When is adaptive better than optimal?

Given a stationary process, let us predict it using a first-order predictor whose single coefficient is adapted to the current observations using a constant gain identification algorithm. We investigate the prediction error variance as a function of the adaptation gain i.e., the length of the memory (the number of observations) of the identification scheme. An infinite-memory corresponds to the asymptotically constant optimal predictor and a finite memory to a locally adaptive time varying predictor. We show that, in some specified situations, the prediction error variance associated with the finite memory adaptation scheme is smaller that the optimal variance. This can only occur if the model is misspecified i.e., the structure of the optimal predictor is too simple     

Nonlinear black-box models in system identification: Mathematical foundations

We discuss several aspects of the mathematical foundations of the nonlinear black-box identification problem. We shall see that the quality of the identification procedure is always a result of a certain trade-off between the expressive power of the model we try to identify (the larger the number of parameters used to describe the model, the more flexible is the approximation), and the stochastic error (which is proportional to the number of parameters). A consequence of this trade-off is the simple fact that a good approximation technique can be the basis of a good identification algorithm. From this point of view, we consider different approximation methods, and pay special attention to spatially adaptive approximants. We introduce wavelet and ‘neuron’ approximations, and show that they are spatially adaptive. Then we apply the acquired approximation experience to estimation problems. Finally, we consider some implications of these theoretical developments for the practically implemented versions of the ‘spatially adaptive’ algorithms.     

Nonlinear black-box modeling in system identification: a unified overview

A nonlinear black-box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area, with structures based on neural networks, radial basis networks, wavelet networks and hinging hyperplanes, as well as wavelet-transform-based methods and models based on fuzzy sets and fuzzy rules. This paper describes all these approaches in a common framework, from a user's perspective. It focuses on what are the common features in the different approaches, the choices that have to be made and what considerations are relevant for a successful system-identification application of these techniques. It is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space. These mappings are discussed separately. The latter mapping is usually formed as a basis function expansion. The basis functions are typically formed from one simple scalar function, which is modified in terms of scale and location. The expansion from the scalar argument to the regressor space is achieved by a radial- or a ridge-type approach. Basic techniques for estimating the parameters in the structures are criterion minimization, as well as two-step procedures, where first the relevant basis functions are determined, using data, and then a linear least-squares step to determine the coordinates of the function approximation. A particular problem is to deal with the large number of potentially necessary parameters. This is handled by making the number of ‘used’ parameters considerably less than the number of ‘offered’ parameters, by regularization, shrinking, pruning or regressor selection.     

Adaptive control of a simple time-varying system

Adaptive control of a first-order randomly varying stochastic process is considered. Several authors have treated this problem using either ergodic theory for Markov processes or martingale limit theorems. In both cases some initial variance calculations and bounds evaluations must be performed in order to apply these techniques and establish the sample path results generally considered in adaptive control. These prior calculations are performed in the more general case where the single parameter follows a random walk and for an identification algorithm requiring no a priori knowledge of noise characteristics     

Remarks on linear and nonlinear filtering

This communication tries to give some insight into relationships existing between Viterbi and the forward-backward algorithm (used in the context of hidden Markov models) on the one hand and Kalman filtering and Rauch-Tung Striebel smoothing on the other. We give a unifying view which shows how those algorithms are related and give an example of a nonlinear hybrid system that can be filtered through a mixed algorithm     

On the projection algorithm and delay of peaking in adaptivecontrol

Adaptive deadbeat control is considered for the deterministic linear plant without any persistent excitation assumption imposed. The upper bound on the rate of convergence is provided for the control algorithm that uses the projection identification algorithm. On the other hand, it is shown that convergence is delayed if the regularized identification algorithm is used. It is shown that arbitrarily large delays can precede an overshoot under certain choices of initial conditions. An estimate is also provided for the amplitude of the peak     

Stable adaptive control for a plant with randomly varying parameters

Adaptive control is considered for a two-dimensional linear discrete-time plant with randomly drifting parameters. The certainty equivalent minimum variance control law along with the projection-like identification algorithm are used. The stability of the parameter estimates and exponential stability of the closed-loop system are proved in the absence of any persistent excitation assumption.     

Minimal stochastic complexity image partitioning with unknown noise model.

We present a generalization of a new statistical technique of image partitioning into homogeneous regions to cases where the family of the probability laws of the gray-level fluctuations is a priori unknown. For that purpose, the probability laws are described with step functions whose parameters are estimated. This approach is based on a polygonal grid which can have an arbitrary topology and whose number of regions and regularity of its boundaries are obtained by minimizing the stochastic complexity of the image. We demonstrate that efficient homogeneous image partitioning can be obtained when no parametric model of the probability laws of the gray levels is used and that this approach leads to a criterion without parameter to be tuned by the user. The efficiency of this technique is compared to a statistical parametric technique on a synthetic image and is compared to a standard unsupervised segmentation method on real optical images.     

Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images

In many imaging applications, the measured optical images are perturbed by strong fluctuations or boise. This can be the case, for example, for coherent-active or low-flux imagery. In such cases, the noise is not Gaussian additive and the definition of a contrast parameter between two regions in the image is not always a straightforward task. We show that for noncorrelated noise, the Bhattacharyya distance can be an efficient candidate for contrast definition when one uses statistical algorithms for detection, location, or segmentation. We demonstrate with numerical simulations that different images with the same Bhattacharyya distance lead to equivalent values of the performance criterion for a large number of probability laws. The Bhattacharyya distance can thus be used to compare different noisy situations and to simplify the analysis and the specification of optical imaging systems.