On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

Abstract

Abstract-Bayesian estimation problems involving Gaussian distributions often result in linear estimation techniques. Nevertheless, there are no general statements as to whether the linearity of the Bayesian estimator is restricted to the Gaussian case. The two common strategies for non-Gaussian models are either finding the best linear estimator or numerically evaluating the Bayesian estimator by Monte Carlo methods. In this paper, we focus on Bayesian interpolation of non-Gaussian first-order autoregressive (AR) processes where the driving innovation can admit any symmetric infinitely divisible distribution characterized by the Lévy-Khintchine representation theorem. We redefine the Bayesian estimation problem in the Fourier domain with the help of characteristic forms. By providing analytic expressions, we show that the optimal interpolator is linear for all symmetric -stable distributions. The Bayesian interpolator can be expressed in a convolutive form where the kernel is described in terms of exponential splines. We also show that the limiting case of Lévy-type AR(1) processes, the system of which has a pole at the origin, always corresponds to a linear Bayesian interpolator made of a piecewise linear spline, irrespective of the innovation distribution. Finally, we show the two mentioned cases to be the only ones within the family for which the Bayesian interpolator is linear.

Statistical optimality of Hermite splines

“…The spline interpolation algorithm is therefore optimal for the estimation of a wide family of stationary random signals. Similar results have been demonstrated by [7], [8]: the optimal (MMSE) interpolator for first-order Lévy processes is the piecewise linear spline. Note that these processes, defined as s such that Ds = w, are non-stationary but (wide-sense) self-similar.…”

“…Our contribution however differs in two ways: first, from the fact that we know the samples s (k) in addition to the s(k), and then from the class of processes we study which are non-stationary (unlike [6], where the stationary case is investigated) and second-order (unlike [7], where first-order Lévy processes are considered).…”

Compressibility of symmetric-α-stable processes

“…When L = D, s is a Lévy process. In general, when L is of the form (22), s is an AR(N ) process (autoregressive process of order N ) [15], [16].…”

WNN-Based Prediction of Security Situation Awareness for the Civil Aviation Network

“…Hwang et al [7] combined the Bayesian network with genetic algorithm (GA) to deal with uncertainty and dynamic properties in the real world. Amini et al [2] redefined the Bayesian estimation problem in the Fourier domain with the help of characteristic forms. The backpropagation (BP) algorithm is proposed to solve the training of a multilayer neural network.…”