Detecting and interpreting distortions in hierarchical organization of complex time series

TL;DR

This paper investigates the asymmetry in singularity spectra of complex time series across various domains, revealing insights into their hierarchical organization and potential implications for understanding underlying dynamics.

Contribution

It provides a unified explanation for the asymmetry in singularity spectra and demonstrates its significance in analyzing diverse empirical time series.

Findings

Asymmetry in singularity spectra is common in empirical data.

Sunspot Number variability shows notable spectral asymmetry.

Asymmetry may indicate different underlying generative mechanisms.

Abstract

Hierarchical organization is a cornerstone of complexity and multifractality constitutes its central quantifying concept. For model uniform cascades the corresponding singularity spectra are symmetric while those extracted from empirical data are often asymmetric. Using the selected time series representing such diverse phenomena like price changes and inter-transaction times in the financial markets, sentence length variability in the narrative texts, Missouri River discharge and Sunspot Number variability as examples, we show that the resulting singularity spectra appear strongly asymmetric, more often left-sided but in some cases also right-sided. We present a unified view on the origin of such effects and indicate that they may be crucially informative for identifying composition of the time series. One particularly intriguing case of this later kind of asymmetry is detected in the daily reported Sunspot Number variability. This signals that either the commonly used famous Wolf formula distorts the real dynamics in expressing the largest Sunspot Numbers or, if not, that their dynamics is governed by a somewhat different mechanism.