A Stieltjes transform approach for analyzing the RLS adaptive Filter

Although the RLS filter is well-known and various algorithms have been developed for its implementation, analyzing its performance when the regressors are random, as is often the case, has proven to be a formidable task. The reason is that the Riccati recursion, which propagates the error covariance matrix, becomes a random recursion. The existing results are approximations based on assumptions that are often not very realistic. In this paper we use ideas from the theory of large random matrices to find the asymptotic (in time) eigendistribution of the error covariance matrix of the RLS filter. Under the assumption of a large dimensional state vector (in most cases n = 10-20 is large enough to get quite accurate predictions) we find the asymptotic eigendistribution of the error covariance for temporally white regressors, shift structured regressors, and for the RLS filter with intermittent observations

Adaptive Quantizers for Estimation

In this paper, adaptive estimation based on noisy quantized observations is studied. A low complexity adaptive algorithm using a quantizer with adjustable input gain and offset is presented. Three possible scalar models for the parameter to be estimated are considered: constant, Wiener process and Wiener process with deterministic drift. After showing that the algorithm is asymptotically unbiased for estimating a constant, it is shown, in the three cases, that the asymptotic mean squared error depends on the Fisher information for the quantized measurements. It is also shown that the loss of performance due to quantization depends approximately on the ratio of the Fisher information for quantized and continuous measurements. At the end of the paper the theoretical results are validated through simulation under two different classes of noise, generalized Gaussian noise and Student's-t noise

Adaptive Stochastic Systems: Estimation, Filtering, And Noise Attenuation

This dissertation investigates problems arising in identification and control of stochastic systems. When the parameters determining the underlying systems are unknown and/or time varying, estimation and adaptive filter- ing are invoked to to identify parameters or to track time-varying systems. We begin by considering linear systems whose coefficients evolve as a slowly- varying Markov Chain. We propose three families of constant step-size (or gain size) algorithms for estimating and tracking the coefficient parameter: Least-Mean Squares (LMS), Sign-Regressor (SR), and Sign-Error (SE) algorithms. The analysis is carried out in a multi-scale framework considering the relative size of the gain (rate of adaptation) to the transition rate of the Markovian system parameter. Mean-square error bounds are established, and weak convergence methods are employed to show the convergence of suitably interpolated sequences of estimates to solutions of systems of ordinary and stochastic differential equations with regime switching. Next we consider problems in noise attenuation in systems with unmodeled dynamics and stochastic signal errors. A robust two-phase design procedure is developed which first estimates the signal in a simplified form, and then applies a control to tune out the noise. Worst-case error bounds are derived in terms of the unmodeled dynamics and variances of the disturbance and measurement errors

Nonasymptotic analysis of adaptive and annealed Feynman-Kac particle models

Sequential and quantum Monte Carlo methods, as well as genetic type search algorithms can be interpreted as a mean field and interacting particle approximations of Feynman-Kac models in distribution spaces. The performance of these population Monte Carlo algorithms is strongly related to the stability properties of nonlinear Feynman-Kac semigroups. In this paper, we analyze these models in terms of Dobrushin ergodic coefficients of the reference Markov transitions and the oscillations of the potential functions. Sufficient conditions for uniform concentration inequalities w.r.t. time are expressed explicitly in terms of these two quantities. We provide an original perturbation analysis that applies to annealed and adaptive Feynman-Kac models, yielding what seems to be the first results of this kind for these types of models. Special attention is devoted to the particular case of Boltzmann-Gibbs measures' sampling. In this context, we design an explicit way of tuning the number of Markov chain Monte Carlo iterations with temperature schedule. We also design an alternative interacting particle method based on an adaptive strategy to define the temperature increments. The theoretical analysis of the performance of this adaptive model is much more involved as both the potential functions and the reference Markov transitions now depend on the random evolution on the particle model. The nonasymptotic analysis of these complex adaptive models is an open research problem. We initiate this study with the concentration analysis of a simplified adaptive models based on reference Markov transitions that coincide with the limiting quantities, as the number of particles tends to infinity.Comment: Published at http://dx.doi.org/10.3150/14-BEJ680 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm