The Estimation of High-contrast Spectra via Iterated Whitening

Abstract

Power spectra are a fundamental tool in data analysis, signal processing, and linear prediction and control. Many who seek to estimate power spectrum are obliged to do so with very little theoretical information of the underlying process and prefer accurate estimates which require as little effort as possible. This work focuses on estimation of the power spectrum of time series data from dynamical systems and stochastic differential equations (SDEs) and attempts to satisfy the preferences above. The method, called iterated whitening (IW) spectral estimation, iteratively builds inexpensive filters that progressively ameliorate the data until the resulting modified data is suitable for accurate spectral estimation. This spectral estimate is then post-processed to return an accurate estimate of the original data. Time series from dynamical systems and SDEs often possess a very large dynamic spectral range which makes them difficult to estimate cheaply and accurately. IW provides a solution to this difficulty. In this dissertation, I discuss some of the issues that the Bartlett estimator has in approximating these ``high-contrast'' spectra by deriving expanded bias and variance formulas. I also showcase another technique for improving a spectral estimator using the method on control variates from Monte Carlo Markov chain theory.

I apply these methods to some time series from a solution to the Kuramoto-Sivashinsky equation which is in spatiotemporal chaos.