Rls Wiener Predictor with Uncertain Observations in Linear Discrete-time Stochastic Systems

This paper proposes recursive least-squares (RLS) l-step ahead predictor and filtering algorithms with uncertain observations in linear discrete-time stochastic systems. The observation equation is given by         y k k z k v    k  , , where is a binary switching sequence with conditional probability. The estimators require the information of the system state-transition matrix 1 p k P k    that the signal exists in the uncertain observation equation and the   2, 2 element   2, | P k j 2     of the conditional probability of   k  , given .


Introduction
The estimation problem given uncertain observations is an important research topic in the area of detection and estimation problems in communication systems [1].Nahi [2], assuming that the state-space model is given, proposes the RLS estimation method with uncertain observations, when the uncertainty is modeled in terms of independent random variables, and the probability that the signal exists in each observation is available.The term uncertain observations refers to the fact that some observations may not contain the signal and consist only of observation noise.In Hadidi and Schwartz [3], Nahi's results are extended to the case where the variables modeling the uncertainty are not necessarily independent.
In the above studies, it is assumed that the state-space model for the signal is given.However, in real applications, the state-space modeling errors might degrade the estimation accuracy.Nakamori [4] derived the RLS Wiener fixed-point smoothing and filtering algorithms, based on the invariant imbedding method, from uncertain observations with uncertainty modeled by independent random variables.In the derivation of such RLS Wiener estimators, the state-transition matrix Φ, the observation matrix H , the variance  ,  K k k of the state vector

 
x k , the variance   R k of the observation noise   v k and the observed values   y k are used.Moreover, Nakamori et al. [5], based on the innovation approach, proposed the RLS Wiener fixed-point smoother and filter in linear discrete-time stochastic systems.Here, the observation equation is given by ,where is a binary switching sequence with conditional probability.The innovation process is given by   in terms of the element ,2 of the conditional probability of , given (2, 2) (see Nakamori et al. [5,6] for details).
In the current paper, under the same assumptions for the observation equation as in Nakamori et al. [5], an algorithm for the RLS Wiener ahead predictor is derived, based on the invariant imbedding method.Thus, the observation equation is given by where is a binary switching sequence with conditional probability.The observation equation adopted in this paper is suitable, for example, to model remote sensing situations with data transmission in multichannels, where the independence assumption of the variables describing the uncertainty in the observations is not realistic.
The estimators require the information of the system state-transition matrix , the observation matrix  H , the variance  ,  K k k of the state vector   x k , the variance of the observation noise, the probability that the signal exists in the uncertain observation equation and the element . The RLS Wiener prediction and filtering algorithms are summarized in Theorem 1 and its proof is deferred to the Appendix.The main issues in this paper which are different from those in Nakamori et al. [5] are concerned with the algorithm derivation; namely: 1) The prediction estimate is given as a linear transformation of the observed values.
2) The prediction algorithms are derived on the basis of the invariant imbedding method.
The current paper's main contribution is the derivation of a recursive least-squares algorithm for the predictor and filter design in systems with non-independent uncertain observations, using covariance information.Without making use of the state-space model, the algorithm is obtained from the autocovariance functions of the signal and the observation noise, the probability that the signal exists in the observed values and the (2,2) element of the conditional probability matrices of the sequence which describes the uncertainty in the observations.This approach is suitable in many practical situations where the equation generating the signal process is unknown, thus being not possible to use the state-space model to address the estimation problem.The deduction of the algorithm is mainly based on an invariant imbedding method.

