Failure in distinguishing colored noise from chaos using the "noise titration" technique

Contrasting chaotic with stochastic dynamics via ordinal transition networks

We introduce a representation space to contrast chaotic with stochastic dynamics. Following the complex network representation of a time series through ordinal pattern transitions, we propose to assign each system a position in a two-dimensional plane defined by the permutation entropy of the network (global network quantifier) and the minimum value of the permutation entropy of the nodes (local network quantifier). The numerical analysis of representative chaotic maps and stochastic systems shows that the proposed approach is able to distinguish linear from non-linear dynamical systems by different planar locations. Additionally, we show that this characterization is robust when observational noise is considered. Experimental applications allow us to validate the numerical findings and to conclude that this approach is useful in practical contexts.

A simple method for detecting chaos in nature

Chaos, or exponential sensitivity to small perturbations, appears everywhere in nature. Moreover, chaos is predicted to play diverse functional roles in living systems. A method for detecting chaos from empirical measurements should therefore be a key component of the biologist's toolkit. But, classic chaos-detection tools are highly sensitive to measurement noise and break down for common edge cases, making it difficult to detect chaos in domains, like biology, where measurements are noisy. However, newer tools promise to overcome these limitations. Here, we combine several such tools into an automated processing pipeline, and show that our pipeline can detect the presence (or absence) of chaos in noisy recordings, even for difficult edge cases. As a first-pass application of our pipeline, we show that heart rate variability is not chaotic as some have proposed, and instead reflects a stochastic process in both health and disease. Our tool is easy-to-use and freely available.

Is human atrial fibrillation stochastic or deterministic?-Insights from missing ordinal patterns and causal entropy-complexity plane analysis

The mechanism of atrial fibrillation (AF) maintenance in humans is yet to be determined. It remains controversial whether cardiac fibrillatory dynamics are the result of a deterministic or a stochastic process. Traditional methods to differentiate deterministic from stochastic processes have several limitations and are not reliably applied to short and noisy data obtained during clinical studies. The appearance of missing ordinal patterns (MOPs) using the Bandt-Pompe (BP) symbolization is indicative of deterministic dynamics and is robust to brief time series and experimental noise. Our aim was to evaluate whether human AF dynamics is the result of a stochastic or a deterministic process. We used 38 intracardiac atrial electrograms during AF from the coronary sinus of 10 patients undergoing catheter ablation of AF. We extracted the intervals between consecutive atrial depolarizations (AA interval) and converted the AA interval time series to their BP symbolic representation (embedding dimension 5, time delay 1). We generated 40 iterative amplitude-adjusted, Fourier-transform (IAAFT) surrogate data for each of the AA time series. IAAFT surrogates have the same frequency spectrum, autocorrelation, and probability distribution with the original time series. Using the BP symbolization, we compared the number of MOPs and the rate of MOP decay in the first 1000 timepoints of the original time series with that of the surrogate data. We calculated permutation entropy and permutation statistical complexity and represented each time series on the causal entropy-complexity plane. We demonstrated that (a) the number of MOPs in human AF is significantly higher compared to the surrogate data (2.7 ± 1.18 vs. 0.39 ± 0.28, p < 0.001); (b) the median rate of MOP decay in human AF was significantly lower compared with the surrogate data (6.58 × 10^-3 vs. 7.79 × 10^-3, p < 0.001); and (c) 81.6% of the individual recordings had a rate of decay lower than the 95% confidence intervals of their corresponding surrogates. On the causal entropy-complexity plane, human AF lay on the deterministic part of the plane that was located above the trajectory of fractional Brownian motion with different Hurst exponents on the plane. This analysis demonstrates that human AF dynamics does not arise from a rescaled linear stochastic process or a fractional noise, but either a deterministic or a nonlinear stochastic process. Our results justify the development and application of mathematical analysis and modeling tools to enable predictive control of human AF.

