Wavelets from a Statistical Perspective

Wavelets from a Statistical Perspective provides a modern, 2nd generation approach to wavelets, extending their applicability to nonlinear, nonparametric processing for piecewise smooth data. It constructs B-spline wavelets on nonequispaced knots and multiscale local polynomial transforms, contrasts the discrete wavelet transform with the continuous version, and finds the right balance between bias and variance in designing a wavelet analysis.

Format: Hardback
Length: 326 pages
Publication date: 23 March 2022
Publisher: Taylor & Francis Ltd

Wavelets, from a Statistical Perspective, presents a modern and second-generation perspective on wavelets, extending far beyond the rigid setting of equispaced, dyadic wavelets in their early days. This book offers methods based on the lifting scheme, enabling researchers to establish wavelets or other multiresolution analyses tailored to their data, spanning from images to scattered data or other irregularly spaced observations. While classical wavelets may differ slightly from other nonparametric methods, this book introduces a multiscale touch to spline, kernel, or local polynomial smoothing procedures, expanding their applicability to nonlinear, nonparametric processing for piecewise smooth data.

Within the book, one chapter constructs B-spline wavelets on nonequispaced knots and multiscale local polynomial transforms. Another chapter explores the link between wavelets and Fourier analysis, a common approach in the classical approach, but without being inevitable. Subsequent chapters contrast the discrete wavelet transform with the continuous version, the nondecimated (or maximal overlap) transform occupying an intermediate position. Finding the right balance between bias and variance is a crucial principle in designing a wavelet analysis through the lifting scheme. Bias and variance also play pivotal roles in nonparametric smoothing within a wavelet framework, particularly in determining effective thresholds or other smoothing parameters.

Numerous illustrations are provided throughout the book, which can be reproduced using the accompanying software available online. The software and exercises serve as valuable starting points for further exploration of the material.

Dimension: 234 x 156 (mm)
ISBN-13: 9781032200675