New design of estimators using covariance information with uncertain observations in linear discrete-time systems

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1 Applied Mathematics and Computation 135 (2003) New design of estimators using covariance information with uncertain observations in linear discrete-time systems Seiichi Nakamori a, *, Raquel Caballero- Aguila b, Aurora Hermoso-Carazo c, Josefa Linares-Perez c a Faculty of Education, Department of Technology, Kagoshima University, Kohrimoto, Kagoshima , Japan b Departamento de Estadıstica e Investigacion Operativa, Universidad de Jaen, Campus Las Lagunillas, s/n, Jaen, Spain c Departamento de Estadıstica e Investigacion Operativa, Universidad de Granada, Campus Fuentenueva, s/n, Granada, Spain Abstract This paper proposes recursive least-squares (RLS) filtering and fixed-point smoothing algorithms with uncertain observations in linear discrete-time stochastic systems. The estimators require the information of the auto-covariance function in the semi-degenerate kernel form, the variance of white observation noise, the observed value and the probability that the signal exists in the observed value. The autocovariance function of the signal is factorized in terms of the observation vector, the system matrix and the cross-variance function of the state variable, that generates the signal, with the signal. These quantities are obtained from the auto-covariance data of the signal. It is shown that the semi-degenerate kernel is expressed in terms of these quantities. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Wiener Hopf equation; Linear discrete-time systems; Recursive estimation; Covariance information; Stochastic process * Corresponding author. address: nakamori@edu.kagoshima-u.ac.jp (S. Nakamori) /02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S (02)

2 430 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) Introduction The estimation problem given the uncertain observations has been seen as an important research in the area of the detection and estimation problems for communication systems. Nahi [1] proposes the recursive least-squares (RLS) estimation technique for the state variable of dynamical systems with the uncertain observations. By the uncertain observations we mean that some observations do not contain the signal and consist of observation noise. The probability for the existence of the signal in the observation is available. This paper, using the covariance information, considers an alternative estimation method for the signal with the uncertain observations in linear discrete-time stochastic systems. The estimation technique using the covariance information has been studied in the context of the detection and estimation of the signal [2]. The recursive Wiener filter [3,4] using the covariance information is devised in linear stochastic systems. Also, the estimation technique with the uncertain observations by use of the covariance information is proposed. However, in [5], the stochastic property for the uncertain probability is not clearly taken into account. Hence, this paper proposes the RLS filtering and fixed-point smoothing algorithms precisely based on the same stochastic assumptions for the uncertain probability with those in [5]. The estimators require the information of the auto-covariance function in the semi-degenerate kernel form, the variance of white observation noise, the observed value and the probability that the signal exists in the observed value. The auto-covariance function of the signal is factorized in terms of the observation vector, the system matrix and the cross-variance function of the state variable, that generates the signal, with the signal. These quantities are obtained from the autocovariance data of the signal. It is shown that the semi-degenerate kernel is expressed in terms of these quantities. The algorithms are derived based on the invariant imbedding method [5]. 2. Estimation problem with uncertain observations Let an observation equation be given by yðkþ ¼U ðkþzðkþþvðkþ; ð1þ where zðkþ is a scalar signal at time k, UðkÞ is a sequence of independent Bernoulli variables, and vðkþ is white noise. It is assumed that the signal process fzðkþg and the noises fu ðkþg, fvðkþg are mutually independent. It is assumed that zðkþ and vðkþ are zero mean and the variance of vðkþ is RðkÞ: E½vðkÞvðsÞŠ ¼ RðkÞd K ðk sþ: ð2þ

3 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) Here, d K ðþ denotes the Kronecker d function. For UðkÞ ¼0 in (1), the observed value consists of the observation noise alone [1]. Let pðkþ represent the probability that the observation at time k contains the signal: pðkþ ¼PrfU ðkþ ¼1g: ð3þ Hence, E½U ðkþš ¼ pðkþ: ð4þ Also, the following stochastic properties are assumed [1]: E½UðkÞUðsÞŠ ¼ pðkþpðsþ; k 6¼ s; E½U 2 ðkþš ¼ pðkþ: ð5þ Let the fixed-point smoothing estimate ^zðk; LÞ of zðkþ at the fixed point k be given by ^zðk; LÞ ¼ XL hðk; i; LÞyðiÞ ð6þ as a linear transform of the observation set fyðiþ; 1 6 i 6 Lg, where hðk; s; LÞ is referred to as an impulse response function. Let us consider the linear least-squares estimation problem which minimizes the cost function J ¼ E½kzðkÞ ^zðk; LÞk 2 Š: ð7þ Minimizing J, we obtain the Wiener Hopf equation E½zðkÞyðsÞŠ ¼ XL hðk; i; LÞE½yðiÞyðsÞŠ ð8þ by an orthogonal projection lemma [6] zðkþ XL hðk; i; LÞyðiÞ?yðsÞ; 0 6 i; s; k 6 L: ð9þ Here, Ô?Õ denotes the notation of the orthogonality. Substituting (1) into (8), and using the statistical properties (2) (5), we have hðk; s; LÞW ðsþ ¼pðsÞKðk; sþ XL hðk; i; LÞpðiÞKði; sþpðsþ; W ðsþ ¼RðsÞþpðsÞKðs; sþ p 2 ðsþkðs; sþ: ð10þ

