Abstract
Correlation matrices are a major type of multivariate data. To examine properties of a given correlation matrix, a common practice is to compare the same quantity between the original correlation matrix and reference correlation matrices, such as those derived from random matrix theory, that partially preserve properties of the original matrix. We propose a model to generate such reference correlation and covariance matrices for the given matrix. Correlation matrices are often analyzed as networks, which are heterogeneous across nodes in terms of the total connectivity to other nodes for each node. Given this background, the present algorithm generates random networks that preserve the expectation of total connectivity of each node to other nodes, akin to configuration models for conventional networks. Our algorithm is derived from the maximum entropy principle. We will apply the proposed algorithm to measurement of clustering coefficients and community detection, both of which require a null model to assess the statistical significance of the obtained results.
Highlights
Correlation matrices are a major form of multivariate data in various domains
A correlation matrix is a covariance matrix
When the input is a correlation matrix, denoted by ρorg, our aim is to ensure that the expected strength of each node of the correlation matrix generated by the configuration model, denoted by ρcon, is similar to that of ρ org
Summary
Correlation matrices are a major form of multivariate data in various domains. Major analysis tools for correlation matrix data include principal component analysis [11], factor analysis [12], Markowitz’s portfolio theory in mathematical finance [13], and random matrix theory [1,2]. A more recent approach to correlational data is network analysis. With this approach, the first task is usually to either threshold on the value of the pairwise correlation to define an unweighted (i.e., binary) network or adopt the value of the pairwise correlation as the edge weight to define a weighted network. Network analysis of correlation matrices is common across disciplines [3,4,5,6,9,10,17,18,19,20,21,22,23]
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