Abstract
The problem of identification of nonstationary stochastic processes (systems or signals) is considered and a new class of identification algorithms, combining the basis functions approach with local estimation technique, is described. Unlike the classical basis function estimation schemes, the proposed regularized local basis function estimators are not used to obtain interval approximations of the parameter trajectory, but provide a sequence of point estimates corresponding to consecutive instants of time. Based on the results of theoretical analysis, the paper addresses and solves all major problems associated with implementation of the new class of estimators, such as optimization of the regularization matrix, adaptive selection of the number of basis functions and the width of the local analysis interval, and reduction of complexity of the computational algorithms.
Highlights
L INEAR time-varying models of nonstationary processes are the basis of many real-life applications in various disciplines such as telecommunications [1], biomedicine [2], [3], geophysics [4], [5] and control science [6], to name just a few
From the qualitative point of view, the local basis function (LBF) and regularized local basis function (RLBF) approaches can be regarded as an extension, to the process identification case, of the signal smoothing technique known as Savitzky-Golay filtering [39]
Since LBF and RLBF estimates are evaluated in the sliding window mode, i.e., computations are repeated for every new location t of the analysis window Tk(t), the computational burden is high
Summary
L INEAR time-varying models of nonstationary processes (signals or systems) are the basis of many real-life applications in various disciplines such as telecommunications [1], biomedicine [2], [3], geophysics [4], [5] and control science [6], to name just a few. In some applications the main purpose of regularization is to prevent the identified models from overfitting, i.e., from including in the model more parameters than can be justified by the data To this end the L1 regularization is more effective than the L2 one [32], leading to such well-known techniques as LASSO (least absolute shrinkage and selection operator) in statistics [33], or basis pursuit in signal processing [34]. In the second part of the paper, which is based on the concept of parameter preestimation [19], the computationally fast and numerically robust version of the regularized basis function scheme is proposed. Since the well-tuned noncausal estimation algorithms provide a better bias-variance trade-off than their comparable causal counterparts, the achievable parameter tracking performance is usually considerably better
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