4 432 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) Here, Kðk; sþ denotes the cross-covariance function of zðkþ. Let the auto-covariance function Kðk; sþ of the signal zðkþ be expressed by the semi-degenerate kernel form Kðk; sþ ¼ AðkÞBT ðsþ; 0 6 s 6 k; BðkÞA T ð11þ ðsþ; 0 6 k 6 s: Here, AðkÞ and BðsÞ are 1 m vector functions. Let us introduce the system matrix U in the state-space model for the state variable xðkþ of dimension n. By introducing the 1 n observation vector H with zðkþ ¼HxðkÞ and the cross-variance function K xz ðs; sþ of xðsþ with zðsþ, the auto-covariance function Kðk; sþ of the signal zðkþ is factorized as Kðk; sþ ¼ HUk s K xz ðs; sþ; 0 6 s 6 k; Kxz Tðk; kþðut Þ s k H T ð12þ ; 0 6 k 6 s: In Theorem 1, under these assumptions, the RLS algorithms for the filtering and fixed-point smoothing estimates of the signal zðkþ are proposed. 3. RLS algorithms for filtering and fixed-point smoothing estimates The filtering and fixed-point smoothing algorithms are derived, starting with (10), based on the invariant imbedding method [5]. Theorem 1. Let the probability for UðLÞ ¼1 be pðlþ in the observation Eq. (1). Let Kðk; sþ represent the auto-covariance function of the signal zðkþ. Let RðLÞ represent the variance of the white observation noise. Then the RLS algorithms for the filtering and fixed-point smoothing estimates of the signal zðkþ consist of (13) (20). Fixed-point smoothing estimate of the signal zðkþ, ^zðk; LÞ: ^zðk; LÞ ¼^zðk; L 1Þþhðk; L; LÞðyðLÞ pðlþ^zðl; L 1ÞÞ: ð13þ One-step ahead prediction estimate of zðlþ :^zðl; L 1Þ ^zðl; L 1Þ ¼AðLÞOðL 1Þ: ð14þ Smoother gain: hðk; L; LÞ hðk; L; LÞ ¼ðBðkÞ Sðk; L 1ÞÞA T ðlþpðlþ½rðlþþpðlþðaðlþb T ðlþ pðlþaðlþrðl 1ÞA T ðlþþš 1 ; Sðk; LÞ ¼Sðk; L 1Þþhðk; L; LÞpðLÞðBðLÞ AðLÞrðL 1ÞÞ; SðL; LÞ ¼AðLÞrðLÞ; ð15þ ð16þ

5 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) JðL; LÞ ¼ðB T ðlþpðlþ rðl 1ÞA T ðlþpðlþþ½rðlþþpðlþðaðlþb T ðlþ pðlþaðlþrðl 1ÞA T ðlþþš 1 ; ð17þ rðlþ ¼rðL 1ÞþJðL; LÞpðLÞðBðLÞ AðLÞrðL 1ÞÞ; rð0þ ¼0; ð18þ OðLÞ ¼OðL 1ÞþJðL; LÞðyðLÞ pðlþaðlþoðl 1ÞÞ; Oð0Þ ¼0: ð19þ Filtering estimate of zðlþ :^zðl; LÞ ^zðl; LÞ ¼AðLÞOðLÞ: ð20þ Innovations process for the uncertain observations: yðlþ pðlþaðlþoðl 1Þ ð¼yðlþ pðlþ^zðl; L 1ÞÞ: Proof. If we subtract Eq. (10) for hðk; s; LÞ from the equation obtained by putting L! L 1 in (10), we have ðhðk; s;lþ hðk;s; L 1ÞÞW ðsþ ¼ hðk; L; LÞpðLÞKðL;sÞpðsÞ XL 1 ðhðk; i; LÞ hðk; i; L 1ÞÞpðiÞKði;sÞpðsÞ: Introducing a function Kðs; L 1Þ satisfying ð21þ Kðs; L 1ÞW ðsþ ¼KðL; sþpðsþ XL 1 Kði; L 1ÞpðiÞKði; sþpðsþ; ð22þ we obtain the difference equation for hðk; s; LÞ as hðk; s; LÞ hðk; s; L 1Þ ¼ hðk; L; LÞpðLÞKðs; L 1Þ ð23þ from (21) and (22). By noting KðL; sþ ¼AðLÞB T ðsþ; 0 6 s 6 L, from (11), we rewrite (22) as Kðs; L 1ÞW ðsþ ¼AðLÞB T ðsþpðsþ XL 1 Kði; L 1ÞpðiÞKði; sþpðsþ: ð24þ Introducing a function Jðs; L 1Þ satisfying Jðs; L 1ÞW ðsþ ¼B T ðsþpðsþ XL 1 Jði; L 1ÞpðiÞKði; sþpðsþ; ð25þ we obtain Kðs; L 1Þ ¼AðLÞJðs; L 1Þ ð26þ from (24) and (25).

