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7. Transforms in Statistics

Brani Vidakovic

It is not an overstatement to say that statistics is based on various transformations of data. Basic statistical summaries such as the sample mean, variance, z-scores, histograms, etc., are all transformed data. Some more advanced summaries, such as principal components, periodograms, empirical characteristic functions, etc., are also examples of transformed data. To give a just coverage of transforms utilized in statistics will take a size of a monograph. In this chapter we will focus only on several important transforms with the emphasis on novel multiscale transforms (wavelet transforms and its relatives).

Transformations in statistics are utilized for several reasons, but unifying arguments are that transformed data

(i)
are easier to report, store, and analyze,
(ii)
comply better with a particular modeling framework, and
(iii)
allow for an additional insight to the phenomenon not available in the domain of non-transformed data.

For example, variance stabilizing transformations, symmetrizing transformations, transformations to additivity, Laplace, Fourier, Wavelet, Gabor, Wigner-Ville, Hugh, Mellin, transforms all satisfy one or more of points listed in (i-iii).

We emphasize that words transformation and transform are often used interchangeably. However, the semantic meaning of the two words seem to be slightly different. For the word transformation, the synonyms are alteration, evolution, change, reconfiguration. On the other hand, the word transform carries the meaning of a more radical change in which the nature and/or structure of the transformed object are altered. In our context, it is natural that processes which alter the data leaving them unreduced in the same domain should be called transformations (for example Box-Cox transformation) and the processes that radically change the nature, structure, domain, and dimension of data should be called transforms (for example Wigner-Ville transform).

In this chapter we focus mainly on transforms providing an additional insight on data. After the introduction discussing three examples, several important transforms are overviewed. We selected discrete Fourier, Hilbert, and Wigner-Ville transforms, discussed in Sect. 7.2, and given their recent popularity, continuous and discrete wavelet transforms discussed in Sects. 7.3 and 7.4.



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