The Bandt-Pompe permutation entropy framework, alongside the complexity-entropy causality plane, has become a standard tool for characterizing the dynamical properties of time series. However, observational noise distorts ordinal pattern probability distributions in ways that can systematically misplace time series within the causality plane, compromising dynamical classification. This effect is particularly relevant for geophysical signals, which are typically poorly and irregularly sampled, and have a low signal-to-noise level. In this work, we characterize the distortions on ordinal pattern statistics using the formalism of majorization. We provide theoretical results and propose corrective strategies that restore discriminability under realistic measurement conditions. To achieve this, we introduce methodology that allows the characterization of noisy dynamical series and further allows the quantification of observational noise without the need of a fitting procedure. Finally, we illustrate our methodology by analyzing paleomagnetic records to determine if the geological evolution of the Earth dipole is better described by a stochastic or chaotic system.
Ordinal patterns are widely used to characterize temporal organization in time series, yet they are often considered insensitive to the amplitude distribution of the data. In this work, we show that this limitation can be overcome by considering the ordinal structure of integrated time series. We investigate random walks generated from independent non-Gaussian increments and derive analytical expressions for the ordinal pattern probabilities associated with them. We show that for symmetric distributions, the probabilities of some ordinal patterns are fully determined by symmetry arguments, while those for the remaining patterns depend explicitly on the shape of the increments distribution. Numerical simulations based on $q$-Gaussian increments validate the theoretical predictions. We further show that the construction of the random walk itself plays a fundamental role in the accurate characterization of non-Gaussian fluctuations, as different centering procedures may significantly affect the resulting ordinal statistics. Finally, we validate the proposed framework using financial time series, showing that the ordinal distributions of integrated logarithmic returns capture non-Gaussian features consistent with a cubic law.
A. M. Roig, Luciano Zunino, F. Olivares· 0 citations
It is demonstrated that the squared Pearson correlation coefficient provides a simple quantitative criterion for distinguishing chaos from noise directly from observed time-series data.
This manuscript develops a non-parametric and robust framework for estimating the scale of additive noise in weakly sparse systems. The method does not require independence, prescribed dependence, or temporal regularity of the noise sequence. We introduce a class of order-statistic estimators based on comparing the sorted observations with deterministic or random proxies generated from a reference noise distribution. This purely spatial approach avoids preliminary filtering or temporal decorrelation, and therefore preserves the sparsity structure of the latent signal. We establish non-asymptotic concentration inequalities for weighted loss functions, with bounds that separate the contribution of the signal from the discrepancy between the ordered noise and the proxy. We then control this proxy discrepancy in independent and correlated regimes, including heavy-tailed reference laws. Finally, we apply the method to high-frequency observations of continuous-time stochastic processes, obtaining scale estimators for fractional Brownian motion and stable L\'evy noise in the presence of lower-variation additive perturbations.
Detecting when the dependence between two components of a multivariate time series changes, while the marginals drift freely, requires a dependence-specific statistic. We take the inferential object to be a density operator -- the trace-normalised second moment of unit-norm random Fourier features of ranks -- rather than a probability distribution. Partial traces recover the marginal operators exactly, so von Neumann entropies yield a quantum mutual information (QMI) statistic computed from prefix sums of small matrices, without density estimation, matrix inversion, or a tuned parameter. We develop the inference it needs: a segment-separable cost that drives penalised optimal partitioning, its split gain a Holevo information; finite-sample exact calibration by joint pair permutation, a block-permutation form for serially dependent series, and an exact, provably consistent exchangeability diagnostic that selects between them. We also prove a weighted chi-square boundary law, at the segment length and not its square root, for the rank-based statistic exactly as computed. In 500 replicates QMI detects nonlinear, correlation-free dependence changes with more power than the Hilbert-Schmidt independence criterion, distance correlation, Spearman, and empirical-copula statistics on the same ranks, by at least 15 percentage points wherever any statistic detects the change. Its false-alarm rate stays near nominal under marginal drift, where the empirical-copula statistic reaches 0.87. On eight years of hourly Korean weather observations, a two-stage segment-and-certify procedure finds dependence-change candidates above chance (five of 27 at p $\le$ 0.05 against 1.4 expected); stage two certifies one as a pure coupling change and reclassifies eight as marginal-driven.
Classical ordinal-pattern methods quantify complexity from a scalar time series without requiring a model, but they are usually reported as global statistics and say little about where along a chaotic trajectory ordinal regimes change. For a smooth flow x˙=F(x) observed through a scalar ϕ(x), we define the first two material derivatives ϕ˙=LFϕ and ϕ¨=LF2ϕ and the associated organizer sets M1(ϕ)={x:ϕ˙=0} and M0(ϕ)={x:ϕ¨=0}. An exact integral criterion shows that if LFϕ keeps a strict sign on an ordinal window, then the corresponding ordinal pattern is monotone; thus, M1 marks where that sufficient sign-persistence mechanism can fail, while M1∩M0 identifies tangencies of M1 at its regular points. A small-delay expansion further shows that ordinal ranks in delay embeddings are controlled at leading orders by (ϕ˙,ϕ¨). We quantify organizer effects with an ordinal sensitivity index (OSI), defined as the Jensen-Shannon divergence between ordinal-pattern distributions in near and far sets. For Lorenz-63 with ϕ = x, we derive a closed-form expression for M0(x). Across Lorenz-63 and the Aizawa system, OSI remains large over broad embedding and delay ranges; near-M1 conditioning increases switching and transition entropy in 98.8%-100% of tested configurations, whereas M0 produces strong but delay-dependent separation and often increased monotone-pattern prevalence. As a secondary delay-space refinement, a boundary-clearance score improves switching prediction beyond |ϕ˙| in all tested configurations.
Real-world systems, from climate models to power grids, often fluctuate due to sensor noise, drift, or environmental variability, yet standard chaos diagnostics assume fixed parameters and asymptotic horizons. We introduce a finite-time framework for two-dimensional maps under independent, identically distributed parameter noise. First, we prove that the maximal finite-time Lyapunov exponent converges, after centering and scaling, to a Gaussian law whose mean and variance depend explicitly on the map's Jacobian statistics. Second, we develop an attractor separation algorithm that uses FTLE histograms and a geometry-based classifier to partition phase space into chaotic and periodic regions under noise. Third, we validate our theory numerically on the noisy Domenicali map, demonstrating Gaussian FTLE distributions, predictable shifts in Kaplan-Yorke dimension, and a sharp noise threshold for basin escape. Finally, we estimate the critical noise level $\sigma_c$ at which attractor coalescence occurs, using three complementary numerical methods to bound it above and to pinpoint an estimate.
Zubeyr Barre, M. Bashir, Martino Domenicali· 0 citations
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