Spectral representation and estimation for locally stationary wavelet processes

Abstract

This article develops a wavelet decomposition of a stochastic process which parallels a time{localized Cram er (Fourier) spectral representation. We provide a time{scale instead of a time{frequency decomposition and, hence, instead of thinking as scale in terms of \inverse frequency" we start from genuine time{scale building blocks or \atoms". Using this class of locally stationary wavelet (LSW) processes, a doubly{indexed array of processes fX t;T g t=1;:::;T ; T 1, we develop a theory for the estimation of the \evolutionary wavelet spectrum". Our asymptotics are based on rescaling in time{ location which allows us to perform rigorous estimation theory starting from a single stretch of observations of fX t;T g. This evolutionary wavelet spectrum measures the local power in the variance{covariance decomposition of the process fX t;T g at a certain

Wavelet Processes and Adaptive Estimation of the Evolutionary Wavelet Spectrum

“…Our LSW model (7) therefore delivers a time±scale decomposition which parallels the time±frequency decomposition of Dahlhaus (1997). (See remark 4.18 in von Sachs et al (1997) for further connections). Our model is similar to Dahlhaus's model in two other ways: the EWS is de®ned only for z P 0, 1, as boundaries do not make sense in this model and the EWS is uniquely de®ned (in terms of localized autocovariance, shown by theorem 1).…”

“…For convenience our indices will now run from 1 to I rather than from À1 to ÀI. Using straightforward algebra we can explicitly derive formulae for entries of A from É j given by expression (13) (see von Sachs et al (1997) for more detail). The elements of A are A jj 2 2j 5 3 Â 2 j ,…”

“…There is strong evidence that the above Haar proof can be extended to other Daubechies compactly supported wavelets with orders N > 1. We brie¯y outline this extension; see von Sachs et al (1997) for further details. The key to the extension is the relationship between A and B to equivalent A* and B* arising from the continuous autocorrelation function built from time continuous Daubechies wavelets.…”

Blind Image Deconvolution

“…(7.14) is not valid in the transform domain since the convolution matrix and DWT matrix do not commute, i.e., W y ¤ K W x C n; where W is the DWT matrix. This problem can be overcome by using the stationary wavelet transform (SWT) [138]. SWT is an undecimated version of DWT which involves filtering the rows and columns using high pass and low pass filters.…”

The locally stationary dual-tree complex wavelet model

“…The increased number of wavelet coefficients permits a multiscale sample autocovariance of sorts to be computed about any point in space. This is problematic in the decimated case since the periodogram is computed over the dyadic grid, and as Nason et al (2000) notes, the decimated LSW model fails to accommodate all stationary processes (von Sachs et al 1998). This contrasts with the non-decimated LSW model, which can describe any stationary process with finite integrated autocovariance.…”