Bayesian nonparametric analysis of multivariate time series: A matrix Gamma Process approach

Abstract

While there is an increasing amount of literature about Bayesian time series analysis, only few Bayesian nonparametric approaches to multivariate time series exist. Most methods rely on Whittle's Likelihood, involving the second order structure of a stationary time series by means of its spectral density matrix. This is often modeled in terms of the Cholesky decomposition to ensure positive definiteness. However, asymptotic properties such as posterior consistency or posterior contraction rates are not known. A different idea is to model the spectral density matrix by means of random measures. This is in line with existing approaches for the univariate case, where the normalized spectral density is modeled similar to a probability density, e.g. with a Dirichlet process mixture of Beta densities. In this work we present a related approach for multivariate time series, with matrix-valued mixture weights induced by a Hermitian positive definite Gamma process. The proposed procedure is shown to perform well for both simulated and real data. Posterior consistency and contraction rates are also established.MSC 2010 subject classifications: Primary 62G20, 62M10, 60G57; secondary 60G15.

Posterior consistency for the spectral density of non‐Gaussian stationary time series

“…Besides the above examples, Shen and Ghosal (2015) also discussed priors based on other basis functions such as Fourier trigonometric basis and polynomial basis. Furthermore, our future research will also aim to extend the consistency proof to the estimation of the Hermitian positive definite spectral density matrix of a non‐Gaussian multivariate time series using a matrix‐Gamma process prior (Meier, 2018).…”

“…Remark The practice of using a (relatively) compact set as the parameter space in (frequentist/Bayesian/parametric/ nonparametric) inference of spectral density functions has a long history (e.g. Hannan, 1973; Taniguchi, 1981; Tamaki, 2008; Bardet et al, 2008; Meier et al, 2020). The reason we adopt this practice is that the proof of the uniform LLNs (Assumption 3) on an infinite‐dimensional space will benefit substantially from the application of some Arzelà‐Ascoli type theorems (e.g.…”

“…The practice of using a (relatively) compact set as the parameter space in (frequentist/Bayesian/parametric/ nonparametric) inference of spectral density functions has a long history (e.g. Hannan, 1973;Taniguchi, 1981;Tamaki, 2008;Bardet et al, 2008;Meier et al, 2020).…”

“…The difficulty of extending the consistency result to non‐Gaussian time series is due to the fact that the contiguity argument, on which prior works relied heavily, is no longer viable for general non‐Gaussian time series as shown in Meier (2018, Section 4.3). Therefore, a novel method is needed to establish posterior consistency under mild conditions on the time series.…”

On Quasi-Infinitely Divisible Random Measures

“…In comparison with usual truncation procedures (see Argiento et al (2016) 2021)) our procedure takes into account the fixed atomic component in a non-trivial way. In particular, by Theorem 4.2 the fixed atomic component (and so the whole CRM) has a simple Lévy-Khintchine formulation, and in some cases it also has an explicit Laplace transform (see Corollary 4.4), which apart from providing flexibility and improving computability it is also useful in nonparametric Bayesian spectral estimation, see Tobar (2018) and Meier et al (2020), where the Fourier transform of the law of processes and/or random measures plays a key role, and in moment-matching criterion for quantifying approximations, see Arbel and Prünster (2017).…”