Generating Signals with Multiscale Time Irreversibility: The Asymmetric Weierstrass Function

• Type of Material: Article
• Authors: Anton Burykin, Emory University; Madalena D. Costa, Harvard University; Chung-Kang Peng, Harvard University; Ary L. Goldberger, Harvard University; Timothy George Buchman, Emory University
• Language: English
• Date: 2011-03-01
• Publisher: Hindawi: Wiley Hindawi Partnership
• Publication Version: Final Published Version
• Copyright Statement: © 2010 Wiley Periodicals, Inc.
• Final Published Version (URL): http://dx.doi.org/10.1002/cplx.20333
• Title of Journal or Parent Work: Complexity
• ISSN: 1076-2787
• Volume: 16
• Issue: 4
• Start Page: 29
• End Page: 38
• Grant/Funding Information: This work was generously supported by grants from the DARPA (49533-LS-DRP, HR0011-05-1-0057, and FA9550-08-1-0364); the NIH (1 K99 AG030677-01A2 and U01EB008577); the James S. McDonnell Foundation; the Ellison Medical Foundation; and the Mathers Charitable Foundation.
• Keywords: REVERSIBILITY; Mathematics; Science & Technology - Other Topics; CHAOS; multiscale time asymmetry; Physical Sciences; SERIES; Weierstrass function; Multidisciplinary Sciences; Mathematics, Interdisciplinary Applications; HEART; sawtooth function; irreversibility; multiscale analysis of time series; arrow of time; LOCAL REGULARITY; Science & Technology
• Research Categories: Biophysics, Medical; Health Sciences, Medicine and Surgery

Abstract

Time irreversibility (asymmetry with respect to time reversal) is an important property of many time series derived from processes in nature. Some time series (e.g., healthy heart rate dynamics) demonstrate even more complex, multiscale irreversibility, such that not only the original but also coarse-grained time series are asymmetric over a wide range of scales. Several indices to quantify multiscale asymmetry have been introduced. However, there has been no simple generator of model time series with "tunable" multiscale asymmetry to test such indices. We introduce an asymmetric Weierstrass function WA (constructed from asymmetric sawtooth functions instead of cosine waves) that can be used to construct time series with any given value of the multiscale asymmetry. We show that multiscale asymmetry appears to be independent of other multiscale complexity indices, such as fractal dimension and multiscale entropy. We further generalize the concept of multiscale asymmetry by introducing time-dependent (local) multiscale asymmetry and provide examples of such time series. The WA function combines two essential features of complex fluctuations, namely fractality (self-similarity) and irreversibility (multiscale time asymmetry); moreover, each of these features can be tuned independently. The proposed family of functions can be used to compare and refine multiscale measures of time series asymmetry.