Autonomous State Space Models for Recursive Signal Estimation Beyond Least Squares

Transcription

1 Autonomous State Space Modes for Recursve Sgna Estmaton Beyond Least Suares Nour Zama, Reto A Wdhaber, and Hans-Andrea Loeger ETH Zurch, Dept of Informaton Technoogy & Eectrca Engneerng ETH Zurch & Bern Unversty of Apped Scences, Be, Swtzerand {zama, wdhaber, oeger}@seeethzch Abstract The paper addresses the probem of fttng, at any gven tme, a parameterzed sgna generated by an autonomous near state space mode (LSSM to dscrete-tme observatons When the cost functon s the suared error, the fttng can be accompshed based on effcent recursons In ths paper, the suared error cost s generazed to more advanced cost functons whe preservng recursve computatons: frst, the standard sampe-wse suared error s augmented wth a sampedependent poynoma error; second, the sampe-wse errors are ocazed by a wndow functon that s tsef descrbed by an autonomous LSSM It s further demonstrated how such a sgna estmaton can be extended to hande unnown addtve and/or mutpcatve nterference A these resuts rey on two facts: frst, the correaton functon between a gven dscrete-tme sgna and a LSSM sgna can be computed by effcent recursons; second, the set of LSSM sgnas s a rng I INTRODUCTION Fttng a parameterzed sgna to dscrete-tme measurements s a very cassca probem About two hundred years ago, Gauss nvented both the east-suares method and ts recursve verson [], and successfuy apped t to predct the orbt of the newy dscovered asterod Ceres Recursve east suares (whch may be consdered as a speca case of Kaman fterng [2], [3], contnues to be a ey agorthm n dgta sgna processng However, the assumptons of nearty and of uadratc costs (or, euvaenty, Gaussan nose are not sutabe for some appcatons, whch has motvated nonnear fters such as the extended Kaman fter (EKF, the unscented Kaman fter (UKF [4], partce fters, and exact recursve fters [5], [6] In ths paper, we focus on recursve sgna estmaton rather than fterng We consder parameterzed sgnas that are generated by an autonomous near state space mode (LSSM wth unnown nta state Such LSSM sgnas are hghy versate for modeng, and are of great practca use by vrtue of two charmng propertes (see Sec II: frst, the correaton functon between any such sgna and any gven dscrete-tme sgna can be computed by effcent recursons; second, the eement-wse mutpcaton of two LSSM sgnas s agan a LSSM sgna These two propertes are smpe but yet extremey vauabe Indeed, n Sec III, we ntroduce a genera cost functon that can st be recursvey computed Ths cost s obtaned by repacng the standard sampe-wse suared error by any sampe-dependent poynoma cost and by weghtng sampe-wse errors wth a LSSM wndow In Sec IV, we agan expot those two propertes to hande sgna estmaton n the presence of an unnown addtve and/or mutpcatve nterference that can be we modeed wth a LSSM Fnay, n Sec V, we present two ustratve appcatons of recursve sgna estmaton beyond east suares II DEFINITION AND PROPERTIES OF LSSM SIGNALS Defnton (LSSM sgna: A dscrete-tme sgna s j R, j Z, s a LSSM sgna (e, generated by a two-sded autonomous LSSM f and ony f there exsts C R n, A R n n, x R n, and C r R nr, A r R nr nr, x r R nr, for some, n, n r N such that { C A j s j x for j ( C r A j r x r for j > The changng pont of ths two-sded mode s defned to be at tme j However, when performng sgna estmaton, ths s not a restrcton snce such LSSM sgna w be shfted by a tme of nterest The sgna mode can aso be made eft-sded or rght-sded by settng x r or x The set of LSSM sgnas, whch s a vector space, conssts