Wiener–kolmogorov Filtering, Frequency-Selective Filtering, and Polynomial Regression

Abstract

Adaptations of the classical Wiener-Kolmogorov filters are described that enable them to be applied to short nonstationary sequences+ Alternative filtering methods that operate in the time domain and the frequency domain are described+ The frequency-domain methods have the advantage of allowing components of the data to be separated along sharp dividing lines in the frequency domain, without incurring any leakage+ The paper contains a novel treatment of the start-up problem that affects the filtering of trended data sequences+

Trend filtering via empirical mode decompositions

“…This turns the trend filtering problem into a bona fide filtering problem. Viewed in this way, it may be profitable to use Wiener-Kolmogorov filtering (Pollock, 2006) to solve the trend filtering problem. Finally, it is worthwhile to mention that generalized "trend cycles," defined as a "short-term trend [that] generally includes cyclical fluctuations," have also been studied (Alexandrov et al, 2008).…”

Filters, Waves and Spectra

“…Such an approach has been expounded in detail by Pollock (2007). The minimum mean-square-error filters can be derived, variously, according a least-squares criterion, a maximum-likelihood criterion and a conditional expectations criterion, which are equivalent.…”

Bayesian non-parametric signal extraction for Gaussian time series

“…In Pollock (2006) the frequency domains version of the Wiener-Kolmogorov filter is proposed. Assuming y = x + ε where x and ε are two uncorrelated zero-mean Gaussian processes, it is shown that…”

“…The non-parametric attitude of this proposal is naturally inspired by and related to the works of Baxter and King (1999), Christiano and Fitzgerald (2003) and Pollock (2006), where the filters to extract the unobserved components are directly derived from the spectral densities.…”