6 434 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) Subtracting the equation obtained by putting L! L 1 in (25) from (25), we have ðjðs; L 1Þ Jðs; L 2ÞÞW ðsþ ¼ pðl 1ÞJðL 1; L 1ÞKðL 1; sþpðsþ XL 2 ðjði; L 1Þ Jði; L 2ÞÞpðiÞKði; sþpðsþ: From (22) and (27), we obtain the difference equation for Jðs; L 1Þ as Jðs; L 1Þ Jðs; L 2Þ ¼ JðL 1; L 1ÞpðL 1ÞKðs; L 2Þ: ð27þ ð28þ If we put s ¼ L 1 in (25), we have JðL 1; L 1ÞW ðl 1Þ ¼ B T ðl 1ÞpðL 1Þ XL 1 Jði; L 1ÞpðiÞKði; L 1ÞpðL 1Þ: ð29þ Since Kði; L 1Þ ¼BðiÞA T ðl 1Þ, 06 i 6 L 1, from (11), we rewrite (29) as JðL 1; L 1ÞW ðl 1Þ ¼ B T ðl 1ÞpðL 1Þ XL 1 Jði; L 1ÞpðiÞBðiÞA T ðl 1ÞpðL 1Þ: ð30þ Introducing a function we obtain rðl 1Þ ¼ XL 1 Jði; L 1ÞpðiÞBðiÞ; JðL 1; L 1ÞW ðl 1Þ ¼ B T ðl 1ÞpðL 1Þ rðl 1ÞA T ðl 1ÞpðL 1Þ: ð31þ ð32þ Subtracting the equation obtained by putting L! L 1 in (31) from (31), we have rðl 1Þ rðl 2Þ ¼JðL 1; L 1ÞpðL 1ÞBðL 1Þ þ XL 2 ðjði; L 1Þ Jði; L 2ÞÞpðiÞBðiÞ: Substituting (28) into (33) and using (26), we have ð33þ rðl 1Þ rðl 2Þ ¼JðL 1; L 1ÞpðL 1ÞBðL 1Þ JðL 1; L 1Þ pðl 1Þ XL 2 AðL 1ÞJði; L 2ÞpðiÞBðiÞ: ð34þ

7 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) From (34) with (31), we obtain (18) for updating rðlþ. The initial condition of rðlþ at L ¼ 0isrð0Þ ¼0 from (31). From (18) and (32), we obtain (17) after some manipulations. In (17) J ðl; LÞ is calculated from rðl 1Þ. If we put s ¼ L in (10), we have hðk; L; LÞW ðlþ ¼pðLÞKðk; LÞ XL hðk; i; LÞpðiÞKði; LÞpðLÞ: Using Kði; LÞ ¼BðiÞA T ðlþ, 06 i 6 L, from (11) and introducing a function ð35þ Sðk; LÞ ¼ XL hðk; i; LÞpðiÞBðiÞ; ð36þ we obtain hðk; L; LÞW ðlþ ¼ðBðkÞ Sðk; LÞÞA T ðlþpðlþ: ð37þ Subtracting the equation obtained by putting L! L 1 in (36) from (36), we have Sðk; LÞ Sðk; L 1Þ ¼hðk; L; LÞpðLÞBðLÞ þ XL 1 ðhðk; i; LÞ hðk; i; L 1ÞÞpðiÞBðiÞ: ð38þ Substituting (23) into (38) and using (26), we have Sðk; LÞ Sðk; L 1Þ ¼hðk; L; LÞpðLÞBðLÞ hðk; L; LÞpðLÞ XL 1 AðLÞJði; L 1ÞpðiÞBðiÞ: ð39þ From (31) and (39), we obtain the equation for updating Sðk; LÞ in (16). The value of Sðk; LÞ at k ¼ L is given by SðL; LÞ ¼ P L hðl; i; LÞpðiÞBðiÞ. Putting k ¼ L in (10), we have hðl; s; LÞW ðsþ ¼pðsÞKðL; sþ XL hðl; i; LÞpðiÞKði; sþpðsþ: ð40þ From the equation obtained by substituting KðL; sþ ¼AðLÞB T ðsþ,06 s 6 L, into (40) and (25), we obtain hðl; s; LÞ ¼AðLÞJ ðs; LÞ: ð41þ

