The Wavelet Transform in Multivariate Data Analysis
Fionn D. Murtagh, Alexandre Aussem, O. J. W. F. Kardaun · COMPSTAT · 1996
Data analysis, for exploratory purposes, or prediction, is usually preceded by various data transformations and recoding. The wavelet transform offers a particularly appealing data transformation, as a preliminary to data analysis, for de-noising, smoothing, etc., in a natural and integrated way. For an introduction to the orthogonal wavelet transform, see e.g. Strang (1989), Daubechies (1992). We consider the signal’s detail signal, ξ m , at resolution levels, m. With the residual, smoothed image, x 0, we have the wavelet transform of an input signal x as follows. Define ξ as the row-wise juxtaposition of all {ξ m } and x 0, and consider W given by $${W_x} = \xi = {\left\{ {{\xi _{N - 1}},...,{\xi _0},{x_0}} \right\}^T}$$ (1) with W T W = I (the identity matrix). Examples of these orthogonal wavelets are the Daubechies family, and the Haar wavelet transform (see Press et al., 1992; Daubechies, 1992). Computational time is O(n) for an n-length input data set.