of near combnatons of (two-sded exponentas, poynomas, snusods, fnte-ength sgnas, and products of those The LSSM parameters {C, A, x} have to be understood as {C, A, x } {C r, A r, x r } In the foowng, the parameters {C, A, C r, A r } are assumed to be nown whe the states x {x, x r } are to be estmated Thus, a LSSM sgna s j (x, j Z, as n ( s a functon of x A Inner Product wth LSSM Sgnas Gven parameters {C, A} and any dscrete-tme sgna y (y,, y K R K of duraton K N, we defne the uantty [ ξ ] ξ (y, C, A (y, C, A R n, (2 ξ (y, C r, A r wth n n + n r and ξ (y, C, A ξ (y, C r, A r (A T C T y R n (3 + (A T r C T r y R nr, (4 ISBN EURASIP 27 36

2 for {,, K} and ξ (y, C, A, otherwse Note that ξ (y, C, A s a near functon of y and can be nterpreted as the output of n near fters The uantty (2 s effcenty computed for a {,, K} usng the forward recurson ξ (y, C, A A T ξ (y, C, A + C T y, (5 ntazed wth ξ (y, C, A and the bacward recurson ξ (y, C r, A r A T ( r ξ + (y, C r, A r + Cr T y +, (6 ntazed wth ξ K (y, C r, A r Proposton (Inner Product wth a LSSM sgna: The nner product between y and a LSSM sgna s(x, for any x, as n ( shfted by a tme and denoted by s (x s y, s (x x T ξ (y, C, A, (7 wth the conventon that x T [x T, xt r ] Proof of Proposton : Ths proposton foows from y, s (x y s (x (8 C A x y + + C r A r x r y (9 x T ξ (y, C, A + x T r ξ (y, C r, A r ( x T ξ (y, C, A ( Proposton has severa mportant conseuences Frst, computng the correaton functon between a sgna of ength K and a LSSM sgna s(x has a compexty of O(Kn 2, wth n max(n, n r Secondy, the nner product between y and s (x can be expressed as a standard nner product n R n between x and ξ (y, C, A Fnay, snce ξ (y, C, A s ndependent of x, the computatona effort to obtan the nner product between y and s (x for any x, s of O(n ony, after havng computed ξ (y, C, A B Eement-wse Product of LSSM sgnas Proposton 2 (Product of LSSM sgnas: Let s ( j and s (2 j, j Z, be two LSSM sgnas wth respectve parameters {C, A, x } and {C 2, A 2, x 2 } Then, s ( j s (2 j, j Z, s aso a LSSM sgna wth parameters {C C 2, A A 2, x x 2 } To eep the notaton concse, C C 2 (and anaogousy for A A 2 and x x 2 means that the Kronecer product s apped ndependenty for the eft-sded part (C, C,2 and for the rght-sded part (C r, C r,2 We use ths conventon a aong Proof of Proposton 2: For j >, we have s ( j s (2 j (C r, A j r, x r,(c r,2 A j r,2 x r,2 (2 (C r, A j r, x r, (C r,2 A j r,2 x r,2 (3 (C r, C r,2 (A r, A r,2 j (x r, x r,2 (4 An anaog reaton hods for j wth the eft-sded parameters, whch then concudes the proof Aong wth the fact that the constant sgna s a LSSM sgna (generated wth C A x C r A r x r and denoted by C A x, ths proposton mpes that the set of LSSM sgnas s a rng Ths property w be extremey usefu n ths paper III RECURSIVE SIGNAL ESTIMATION BEYOND LEAST SQUARES Let y (y,, y K R K be dscrete-tme observatons of duraton K N At any gven tme ndex {,, K}, we wsh to ft a LSSM sgna s(x wth parameters {C, A, x} and unnown x to the observatons It s we-nown that the suared error functon ( J (x y s (x 2 (5 can be computed effcenty wth recursons as n Kaman fterng Indeed, usng the functon n (2 and Propostons & 2, we have J (x ξ (y 2,, 2x T ξ (y, C, A +(x x T ξ (y, C C, A A, (6 where y p denotes the sgna y rased eement-wse to the power of p N We now generaze the suared cost n (5 n two dfferent ways whe preservng recursve cost computatons, and thus, computatona effcency A Tme-Dependent Poynoma Cost Assume that each observaton y, {,, K}, comes wth ts own poynoma cost P of maxmum degree d N and coeffcents j R, j {,, d} Suppose we wsh to perform sgna estmaton at any tme by mnmzng the cost functon ( J (x P y s (x (7 The poynomas P w normay be chosen such that P (u, for a u R, but ths s actuay not a restrcton An mportant speca case of (7 conssts of poynomas P ndependent of (e, P P, for a, whch eads to J (x P ( y s (x (8 When P (u u 2, the cost (8 becomes the one of (5 It turns out that the cost functon (7 can be recursvey and effcenty computed thans to the reaton J (x ( ( x T ξ (ỹ (, C, A, (9 wth ỹ ( R K such that ỹ ( j ( j j yj, (2 ISBN EURASIP

3 for {,, K}, {,, d}, and wth C C C, (2 }{{} tmes for and C The proof of (9 foows from the proof of (23, whch s gven n the next secton In partcuar, (9 concdes wth (6 when P (u u 2, for a B LSSM-Wndowed and Tme-Dependent Poynoma Cost Often, we further want to mt the horzon of the actua sgna estmaton For that purpose, we ocaze the cost functon (7 usng a LSSM wndow w j, j Z, centered at tme ndex Thus, gven the poynoma costs P, {,, K}, of maxmum degree d N, we wsh to perform sgna estmaton at any tme by mnmzng the wndowed cost functon J (x ( w P y s (x, (22 where w j, j Z, s a LSSM sgna wth fxed parameters {C w, A w, x w } For nstance, when C w x w and A,w A r,w γ for some γ (,, the cost J (x s computed on a symmetrc exponentay-decayng wndow centered at tme Another exampe s a fnte-ength wndow, whch s obtaned by choosng A,w and A r,w to be npotent Note that (7 s a speca case of (22 wth C w A w x w (e, a constant nfnte-ength wndow The cost (22 s recursvey and effcenty computed thans to the reaton J (x ( (( x x w T ξ (, (23 wth, for {,, d}, ξ ( ξ (ỹ (, ( C C w, ( A A w, (24 where ỹ ( s defned n (2 The uantty ξ ( s a near functon of ỹ ( but no onger of y A graphca representaton of formua (23 s gven n Fg Ths formua foows from J (x w j j j ( ( ( j y s (x j ( j (25 j yj w (s (x (26 j yj w (s (x (27 ỹ ( w (s (x, (28 from whch we deduce (23 usng Proposton snce for any {,, d}, w (s (x, Z, s a LSSM sgna shfted by a tme and wth LSSM parameters {( C C w, ( A A w, ( x x w } accordng to Proposton 2 The formua (23 aso proves that {ξ ( : {,, d}} s a fnte-dmensona suffcent statstc for x The compexty of computng J, for a {,, K}, s ony of O(Kn 2d n 2 w (wth n max(n, n r and n w max(n,w, n r,w, whch bascay corresponds to the compexty of computng ξ (d In partcuar, whether the poynomas P are tme-dependent or not, the computatona compexty remans of the same order Note aso the suared dependency of the compexty wth respect to the order of the LSSM wndow The cost functon (23 s a mutvarate poynoma n x Thus, ts mnmzaton can be done usng exact agebrac methods such as Gröbner bases or usng a reaxaton method such as a sum of suare formuaton soved by semdefnte programmng [7] IV SIGNAL ESTIMATION IN THE PRESENCE OF INTERFERENCES In severa practca appcatons, sgnas of nterest are atered by some nterference sgna, whch needs to be taen nto consderaton n the sgna estmaton probem For that purpose, n addton to modeng a sgna of nterest wth a LSSM wth parameters {C, A, x} wth unnown x, we aso mode nterferences wth a LSSM sgna g j (x g, j Z, wth parameters {C g, A g, x g } wth unnown x g In the foowng, we propose three cost functons J (x, x g based on (22, whch hande nterference sgnas n dfferent ways Actuay, except computatona compexty, nothng prevents from combnng these three ways of deang wth nterferences A Addtve Interference When the nterference sgna s addtve, sgna estmaton at any tme can be done by mnmzng the cost functon J (x, x g ( w P y (g (x g + s (x (29 Snce g j (x g + s j (x, j Z, s aso a LSSM sgna obtaned by stacng (n