Withdrawn - 13. Decomposing the Spectral Density of ARMA Models to Describe Quasi-Periodic Oscillations in X-ray Binary Systems

Astronomers aim to model the power spectral density (PSD) of X-ray binary systems, focusing on quasi-periodic oscillations (QPOs), narrow peaks in the frequency domain that reveal processes near compact objects such as black holes or neutron stars. A common method involves computing the periodogram and fitting a sum of Lorentzian functions to estimate spectral peaks (e.g., Uttley et al., 2014; Pawar et al., 2015; Malzac et al., 2018). While widely used, this two-step approach separates estimation from modeling, limiting coherence and interpretability.

We propose a statistically grounded alternative using autoregressive moving average (ARMA) models to represent the PSD directly. We show that the spectral density of an ARMA(p,q) process can be analytically decomposed into s component functions, where s depends on the nature of the roots of the autoregressive polynomial-real or complex. Each component corresponds to a spectral peak, and we derive closed-form expressions for their frequencies, offering an interpretable alternative to Lorentzian fitting. To illustrate this methodology, we estimate an ARMA model using its state-space form and the Kalman filter. We apply it to data from the Rossi X-ray Timing Explorer (RXTE) light curve of the binary system XTE J1550–564. The resulting decomposition captures both broadband variability and QPO features, showing the model's ability to represent complex astrophysical signals in a coherent way.

Though motivated by astrophysics, this framework applies to fields where spectral peaks matter-such as neuroscience, geophysics, and engineering-offering a flexible, analytically grounded approach to frequency-domain analysis. This work, developed with Dr. Giovanni Motta (Columbia University) and Dr. Malgorzata Sobolewska (Smithsonian Astrophysical Observatory), reflects a multidisciplinary, women-led collaboration advancing statistical innovation at the interface of data science and astronomy.

Time Series Analysis

Spectral Density

ARMA Models

Frequency-Domain Analysis

Quasi-Periodic Oscillations

Statistical Signal Processing 

Expert

Knowledge

Women in Statistics and Data Science 2025