PRINCIPAL COMPONENT ANALYSIS AND ITS GENERALIZATIONSFOR ANY TYPE SEQUENCE (PCA-SEQ)

Abstract

In the 40s of the last century, Karhunen and Loeve proposed a method for processing of one-dimensional numeric time series by converting it into a multidimensional by shifts. In fact, a one-dimensional number series was decomposed into several orthogonal time series. This method has many times been independently developed and applied in practice under various names (EOF, SSA, Caterpillar, etc.). Nowadays, the name SSA (the Singular Spectral Analysis) is most often used. It turned out that it is universal, applicable to any time series without requiring stationary assumptions, automatically decomposes time series into a trend, cyclic components and noise. By the beginning of the 1980s Takens showed that for a dynamical system such a method makes it possible to obtain an attractor from observing only one of these variables, thereby bringing the method to a powerful theoretical basis. In the same years, the practical benefits of phase portraits became clear. In particular, it was used in the analysis and forecast of the animal abundance dynamics. In this paper we propose to extend SSA to one-dimensional sequence of any type elements, including numbers, symbols, figures, etc., and, as a special case, to molecular sequence. Technically, the problem is solved almost the same algorithm as the SSA. The sequence is cut by a sliding window into fragments of a given length. Between all fragments, the matrix of Euclidean distances is calculated. This is always possible. For example, the square root from the Hamming distance between fragments is the Euclidean distance. For the resulting matrix, the principal components are calculated by the principal-coordinate method (PCo). Instead of a distance matrix one can use a matrix of any similarity/dissimilarity indexes and apply methods of multidimensional scaling (MDS). The result will always be PCs in some Euclidean space. We called this method PCA-Seq. It is certainly an exploratory method, as its particular case SSA. For any sequence, including molecular, PCA-Seq without any additional assumptions allows to get its principal components in a numerical form and visualize them in the form of phase portraits. Long-term experience of SSA application for numerical data gives all reasons to believe that PCA-Seq will be not less useful in the analysis of non-numerical data, especially in hypothesizing. PCA-Seq is implemented in the freely distributed Jacobi 4 package (http://mrherrn.github.io/JACOBI4/).