a sutabe manner the LSSM parameters {C, A, x} wth {C g, A g, x g }, the reaton (23 st appes to compute J (x, x g by repacng the parameters C and A n (24 wth the parameters of the staced mode B Mutpcatve Interference In case of mutpcatve nterference, sgna estmaton at any tme can be done by mnmzng the cost functon J (x, x g ( w P y g (x g s (x (3 Snce g j (x g s j (x, j Z, s aso a LSSM sgna (accordng to Proposton 2 wth LSSM parameters {C C g, A A g, x x g }, the reaton (23 st appes to compute J (x, x g wth the substtuton C C C g, A A A g, and x x x g ISBN EURASIP

4 y y y y d ỹ ( ỹ ( ỹ (d {( C C w, ( A A w } {( C C w, ( A A w } {( d C C w, ( d A A w } ξ ( ξ ( ξ (d (( x x w T (( x x w T (( d x x w T + J (x Fg Graphca representaton of the cost computaton accordng to (23 and usng (2, (24, and (2 C Inverse Mutpcatve Interference In some cases, a mutpcatve nterference s better modeed as the (eement-wse nverse of a LSSM sgna (e, /g j (x g, j Z rather than a LSSM sgna drecty Unfortunatey, the eement-wse nverse of a LSSM sgna s not n genera a LSSM sgna However, nstead of mutpyng the sgna of nterest wth the nterference sgna n the cost functon, an aternatve s to mutpy the observatons wth the eement-wse nverse nterference Thus, sgna estmaton at any tme can be done by mnmzng the cost functon ( J (x, x g w P g (x g y s (x (3 Agan, J (x, x g can st be recursvey computed Indeed, usng smar expansons as n (25 and (26, we have j J (x ( j j j yj (g (x g j w (s (x (32 j ( ( x (,j T (ỹ(,j ξ, C (,j, Ã(,j, (33 wth ỹ (,j R K and such that for (, j {,, d} 2, j, ( ỹ (,j j j, {,, K} (34 C (,j ( C ( j C g C w (35 Ã (,j ( A ( j A g A w (36 x (,j ( x ( j x g x w (37 Once more, usng Proposton 2, those parameters foow from the fact that (g (x g j w (s (x, Z, (cf (32 s a LSSM sgna shfted by a tme and wth parameters { C (,j, Ã(,j, x (,j } V EXPERIMENTAL RESULTS A Detecton of a Moduated Sgna We want to detect puses of snusoda shape of freuency Ω R + n an amptude-moduated carrer sgna of freuency Ω g R + and bured wth addtve whte Gaussan nose A typca observed sgna s dspayed n Fg 2, upper pot y w g (ˆx g g (ˆx gs (ˆx Tme ndex LLR Fg 2 Synthetc exampe of amptude-moduated puse detecton For ω R, we denote [ ] cos ω sn ω R(ω R 2 2 (38 sn ω cos ω The carrer sgna s seen as an nterference sgna g(x g wth LSSM parameters C r,g C,g [, ], A r,g A,g R(Ω g, and unnown x g R 2 R 2 The sgna of nterest s(x, consstng of puses of snusoda shape, s generated wth the LSSM parameters C r C [, ], A r A R(Ω, and unnown x R 2 R 2 For a tme ndces, we recursvey compute a cost J (x, x g as n (3 usng P (u u 2, for a (e, standard suared error, but wth a two-sded exponenta wndow wth parameters C w x w and A,w A r,w γ for some γ (, In Fg 2, we ustrate the resuts of our sgna estmaton method In the ower pot, we dspay the og-ehood rato LLR mn J (x, x g 2 n x,x g, (39 J (x, x g whch ndcates how ey the presence of a sgna of nterest s In the mdde part of Fg 2, we pot the estmated sgna obtaned at ndex 35 where LLR s maxmum Note the actua separaton of the carrer sgna from the sgna of nterest ISBN EURASIP