Using missing ordinal patterns to detect nonlinearity in time series data

The number of missing ordinal patterns (NMP) is the number of ordinal patterns that do not appear in a series after it has been symbolized using the Bandt and Pompe methodology. In this paper, the NMP is demonstrated as a test for nonlinearity using a surrogate framework in order to see if the NMP for a series is statistically different from the NMP of iterative amplitude adjusted Fourier transform (IAAFT) surrogates. It is found that the NMP works well as a test statistic for nonlinearity, even in the cases of very short time series. Both model and experimental time series are used to demonstrate the efficacy of the NMP as a test for nonlinearity.

Local noise sensitivity: Insight into the noise effect on chaotic dynamics

Noise contamination in experimental data with underlying chaotic dynamics is one of the significant problems limiting the application of many nonlinear time series analysis methods. Although numerous studies have been devoted to the investigation of different aspects of noise-nonlinear dynamics interactions, the effects produced by noise on chaotic dynamics are not fully understood. This study sought to analyze the local effects produced by noise on chaotic dynamics with a smooth attractor. Local Wayland test translation errors were calculated for noise-induced Lorenz and Rössler chaotic models, and for experimental green light photoplethysmogram data. Results demonstrated that under noise induction, local regions on the chaotic attractor with high values of local translation error can be observed. This phenomenon was defined as the local noise sensitivity. It was found that for both models, local noise-sensitive regions were located close to the system's equilibrium points. Additionally, it was found that the reconstructed dynamics represent well the local noise sensitivity of the original dynamics. The concept of local noise sensitivity is expected to contribute to various applied studies, as it reveals regions of chaotic attractors that are sensitive to the presence of noise.

Generation of 2N + 1-scroll existence in new three-dimensional chaos systems

We propose a systematic methodology for creating 2N + 1-scroll chaotic attractors from a simple three-dimensional system, which is named as the translation chaotic system. It satisfies the condition a12a21 = 0, while the Chua system satisfies a12a21 > 0. In this paper, we also propose a successful (an effective) design and an analytical approach for constructing 2N + 1-scrolls, the translation transformation principle. Also, the dynamics properties of the system are studied in detail. MATLAB simulation results show very sophisticated dynamical behaviors and unique chaotic behaviors of the system. It provides a new approach for 2N + 1-scroll attractors. Finally, to explore the potential use in technological applications, a novel block circuit diagram is also designed for the hardware implementation of 1-, 3-, 5-, and 7-scroll attractors via switching the switches. Translation chaotic system has the merit of convenience and high sensitivity to initial values, emerging potentials in future engineering chaos design.

Using forbidden ordinal patterns to detect determinism in irregularly sampled time series

It is known that when symbolizing a time series into ordinal patterns using the Bandt-Pompe (BP) methodology, there will be ordinal patterns called forbidden patterns that do not occur in a deterministic series. The existence of forbidden patterns can be used to identify deterministic dynamics. In this paper, the ability to use forbidden patterns to detect determinism in irregularly sampled time series is tested on data generated from a continuous model system. The study is done in three parts. First, the effects of sampling time on the number of forbidden patterns are studied on regularly sampled time series. The next two parts focus on two types of irregular-sampling, missing data and timing jitter. It is shown that forbidden patterns can be used to detect determinism in irregularly sampled time series for low degrees of sampling irregularity (as defined in the paper). In addition, comments are made about the appropriateness of using the BP methodology to symbolize irregularly sampled time series.

Discriminating chaotic and stochastic dynamics through the permutation spectrum test

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.

Two chaotic global models for cereal crops cycles observed from satellite in northern Morocco

The dynamics underlying cereal crops in the northern region of Morocco is investigated using a global modelling technique applied to a vegetation index time series derived from satellite measurements, namely, the normalized difference vegetation index from 1982 to 2008. Two three-dimensional chaotic global models of reduced size (14-term and 15-term models) are obtained. The model validation is performed by comparing their horizons of predictability with those provided in previous studies. The attractors produced by the two global models have a complex foliated structure-evidenced in a Poincaré section-rending a topological characterization difficult to perform. Thus, the Kaplan-Yorke dimension is estimated from the synthetic data produced by our global models. Our results suggest that cereal crops in the northern Morocco are governed by a weakly dissipative three-dimensional chaotic dynamics.

