 tme 1 1 tme num sample 1434 (tran3, test1 flter num sample 1434 (tran3, test flter flter flter 1 1 tme 1 1 tme num sample 1673 (tran4, test1 flter num sample 1673 (tran4, test flter flter flter 1 1 tme 1 1 tme Fgure : Performance of the nkb-flter and the nkb-er n dfferent tranng and test data on the clutter problem Ths fgure shows 8 (4 epermental results The upper-left two fgures show the performance on the dmenson and when the tranng sample sze s n 96, respectvel The lower fgures show the results when the tranng sample sze s ncreased to n 96, 119, 1434, 1673 It s that the performance s ncreased The rght eght fgures show results on dfferent test data where 1 ρ and ρ are probabltes of measurements from the actual target and clutter, respectvel The measurement from the actual target s a Gaussan N(z t H t,r wth the measurement model matr H and
4 Yu Nshama, Amr Hossen Afsharnejad, Shunsuke Naruse nose covarance matr R The measurement from the clutter s unform on the area S Weusedthesame parameter settng as the RBMCDA s demo used, e, the sze of tme step t 1, q 1, ρ 1/, S [, ] [ 4, 4], and H ( 1 1 (,R nkb-er settng: We used Gaussan kernels k X ( 1, e 1 /σ X and kz (z 1, z e z1 z /σ Z for target s states and measurements, respectvel, where σx σ Z 1 We set regularzaton constants ϵ n δ n ϵ n δ n 1 Note ϵ n and δ n are new regularzaton constants ntroduced for KB-er Asupplementarvdeo:We present an anmaton whch shows results of the nkb-flter [1] and the nkber (Algorthm 1 n the clutter problem Please see a supplementar move fle (mov Fgure 4 presents a snapshot of the anmaton at tme step t 111 Supplementar results: Fgure shows other results n dfferent tranng and test data on the clutter problem 63 Margnal Kernel Mean Computaton on Tree Graphs In ths secton, we present margnal kernel mean computaton on general tree graphs b usng the nkb-flter and the nkb-er, as the etenson of state space models 631 The nkb-flter & nkb-er on N Branch Cases For ease of understandng, we begn wth the two branch case shown n Fgure 6 (left Let : ( 1:T, t+1: T be hdden varables and z : (z 1:T, z t+1: T be measurement varables The jont probablt denst functon (pdf p(, z of Fgure 6 (left s gven b 11 ( T 1 p(, z ( T p( +1 p(z T 1 p( +1 t T t+1 p( z, where p( 1 : p( 1 and t : t For ease of presentaton, we assume that the transton process {p( +1 } T 1 and {p( +1 } T 1 t follow the same condtonal pdf p( We also assume that the measurement process {p(z } T and {p( z } T t+1 follow the same condtonal pdf p(z It s not dffcult to etend ths to general nhomogenous cases, f there s a tranng sample for learnng each of them We assume that there are tranng data { X, X }l and {X,Z } n for p( andp(z, respectvel The objectve here s to compute the kernel means {m Xτ z} T τ 1 and {m X τ z} T τ t+1 of condtonal dstrbutons {p( τ z} T τ 1 and {p( τ z} T τ t+1 gven measurements z, respectvel We begn wth gvng an order to the two branches Wlog, we set ( t+1:t,z t+1:t > ( t+1: T, z t+1: T We have outputs of the nkb-flter and the nkb-er on chan ( 1:T,z 1:T as ˆm Xt z 1:t ˆm Xt z 1:T n α (t k X (,X, t 1,,T, w (t k X (, X, t 1,,T 1 11 For smplct, we omtted llustratons of observable varables z n Fgure 6
![or Song et al [] to](/docs-images/96/127569252/images/5-13.jpg "ˆm Xt z 1:T,wehave")
5 Flterng Smoothng 1 t 1 (1 ( (1 (9 t+11 ( (6 (7 (3 (4 t+1 T (8 T 1 1 (1 (1 T (1 (4 ( (3 ( T ( ( (6 (3 t T (3 (N+ (7 (N+1 (N (N T ( N 1 1 (17 (1 ( 11 (18 (3 ( (11 ( 9 (13 8 (7 6 6 Fgure 6: Margnal kernel mean computaton on tree graphs usng the nkb-flter and the nkb-er; (left the smple two branch case, (mddle general N branch case, and (rght a tree eample B applng the nonparametrc kernel sum rule 1 (Secton or Song et al [] to ˆm Xt z 1:T,wehave ˆm Xt+1 z 1:T Û X t+1 X t ˆm Xt z 1:T η (t+1 k X (, X, where Û X t+1 X t s the nksr operator to obtan the estmator ˆm Xt+1 z 1:T Net, we appl the KB-flter to the other chan ( t+1: T, z t+1: T wth the ntal belef ˆm Xt+1 z 1:T,sothattheoutputsare n ˆm Xτ z 1:T, z t+1:τ ᾱ (τ k X (,X, τ t +1,, T Then, we appl the nkb-er to the chan ( 1:t,z 1:t ( t+1: T, z t+1: T backward wth the ntal kernel mean m X T z,sothattheoutputsare ˆm Xτ z ˆm Xτ z w (τ k X (, X τ t +1,, T 1 w (τ k X (, X τ 1,,t The numbers wrtten n Fgure 6 (left show the order of nference of KB-flter and KB-er B nducton, the same apples to the N branch case n Fgure 6 (mddle Frst, gve an order to the N branches Then, appl KB-flter and KB-er to one branch b one branch As an eample, Fgure 6 (rght shows the order of KB-flter and KB-er n a tree graph Thus, the margnal kernel mean computaton on a general tree graph s obtaned 1 B the Markov propert, the condtonal pdf p( t+1 z 1:T hasthesumruleepresson: p( t+1 z 1:T p(, z t+1: T z 1:T δ( t+1 t+1d z t+1: T d p( t+1 tp( t z 1:T d t, where δ( t+1 t+1 sthedrac sdeltafuncton
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