# Correlations in magnitude series to assess nonlinearities: application to multifractal models and heartbeat fluctuations

## TL;DR

This paper analytically links magnitude series correlations to nonlinearities in time series, demonstrating their application in distinguishing heart rate variability during rest and exercise, revealing increased non-linearity during rest.

## Contribution

It provides analytical expressions for magnitude autocorrelation in linear Gaussian noise and introduces a non-linearity index applicable to short records without scaling.

## Key findings

- Higher non-linearity in heartbeat signals during rest compared to exercise
- Method applicable to short time series without requiring scaling
- Supports link between non-linearity and autonomic nervous system activity

## Abstract

The correlation properties of the magnitudes of a time series (sometimes called volatility) are associated with nonlinear and multifractal properties and have been applied in a great variety of fields. Here, we have obtained analytically the expression of the autocorrelation of the magnitude series of a linear Gaussian noise as a function of its correlation as well as several analytical relations involving them. For both, models and natural signals, the deviation from these equations can be used as an index of non-linearity that can be applied to relatively short records and that does not require the presence of scaling in the time series under study. We apply this approach to show that the heart-beat records during rest show higher non-linearities than the records of the same subject during moderate exercise. This behavior is also achieved on average for the analyzed set of 10 semiprofessional soccer players. This result agrees with the fact that other measures of complexity are dramatically reduced during exercise and can shed light on its relationship with the withdrawal of parasympathetic tone and/or the activation of sympathetic activity during physical activity.