Multivariate Bayesian P-spline estimation of spectral density matrices, with application to LISA TDI noise

2026-07-06

AI Summary

This study proposes a Bayesian approach for estimating the frequency-dependent cross-spectral density matrix of stationary multivariate time series, guaranteeing Hermitian positive definiteness at all frequencies. The method parameterizes the inverse spectral matrix via a frequency-varying Cholesky decomposition and imposes penalized B-spline priors on both diagonal and off-diagonal elements to control smoothness. Robust inference is achieved by combining a coarse-grained Whittle likelihood with safe-Bayes η-tempering, while variational initialization enhances posterior stability. This work represents the first integration of Bayesian P-splines with Cholesky-based parameterization for multivariate spectral estimation, enabling flexible smooth modeling while preserving positive definiteness. Experiments demonstrate accurate recovery of spectral structure in synthetic VAR(2) data with credible intervals achieving nominal coverage, and on LISA TDI noise data, the full multivariate model substantially outperforms diagonal assumptions—reducing relative integrated squared error (RISE) by over an order of magnitude, especially under asymmetric noise conditions.

Abstract

We present a Bayesian P-spline method for estimating the frequency-dependent cross-spectral density matrix of stationary multivariate time series. The inverse spectral matrix is parametrised through its frequency-varying Cholesky decomposition, which guarantees Hermitian positive definiteness at every frequency. Each real log-diagonal entry and each real and imaginary off-diagonal entry is given an independent penalised B-spline prior that controls smoothness. Inference uses a blocked, coarse-grained Whittle likelihood with safe-Bayes $η$-tempering to stabilise posterior calibration, sampled by the No-U-Turn Sampler from a variational initialisation. On synthetic VAR(2) benchmarks with known ground truth, the method recovers both diagonal and cross-spectral structure, attains near-nominal credible-interval coverage, and achieves a relative integrated squared (Frobenius) error (RISE) that decreases with sample size. We then apply the method to publicly released simulated LISA time-delay interferometry (TDI) data in two noise configurations. In the idealised symmetric case, the full multivariate model and a reduced model that assumes a diagonal AET noise covariance agree to within $\sim10^{-3}$ in RISE. Under realistic noise that is asymmetric across the six Movable Optical Sub-Assemblies (MOSAs), the AET-diagonal assumption fails by more than an order of magnitude in RISE ($\sim\!3.3\!\times\!10^{-2}$ versus $\sim\!10^{-3}$), whereas the full multivariate model recovers the cross-spectral structure.

Problem

• spectral density matrix
• multivariate time series
• cross-spectral estimation
• LISA TDI noise
• non-diagonal covariance

Innovation

• Bayesian P-splines
• spectral density matrix
• Cholesky decomposition
• multivariate time series
• LISA TDI noise