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Accurate identification of stochastic systems with fast-varying parameters is a challenging task which cannot be accomplished using model-free estimation methods, such as weighted least squares, which assume only that system coefficients can be regarded as locally constant. The current state-of-the-art solutions are based on the assumption that system parameters can be locally approximated by a linear combination of appropriately chosen basis functions. The paper shows that tracking performance of the resulting local basis function estimation algorithms can be further improved by means of regularization. The method is illustrated by an important recent application - identification of fast time-varying acoustic channels used in underwater communication.

The problem of identification of multivariate autoregressive processes (systems or signals) with unknown and possibly time-varying model order and time-varying rate of parameter variation is considered and solved using parallel estimation approach. Under this approach, several local estimation algorithms, with different order and bandwidth settings, are run simultaneously and compared based on their predictive performance. First, the competitive decision schemes are considered. It is shown that the best parameter tracking results can be obtained when the order is selected based on minimization of the appropriately modified Akaike's final prediction error statistic, and the bandwidth is chosen using the localized version of the Rissanen's predictive least squares statistic. Next, it is shown that estimation results can be further improved if a collaborative decision is made by means of applying the Bayesian model averaging technique.

The problem of identification of a nonstationary autoregressive signal using non-causal estimation schemes is considered. Noncausal estimators can be used in applications that are not time-critical, i.e., do not require real-time processing. A new adaptive estimation bandwidth selection rule based on evaluation of pseudoprediction errors is proposed, allowing one to adjust tracking characteristics of noncausal estimators to unknown and/or time-varying degree of signal nonstationary. The new rule is compared with the previously proposed one, based on the generalized Akaike’s final prediction error criterion.

Savitzky–Golay (SG) filtering is a classical signal smoothing technique based on the local least squares approximation of the analyzed signal by a linear combination of known functions of time (originally — powers of time, which corresponds to polynomial approximation). It is shown that the regularized version of the SG algorithm can be successfully applied to identification of time-varying finite impulse response (FIR) systems. Such a solution is possible owing to the recently proposed preestimation technique, which converts the problem of identification of a time-varying FIR system into the problem of smoothing of the appropriately generated preestimates of system parameters. The resulting fast regularized local basis function estimators, optimized using the empirical Bayes approach, have very good parameter tracking capabilities, favorably comparing with the state-of-the-art in terms of accuracy, computational complexity and numerical robustness.

When estimating the correlation/spectral structure of a locally stationary process, one has to make two important decisions. First, one should choose the so-called estimation bandwidth, inversely proportional to the effective width of the local analysis window, in the way that complies with the degree of signal nonstationarity. Too small bandwidth may result in an excessive estimation bias, while too large bandwidth may cause excessive estimation variance. Second, but equally important, one should choose the appropriate order of the spectral representation of the signal so as to correctly model its resonant structure–when the order is too small, the estimated spectrum may not reveal some important signal components (resonances), and when it is too high, it may indicate the presence of some nonexistent components. When the analyzed signal is not stationary, with a possibly time-varying degree of nonstationarity, both the bandwidth and order parameters should be adjusted in an adaptive fashion. The paper presents and compares three approaches allowing for unified treatment of the problem of adaptive bandwidth and order selection for the purpose of identification of nonstationary vector autoregressive processes: the cross-validation approach, the full cross-validation approach, and the approach that incorporates the multivariate version of the generalized Akaike’s final prediction error criterion. It is shown that the latter solution yields the best results and, at the same time, is very attractive from the computational viewpoint.

The problem of off-line identification of a nonstationary autoregressive process with a time-varying order and a time-varying degree of nonstationarity is considered and solved using the parallel estimation approach. The proposed parallel estimation scheme is made up of several bidirectional (noncausal) exponentially weighted lattice algorithms with different estimation memory and order settings. It is shown that optimization of both settings can be carried out by means of minimization of the locally evaluated accumulated forward/backward prediction error statistic.

The paper presents a new, two-stage approach to identification of linear time-varying stochastic systems, based on the concepts of preestimation and postfiltering. The proposed preestimated parameter trajectories are unbiased but have large variability. Hence, to obtain reliable estimates of system parameters, the preestimated trajectories must be further filtered (postfiltered). It is shown how one can design and optimize such postfilters using the basis function framework. The proposed solution to adaptive tuning of postfilter settings is based on parallel estimation and cross-validatory analysis. When compared with the classical solutions to the problem of parameter tracking, the new approach offers, without compromising good tracking performance, significant computational savings, higher numerical robustness and greater flexibility.

In this paper we consider the problem of noncausal identification of nonstationary, linear stochastic systems, i.e., identification based on prerecorded input/output data. We show how several competing weighted least squares parameter smoothers, differing in memory settings, can be combined together to yield a better and more reliable smoothing algorithm. The resulting parallel estimation scheme automatically adjusts its smoothing bandwidth to the unknown, and possibly time-varying, rate of nonstationarity of the identified system. It also allows one to account for the distribution of measurement noise, and in particular – to cope with heavy-tailed disturbances, such as Laplacian noise, or light-tailed disturbances, such as uniform noise.

The problem of either elimination or extraction of an unknown number of nonstationary complex sinusoidal signals burried in noise is considered. We derive a class of adaptive notch filtering algorithms which can be efficiently (almost transientless) started, or restarted, using the results of local nonparametric, periodogram-based signal analysis.

The problem of elimination of a measurement noise and impulsive disturbances from time-varying autoregressive signals is considered. The combined signal filtering / parameter tracking / outlier elimination algorithms proposed in the paper borrow their form from the "prototype" algorithm based on extended Kalman filtering (EKF). While the original EKF solution is computationally quite involved the EKF-like algorithms are much simpler and easier to implement. The new approach was successfully applied to restoration of archive audio recordings.

This paper presents a novel method for dynamic weighing of cars making up a freight train. The proposed algorithms allow for automatic and accurate estimation of weight of separate cars without the need to uncouple them nor to stop them on the measuring platform. The new approach is illustrated with both the results of computer simulations and real experiments.

The new parametric spectrum estimator for the purpose of nonstationary autoregressive process analysis is presented. The proposed estimator is obtained by minimization of the Bayesian risk function corresponding to the normalized mean square spectral error measure. The obtained results concern the two most frequently used models of process parameters' variation : the Kalman filter model and the fadding memory (exponential forgetting) one. The efficient computational algorithms are indicated and the results of computer simulation are presented.