Decomposing the spectral density function of an ARMA process
Abstract
Quasi-periodic oscillations are characterized by fitting Lorentzian functions to a nonparametric estimate of the power spectral density, a two-step procedure that separates estimation from modeling and does not accommodate measurement error. We develop a parametric alternative based on autoregressive moving-average (ARMA) processes observed with measurement error. We derive an exact decomposition of the ARMA spectral density into real-valued components, one for each class of autoregressive roots under complex conjugation, plus a polynomial term when the moving-average order is at least the autoregressive order. Each component is a Wrapped Cauchy kernel, and for complex roots it is accompanied by a conjugate-Poisson term that makes the peak asymmetric and displaces its maximum from the frequency given by the argument of the root. We obtain closed-form expressions for the location, height, width, and asymmetry of each peak, and a matching variance decomposition. Parameters are estimated by maximum likelihood through the Kalman filter, and all descriptors follow by plug-in. Simulations show that the closed-form peak location is recovered without appreciable bias. An X-ray binary light curve illustrates the methodology.
Related
- The pseudo-Riemannian spectral zeta function density on scattering manifolds
- Non-equilibrium density fluctuations for the stirring process with weak boundary dynamics
- Spectral comparison on Specht modules and mixing of the interchange process
- Logistic Gaussian process density regression: a generalized Bayesian approach
- Malliavin smoothness and density estimates for the third-order Hermite process
BibTeX
@misc{sotovasquez2026decomposing,
title = {{Decomposing the spectral density function of an ARMA process}},
author = {Darlin Soto-Vásquez and Francisco Cuevas-Pacheco and Giovanni Motta and Malgorzata Sobolewska},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14384},
eprint = {2610.14384},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14384},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Decomposing the spectral density function of an ARMA process
AU - Darlin Soto-Vásquez
AU - Francisco Cuevas-Pacheco
AU - Giovanni Motta
AU - Malgorzata Sobolewska
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14384
UR - https://arxiv.org/abs/2610.14384
AB - Quasi-periodic oscillations are characterized by fitting Lorentzian functions to a nonparametric estimate of the power spectral density, a two-step procedure that separates estimation from modeling and does not accommodate measurement error. We develop a parametric alternative based on autoregressive moving-average (ARMA) processes observed with measurement error. We derive an exact decomposition of the ARMA spectral density into real-valued components, one for each class of autoregressive roots under complex conjugation, plus a polynomial term when the moving-average order is at least the autoregressive order. Each component is a Wrapped Cauchy kernel, and for complex roots it is accompanied by a conjugate-Poisson term that makes the peak asymmetric and displaces its maximum from the frequency given by the argument of the root. We obtain closed-form expressions for the location, height, width, and asymmetry of each peak, and a matching variance decomposition. Parameters are estimated by maximum likelihood through the Kalman filter, and all descriptors follow by plug-in. Simulations show that the closed-form peak location is recovered without appreciable bias. An X-ray binary light curve illustrates the methodology.
ER -