In recent years there has been much interest in robustness issues in general and in robust signal processing schemes in particular. Robust schemes are useful in situations where imprecise a priori knowledge of input characteristics makes the sensitivity of performance to deviations from assumed conditions an important factor in the design of good signal processing schemes. In this survey we discuss the minimax approach for the design of robust methods for signal processing. This has proven to be a very useful approach because it leads to constructive procedures for designing robust schemes. Our emphasis is on the contributions which have been made in robust signal processing, although key results of other robust statistical procedures are also considered. Most of the results we survey have been obtained in the past fifteen years, although some interesting earlier ideas for minimax signal processing are also mentioned. This survey is organized into five main parts, which deal separately with robust linear filters for signal estimation, robust linear filters for signal detection and related applications, nonlinear methods for robust signal detection, nonlinear methods for robust estimation, and robust data quantization. The interrelationships among many of these results are also discussed in the survey.

Robust techniques for signal processing: A survey

Under the usual assumptions of normality, the recursive estimator known as the Kalman filter gives excellent results and has found an extremely broad field of application--not only for estimating the state of a stochastic dynamic system, but also for estimating model parameters as well as detecting abrupt changes in the states or the parameters. It is well known, however, that significantly nonnormal noise, and particularly the presence of outliers, severely degrades the performance of the Kalman filter. This results in poor state estimates, nonwhite residuals and invalid inference. A first-order approximation is derived for the conditional prior distribution of the state of a discrete-time stochastic linear dynamic system in the presence of $\varepsilon$-contaminated normal observation noise. This distribution is then used to derive a first-order approximation of the conditional mean (minimum-variance) estimator. If the observation noise distribution can be represented as a member of the $\varepsilon$-contaminated normal neighborhood, then the conditional prior is also, to first order, an analogous perturbation from a normal distribution whose first two moments are given by the Kalman filter. Moreover, the perturbation is itself of a special form, combining distributions whose parameters are given by banks of parallel Kalman filters and optimal smoothers.

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Robust Recursive Estimation in the Presence of Heavy-Tailed Observation Noise

Paper: An adaptive robustizing approach to kalman filtering

The problem of estimating the prior probabilities q = (q_{1} \cdots q_{m-1}) of m statistical classes with known probability density functions F_{1}(X) \cdots F_{m}(x) on the basis of n statistically independent observations (X_{l} \cdots x_{n}) is considered. The mixture density g(x|q) = \sum^{m-1}_{j-1}q_{j}F_{j}(x) + (1 - \sum^{m-1}_{\tau = 1}q_{\tau})F_{m(x) is used to show that the maximum likelihood estimate of q is asymptotically efficient and weakly consistent under very mild constraints on the set of density functions. A recursive estimate is proposed for q . By using stochastic approximation theory and optimizing the gain sequence, it is shown that the recursive estimate is asymptotically efficient for the m = 2 class case. For m > 2 classes, the rate of convergence is computed and shown to be very close to asymptotic efficiency.

Recursive estimation of prior probabilities using a mixture

Robustized vector Robbins-Monro algorithm with applications to m-interval detection

In this paper, a bivariate extension of the m-Interval Partition Statistic is introduced for the problem of robust detection of a shift of mean in bivariate data. This extension is obtained in a manner similar to Hodges bivariate extension of the sign test. The resulting Bivariate m-Interval Partition Statistic is shown to be robust and amenable to digital implementation. To use the Bivariate m-Interval Partition Statistic, a finite number of quantiles of a specified number of cumulative distribution functions are required. A computationally simple algorithm based upon the Robbins—Monro procedure is presented for estimating these parameters in most realistic noise environments. Simulations of this procedure illustrate its practicality and add considerable insight as to the nature of these quantiles. The performance of the Bivariate m-Interval Partition Statistic as a classifier is discussed in term of its type I error and power and simulated curves for several different noise distributions are included. Two sets of scores are discussed and then incorporated in an application of the Bivariate m-Interval Partition Statistic to edge detection. This example illustrates both the performance and robustness of this classifier in severe noise environments.

Paul Kersten

Papers

Robustized vector Robbins-Monro algorithm with applications to m-interval detection

Bivariate m-interval classifiers with application to edge detection

Robustized recursive estimation with applications