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Volume 2017, Article ID 1768264, 13 pages
Research Article

Efficient Computation of Multiscale Entropy over Short Biomedical Time Series Based on Linear State-Space Models

1BIOtech, Department of Industrial Engineering, University of Trento, Trento, Italy
2Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici (DEIM), University of Palermo, Palermo, Italy
3Department of Biomedical Sciences for Health, University of Milan, Milan, Italy
4Department of Cardiothoracic-Vascular Anesthesia and Intensive Care, IRCCS Policlinico San Donato, San Donato Milanese, Milan, Italy
5Department of Physiology, Jessenius Faculty of Medicine, Comenius University in Bratislava, Mala Hora 4C, 03601 Martin, Slovakia
6Biomedical Center Martin, Jessenius Faculty of Medicine, Comenius University in Bratislava, Mala Hora 4C, 03601 Martin, Slovakia
7Bruno Kessler Foundation, Trento, Italy

Correspondence should be addressed to Luca Faes; moc.liamg@acul.seaf

Received 18 September 2017; Accepted 13 November 2017; Published 7 December 2017

Academic Editor: Anne Humeau-Heurtier

Copyright © 2017 Luca Faes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.