Locally adaptive estimation of evolutionary wavelet spectra

TL;DR

This paper introduces a wavelet-based model for local stationarity that captures rapidly changing spectral densities and proposes an adaptive estimator for the time-varying spectrum, enhancing analysis of non-stationary processes.

Contribution

It extends the class of locally stationary wavelet processes and defines a novel wavelet spectrum for processes with abrupt spectral changes, along with an adaptive estimation method.

Findings

The estimator performs well in both homogeneous and inhomogeneous spectral regions.

The model captures sudden changes in spectral density effectively.

The wavelet spectrum provides a localized representation of autocovariance.

Abstract

We introduce a wavelet-based model of local stationarity. This model enlarges the class of locally stationary wavelet processes and contains processes whose spectral density function may change very suddenly in time. A notion of time-varying wavelet spectrum is uniquely defined as a wavelet-type transform of the autocovariance function with respect to so-called autocorrelation wavelets. This leads to a natural representation of the autocovariance which is localized on scales. We propose a pointwise adaptive estimator of the time-varying spectrum. The behavior of the estimator studied in homogeneous and inhomogeneous regions of the wavelet spectrum.