A wavelet denotes a wave-like, generally localized, oscillating function, equipped with certain relationships across scales. A wavelet transformation describes a representation of data, decomposed onto a set of different wavelet functions, often forming a basis or a frame.
Wavelets can be informally described as localised, oscillatory functions designed to possess attractive properties that Fourier-related functions (sines, cosines) do not enjoy. They are generated by dilation, translation, rotation of a reduced set of functions. Their discretization yields sets of vectors forming bases or frames. Their design permits the decomposition of data at different scales or resolutions, with some attractive features for time-series statistical analysis, e.g. sparse representations, separation of polynomial trends, change detection, decorrelation, noise whitening. They have been used for nonparametric regression, in conjunction with efficient estimation techniques, notably the Stein's Unbiased Risk Estimator (SURE) procedure.
- Wavelet Methods in Statistics with R, G. Nason, 2008, Springer
- Wavelet methods in statistics: Some recent developments, A. Antoniadis, 2007
- Wavelets and Statistics, A. Antoniadis, G. Oppenheim, 1995, Springer
