Threshold-free estimation of entropy from a Pearson matrix

¹ Department of Physics, Federal University of Rio Grande do Norte - Natal, RN, 59078-970, Brazil
² Department of Network and Data Science, Central European University - Vienna, 1100, Austria
³ Cognitive Neuroscience, Scuola Internazionale Superiore di Studi Avanzati - Trieste, 34136, Italy
⁴ Brain Institute, Federal University of Rio Grande do Norte - Natal, RN, 59076-550, Brazil
⁵ Departamento de Física, Universidade Federal do Paraná - Curitiba, PR, 81531-980, Brazil
⁶ Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco Recife, PE, 50670-901, Brazil
⁷ National Institute of Science and Technology of Complex Systems, Federal University of Rio Grande do Norte Natal, RN, 59078-970, Brazil

^(a) E-mail: h.felippe@fisica.ufrn.br (corresponding author)

Received: 8 August 2022
Accepted: 24 January 2023

Abstract

There is demand in diverse fields for a reliable method of estimating the entropy associated with correlations. The estimation of a unique entropy directly from the Pearson correlation matrix has remained an open problem for more than half a century. All existing approaches lack generality insofar as they require thresholding choices that arbitrarily remove possibly important information. Here we propose an objective procedure for directly estimating a unique entropy of a general Pearson matrix. We show that upon rescaling the Pearson matrix satisfies all necessary conditions for an analog of the von Neumann entropy to be well defined. No thresholding is required. We demonstrate the method by estimating the entropy from neuroimaging time series of the human brain under the influence of a psychedelic.