lms algorithms for tracking slow markov chains with applications to hidden markov estimation and adaptive multiuser detection this paper analyzes the tracking properties of the least mean squares lms algorithm when the underlying parameter evolves according to a finite state markov chain with infrequent jumps first using perturbed liapunov function methods mean square error estimates are obtained for the tracking error then using recent results on two time scale markov chains mean ordinary differential equation and diffusion approximation results are obtained it is shown that a sequence of the centered tracking errors converges to an ordinary differential equation moreover a suitably scaled sequence of the tracking errors converges weakly to a diffusion process it is also shown that iterate averaging of the tracking algorithm results in optimal asymptotic convergence rate in an appropriate sense two application examples analysis of the performance of an adaptive multiuser detection algorithm in a direct sequence code division multiple access ds cdma system and tracking analysis of the state of a hidden markov model hmm with infrequent jumps are presented

fast recursive basis function estimators for identification of time varying processes when system parameters vary rapidly with time the weighted least squares filters are not capable of following the changes satisfactorily some more elaborate estimation schemes based on the method of basis functions have to be used instead the basis function estimators have increased tracking capabilities but are computationally very demanding the paper introduces a new class of adaptive filters based on the concept of postfiltering which have improved parameter tracking capabilities that are typical of the basis function algorithms but at the same time have pretty low computational requirements which is typical of the weighted least squares algorithms

exponential stability of general tracking algorithms tracking and adaptation algorithms are from a formal point of view nonlinear systems which depend on stochastic variables in a fairly complicated way the analysis of such algorithms is thus quite complicated a first step is to establish the exponential stability of these systems this is of interest in its own right and a prerequisite for the practical use of the algorithm it is also a necessary starting point to analyze the performance in terms of tracking and adaptation because that is how close the estimated parameters are to the time varying true ones in this paper we establish some general conditions for the exponential stability of a wide and common class of tracking algorithms this includes least mean squares recursive least squares and kalman filter based adaptation algorithms we show how stability of an averaged linear and deterministic equation and stability of the actual algorithm are linked to each other under weak conditions on the involved stochastic processes we also give explicit conditions for exponential stability of the most common algorithms the tracking performance of the algorithms is studied in a companion paper

lattice filter based multivariate autoregressive spectral estimation with joint model order and estimation bandwidth adaptation the problem of parametric autoregressive model based estimation of a time varying spectral density function of a multivariate nonstationary process is considered it is shown that estimation results can be considerably improved if identification of the autoregressive model is carried out using the two sided doubly exponentially weighted lattice algorithm which combines results yielded by two one sided lattice algorithms running forward in time and backward in time respectively it is also shown that the model order and the most appropriate estimation bandwidth can be efficiently selected using the suitably modified akaike s final prediction error criterion

recursive functional series modeling estimators for identification of time varying plants more bad news than good the properties of a class of recursive estimators the exponentially weighted functional series modeling estimators are discussed these estimators can be used e g in adaptive prediction or control applications it is argued that there exists a relationship between the amount of information about time varying system parameters which is available a priori and the robustness of the identification algorithm based on such prior knowledge the more specialized the estimation algorithm is the less reliable it might be under nonstandard conditions this is the reason why simple algorithms such as exponentially weighted least squares have to be recommended as if no information about the system nonstationarity is available in advance

analysing the efficiency of some modifications of the secant method some modifications of the secant method for solving nonlinear equations are revisited and the local order of convergence is found in a direct symbolic computation to do this a development of the inverse of the first order divided differences of a function of several variables in two points is presented a generalisation of the efficiency index used in the scalar case to several variables is also analysed in order to use the most competitive algorithm

performance analysis of multi innovation gradient type identification methods it is well known that the stochastic gradient sg identification algorithm has poor convergence rate in order to improve the convergence rate we extend the sg algorithm from the viewpoint of innovation modification and present multi innovation gradient type identification algorithms including a multi innovation stochastic gradient misg algorithm and a multi innovation forgetting gradient mifg algorithm because the multi innovation gradient type algorithms use not only the current data but also the past data at each iteration parameter estimation accuracy can be improved finally the performance analysis and simulation results show that the proposed misg and mifg algorithms have faster convergence rates and better tracking performance than their corresponding sg algorithms