Sparse model from optimal nonuniform embedding of time series

An approach to obtaining a parsimonious polynomial model from time series is proposed. An optimal minimal nonuniform time series embedding schema is used to obtain a time delay kernel. This scheme recursively optimizes an objective functional that eliminates a maximum number of false nearest neighbors between successive state space reconstruction cycles. A polynomial basis is then constructed from this time delay kernel. A sparse model from this polynomial basis is obtained by solving a regularized least squares problem. The constraint satisfaction problem is made computationally tractable by keeping the ratio between the number of constraints to the number of variables small by using fewer samples spanning all regions of the reconstructed state space. This helps the structure selection process from an exponentially large combinatorial search space. A forward stagewise algorithm is then used for fast discovery of the optimization path. Results are presented for the Mackey-Glass system.

Highly comparative time-series analysis: the empirical structure of time series and their methods

The process of collecting and organizing sets of observations represents a common theme throughout the history of science. However, despite the ubiquity of scientists measuring, recording and analysing the dynamics of different processes, an extensive organization of scientific time-series data and analysis methods has never been performed. Addressing this, annotated collections of over 35 000 real-world and model-generated time series, and over 9000 time-series analysis algorithms are analysed in this work. We introduce reduced representations of both time series, in terms of their properties measured by diverse scientific methods, and of time-series analysis methods, in terms of their behaviour on empirical time series, and use them to organize these interdisciplinary resources. This new approach to comparing across diverse scientific data and methods allows us to organize time-series datasets automatically according to their properties, retrieve alternatives to particular analysis methods developed in other scientific disciplines and automate the selection of useful methods for time-series classification and regression tasks. The broad scientific utility of these tools is demonstrated on datasets of electroencephalograms, self-affine time series, heartbeat intervals, speech signals and others, in each case contributing novel analysis techniques to the existing literature. Highly comparative techniques that compare across an interdisciplinary literature can thus be used to guide more focused research in time-series analysis for applications across the scientific disciplines.

Individuality of breathing patterns in patients under noninvasive mechanical ventilation evidenced by chaotic global models

Autonomous global models based on radial basis functions were obtained from data measured from patients under noninvasive mechanical ventilation. Some of these models, which are discussed in the paper, turn out to have chaotic or quasi-periodic solutions, thus providing a first piece of evidence that the underlying dynamics of the data used to estimate the global models are likely to be chaotic or, at least, have a chaotic component. It is explicitly shown that one of such global models produces attractors characterized by a Horseshoe map, two models produce toroidal chaos, and one model produces a quasi-periodic regime. These topologically inequivalent attractors evidence the individuality of breathing profiles observed in patient under noninvasive ventilation.

Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach

In this paper we introduce a multiscale symbolic information-theory approach for discriminating nonlinear deterministic and stochastic dynamics from time series associated with complex systems. More precisely, we show that the multiscale complexity-entropy causality plane is a useful representation space to identify the range of scales at which deterministic or noisy behaviors dominate the system's dynamics. Numerical simulations obtained from the well-known and widely used Mackey-Glass oscillator operating in a high-dimensional chaotic regime were used as test beds. The effect of an increased amount of observational white noise was carefully examined. The results obtained were contrasted with those derived from correlated stochastic processes and continuous stochastic limit cycles. Finally, several experimental and natural time series were analyzed in order to show the applicability of this scale-dependent symbolic approach in practical situations.

