Correlation Integral vs. second order Factorial Moments and an efficient computational technique

TL;DR

This paper introduces a novel, efficient computational method linking correlation integrals and factorial moments, enabling rapid analysis of large datasets and detection of weak signals in noisy environments.

Contribution

It presents a new mapping and a fast computation technique for second order factorial moments and correlation integrals, especially effective in high-dimensional data analysis.

Findings

The technique is more efficient than conventional methods for low pair counts.

It becomes increasingly effective as the embedding space dimension grows.

Allows analysis of large datasets with very low scales or high partitions.

Abstract

We develop a mapping between the factorial moments of the second order $F_2$ and the correlation integral $C$. We formulate a fast computation technique for the evaluation of both, which is more efficient, compared to conventional methods, for data containing number of pairs per event which is lower than the estimation points. We find the effectiveness of the technique to be more prominent as the dimension of the embedding space increases. We are able to analyse large amount of data in short computation time and access very low scales in $C$ or extremely high partitions in $F_2$. The technique is an indispensable tool for detecting a very weak signal hidden in strong noise.