Fourier analysis of symbolic data: A brief review

We overview and discuss several methods for the Fourier analysis of symbolic data, such as DNA sequences, emphasizing their mutual connections. We consider the indicator sequence approach, the vector and the symbolic autocorrelation methods, and methods such as the spectral envelope, that for each frequency optimize the symbolic-no-numeric mapping to emphasize any periodic data features. We discuss the equivalence or connections between these methods. We show that it is possible to define the autocorrelation function of symbolic data, assuming only that we can compare any two symbols and decide if they are equal or distinct. The autocorrelation is a numeric sequence, and its Fourier transform can also be obtained by summing the squares of the Fourier transform of indicator sequences (zero/one sequences indicating the position of the symbols). Another interpretation of the spectrum is given, borrowing from the spectral envelope concept: among all symbolic-to-numeric mappings there is one that maximizes the spectral energy at each frequency, and leads to the spectrum. (C) 2004 Elsevier Inc. All rights reserved.