Polynomial search and global modeling: Two algorithms for modeling chaos

Global modeling aims to build mathematical models of concise description. Polynomial Model Search (PoMoS) and Global Modeling (GloMo) are two complementary algorithms (freely downloadable at the following address: http://www.cesbio.ups-tlse.fr/us/pomos_et_glomo.html) designed for the modeling of observed dynamical systems based on a small set of time series. Models considered in these algorithms are based on ordinary differential equations built on a polynomial formulation. More specifically, PoMoS aims at finding polynomial formulations from a given set of 1 to N time series, whereas GloMo is designed for single time series and aims to identify the parameters for a selected structure. GloMo also provides basic features to visualize integrated trajectories and to characterize their structure when it is simple enough: One allows for drawing the first return map for a chosen Poincaré section in the reconstructed space; another one computes the Lyapunov exponent along the trajectory. In the present paper, global modeling from single time series is considered. A description of the algorithms is given and three examples are provided. The first example is based on the three variables of the Rössler attractor. The second one comes from an experimental analysis of the copper electrodissolution in phosphoric acid for which a less parsimonious global model was obtained in a previous study. The third example is an exploratory case and concerns the cycle of rainfed wheat under semiarid climatic conditions as observed through a vegetation index derived from a spatial sensor.

Effects of maturation and acidosis on the chaos-like complexity of the neural respiratory output in the isolated brainstem of the tadpole, Rana esculenta

Human ventilation at rest exhibits mathematical chaos-like complexity that can be described as long-term unpredictability mediated (in whole or in part) by some low-dimensional nonlinear deterministic process. Although various physiological and pathological situations can affect respiratory complexity, the underlying mechanisms remain incompletely elucidated. If such chaos-like complexity is an intrinsic property of central respiratory generators, it should appear or increase when these structures mature or are stimulated. To test this hypothesis, we employed the isolated tadpole brainstem model [Rana (Pelophylax) esculenta] and recorded the neural respiratory output (buccal and lung rhythms) of pre- (n = 8) and postmetamorphic tadpoles (n = 8), at physiologic (7.8) and acidic pH (7.4). We analyzed the root mean square of the cranial nerve V or VII neurograms. Development and acidosis had no effect on buccal period. Lung frequency increased with development (P < 0.0001). It also increased with acidosis, but in postmetamorphic tadpoles only (P < 0.05). The noise-titration technique evidenced low-dimensional nonlinearities in all the postmetamorphic brainstems, at both pH. Chaos-like complexity, assessed through the noise limit, increased from pH 7.8 to pH 7.4 (P < 0.01). In contrast, linear models best fitted the ventilatory rhythm in all but one of the premetamorphic preparations at pH 7.8 (P < 0.005 vs. postmetamorphic) and in four at pH 7.4 (not significant vs. postmetamorphic). Therefore, in a lower vertebrate model, the brainstem respiratory central rhythm generator accounts for ventilatory chaos-like complexity, especially in the postmetamorphic stage and at low pH. According to the ventilatory generators homology theory, this may also be the case in mammals.

Working conditions for safe detection of nonlinearity and noise titration

Even if noise titration cannot be satisfactorily used to prove the presence of chaos, it can still be used to detect nonlinear component in dynamics. Nevertheless, since the technique have the use of nonlinear models for one-step-ahead predictions, it requires an acute choice of modeling parameters, i.e., the number of terms and the nonlinearity degree of the models. Based on illustrative examples, we propose conditions under which the method of noise titration can be reliably applied to characterize nonlinearity in the time series. It is thus possible to compare different time series and state which one is governed by the strongest nonlinearity. For instance, it is shown that, when there is a single nonlinear term in the equations describing the system, the variable on which it acts can be identified among the others.

