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Wavelet methods

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This overview article motivates the use of wavelets in statistics, and introduces the basic mathematics behind the construction of wavelets. Topics covered include the continuous and discrete wavelet transforms, multiresolution analysis and the non-decimated wavelet transform. We describe the basic mechanics of nonparametric function estimation via wavelets, emphasising the concepts of sparsity and thresholding. A simple proof of the mean-square consistency of the wavelet estimator is also included. The article ends with two special topics: function estimation with Unbalanced Haar wavelets, and variance stabilisation via the Haar-Fisz transformation. Wavelets aremathematical functions which, when plotted, resemble “little waves”: that is, they are compactly or almost-compactly supported, and they integrate to zero. This is in contrast to “big waves” – sines and cosines in Fourier analysis, which also oscillate, but the amplitude of their oscillation never changes. Wavelets are useful for decomposing data into “wavelet coefficients”, which can then be processed in a way which depends on the aim of the analysis. One possibly advantageous feature of this decomposition is that in some set-ups, the decomposition will be sparse, i.e. most of the coefficients will be close to zero, with only a few coefficients carrying most of the information about the data. One can imagine obvious uses of this fact, e.g. in image compression. The decomposition is particularly informative, fast and easy to invert if it is performed using wavelets at a range of scales and locations. The role of scale is similar to the role of frequency in Fourier analysis. However, the concept of location is unique to wavelets: as mentioned above, they are localised around a particular point of the domain, unlike Fourier functions. This article provides a self-contained introduction to the applications of wavelets in statistics and attempts to justify the extreme popularity which they have enjoyed in the literature over the past 15 years.

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