Non-equispaced B-spline wavelets

TL;DR

This paper introduces a method for constructing non-equispaced B-spline wavelet transforms, extending existing wavelet families, and provides new insights into their statistical applications, especially regarding bias and variance control.

Contribution

It presents a novel construction of wavelet transforms on non-equispaced grids using lifting steps, and offers new frameworks for bias correction and variance management in statistical contexts.

Findings

Extended wavelet construction to non-equispaced data.

Identified bias issues with non-interpolating wavelets on irregular grids.

Developed a variance control framework for wavelet-based statistical analysis.

Abstract

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.