Counting forbidden patterns in irregularly sampled time series. II. Reliability in the presence of highly irregular sampling

Abstract

We are motivated by real-world data that exhibit severe sampling irregularities such as geological or paleoclimate measurements. Counting forbidden patterns has been shown to be a powerful tool towards the detection of determinism in noisy time series. They constitute a set of ordinal symbolic patterns that cannot be realised in time series generated by deterministic systems. The reliability of the estimator of the relative count of forbidden patterns from irregularly sampled data has been explored in two recent studies. In this paper, we explore highly irregular sampling frequency schemes. Using numerically generated data, we examine the reliability of the estimator when the sampling period has been drawn from exponential, Pareto and Gamma distributions of varying skewness. Our investigations demonstrate that some statistical properties of the sampling distribution are useful heuristics for assessing the estimator's reliability. We find that sampling in the presence of large chronological gaps can still yield relatively accurate estimates as long as the time series contains sufficiently many densely sampled areas. Furthermore, we show that the reliability of the estimator of forbidden patterns is poor when there is a high number of sampling intervals, which are larger than a typical correlation time of the underlying system.

Characterizing stochastic time series with ordinal networks

“…By knowing the exact form of random ordinal networks, we can also estimate the fraction of missing transi- tions in empirical time series. This analysis is somewhat similar to the one of missing permutations or forbidden patterns introduced by Amigó et al [30][31][32] and explored by several works [33][34][35]. These works have observed that some ordinal patterns cannot occur in chaotic systems [for instance, the permutation (2, 1, 0) never appears in logistic map under fully developed chaos] and that even stochastic processes may present missing ordinal patterns depending on the time series length and the choice of embedding dimension.…”

Macrophages use a bet-hedging strategy for antimicrobial activity in phagolysosomal acidification

“…We then compared these distributions to the pH distribution in bead-tion centered at 0 with non-zero values, suggesting each individual phagosome has an independent target phagolysosomal pH (Supplemental Figure 1; supplemental material available online with this article; https://doi.org/10.1172/JCI133938DS1). The ordinal pattern analysis was repeated on several other strains and conditions throughout the project and at no point did we observe forbidden ordinal patterns (Supplemental Figure 2A), again suggesting the process exhibits no signature of deterministic chaos (22)(23)(24)(25). Given that beads are inert and cannot modify pH, we reasoned that their corresponding phagolysosomes would be the closest approximation of a default acidification state and would therefore represent a baseline to which we could compare phagolysosome acidification dynamics in other conditions.…”

“…To determine whether phagolysosomal acidification is a deterministic or stochastic dynamical process, we employed a permutation spectrum test (21) in which the distribution of ordinal patterns occurring in subsets of our full data set was analyzed ( Figure 1C). Ordinal patterns simply refer to the order of each measurement in terms of value, in our case the phagolysosomal pH, within a scanning window, which parses all measurements within each condition, exemplified in Figure 1 (22)(23)(24)(25). Here we found a 4-unit window size to be the most appropriate.…”

Twenty years after “The Impact of Chaos”: what have been achieved and what should be answered

“…I would like to pay attention how the researches of Refs. [74,[90][91][92] will develop in the future.…”