Discriminating chaotic and stochastic dynamics through the permutation spectrum test

Abstract

In this paper, we propose a new heuristic symbolic tool for unveiling chaotic and stochastic dynamics: the permutation spectrum test. Several numerical examples allow us to confirm the usefulness of the introduced methodology. Indeed, we show that it is robust in situations in which other techniques fail (intermittent chaos, hyperchaotic dynamics, stochastic linear and nonlinear correlated dynamics, and deterministic non-chaotic noise-driven dynamics). We illustrate the applicability and reliability of this pragmatic method by examining real complex time series from diverse scientific fields. Taking into account that the proposed test has the advantages of being conceptually simple and computationally fast, we think that it can be of practical utility as an alternative test for determinism. The importance of distinguishing between periodic, chaotic, and stochastic dynamics from time series analysis is well-recognized for understanding the mechanisms that govern the regarded complex systems. In this work, we have introduced a conceptually simple and computationally fast symbolic visual test for discriminating chaotic and stochastic dynamics, called the permutation spectrum test. Because the symbolization is made by implementing the Bandt and Pompe methodology, all the advantages associated with this natural encoding (simplicity, extremely fast calculation, robustness, and invariance with respect to monotonous transformations) are inherited by the permutation spectrum test. We have shown that this pragmatic approach is robust in situations in which other tests fail. We have also confirmed its practical utility by examining several experimental and natural time series.

Lagrangian Cloud Tracking and the Precipitation-Column Humidity Relationship

“…A pattern in between, pattern 7 for example, exhibits a decrease to a minimum, then a significant monotonic increase in <RH>. See Kulp and Zunino [59] for a schematic of all 24 patterns (their Figure 1) or Parlitz et al [60] (their Figure 2b). Broadly, low-numbered ordinal patterns exhibit increases and high-numbered patterns exhibit decreases with time.…”

“…For the permutation spectrum test [59], all possible sequential evolutions of an environmental variable, for example <RH>, are represented by a particular pattern number. Pattern 1 is a monotonic increase in <RH> over the normalized lifetime of the cloud (of 4 units length (while prototypes in Figure 3 have a length of 5, 4 nodes were used here to limit the number of ordinal patterns which goes as the factorial of the length in order to achieve some statistical power with 487 clouds)) (see Figure 4).…”

“…In order to test whether the normalized cloud evolutions are actually reflective of patterns occurring in individual clouds, the permutation spectrum test of Kulp and Zunino [59] is modified and applied. The permutation spectrum test sorts time series data into patterns based on temporal evolution.…”

On the dynamics of ocean ambient noise: Two decades later

“…Recently, PST was introduced by Kulp and Zunino. 57 It is a symbolic based visual test for discriminating chaotic and stochastic dynamics from complex time series. Since this method is inherited from Bandt and Pompe methodology, 54 it is simple, computationally fast, and robust to noise.…”

Chaotic versus stochastic behavior in active-dissipative nonlinear systems

“…To this end, we adopt two time-series analyses in terms of symbolic dynamics, namely the permutation entropy and the permutation spectrum proposed by Kulp and Zunino [56] as a means to distinguish between chaotic and stochastic dynamics. In relation to the Shannon entropy, another useful measure in terms of complex networks, referred to as the network entropy, has been proposed by Luque et al [57].…”