8 436 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) Hence, from (31), (36) and (41), we obtain the initial condition SðL; LÞ in (16) as SðL; LÞ ¼ XL AðLÞJ ði; LÞpðiÞBðiÞ ¼AðLÞrðLÞ: ð42þ From (6), the filtering estimate of the signal zðlþ is given by ^zðl; LÞ ¼ XL hðl; i; LÞyðiÞ: ð43þ Substituting (41) into (43) and introducing a function OðLÞ ¼ XL J ði; LÞyðiÞ; ð44þ we obtain (20). Subtracting the equation obtained by putting L! L 1 in (44) from (44), we have OðLÞ OðL 1Þ ¼JðL; LÞyðLÞþ XL 1 ðjði; LÞ Jði; L 1ÞÞyðiÞ: ð45þ Using (26), (28) and (44), we obtain (19). The initial condition of OðLÞ at L ¼ 0 is given by Oð0Þ ¼0 from (44). Subtracting the equation obtained by putting L! L 1 in (6) from (6), we have ^zðk; LÞ ^zðk; L 1Þ ¼hðk; L; LÞyðLÞþ XL 1 ðhðk; i; LÞ hðk; i; L 1ÞÞyðiÞ: ð46þ Substituting (23) into (46) and using (26) and (44), we obtain (13) for updating ^zðk; LÞ. From (16) and (37), we obtain (15) after some manipulations. Theorem 1 proposes the RLS algorithms for the filtering estimate ^zðl; LÞ, the one-step ahead prediction estimate ^zðl; L 1Þ of the signal zðlþ and the fixedpoint smoothing estimate ^zðk; LÞ of the signal zðkþ at the fixed point k. Based on the filtering theory [6] in linear discrete-time stochastic systems, the innovations process for the uncertain observations is given by mðlþ ¼yðLÞ pðlþ^zðl; L 1Þ. It is noted that the variance of the innovations process is given by RðLÞþpðLÞðAðLÞB T ðlþ pðlþaðlþrðl 1ÞA T ðlþþ.

9 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) Factorization technique of auto-covariance function of signal [7] The auto-covariance function of the signal zðkþ is expressed by (12). In this section, the estimation method of H, U and K xz ðs; sþ from Kðk; sþ is shown. Here, the wide-sense stationarity for the covariance functions is assumed as K xz ðk; sþ ¼K xz ðk sþ and Kðk; sþ ¼Kðk sþ. Let H represent the Hankel matrix consisting of the auto-covariance data of the signal 2 3 Kð0Þ Kð1Þ Kð2Þ Kð1Þ Kð2Þ Kð3Þ Kð2Þ Kð3Þ Kð4Þ H ¼ : ð47þ The 1 n observation vector H, the n 1 cross-variance function K xz ðl; LÞ and the n n system matrix U are estimated as follows: H ¼ ½Kð0Þ Kð1Þ Kðn 1Þ Š 2 3 Kð0Þ Kð1Þ Kðn 1Þ Kð1Þ Kð0Þ Kðn 2Þ Kðn 2Þ Kð0Þ Kð1Þ 5 Kðn 1Þ Kðn 2Þ Kð0Þ 1 ; ð48þ K xz ðl; LÞ ¼½Kð0Þ Kð1Þ Kðn 2Þ Kðn 1Þ Š T ; ð49þ 2 3 Kð1Þ Kð0Þ Kðn 2Þ Kð2Þ Kð1Þ Kðn 3Þ U ¼ Kðn 1Þ Kð1Þ Kð0Þ 5 KðnÞ Kðn 1Þ Kð1Þ 2 3 Kð0Þ Kð1Þ Kðn 1Þ Kð1Þ Kð0Þ Kðn 2Þ Kðn 2Þ Kð0Þ Kð1Þ 5 Kðn 1Þ Kðn 2Þ Kð0Þ 1 : ð50þ