Problem Formulation
Consider the following observation equation where is a signal, x k is the zero-mean state vector and is a white noise with zero mean and the variance of where denotes the Kronecker delta function.
, which describes the uncertainty in the observations, has the following stochastic properties [3]: (2, .The element of the conditional probability matrix of 2) and the sequences v k are mutually independent.Let us introduce the system matrix in the statespace model for the state vector , , The purpose of this paper is to design a covariancebased recursive algorithm to obtain the ahead prediction estimate of Due to the presence of a multiplicative noise component in the observation Equation (1), even if the additive noise is Gaussian, the conditional , which provides the least-squares estimator, is not a linear function of the observations and its computation can be very complicated requiring, in general, an exponentially growing memory.For this reason, our attention is focused on the least-squares linear estimation problem.Specifically, we are interested in obtaining the leastsquares linear estimator of the state vector as a linear transformation of the observed values  , Copyright © 2011 SciRes.JSIP impulse-response function.
Let us consider the least-squares prediction problem, which minimizes the criterion The orthogonal projection lemma [7] assures that  , x k l k  is the only linear combination of the observations   y i , such that the estimation error is orthogonal to them, that is, This condition is equivalent to the Wiener-Hopf equation , which minimizes the cost function (6).From , the left-hand side of (7) is written as . Then, from the observation equation (1) and the covariance function (2) for white observation noise , is reduced to Substituting ( 9) and ( 10) into (8), we have Under these conditions, in Section 3 the RLS Wiener prediction and filtering algorithms are presented.

RLS Wiener Prediction and Filtering Algorithm
Nakamori et al. [5,6], based on the innovation approach, proposed the algorithms for the fixed-point smoothing estimate and the filtering estimate.These algorithms are derived taking into account that the innovation process is expressed as Under the preliminary assumptions made in Section 2, Theorem 1 proposes the RLS Wiener algorithms for the step l  ahead prediction estimates of the signal and the state vector   x k l  .These algorithms are derived, starting with (11), by iterative use of the invariant imbedding method.
Theorem 1.Consider the observation equation described in (1) and assume that the probability   l  step ahead prediction estimate of the state vector Proof of Theorem 1 is detailed in the Appendix.
Clearly, the algorithms for the filtering estimate are the same as those proposed in Nakamori et al. [5].From Theorem 1, the innovation process   k  is represented by

A Numerical Simulation Example
In order to illustrate the application of the RLS Wiener prediction algorithm proposed in Theorem 1, we consider a scalar signal   z k whose autocovariance function where and

 
The covariance function (19) corresponds to a signal process generated by a second-order AR model.Therefore, according to Nakamori [4], the observation vector , H the variance of the state vector x k and the system matrix in the state equation are as follows: As in Nakamori et al. [5], we consider that the signal is transmitted through one of two channels, characterized by its observation equation as follows: where is a zero-mean white observation noise and is a sequence of independent random variables taking values or 1 with is described by , for all .k We assume that channel 1 is chosen at random with probability and, hence, channel 2 is selected with probability .Then, the observation equation is a sequence of rand h take valu 1 with learly om variables whic , es 0 or for all , and conditional probability matrix k for all , 0, ,  From F e 2, it is deduced that, as l becomes larger, the prediction accuracy worse in and the certain observations cases, with each different observation noise.It might also be noticed that the MSVs with uncertain observations are almost equal to those with certain observations except for the observation noise with variance 2 0.1 .For the observation noise with variance 2 0.1 , the MSVs of the prediction errors with the certain observati are smaller than those with the uncertain ob ervations, particularly for the 2 and 4-step ahead predictions.
For reference, the autoregressive (AR) model used to generate the signal , given   j  .A numerical simulation example in Section 4 shows that the prediction algorithm proposed in this paper is feasible.

REFERENCES
et us introduce the equation concerned with the function Appendix A. Proof of Theorem 1 From ( 11) and (A-1) it follows that Subtracting the equation obtained by From (A-1), (A-3) and the relationship , it follows that .
Here, the relationship

  
Subtracting the equation obtained by putting in (A-6) from (A-6) yields Here, (A-5) and (A-7) have been used.Clearly, from (A-6), the initial condition for the recursive Eq x k l  is given by (5).From ( 5) and (A ws -2), it follo that

T T T T h k k k p k K k k H P k S k H R k k HK k k H P k H S k
are given.Let the system statetransition matrix  the observation matrix , H the autovariance function   , K s s of the state vector   x s , the variance   R k of the white observation noise   v k and the observed value   y k l  be given.Then the RLS Wiener algorithms for the ahead predic of Theo m 1, the predicttion estimate of the signal has been calculated recursively.

Figure 1 3 .
Figure 1 illustrates the signal   z k and its prediction es