Locally adaptive estimation of evolutionary wavelet spectra

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.Comment: Published in at http://dx.doi.org/10.1214/07-AOS524 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Costationarity of Locally Stationary Time Series

“…Further explanation, including wavelets, can be found in Nason (2008). Jumps in the spectrum are permitted by the generalized extension developed by Van Bellegem and von Sachs (2008) by extending S = p to functions of bounded variation. We adopt this here.…”

“…be a LSW process with spectrum p X (z, j) = |W j (z)| 2 , discrete nondecimated wavelets {ψ j,k }, and innovations {ξ j,k }, satisfying the conditions from Van Bellegem and von Sachs (2008), page 1883, Definition 1. Let a : [0, 1] → R be a function of bounded total variation with constant R 3 , and let (a(t), a t,T ) be a close pair with constant R 2 .…”

“…Let a : [0, 1] → R be a function of bounded total variation with constant R 3 , and let (a(t), a t,T ) be a close pair with constant R 2 . Let the quantity p Y (z, j) = |a(z)| 2 p X (z, j) satisfy the LSW spectrum conditions from Van Bellegem and von Sachs (2008) for all j = 1, 2, . .…”

The locally stationary dual-tree complex wavelet model

“…For example, work has focussed on: variants of the smoothness constraints placed on the local spectrum function, such as the work by van Bellegem and von Sachs (2008), Fryzlewicz and Nason (2006), and Nason and Stevens (2015); confidence intervals for the empirical local covariance as recently derived by Nason (2013); and changepoint estimation, such as the work of Killick et al (2013) and Cho and Fryzlewicz (2015).…”

Frequency Domain Tests of Semiparametric Hypotheses for Locally Stationary Processes

“…We mention here among others the contributions by Neumann and von Sachs (1997), Dahlhaus et al (1999), Chang and Morettin (1999), van Bellegem and Dahlhaus (2006), Dahlhaus and Polonik (2006) and van Bellegem and von Sachs (2008). Forecasting problems for non-stationary time series have been considered by Fryzlewicz et al (2003).…”