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The least-squares quadratic estimation problem of signals from observations coming from multiple sensors is addressed when there is a nonzero probability that each observation does not contain the signal to be estimated. We assume that, at each sensor, the uncertainty about the signal being present or missing in the observation is modelled by correlated Bernoulli random variables, whose probabilities are not necessarily the same for all the sensors. A recursive algorithm is derived without requiring the knowledge of the signal state-space model but only the moments (up to the fourth-order ones) of the signal and observation noise, the uncertainty probabilities, and the correlation between the variables modelling the uncertainty. The estimators require the autocovariance and cross-covariance functions of the signal and their second-order powers in a semidegenerate kernel form. The recursive quadratic filtering algorithm is derived from a linear estimation algorithm for a suitably defined augmented system.

In many real systems the signal to be estimated can be randomly missing in the observations due, for example, to intermittent failures in the observation mechanism, fading phenomena in propagation channels, target tracking, accidental loss of some measurements, or data inaccessibility during certain times. Usually, these situations are characterized by including in the observation equation not only an additive noise, but also a multiplicative noise consisting of a sequence of Bernoulli random variables taking the value one if the observation is state plus noise, or the value zero if it is only noise (

On the other hand, in some practical situations the state-space model of the signal is not available and another type of information must be processed for the estimation. In the last years, the estimation problem from uncertain observations has been investigated using covariance information, and algorithms with a simpler structure than those obtained when the state-space model is known have been derived (see, e.g., [

Recently, the least-squares linear estimation problem using uncertain observations transmitted by multiple sensors, whose statistical properties are assumed not to be the same, has been studied by several authors under different approaches and hypotheses on the processes (see, e.g., [

In this paper, using covariance information, recursive algorithms for the least-squares quadratic filtering problem from correlated uncertain observations coming from multiple sensors with different uncertainty characteristics are proposed. This paper extends the results in [

To address the quadratic estimation problem, augmented signal and observation vectors are introduced by assembling the original vectors with their second-order powers defined by the Kronecker product, thus obtaining a new augmented system and reducing the quadratic estimation problem in the original system to the linear estimation problem in the augmented system. By using an innovation approach, the linear estimator of the augmented signal based on the augmented observations is obtained, thus providing the required quadratic estimator.

The performance of the proposed filtering algorithms is illustrated by a numerical simulation example where the state of a first-order autoregressive model is estimated from uncertain observations coming from two sensors with different uncertainty characteristics correlated at times

The problem at hand is to determine the least-squares (LS) quadratic estimator of an

Consider

To simplify the notation, the observation equation (

It is known that if the signal

The

For

For

The signal process,

Given the observation model (

To obtain this linear estimator, the first- and second-order statistical properties of the augmented vectors

By using the Kronecker product properties and denoting

Note that the LS linear estimator of

The signal and noise processes

If the signal process

It is immediate from hypothesis (H1) about the covariance functions of the signal and its second-order moments.

Under (H1)–(H4), the noise

It is obvious that

Indeed, since

On the other hand,

The uncorrelation between

As indicated above, to obtain the LS quadratic estimators of the signal

Due to the fact that the observations are generally nonorthogonal vectors, we will use an innovation approach, consisting of transforming the observation process

The innovation process is constructed by the

Note that the projection

Next, taking into account that the innovations constitute a white process, we derive a general expression for the LS linear estimator of the augmented signal,

Using the properties of the processes involved in (

The quadratic filter,

We start by obtaining an explicit formula for the innovations,

Next, expression (

Expression (

Finally, we obtain expression (

To conclude, as a measure of the estimation accuracy, we have calculated the filtering error covariance matrices,

The observation model considered in Section

For

This correlation model allows us to consider certain situations where the signal cannot be missing at

Similar considerations to those made in Section

The quadratic filter,

It is analogous to that of Theorem

To illustrate the application of the proposed filtering algorithm a numerical simulation example is shown now. To check the effectiveness of the proposed quadratic filter, we ran a program in MATLAB which, at each iteration, simulates the signal and the observed values and provides the linear and quadratic filtering estimates, as well as the corresponding error covariance matrices.

For the simulations, this program has been applied to a scalar signal

The autocovariance functions of the signal and their second-order powers are given in a semidegenerate kernel form, specifically,

and the covariance function of the signal and their second-order powers is given by

Consider two sensors whose measurements, according to our theoretical study, are perturbed by sequences of Bernoulli random variables

Assume that the additive noises are independent and have the following probability distributions:

Now, in accordance with the proposed uncertain observation model, we assume that the uncertainty at any time

To model the uncertainty in this way, we can consider two independent sequences of independent Bernoulli random variables,

For the application, we have assumed that the variables

Since

For

For

Summarizing, the correlation function of

and, hence, the measurements above described are in accordance with the proposed correlation model.

To analyze the performance of the proposed estimators, the linear and quadratic filtering error variances have been calculated for different values of

First, considering

Linear and quadratic filtering error variances for

Next, we compare the performance of the linear and quadratic filtering estimators for the values

In Figure

Linear and quadratic filtering error variances at

Linear and quadratic filtering error variances at

Finally, for

Linear and quadratic filtering error variances at

A recursive quadratic filtering algorithm is proposed from correlated uncertain observations coming from multiple sensors with different uncertainty characteristics. This is a realistic assumption in situations concerning sensor data that are transmitted over communication networks where, generally, multiple sensors with different properties are involved. The uncertainty in each sensor is modelled by a sequence of Bernoulli random variables which are correlated at times

Using covariance information, the algorithm is derived by applying the innovation technique to suitably defined augmented signal and observation vectors, and the LS quadratic estimator of the signal is obtained from the LS linear estimator of the augmented signal based on the augmented observations.

The performance of the proposed filtering algorithm is illustrated by a numerical simulation example where the state of a first-order autoregressive model is estimated from uncertain observations coming from two sensors with different uncertainty characteristics correlated at times

This paper is supported by Ministerio de Educación y Ciencia (Grant no. MTM2008-05567) and Junta de Andalucía (Grant no. P07-FQM-02701).