5 [mv] [mm] m9 m8 m7 m6 m5 m4 m3 m2 m 5 Tme [s] u (m ˆr s (ˆx Fg 3 Mut-channe esophagea ECG sgna (bue nes, catheter dspacement estmate (green dots and estmated LSSM sgna (bac dashed ne B Estmaton of a Catheter Movement Une surface eectrocardogram (ECG recordngs, esophagea ECG recordngs, obtaned usng eectrodes paced n the esophagus, are not commony used However, snce the esophagus s much coser to the heart than the chest surface, esophagea ECG recordngs contan vauabe nformaton provded that we can actuay extract t In such recordngs the catheter contanng the eectrodes typcay moves due to, among others, the patent s breathng Gven a M-channe esophagea ECG recordng, we want to estmate the reatve vertca movement of the catheter n the esophagus The ey dea to expot s that when the catheter sowy moves, the sgna shape produced by a heart beat n a gven channe s ute smar to the sgna shape produced by the prevous heart beat n another channe [8] We consder a catheter that hods M + rng-shaped eectrodes whch are ocated at dstances d < d < < d M from the catheter tp For m {,, M}, et u (m n R, n {,, N} be the votage measured between eectrodes m and m Wthn these N sampes, we observe K + heart beats wth correspondng R peas at tme ndces N and correspondng unnown catheter postons r R, {,, K} The frst beat (e, for s consdered as reference beat wth r Each beat effectvey produces sgna shapes n the nterva { + a,, + b}, (a, b Z 2, a < b In order to compare sgna shapes produced by the th heart beat wth the ones produced by the reference beat, we defne the cost functon b dm P (r (ϕ +n(z r ϕ +n(z 2 dz, (4 d na whch depends on the dspacement r and where ϕ n (z, z [d, d M ], s a rea poynoma of degree Q N, whch nterpoates the votage measurements (u ( n,, u (M n across channes at tme ndex n It foows that P (r (potted n Fg 4 s aso a poynoma n r of degree 2Q Fg 4 Seecton of poynomas P (r for few ndces Then, we mode the dspacement of the catheter wth a LSSM sgna s(x wth parameters C and A Fnay, we perform sgna estmaton by mnmzng the cost functon J (x w P (s (x, (4 where w corresponds to an exponenta wndow, as an exampe of (22 Fg 3 shows a snpped of a mut-channe esophagea ECG sgna (M 9 and the estmated catheter poston For ths exampe, we chose Q 5, C r C [, ], and A r A R(ω Note that the catheter poston shows a perodc movement, reveang the subject s breathng actvty VI CONCLUSION We have ntroduced a genera cost functon, whch ncudes sampe-dependent poynoma costs aong wth wndow weghts generated by a LSSM, to ft a parameterzed LSSM sgna to dscrete-tme measurements We have shown how ths cost functon can be recursvey computed and can hande unnown addtve and/or mutpcatve nterferences n a sgna estmaton probem The two propertes we used are that the correaton functon between any gven dscrete-tme sgna and a LSSM sgna s recursvey computed and the set of LSSM sgnas s a rng We have aso presented two appcatons, whch, however, hardy suffce to ustrate the versatty of the proposed approach REFERENCES [] C-F Gauss, Theora combnatons observatonum errorbus mnms obnoxae Henrcus Deterch, Gottngae, 823 [2] S Hayn, Adaptve fter theory Pearson Educaton Inda, 28 [3] H-A Loeger, J Dauwes, J Hu, S Kor, L Png, and F R Kschschang, The factor graph approach to mode-based sgna processng, Proceedngs of the IEEE, vo 95, no 6, pp , 27 [4] S J Juer and J K Uhmann, Unscented fterng and nonnear estmaton, Proceedngs of the IEEE, vo 92, no 3, pp 4 422, 24 [5] F Daum, Nonnear fters: beyond the Kaman fter, IEEE Aerospace and Eectronc Systems Magazne, vo 2, no 8, pp 57 69, 25 [6] V Beneš, Exact fnte-dmensona fters for certan dffusons wth nonnear drft, Stochastcs: An Internatona Journa of Probabty and Stochastc Processes, vo 5, no -2, pp 65 92, 98 [7] P A Parro and B Sturmfes, Mnmzng poynoma functons, Agorthmc and uanttatve rea agebrac geometry, DIMACS Seres n Dscrete Mathematcs and Theoretca Computer Scence, vo 6, pp 83 99, 23 [8] D Bruegger, 3D Reconstructon and Smuaton of Heart Potentas n the Esophagus, Master s thess, Bern Unversty of Apped Scences, Be, Swtzerand, 26 ISBN EURASIP