MA estimation in polynomial time

Abstract

The parameter estimation of moving-average (MA) signals from second-order statistics was deemed for a long time to be a di cult nonlinear problem for which no computationally convenient and reliable solution was possible. In this paper we show how the problem of MA parameter estimation from sample covariances can be formulated as a semide nite program which can be solved in polynomial time as e ciently as a linear program. Two methods are proposed which rely on two speci c (over)parametrizations of the MA covariance sequence, whose use makes the minimization of the covariance tting criterion a convex problem. The MA estimation algorithms proposed here are computationally fast, statistically accurate, and reliable (i.e. they \never" fail). None of the previously available algorithms for MA estimation (methods based on higher-order statistics included) shares all these desirable properties. Our methods can also be used to obtain the optimal least squares approximant of an invalid (estimated) MA spectrum (that takes on negative values at some frequencies), which was another long-standing problem in the signal processing literature awaiting a satisfactory solution.

Jeffrey's divergence between autoregressive moving-average processes

“…In this latter case and for the noisefree case, the AR parameters can be estimated in many ways: YuleWalker (YW) equations, correlation method, adaptive filtering, etc. Concerning the MA parameter estimations, one can use the Durbin algorithm, the approach combining the inverse Fourier transform of the inverse of the MA power spectral density (PSD) and the YW equations [6], the "vocariance" methods [7] [8], the one based on higher-order statistics [9], the covariance fitting approaches [10] and the spectral factorization based on the estimation of the outer factor in the PSD such as [11]. In the noisy case, adaptive filters such as the γ-LMS [12] and the ρ-LMS [13] and extended Kalman filter can estimate the AR parameters from noisy observations.…”

“…al. suggest combining approaches that were initially proposed by Davila [15] for the AR-parameter estimations and Stoica [10] for the MA-parameter estimations.…”

Modified Durbin Method for Accurate Estimation of Moving-Average Models

“…However, the approximate nonlinear maximum likelihood estimators failed to converge in some situations [11]. Recently, the drawbacks of all the existing methods for MA estimation have been summarized [12]. For a long time, there has been no decisive answer in the literature about which MA method is to be preferred.…”

“…In practice, estimates of the finite-order AR models have to be used. A common choice has been to use the parameters of the best predicting AR model order, or an AR model order that depends on the number of MA parameters that is estimated [12]. The asymptotic consequences of using a finite-order AR model have been investigated for a MA(1) process [14].…”

“…Different AR methods and intermediate AR orders have been used in Durbin's method [12], [15], [16]. The best value to be chosen for the intermediate AR order has been investigated [15].…”