Multiscale information storage of linear long-range correlated stochastic processes

Information storage, reflecting the capability of a dynamical system to keep predictable information during its evolution over time, is a key element of intrinsic distributed computation, useful for the description of the dynamical complexity of several physical and biological processes. Here we introduce a parametric approach which allows one to compute information storage across multiple timescales in stochastic processes displaying both short-term dynamics and long-range correlations (LRC). Our analysis is performed in the popular framework of multiscale entropy, whereby a time series is first “coarse grained” at the chosen timescale through low-pass filtering and downsampling, and then its complexity is evaluated in terms of conditional entropy. Within this framework, our approach makes use of linear fractionally integrated autoregressive (ARFI) models to derive analytical expressions for the information storage computed at multiple timescales. Specifically, we exploit state space models to provide the representation of lowpass filtered and downsampled ARFI processes, from which information storage is computed at any given timescale relating the process variance to the prediction error variance. This enhances the practical usability of multiscale information storage, as it enables a computationally reliable quantification of a complexity measure which incorporates the effects of LRC together with that of short-term dynamics. The proposed measure is first assessed in simulated ARFI processes reproducing different types of autoregressive dynamics and different degrees of LRC, studying both the theoretical values and the finite sample performance. We find that LRC alter substantially the complexity of ARFI processes even at short timescales, and that reliable estimation of complexity can be achieved at longer timescales only when LRC are properly modeled. Then, we assess multiscale information storage in physiological time series measured in humans during resting state and postural stress, revealing unprecedented responses to stress of the complexity of heart period and systolic arterial pressure variability, which are related to the different role played by LRC in the two conditions.

Figure 1

Theoretical profiles of multiscale information storage for simulated ARFI processes with varying amplitude of stochastic oscillations. Plots depict the information storage $S_X$ computed as a function of the cutoff frequency $f_{\tau }$ of the lowpass filter used to change the timescale for an ARFI process characterized by two complex conjugate poles with fixed phase $\varphi =2\pi 0.1$ and variable modulus $\rho =0$ (a), $\rho =0.5$ (b), $\rho =0.8$ (c), and $\rho =0.9$ (d), and variable differencing parameter $d=0,0.05,0.4,0.7$.

Figure 2

Theoretical profiles of multiscale information storage for simulated ARFI processes with varying frequency of stochastic oscillations. Plots depict the information storage $S_X$ computed as a function of the cutoff frequency $f_{\tau }$ of the lowpass filter used to change the timescale for an ARFI process characterized by a pair of complex conjugate poles with fixed modulus and phase (${\rho }_1=0.8,{\varphi }_1=2\pi 0.1$), and another pair of complex conjugate poles with fixed modulus ${\rho }_2=0.8$ and variable phase ${\varphi }_2=2\pi f_2$, where $f_2=0.15$ (a), $f_2=0.2$ (b), $f_2=0.25$ (c), and $f_2=0.3$ Hz (d), as well as variable differencing parameter $d=0,0.05,0.4,0.7$.

Figure 3

Dependence of the theoretical profiles of multiscale information storage on the approximation of a simulated ARFI process with a finite-order AR process. Plots depict the information storage $S_X$ computed as a function of the cutoff frequency $f_{\tau }$ of the lowpass filter used to change the timescale for an ARFI process characterized by two complex conjugate poles with phase $\varphi =2\pi 0.1$ and modulus $\rho =0$ (a) or $\rho =0.8$ (b), or by two pairs of complex conjugate poles with modulus ${\rho }_1={\rho }_2=0.8$ and phases ${\varphi }_1=2\pi 0.1,{\varphi }_2=2\pi 0.15$ (c) or ${\varphi }_1=2\pi 0.1,{\varphi }_2=2\pi 0.3$ (d). In each panel, results are plotted for values of the differencing parameter $d=0,0.4,0.7$ and two values for the truncation parameter: $q=10$ (dashed lines) and $q=50$ (solid lines).

Figure 4

Estimation of multiscale information storage over finite length realizations of simulated ARFI processes. Plots depict the theoretical values (red) and the distributions [median and $10\text{th}–90\text{th}$ percentiles (dispersion interval, D.I.) over 100 realizations] of the information storage $S_X$ computed as a function of the cutoff frequency $f_{\tau }$ of the lowpass filter used to change the timescale for an ARFI process characterized by two complex conjugate poles with modulus and phase $\rho =0.8,\varphi =2\pi 0.1$ for values of the differencing parameters $d=0$ [(a)–(c)], $d=0.4$ [(d)–(f)], and $d=0.7$ [(g)–(i)], using the proposed ARFI method [(a), (d), (g)], the linear multiscale AR method [25] [(b), (e), (h)], and the refined multiscale complexity approach [26] [(c), (f), (i)].

Figure 5

Estimation of multiscale information storage as a function of the length of simulated ARFI processes. Plots depict the theoretical values (red) and the average estimated values (median over 100 realizations, other colors) of the information storage $S_X$ computed as a function of the cutoff frequency $f_{\tau }$ of the lowpass filter used to change the timescale for an ARFI process characterized by two complex conjugate poles with modulus and phase $\rho =0.8,\varphi =2\pi 0.1$ for values of the differencing parameters $d=0$ [(a)–(c)], $d=0.4$ [(d)–(f)], and $d=0.7$ [(g)–(i)], using the proposed ARFI method [(a), (d), (g)], the linear multiscale AR method [25] [(b), (e), (h)], and the refined multiscale complexity approach [26] [(c), (f), (i)].

Figure 6

Multiscale information storage for the heart period time series. Plots report the confidence interval (C.I.) for the mean of the index of information storage computed across subjects using the eAR approach (a), the eARd method (c), and the eARFI method (e) as a function of the cutoff frequency of the rescaling filter in the supine (SU) and upright (UP) body positions. For each estimation method, the paired C.I. of $\mathrm{UP}-\mathrm{SU}$ are also plotted in panels (b), (d), and (f).

Figure 7

Multiscale information storage for the systolic pressure time series. Plots report the confidence interval (C.I.) for the mean of the index of information storage computed across subjects using the eAR approach (a), the eARd method (c), and the eARFI method (e) as a function of the cutoff frequency of the rescaling filter in the supine (SU) and upright (UP) body positions. For each estimation method, the paired C.I. of $\mathrm{UP}-\mathrm{SU}$ are also plotted in panels (b), (d), and (f).

Figure 8

Multiscale information storage for the respiratory time series. Plots report the confidence interval (C.I.) for the mean of the index of information storage computed across subjects using the eAR approach (a), the eARd method (c), and the eARFI method (e) as a function of the cutoff frequency of the rescaling filter in the supine (SU) and upright (UP) body positions. For each estimation method, the paired C.I. of $\mathrm{UP}-\mathrm{SU}$ are also plotted in panels (b), (d), and (f).

Figure 9

Values of the differencing parameter $d$ estimated for each subject (index 1,...,61) for the heart period [(a), (b)], systolic pressure [(c), (d)], and respiration [(e), (f)] time series in the supine [SU; (a), (c), (e)] and upright [UP; (b), (d), (f)] body positions. Each panel reports also the $95\text{th}$ confidence intervals (solid lines) of the distribution of $d$ computed around the zero level (dashed line).