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The most common approach to assess the dynamical complexity of a time series across multiple temporal scales makes use of the multiscale entropy (MSE) and refined MSE (RMSE) measures. In spite of their popularity, MSE and RMSE lack an analytical framework allowing their calculation for known dynamic processes and cannot be reliably computed over short time series. To overcome these limitations, we propose a method to assess RMSE for autoregressive (AR) stochastic processes. The method makes use of linear state-space (SS) models to provide the multiscale parametric representation of an AR process observed at different time scales and exploits the SS parameters to quantify analytically the complexity of the process. The resulting linear MSE (LMSE) measure is first tested in simulations, both theoretically to relate the multiscale complexity of AR processes to their dynamical properties and over short process realizations to assess its computational reliability in comparison with RMSE. Then, it is applied to the time series of heart period, arterial pressure, and respiration measured for healthy subjects monitored in resting conditions and during physiological stress. This application to short-term cardiovascular variability documents that LMSE can describe better than RMSE the activity of physiological mechanisms producing biological oscillations at different temporal scales.

An intrinsic feature of almost all physiological systems, which is clearly visible in the time course of the variables measured from these systems, is their dynamical complexity. It is indeed widely acknowledged that physiological systems such as the brain, the cardiovascular system, and the muscular system produce output signals that exhibit highly complex dynamics, resulting from the combined activity of several mechanisms of physiological regulation which are coupled with each other through structural and functional pathways [

In spite of its acknowledged usefulness, MSE has been shown to present some shortcomings which have led many authors to propose improvements and modifications of the original algorithm [

Even though RMSE and other refinements and extensions [

The present study introduces a new approach for the evaluation of multiscale complexity that is explicitly designed to address the limitations of existing MSE methods described above. The approach builds on recent theoretical advances providing exact techniques for the analytical computation of information-theoretic measures, including entropy rates, for linear autoregressive (AR) stochastic processes [

Multiscale entropy (MSE), originally proposed by Costa et al. [

Specifically, let us consider a discrete-time, stationary stochastic process

The MSE measure resulting from the above procedure, which we denote as

The original MSE formulation suffers from two main limitations: the suboptimal procedure for elimination of the fast temporal scales implemented by (

In fact, the change of scale of a process_{d}(

Besides the type of filter, another crucial difference exists between the original and refined formulations of MSE. Whereas in MSE the parameter

In this section we present a method to assess the multiscale complexity of linear Gaussian stochastic processes. The method is based on the fact that if the variables obtained sampling the considered process

Given the AR representation, the variance of the process and of the innovations can be used to derive an information-theoretic description of the statistical structure of the process [

Now we turn to show how to compute analytically the variance of the AR innovations after rescaling the original process, in a way such that this variance can be used as in (

The next step of the procedure is to provide a representation for the downsampled process

The derivations above allow computing analytically all the parameters of the state-space process (see (_{s} in order to provide fine-tuning of the filtering process. Moreover we stress that since we filter the original process with a FIR filter that prevents aliasing and we compute at each time the complexity of the normalized process, our state-space MSE measure follows the philosophy of the RMSE method rather than that of the original MSE.

The proposed approach for multiscale complexity analysis is implemented in the LMSE MATLAB® toolbox, which includes the algorithms for computing LMSE and RMSE for the simulated processes and exemplificative realizations of the cardiovascular data studied in this paper. The toolbox is uploaded as supplementary material to this article and is freely available for download from

To investigate the theoretical profiles of the dynamical complexity of a stochastic process as a function of the parameters determining its dynamics, as well as to assess the computational reliability of the proposed estimator of the refined MSE, in this section we consider a set of linear processes simulated with varying oscillatory components, for which we compute exact values of MSE and compare LMSE and RMSE estimates obtained from short process realizations.

Simulations are designed considering AR processes described by

Given the configurations described above, first we determine the theoretical values of the MSE using the procedure described in Section

After theoretical analysis, practical estimation of MSE was performed choosing two representative cases of the parameter setting for Type 1 simulation

Results of the theoretical analysis are reported in Figure

Theoretical profiles of the multiscale entropy (MSE) computed applying the proposed approach to the true parameters of simulated AR processes. Plots depict the exact values of MSE (

Theoretical MSE

Theoretical MSE

Theoretical MSE

Theoretical MSE

Figures

Figures

Figure

Estimation of multiscale entropy (MSE) over finite-length realizations of simulated AR processes. Plots depict the exact values (red lines) and the distributions (median and 10th–90th percentiles) of the MSE estimates (

