A Unified Formulation of Gaussian Versus Sparse Stochastic Processes—Part I: Continuous-Domain Theory

Abstract

We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by some generalized white noise process, which may be Gaussian or not (e.g., Laplace, impulsive Poisson or alpha stable). This allows for a conceptual decoupling between the correlation properties of the process, which are imposed by the whitening operator L, and its sparsity pattern which is determined by the type of noise excitation. The latter is fully specified by a Lévy measure. We show that the range of admissible innovation behavior varies between the purely Gaussian and super-sparse extremes. We prove that the corresponding generalized stochastic processes are well-defined mathematically provided that the (adjoint) inverse of the whitening operator satisfies some Lp bound for p ≥ 1. We present a novel operator-based method that yields an explicit characterization of all Lévy-driven processes that are solutions of constant-coefficient stochastic differential equations (SDE). When the underlying system is stable, we recover the family of stationary CARMA processes, including the Gaussian ones. The approach remains valid when the system is unstable and leads to the identification of potentially useful generalizations of the Lévy processes, which are sparse and non-stationary. Finally, we show that these processes admit a sparse representation in some matched wavelet domain and provide a full characterization of their transform-domain statistics.

MMSE denoising of sparse and non-Gaussian AR(1) processes

“…In this section, we first present our extended class of nonGaussian (or sparse) continuous-time AR(1) processes using the innovation model proposed in [13] and [14]. The model boils down to the linear filtering of a non-Gaussian white noise (innovation) by using a suitable (1st order) shaping filter.…”

“…where w is a continuous-time white Lévy noise [13]. The crucial extension here is that the continuous-time innovation w is not necessarily Gaussian.…”

“…However, in the case that κ = 0 which leads to the generation of Lévy processes, there are some subtle theoretical points that need to be taken care of [13].…”

Interpretation of continuous-time autoregressive processes as random exponential splines

“…We recall here a brief exposition of the construction of CAR processes in the framework of generalized random processes. More details can be found in [9], [14].…”

“…Moreover, it has the following property, allowing for the definition of general CAR processes: Proposition 3 (Theorem 4, [14]). Let w be a white noise with Lévy-Schwartz exponent f (ω) and P a polynomial.…”