Symbolic correlation integral

This paper aims to introduce the concept of symbolic correlation integral SC that is extensively used in many scientific fields. The new correlation integral SC avoids the noisy parameter 𝜀 of the classical correlation integral, defined by Grassberger and Procaccia (1983 Grassberger, P., Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica D: Nonlinear Phenomena 9(1–2):189–208.[Crossref], [Web of Science ®] , [Google Scholar]) and extensively used for constructing correlation-integral-based statistics, as in the BDS test. Once the free parameter 𝜀 disappears, it is possible to construct a nonparametric powerful test for independence that can also be used as a diagnostic tool for model selection. The symbolic correlation integral is also extended to deal with multivariate models, and a test for causality is proposed as an example of the theoretical power of the new concept. With extensive Monte Carlo simulations, the paper shows the good size and power performance of symbolic correlation-integral-based tests.