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

Abstract

The correlation properties of the magnitude of a time series are associated with nonlinear and multifractal properties and have been applied in a great variety of fields. Here we have obtained the analytical expression of the autocorrelation of the magnitude series (C_{|x|}) of a linear Gaussian noise as a function of its autocorrelation (C_{x}). For both, models and natural signals, the deviation of C_{|x|} from its expectation in linear Gaussian noises can be used as an index of nonlinearity that can be applied to relatively short records and does not require the presence of scaling in the time series under study. In a model of artificial Gaussian multifractal signal we use this approach to analyze the relation between nonlinearity and multifractallity and show that the former implies the latter but the reverse is not true. We also apply this approach to analyze experimental data: heart-beat records during rest and moderate exercise. For each individual subject, we observe higher nonlinearities during rest. 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.

Nonlinear correlations in multifractals: Visibility graphs of magnitude and sign series

“…Finally, we introduce a parameter that can well measure the strength of nonlinear correlation, where the multifractal spectrum width ∆α q , that is a typical and widely used measure for such an analysis, is not able to discover such nonlinearities. Our results are in line with findings in recent studies [29,30].…”

“…Recently, it has been shown that for a linearly correlated Gaussian series, such a symmetric relationship is also observed in the behavior of the second order correlation function of the magnitude C |x| (s) versus original series C(s) [29]. We argue that the method used in [29] is strongly dependent on the PDF of original series x, and thus before doing such an analysis one needs to replace rank-wisely the series values with a Gaussian ones, however in our approach no such a replacement is needed. Fig.…”

“…In this article, we apply the horizontal visibility graph algorithm to map fractal and multifratal time series into graphs. By investigating topological characteristics of the resulting graphs, we first show that this approach can well detect linear and nonlinear correlations, even for situations that DFA and MFDFA predict uncorrelatedness, due to some technical issues [23,29,30]. On the other hand, we show that owing to the unique characterisitc of the horizontal visibility graph algorithm, one can calculate linear or nonlinear correlations, without the need to eliminate the impact of non-Gaussianity of the original series.…”

“…It is also indicated that in some situations MFDFA wrongly predicts linearity in spite of the nonlinearity of the studied time series. To correctly discover nonlinearity, they proposed to find the deviation of the second order correlation function of the series magnitude C |x| (s) = |x(t + s)||x(t)| − |x(t)| 2 from its expectation in linear Gaussian series [29]. They argued that this method can be used for short series, and also can be applied even for series that do not show any scaling behavior.…”

Temporal correlation functions of dynamic systems in non-stationary states

“…Naturally, one may also study the auto-correlation of the fluctuations with DFA, by decomposing the time series into the sign series and the magnitude series [1,2,4,33]. Such an approach may provide further understanding associated with nonlinear and multi-fractal properties [34,35].…”