on computational efficiency for multi precision zero finding methods an adapted definition related to multi precision arithmetic zero finding methods is considered and several improvements to iterative methods to compute nonlinear equation solutions which increase the local order of convergence are revisited furthermore a higher efficiency when we use a definition related to adaptive multi precision arithmetic is also given numerical results computed with a floating point system representing 200 and 1000 decimal digits support this theory

improvements of the efficiency of some three step iterative like newton methods an improvement of the local order of convergence is presented to increase the efficiency of the iterative method with an appropriate number of evaluations of the function and its derivative the third and fourth order of known two step like newton methods have been improved and the efficiency has also been increased

coupling and mixing times in a markov chain the derivation of the expected time to coupling in a markov chain and its relation to the expected time to mixing as introduced by the author in mixing times with applications to perturbed markov chains linear algebra appl 417 108 123 2006 are explored the two state cases and three state cases are examined in detail

nonasymptotic results for finite memory wls filters a nonasymptotic analysis of properties of weighted least squares wls adaptive filters used for identification of time varying systems is presented it is shown that the problem of mean square boundedness of wls estimates is closely related to the problem of invertibility in the mean sense of the corresponding regression matrix necessary and sufficient conditions are discussed for such invertibility to hold based on that a number of results are derived paralleling those already obtained for least mean square lms filters and the problem of statistical robustness of the wls estimator is briefly discussed

identification of quasi periodically varying systems using the combined nonparametric parametric approach the problem of identification of quasi periodically varying finite impulse response systems is considered neither the number of system frequency modes nor the initial frequency values are assumed to be known a priori the proposed solution is a blend of the parametric model based and nonparametric discrete fourier transform based approach to system identification it is shown that the results of nonparametric analysis can be used to identify the number of frequency modes and to determine initial conditions needed to smoothly start or restart the model based tracking algorithm such a combined nonparametric parametric approach allows one to preserve advantages of both frameworks leading to an estimation procedure which guarantees global frequency search high frequency resolution fast initial convergence and good steady state tracking capabilities

shaping frequency dependent time resolution when estimating spectral properties with parametric methods the problem of tracking time varying properties of a signal is studied the somewhat contradictory notion of time varying spectrum and how to estimate the current spectrum in an on line fashion is discussed the traditional concepts and relations between time and frequency resolution are crucial for this problem we introduce two definitions for the time resolution of filters essentially measuring the effective number of past data that are used to form the estimate in for example wavelet transform techniques frequency dependent time resolutions are used so that fewer data are used at higher frequencies thus enabling faster tracking of high frequency components at the price of worse frequency resolution the main contribution of the paper is to show how this same feature can be introduced when estimating spectra via a time varying autoregressive model of the signal this is achieved by a special choice of nominal covariance matrix for the underlying parameter changes

some stochastic properties of semi magic and magic markov chains abstract gustafson and styan gustafson and styan superstochastic matrices and magic markov chains linear algebra appl 430 2009 2705 2715 examined the mathematical properties of superstochastic matrices the transition matrices of magic markov chains formed from scaled magic squares this paper explores the main stochastic properties of such chains as well as semi magic chains with doubly stochastic transition matrices stationary distribution generalized inverses of markovian kernels mean first passage times variances of the first passage times and expected times to mixing are considered some general results are developed some observations from the chains generated by matlab are discussed some conjectures are presented and some special cases involving three and four states are explored in detail

tracking of time varying mobile radio channels 1 the wiener lms algorithm adaptation algorithms with constant gains are designed for tracking smoothly time varying parameters of linear regression models in particular channel models occurring in mobile radio communications in a companion paper an application to channel tracking in the is 136 tdma system is discussed the proposed algorithms are based on two key concepts first the design is transformed into a wiener filtering problem second the parameters are modeled as correlated arima processes with known dynamics this leads to a new framework for systematic and optimal design of simple adaptation laws based on prior information the algorithms can be realized as wiener filters called learning filters or as lms newton updates complemented by filters that provide predictions or smoothing estimates the simplest algorithm named the wiener lms is presented all parameters are here assumed governed by the same dynamics and the covariance matrix of the regressors is assumed known the computational complexity is of the same order of magnitude as that of lms for regressors which are either white or have autoregressive statistics the tracking performance is however substantially improved