10 438 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) The necessary and sufficient condition, that the dimension of the state variable xðkþ is n, is that the rank of the Hankel matrix H is n. In Section 5, we apply the estimation technique in this section to an estimation problem of the signal generated by the second-order auto-regressive (AR) model when the uncertain observations are given. 5. A numerical simulation example Let the auto-covariance data KðmÞ of the signal zðlþ be given by Kð0Þ ¼r 2 ; KðmÞ ¼r 2 fa 1 ða 2 2 1Þam 1 =½ða 2 a 1 Þða 2 a 1 þ 1ÞŠ a 2 ða 2 1 1Þam 2 =½ða 2 a 1 Þða 1 a 2 þ 1ÞŠg; 0 < m; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 1 ; a 2 ¼ a 1 a 2 1 4a 2 2; ð51þ a 1 ¼ 0:1; a 2 ¼ 0:8; r ¼ 0:5: Fig. 1. Signal zðkþ and the filtering estimate ^zðk; kþ for white Gaussian observation noise Nð0; 0:5 2 Þ.

11 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) If we substitute KðiÞ; i ¼ 0; 1; 2, into (48) (50) for n ¼ 2, we obtain the observation vector H, the cross-variance function K xz ðl; LÞ and the system matrix U in the state-space model for the state variable xðkþ as follows: H ¼ ½1 0Š; K xz ðl; LÞ ¼ Kð0Þ Kð1Þ Kð0Þ ¼0:25; Kð1Þ ¼0:125: ; U ¼ By comparing (11) with (12), we find the relationships 0 1 a 2 a 1 ; ð52þ AðkÞ ¼HU k ; B T ðsþ ¼U s K xz ðs; sþ: ð53þ Let the probability be pðlþ ¼0:9. Substituting AðkÞ and BðsÞ with the probability pðlþ into the estimation algorithm of Theorem 1, we can calculate the filtering and fixed-point smoothing estimates of the signal. Due to the stochastic property of the Bernoulli random variable UðkÞ in (1), the individual observation consists of only the observation noise at k ¼ 4; 14; 19; 31; 40; 47; 61; 75; 85; 102; 108; 119; 121; 136; 139; 158; 168; 195; 208 and 215 for 0 < k Fig. 1 illustrates the signal zðkþ and the filtering estimate ^zðk; kþ vs. k for Fig. 2. Observed value for white Gaussian observation noise Nð0; 0:5 2 Þ.

12 440 S. Nakamori et al. / Appl. Math. Comput. 135 (2003) Table 1 Mean-square values of the filtering error zðkþ ^zðk; kþ and the fixed-point smoothing error zðkþ ^zðk; LÞ for the observation noises Nð0; 0:3 2 Þ, Nð0; 0:5 2 Þ, Nð0; 0:7 2 Þ and Nð0; 1Þ White Gaussian observation noise MSVs of the filtering error Uncertain observation Certain observation MSVs of the fixed-point smoothing error Uncertain observation Certain observation Nð0; 0:3 2 Þ Nð0; 0:5 2 Þ Nð0; 0:7 2 Þ Nð0; 1Þ the white Gaussian observation noise Nð0; 0:5 2 Þ. Fig. 2 illustrates the observed value yðkþ vs. k for the observation noise Nð0; 0:5 2 Þ. Table 1 summarizes the mean-square values (MSVs) of the filtering and fixed-point smoothing errors of the signal for Nð0; 0:3 2 Þ, Nð0; 0:5 2 Þ, Nð0; 0:7 2 Þ and Nð0; 1Þ in the cases of the uncertain and certain observations. The MSVs are calculated by X 200 ðzðiþ ^zði; iþþ 2 =200 and X 200 X 20 j¼1 ðzðiþ ^zði; i þ jþþ 2 =4000; respectively, where zðiþ ^zði; iþ is the filtering error and zðiþ ^zði; i þ jþ is the fixed-point smoothing error. From Table 1, it is shown that the estimation accuracy of the fixed-point smoother is superior to the filter, and that the MSVs with the uncertain observations are almost equal to those with the certain observations. For references, the sequence of the signal zðkþ with the auto-covariance function (51) is generated by the second-order AR model [8] zðkþ ¼ a 1 zðk 1Þ a 2 zðk 2ÞþeðkÞ; E½eðkÞ 2 Š¼r 2 : ð54þ 6. Conclusions In this paper, the RLS estimation technique with the uncertain observations is proposed in linear discrete-time stochastic systems. It is assumed that the auto-covariance function of the signal is expressed in the semi-degenerate kernel form and the probability that the signal exists in the observed value is known as the a priori information of the estimators. The proposed filter and fixed-point smoother are designed in the relation of the recursive Wiener filter [3,4] and the existing estimators [1] with the uncertain observations.

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