LMSE

RMSE

LMSE

RMSE

To illustrate the application of the proposed approach for the computation of MSE over short biomedical time series, this section reports the analysis of cardiovascular and respiratory variability series. Specifically, we compare the abilities of LMSE and RMSE in detecting the multiscale complexity of heart period (HP), systolic arterial pressure (SAP), and respiration (RESP) time series measured from a large group of healthy subjects in a resting state condition as well as during two types of physiological stress, that is, postural stress and mental stress [

The analyzed time series belong to a database collected to assess the dynamics of HP, SAP, and RESP during two types of physiological stress commonly studied in cardiovascular variability, that is, postural stress induced by head-up tilt (HUT) and mental stress induced by mental arithmetics (MA); we refer to [

The analysis was performed on segments of 300 consecutive points, free of artifacts and satisfying stationarity requirements, extracted from the three time series for each subject and condition. The preprocessing steps consisted in removing the linear trend from each sequence and in reducing the series to zero mean. Then, for each individual time series, a linear AR model was identified using the standard least squares method and using the Bayesian Information Criterion to set the model order within the range

Statistically significant differences among the MSE profiles obtained in the three conditions (i.e., SU, HUT, and MA) were first assessed by means of the multivariate ANOVA. Then, if the null hypothesis that the means of MSE computed across time scales for each condition are the same multivariate vector was rejected, the univariate ANOVA was applied to the three distributions of MSE obtained during SU, HUT, and MA at any assigned time scale. Furthermore, if at a given time scale the null hypothesis that the means of MSE computed in the three conditions are the same number was rejected, a post hoc pairwise test (i.e., the Student

The results of multiscale complexity analysis of HP, SAP, and RESP are depicted, respectively, in Figures

Estimation of multiscale entropy (MSE) for the time series of the heart period. Plots depict the distributions (median and 25th–75th percentiles) of the MSE estimates

Heart period, LMSE

Heart period, RMSE

Estimation of multiscale entropy (MSE) for the time series of the systolic arterial pressure. Plots depict the distributions (median and 25th–75th percentiles) of the MSE estimates

Systolic pressure, LMSE

Systolic pressure, RMSE

Estimation of multiscale entropy (MSE) for the time series of respiration. Plots depict the distributions (median and 25th–75th percentiles) of the MSE estimates

Respiration, LMSE

Respiration, RMSE

The analysis of LMSE and RMSE computed for the HP time series, reported in Figure

The MSE analysis performed for the SAP time series, reported in Figure

The multiscale complexity analysis of respiration variability, reported in Figure

The present study introduces for the first time a multiscale entropy measure that is based on theoretical rather than empirical grounds and can thus be analytically computed from the parametric representation of an observed stochastic process. As a matter of fact, the proposed LMSE method is highly data-efficient because it stems from simple linear parametric modeling and is thus much more reliable than MSE or its modifications [

Our approach can be fruitfully exploited, as done in the present study, to relate the multiscale complexity of a stochastic process to the parameters that establish its dynamical features, or to estimate patterns of multiscale complexity from short process realizations. In this work, we have formalized the dependence of the multiscale complexity of an AR process on the amplitude and frequency of its stochastic oscillatory components and have assessed multiscale patterns of short-term cardiovascular complexity which cannot be fully retrieved using standard MSE methods. Our results emphasize the role of the sympathetic control in driving the increased regularity of low-frequency heart rate oscillations and the increased complexity of low-frequency arterial pressure oscillations during postural stress. LMSE analysis stresses also the importance of dynamics occurring within the low-frequency band in determining the increased complexity of arterial pressure, and even that of respiration, during mental stress.

The main strength of the proposed approach, that is, the linear parametric formulation, constitutes also one of its major limitations. In fact, the computation of LMSE holds exactly only if the observed process has a Gaussian distribution; in such a case, the linear AR description fully captures all of the variability in the process that determines the measured entropy rates, and model-free formulations as the one implemented by SampEn have no additional utility [

The authors declare that they have no conflicts of interest.

Luca Faes and Giandomenico Nollo were partly supported by funding from the Healthcare Research and Implementation Program (IRCS) of the Autonomous Province of Trento (PAT), Italy. Michal Javorka was supported by Grants APVV-0235-12, VEGA 1/0117/17, and VEGA 1/0202/16 and project “Biomedical Center Martin,” ITMS code 26220220187, the project cofinanced from EU sources.

The LMSE MATLAB toolbox is available as supplementary material to this article. The package contains functions for the computation of multiscale entropy based on the linear state-space approach and on the refined approach proposed in [11], as well as scripts that allow performing multiscale complexity analysis for the simulated and real data considered in the present study.