# Bayesian spectral density estimation using P-splines with quantile-based knot placement

## TL;DR

This paper introduces a Bayesian spectral density estimation method using P-splines with a novel, data-driven knot placement strategy, improving computational efficiency while maintaining accuracy in capturing spectral features.

## Contribution

It proposes a new Bayesian spectral density estimation approach with a data-driven knot placement scheme using P-splines, reducing computational costs compared to existing methods.

## Key findings

- Accurately estimates spectral peaks with the new knot placement.
- Reduces computational complexity significantly.
- Maintains flexibility in modeling sharp spectral features.

## Abstract

This article proposes a Bayesian approach to estimating the spectral density of a stationary time series using a prior based on a mixture of P-spline distributions. Our proposal is motivated by the B-spline Dirichlet process prior of Edwards et al. (2019) in combination with Whittle's likelihood and aims at reducing the high computational complexity of its posterior computations. The strength of the B-spline Dirichlet process prior over the Bernstein-Dirichlet process prior of Choudhuri et al. (2004) lies in its ability to estimate spectral densities with sharp peaks and abrupt changes due to the flexibility of B-splines with variable number and location of knots. Here, we suggest to use P-splines of Eilers and Marx (1996) that combine a B-spline basis with a discrete penalty on the basis coefficients. In addition to equidistant knots, a novel strategy for a more expedient placement of knots is proposed that makes use of the information provided by the periodogram about the steepness of the spectral power distribution. We demonstrate in a simulation study and two real case studies that this approach retains the flexibility of the B-splines, achieves similar ability to accurately estimate peaks due to the new data-driven knot allocation scheme but significantly reduces the computational costs.