On the dependency between principal components: Application to determine the rank of a matrix in an evolutionary process

Abstract

AbstractPrincipal component analysis (PCA) is a versatile and well‐established approach to explore the relationship between observations represented by their descriptor variables. The latter relationship can be visualized by score plots, ie, the representation of the observations in low dimensional space. The important hyperparameter of PCA is the choice of the correct dimensionality for representing the observations since it bears information about the number of varying analytes contributing to the studied signals. Therefore, the latter dimension is also called the chemical rank of the system. In this paper, a new method for exploring the dependency between principal components of an evolutionary process is proposed. The dependency was quantitatively characterized by two measures: maximum information coefficient (MIC) and distance correlation (DC). Since the dependency measures could discriminate between chemical and nonchemical factors, this concept was used for rank determination. The performance of MIC‐derived indices in terms of detecting meaningful chemical information was better than the one based on indices derived from DC. The performance of each dependency measure in rank estimation was investigated in several simulations and with various real data sets. From the accuracy results and computation time, the proposed methods were compared with previously published rank determination methods. The results showed that MIC could provide accurate estimation of chemical rank in the reasonable timescale rather than DC and also the published rank estimation methods, in the most situations.