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

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 ($C_{\left|x\right|}$) of a linear Gaussian noise as a function of its autocorrelation ($C_x$). For both, models and natural signals, the deviation of $C_{\left|x\right|}$ 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.

Figure 1

Autocorrelation of the magnitude series $C_{\left|x\right|}$ as a function of the autocorrelation of the series $C_x$. The solid line corresponds to the exact expression given by Eq. (8) and the dashed line to its quadratic approximation given by Eq. (9). The symbols correspond to the autocorrelation at distances $\ell =1,...,20$ for several artificial series generated with linear Gaussian models: diamonds, autorregresive AR(1) model $x_i=c+\varphi x_{i-1}+{\varepsilon }_i$, with $\varphi =0.9$, $c=0$ and $\left\{{\varepsilon }_i\right\}$ a white noise; circles, fractional Gaussian noise (fGn) with Hurst exponent $H=0.95$, and triangles fGn with $H=0.05$. While the two first models (diamonds and circles) generate highly correlated series, the last one (triangles) leads to an anticorrelated series. However, in all cases the correlations of the magnitude series are positive.

Figure 2

Autocorrelation function $C_x$ (triangles) and autocorrelation function of magnitude series $C_{\left|x\right|}$ (circles) as a function of distance $\ell$ for two examples of fractional Gaussian noises (fGn): $H=0.9>0.75$ (full symbols) and $H=0.7<0.75$ (open symbols). We generate series of approximate fGns of length $2^{24}\simeq 1.6\times {10}^7$ using the Fourier Filtering Method [19]. To avoid statistical fluctuations we average over an ensemble of 100 realizations. The dashed line corresponds to $1/\ell$, which is the boundary between the regions of long- and short-range correlations (see text). As expected, $C_{\left|x\right|}$ decays with an exponent double that of $C_x$, and in both cases the magnitude series are positively correlated, but for $H=0.7<0.75C_{\left|x\right|}$ decays faster than $1/\ell$ and lies in the short-range correlations region. Note also that in this case and due to both the fast decay of $C_{\left|x\right|}$ and its small values at $\ell =1$, a noisy behavior is reached relatively soon ($\ell <100$) even for long series. This makes difficult the detection of power-law behaviors in this region when dealing with real data.

Figure 3

Autocorrelation function $C_x$ (full symbols) and autocorrelation function of sign series $C_s$ (open symbols) as a function of distance $\ell$ for two examples of fractional Gaussian noises (fGn): correlated $H=0.9>0.5$ (triangles) and anticorrelated $H=0.3<0.5$ (circles). In this case we plot $|C_x|$ and $|C_s|$ to allow the representation in double logarithmic scale. We generate series of approximate fGns of length $2^{24}\simeq 1.6\times {10}^7$ using the Fourier Filtering Method [19]. To avoid statistical fluctuations we average over an ensemble of 100 realizations. As can be expected from (18), $C_x$ and $C_s$ decay with the same exponent. Note that for the anticorrelated series ($H=0.3$) the anticorrelations decay very fast ($\gamma =1.4$) and an oscillatory behavior can be observed. This oscillation is amplified for $H<0$ (not shown).

Figure 4

Autocorrelation of the magnitude series $C_{\left|x\right|}$ as a function of the autocorrelation of the series $C_x$ for nonlinear series generated using the composition method [11] by multiplying the series of magnitudes and signs of two independent fGns (see text). Equation (8) (dashed line) has been included as a reference for the linear Gaussian noise. Solid lines correspond to the theoretical curves obtained using (21) and assuming that the fGns verify the exact asymptotic formula for their autocorrelation function (10). Equation (21) is exact, the observed deviations from these curves are due to the fact that the series generated by means of the Fourier filtering method are approximate fGns, the expressions used for the autocorrelation are only valid asymptotically and also due to the statistical fluctuations (especially for small values of $C_{\left|x\right|}$).

Figure 5

Record of interbeat intervals $t_{RR}$ during rest and moderate exercise for a semiprofessional soccer player. (a) Full record of 10 minutes resting in supine position on the soccer field pus 20 minutes running at moderate pace. (b, c) Separate records for rest and exercise. (d, e) Series of $\mathrm{\Delta }{t}_{RR}$ for rest and exercise. (f, g) Distributions of $\mathrm{\Delta }{t}_{RR}$. For comparison it has been included the best fit to a Gaussian distribution (thick line).

Figure 6

Autocorrelation $C_x$ and autocorrelation of the magnitude $C_{\left|x\right|}$ for the series of $\mathrm{\Delta }{t}_{RR}$ during rest and exercise shown in Fig. 5. (a) $C_{\left|x\right|}$ vs $C_x$ during rest (circles) and exercise (triangles). The thick line corresponds to the theoretical expectation for a linear Gaussian noise (8). (b) Autocorrelation $C_x\left(\ell \right)$ as a function of the lag $\ell$ during rest and exercise. (c) Autocorrelation of magnitude series $C_{\left|x\right|}\left(\ell \right)$ as a function of the lag $\ell$ during rest and exercise. (d) Difference between $C_{\left|x\right|}$ and the theoretical expectation for a linear Gaussian noise given $C_x$ (see text) as a function of the lag $\ell$ during rest and exercise.

Figure 7

Nonlinearity index $\mathrm{\Delta }$ (${\ell }_{\max }=10$) for 10 semiprofessional soccer players, all males with age $23.8\pm 2.9$ yr. Circles: records of 10 minutes of normal wake rest condition, lying in supine position on the soccer field. Squares: 20 minutes of running at typical warming-up pace. Triangles: average of $\mathrm{\Delta }$ over an ensemble of subseries of the running record with the same size of the corresponding rest record (see text). Error bars indicate $\pm$ standard deviation.