fast algorithms for identification of periodically varying systems the problem of identification tracking of periodically varying systems is considered when system coefficients vary rapidly with time the most frequently used weighted least squares wls and least mean squares lms algorithms are not capable of tracking the changes satisfactorily to obtain good estimation results one has to use more specialized adaptive filters such as the basis function bf algorithms which are based on explicit models of parameter changes unfortunately estimators of this kind are numerically very demanding the paper describes new recursive algorithms that combine low computational requirements which are typical of wls and lms filters with very good tracking capabilities which are typical of bf filters

a convergence improvement factor and higher order methods for solving nonlinear equations abstract a convergence improvement factor is introduced for solving nonlinear equations so as to increase the efficiency of iterative methods many well known iterative methods are covered

wiener design of adaptation algorithms with time invariant gains a design method is presented that extends least mean squared lms adaptation of time varying parameters by including general linear time invariant filters that operate on the instantaneous gradient vector the aim is to track time varying parameters of linear regression models in situations where the regressors are stationary or have slowly time varying properties the adaptation law is optimized with respect to the steady state parameter error covariance matrix for time variations modeled as vector arima processes the design method systematically uses prior information about time varying parameters to provide filtering prediction or fixed lag smoothing estimates for arbitrary lags the method is based on a transformation of the adaptation problem into a wiener filter design problem the filter works in open loop for slow parameter variations whereas a time varying closed loop has to be considered for fast variations in the latter case the filter design is performed iteratively the general form of the solution at each iteration is obtained by a bilateral diophantine polynomial matrix equation and a spectral factorization for white gradient noise the diophantine equation has a closed form solution further structural constraints result in very simple design equations under certain model assumptions the wiener designed adaptation laws reduce to lms adaptation compared with kalman estimators the channel tracking performance becomes nearly the same in mobile radio applications whereas the complexity is in general much lower

generalized inverses and their application to applied probability problems the main aim of this paper is to examine the applicability of generalized inverses to a wide variety of problems in applied probability where a markov chain is present either directly or indirectly through some form of imbedding by characterizing all generalized inverses of i p where p is the transition matrix of a finite irreducible discrete time markov chain we are able to obtain general procedures for finding stationary distributions moments of the first passage time distributions and asymp totic forms for the moments of the occupation time random variables it is shown that all known explicit methods for examining these problems can be expressed in this generalized inverse framework more generally in the context of a markov renewal process setting the aforementioned problems are also examined using generalized inverses of i p as a special case markov chains in continuous time are considered and we show that the generalized inverse technique can be applied directly to the infinitesimal generator of the process instead of to i p where p is the transition matrix of the discrete time jump markov chain

stability of recursive stochastic tracking algorithms first the paper gives a stability study for the random linear equation x n 1 i a n x n it is shown that for a quite general class of random matrices a n of interest the stability of such a vector equation can be guaranteed by that of a corresponding scalar linear equation for which various results are given without requiring stationary or mixing conditions then these results are applied to the main topic of the paper i e to the estimation of time varying parameters in linear stochastic systems giving a unified stability condition for various tracking algorithms including the standard kalman filter least mean squares and least squares with forgetting factor

analysis of mean square error and transient speed of the lms adaptive algorithm for the least mean square lms algorithm we analyze the correlation matrix of the filter coefficient estimation error and the signal estimation error in the transient phase as well as in steady state we establish the convergence of the second order statistics as the number of iterations increases and we derive the exact asymptotic expressions for the mean square errors in particular the result for the excess signal estimation error gives conditions under which the lms algorithm outperforms the wiener filter with the same number of taps we also analyze a new measure of transient speed we do not assume a linear regression model the desired signal and the data process are allowed to be nonlinearly related the data is assumed to be an instantaneous transformation of a stationary markov process satisfying certain ergodic conditions

estimating time varying parameters by the kalman filter based algorithm stability and convergence convergence and stability properties of the kalman filter based parameter estimator are established for linear stochastic time varying regression models the main features are both the variances and sample path averages of the parameter tracking error are shown to be bounded the regression vector includes both stochastic and deterministic signals and no assumptions of stationarity or independence are requires and the unknown parameters are only assumed to have bounded variations in an average sense

