# Spectral Inference under Complex Temporal Dynamics

## TL;DR

This paper introduces a unified framework for inferring evolutionary Fourier power spectra in complex, possibly nonlinear, locally stationary processes, providing statistically valid confidence regions and versatile tools for time-frequency analysis.

## Contribution

It develops a comprehensive theory and bootstrap-based methodology for constructing confidence regions for spectral densities in non-stationary processes, applicable to various inference tasks.

## Key findings

- Constructs asymptotically correct simultaneous confidence regions for spectral densities.
- Proposes a bootstrap method for practical implementation of confidence regions.
- Enables visual and statistical evaluation of time-varying spectral patterns.

## Abstract

We develop unified theory and methodology for the inference of evolutionary Fourier power spectra for a general class of locally stationary and possibly nonlinear processes. In particular, simultaneous confidence regions (SCR) with asymptotically correct coverage rates are constructed for the evolutionary spectral densities on a nearly optimally dense grid of the joint time-frequency domain. A simulation based bootstrap method is proposed to implement the SCR. The SCR enables researchers and practitioners to visually evaluate the magnitude and pattern of the evolutionary power spectra with asymptotically accurate statistical guarantee. The SCR also serves as a unified tool for a wide range of statistical inference problems in time-frequency analysis ranging from tests for white noise, stationarity and time-frequency separability to the validation for non-stationary linear models.