Nonparametric Multivariate Spectral Density Estimation via Basis Expansion

In time series analysis, we may analyze the behavior of a multivariate time series in time domain using the autocovariance matrices or in frequency domain using the spectral density matrices. Today spectral analysis of time series is of considerable interest in a variety of fields. Many natural phenomena have frequency-dependent variability quantified through the power spectrum, which contains important information on the underlying physical mechanisms. The multivariate power spectrum is a positive-definite and Hermitian matrix. So due to these constraints, estimating the spectrum of a multivariate time series is more complicated than the univariate one. There are many nonparametric methods for estimating the spectrum of a multivariate time series.
In this paper, we developed an efficient nonparametric method for estimating the multivariate spectral density using basis expansions. Similar to Dai and Guo (2004), Rosen and Stoffer (2007), and Krafty and Collinge (2013), we applied a Cholesky decomposition to the power spectrum to ensure Hermitian positive definiteness, where the real parts and imaginary parts of the Cholesky elements are modeled by basis expansions. The basis coefficients were estimated by minimizing the penalized Whittle likelihood approximation with two different penalties, based on the second derivative of the basis functions and the difference operator. The penalty parameter is selected by minimizing the well-known Akaike information criterion (AIC). To minimize the penalized Whittle likelihood, we applied a Fisher-Scoring algorithm.
In a simulation study, we illustrated our proposed algorithm (NMSDE) using some simulated data and compared its performance with some other alternative approaches. According to the simulation results, our approach is competitive with the method proposed by Krafty and Collinge (2013) which is the state of the art in the Cholesky-based methods. Finally, we applied our approach to a real-world problem, Elnino cycle.

Multivariate time series

Spectral density

Smoothing spline 

Mid-Level

Computational Statistics

Symposium on Data Science and Statistics (SDSS) 2023