frequency domain analysis of tracking and noise performance of adaptive algorithms in this paper an analysis of the tracking and noise performance of several adaptive algorithms is carried out for the case of model structures with fixed pole positions such structures have previously been proposed as an efficient generalization of the common fir model structure the focus of this work is to analyze the associated tradeoff between noise sensitivity and tracking ability in the frequency domain by illustrating how it is influenced by such things as input and noise spectral densities step size and in particular the choice of the fixed pole locations the latter influence is not described by preexisting analysis but is shown here to be amenable to attack by a particular class of orthonormal bases

variants of a classic traub s result we present a generalization of a modification of a classic traub s result which allows constructing from a given iterative method with order of convergence p efficient iterative methods with order of convergence at least pq 1 q p 1 q n

extension of modal analysis to linear time varying systems linear time varying ltv systems have been often dealt with on a case by case basis many well developed concepts and analytic methods of linear time invariant lti systems cannot be applied to ltv systems for example the conventional definition of modal parameters is invalid for ltv systems the first part of this paper explores the possibility of extending the modal concept of ltv systems the discrete time state space model is used to represent ltv systems by analogy to lti systems the pseudo modal parameters are defined using the eigenvalues of the discrete time state transition matrix the paper shows that the pseudo modal parameters preserve certain properties of the conventional modal parameters defined for lti systems the second part of the paper extends a previously developed algorithm to identify the pseudo modal parameters using forced responses and forcing inputs for a general ltv system the input and output hankel matrices formed by an ensemble of data satisfy a matrix facotrization relation the key step of the method is to modify the output matrix in such a way that the range space of the observability matrix can be extracted a robotic manipulator with varying inertia links is used as an example the first part of the numerical simulation illustrates the applications of the pseudo modal parameters the second part of the simulation tests the identification algorithm under different conditions several practical issues are addressed

second order convergence analysis of stochastic adaptive linear filtering the convergence of an adaptive filtering vector is studied when it is governed by the mean square error gradient algorithm with constant step size we consider the mean square deviation between the optimal filter and the actual one during the steady state this quantity is known to be essentially proportional to the step size of the algorithm however previous analyses were either heuristic or based upon the assumption that successive observations were independent which is far from being realistic actually in most applications two successive observation vectors share a large number of components and thus they are strongly correlated in this work we deal with the case of correlated observations and prove that the mean square deviation is actually of the same order or less than the step size of the algorithm this result is proved without any boundedness or barrier assumption for the algorithm as it has been done previously in the literature to ensure the nondivergence our assumptions are reduced to the finite strong memory assumption and the finite moments assumption for the observation they are satisfied in a very wide class of practical applications

a discretization scheme for some conservative problems we approximate a locally unique solution of an equation in banach spaces using a newton like method of r order three then we apply this method to obtain an existence uniqueness result for a basic conservative problem given by a nonlinear boundary value problem next by means of a discretization method we approximate the solution of the conservative problem

tracking and identification of regime switching systems using binary sensors this work is concerned with tracking and system identification for time varying parameters the parameters are markov chains and the observations are binary valued with noise corruption to overcome the difficulties due to the limited measurement information wonham type filters are developed first then based on the filters two popular estimators namely mean squares estimator msq and maximum posterior map estimator are constructed for the mean squares estimator we derive asymptotic normality in the sense of weak convergence and in the sense of strong approximation the asymptotic normality is then used to derive error bounds when the markov chain is infrequently switching we derive error bounds for map estimators when the markovian parameters are fast varying we show that the averaged behavior of the parameter process can be derived from the stationary measure of the markov chain and that can be estimated using empirical measures upper and lower error bounds on estimation errors are also established

towards understanding the capability of adaptation for time varying systems the information uncertainty coupled with the structural complexity is more pertinent than the rate of parameter changes in characterizing the capability of adaptation for time varying systems

mixing times with applications to perturbed markov chains abstract a measure of the mixing time or time to stationarity in a finite irreducible discrete time markov chain is considered the statistic i j 1 m m ij j where j is the stationary distribution and mij is the mean first passage time from state i to state j of the markov chain is shown to be independent of the initial state i so that i for all i is minimal in the case of a periodic chain yet can be arbitrarily large in a variety of situations an application considering the effects perturbations of the transition probabilities have on the stationary distributions of markov chains leads to a new bound involving for the 1 norm of the difference between the stationary probability vectors of the original and the perturbed chain when is large the stationary distribution of the markov chain is very sensitive to perturbations of the transition probabilities

a class of subspace model identification algorithms to identify periodically and arbitrarily time varying systems abstract subspace model identification algorithms that allow the identification of a linear time varying ltv state space model from an ensemble set of input output measurements are presented in this paper each pair of input and output sequences in this ensemble is recorded when the underlying system to be identified undergoes the same time varying behavior the algorithms operate directly on the available ensemble of input output data and are a generalization of the recently proposed multivariable output error state space moesp class of algorithms to this ensemble type of identification problems a special case is considered in this paper where the repetition of this time varying behavior is intrinsic namely in periodically time varying systems an example of identifying a multirate sampled data system from a recorded input and output sequence demonstrates some of the capabilities of the presented subspace model identification algorithms

performance analysis of general tracking algorithms a general family of tracking algorithms for linear regression models is studied it includes the familiar least mean square gradient approach recursive least squares and kalman filter based estimators the exact expressions for the quality of the obtained estimates are complicated approximate and easy to use expressions for the covariance matrix of the parameter tracking error are developed these are applicable over the whole time interval including the transient and the approximation error can be explicitly calculated

continuous time tracking algorithms involving two time scale markov chains this work is concerned with least mean squares lms algorithms in continuous time for tracking a time varying parameter process a distinctive feature is that the true parameter process is changing at a fast pace driven by a finite state markov chain the states of the markov chain are divisible into a number of groups within each group the transitions take place rapidly among different groups the transitions are infrequent introducing a small parameter into the generator of the markov chain leads to a two time scale formulation the tracking objective is difficult to achieve nevertheless a limit result is derived yielding algorithms for limit systems moreover the rates of variation of the tracking error sequence are analyzed under simple conditions it is shown that a scaled sequence of the tracking errors converges weakly to a switching diffusion in addition a numerical example is provided and an adaptive step size algorithm developed

on adaptive stabilization of time varying stochastic systems the basic stability issue of time varying stochastic systems under adaptive control is studied a difficulty arising from treating the stochastic case as compared to the deterministic case is the lack of an a priori upper bound on the sample paths of the random noise sequence a projected gradient algorithm with small stepsize is used avoiding possible large deviations of the estimates it is shown that if the unknown parameters vary slowly in some sense then an adaptive control law can be designed so that the closed loop system is stable issues of performance and robustness are also discussed

generalized adaptive notch and comb filters for identification of quasi periodically varying systems the problem of identification tracking of quasi periodically varying real valued systems is considered this problem is a generalization to the system case of a classical signal processing task of either elimination or extraction of nonstationary sinusoidal signals buried in noise the solution is based on the exponentially weighted basis function ewbf approach the proposed algorithms are capable of tracking slow changes in system frequencies which means that not only the expansion coefficients in the basis function description of the analyzed system but also the basis functions themselves are adjusted in an adaptive manner first assuming that the system frequencies are known and constant the running basis and fixed basis variants of the ewbf algorithm are derived and their relationship to the classical notch filter with constrained poles and zeros is established next the frequency adaptive versions of both algorithms are obtained using the gradient search and recursive prediction error principles respectively finally the interrelated frequencies case is analyzed and two additional parameter tracking algorithms generalized adaptive comb filters are derived

an improvement of the euler chebyshev iterative method abstract we present a new method for the computation of the solutions of nonlinear equations when it is necessary to use high precision we improve the euler chebyshev iterative method which is a generalization of an improvement of newton s method a symbolic computation allows us to find the best coefficients respect to the local order of convergence the adaptation of the strategy presented here gives an additional iteration function with an additional evaluation of the function it provides a lower cost if we use adaptive multi precision arithmetics the numerical results computed using this system with a floating point representing a maximum of 210 decimal digits support this theory

variances of first passage times in a markov chain with applications to mixing times jm m 1 as a measure of the mixing time or time to stationarity in a finite irreducible discrete time markov chain with stationary distribution pj and mij as the mean first passage time from state i to state j of the markov chain this was shown to be independent of the initial state i with i for all i minimal in the case of a periodic chain yet can be arbitrarily large in a variety of situations in this paper we explore the variance of the mixing time v i starting in state i the v i are shown to depend on i and an exploration of recommended starting states given knowledge of the transition probabilities is considered as a preamble a study of the computation of second moments of the mixing times m ij 2 and the variance of the first passage times in a discrete time markov chain is carried out leading to some new results