Solar Cycle Prediction

A review of solar cycle prediction methods and their performance is given, including forecasts for cycle 24. The review focuses on those aspects of the solar cycle prediction problem that have a bearing on dynamo theory. The scope of the review is further restricted to the issue of predicting the amplitude (and optionally the epoch) of an upcoming solar maximum no later than right after the start of the given cycle. Prediction methods form three main groups. Precursor methods rely on the value of some measure of solar activity or magnetism at a specified time to predict the amplitude of the following solar maximum. Their implicit assumption is that each numbered solar cycle is a consistent unit in itself, while solar activity seems to consist of a series of much less tightly intercorrelated individual cycles. Extrapolation methods, in contrast, are based on the premise that the physical process giving rise to the sunspot number record is statistically homogeneous, i.e., the mathematical regularities underlying its variations are the same at any point of time and, therefore, it lends itself to analysis and forecasting by time series methods. Finally, instead of an analysis of observational data alone, model based predictions use physically (more or less) consistent dynamo models in their attempts to predict solar activity. In their overall performance during the course of the last few solar cycles, precursor methods have clearly been superior to extrapolation methods. Nevertheless, most precursor methods overpredicted cycle 23, while some extrapolation methods may still be worth further study. Model based forecasts have not yet had a chance to prove their skills. One method that has yielded predictions consistently in the right range during the past few solar cycles is that of K. Schatten et al., whose approach is mainly based on the polar field precursor. The incipient cycle 24 will probably mark the end of the Modern Maximum, with the Sun switching to a state of less strong activity. It will therefore be an important testbed for cycle prediction methods and, by inference, for our understanding of the solar dynamo.

Multifractality and heart rate variability

In this paper, we participate to the discussion set forth by the editor of Chaos for the controversy, "Is the normal heart rate chaotic?" Our objective was to debate the question, "Is there some more appropriate term to characterize the heart rate variability (HRV) fluctuations?" We focused on the approximately 24 h RR series prepared for this topic and tried to verify with two different techniques, generalized structure functions and wavelet transform modulus maxima, if they might be described as being multifractal. For normal and congestive heart failure subjects, the h(q) exponents showed to be decreasing for increasing q with both methods, as it should be for multifractal signals. We then built 40 surrogate series to further verify such hypothesis. For most of the series (approximately 75%-80% of cases) multifractality stood the test of the surrogate data employed. On the other hand, series coming from patients in atrial fibrillation showed a small, if any, degree of multifractality. The population analyzed is too small for definite conclusions, but the study supports the use of multifractal series to model HRV. Also it suggests that the regulatory action of autonomous nervous system might play a role in the observed multifractality.

Identifying chaos from heart rate: the right task?

Providing a conclusive answer to the question "is this dynamics chaotic?" remains very challenging when experimental data are investigated. We showed that such a task is actually a difficult problem in the case of heart rates. Nevertheless, an appropriate dynamical analysis can discriminate healthy subjects from patients.

Introduction to controversial topics in nonlinear science: is the normal heart rate chaotic?

In June 2008, the editors of Chaos decided to institute a new section to appear from time to time that addresses timely and controversial topics related to nonlinear science. The first of these deals with the dynamical characterization of human heart rate variability. We asked authors to respond to the following questions: Is the normal heart rate chaotic? If the normal heart rate is not chaotic, is there some more appropriate term to characterize the fluctuations (e.g., scaling, fractal, multifractal)? How does the analysis of heart rate variability elucidate the underlying mechanisms controlling the heart rate? Do any analyses of heart rate variability provide clinical information that can be useful in medical assessment (e.g., in helping to assess the risk of sudden cardiac death)? If so, please indicate what additional clinical studies would be useful for measures of heart rate variability to be more broadly accepted by the medical community. In addition, as a challenge for analysis methods, PhysioNet [A. L. Goldberger et al., "PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals," Circulation 101, e215-e220 (2000)] provided data sets from 15 patients of whom five were normal, five had heart failure, and five had atrial fibrillation (http://www.physionet.org/challenge/chaos/). This introductory essay summarizes the main issues and introduces the essays that respond to these questions.

Stochastic vagal modulation of cardiac pacemaking may lead to erroneous identification of cardiac "chaos"

Fluctuations in the time interval between two consecutive R-waves of electrocardiogram during normal sinus rhythm may result from irregularities in the autonomic drive of the pacemaking sinoatrial node (SAN). We use a biophysically detailed mathematical model of the action potentials of rabbit SAN to quantify the effects of fluctuations in acetylcholine (ACh) on the pacemaker activity of the SAN and its variability. Fluctuations in ACh concentration model the effect of stochastic activity in the vagal parasympathetic fibers that innervate the SAN and produce varying rates of depolarization during the pacemaker potential, leading to fluctuations in cycle length (CL). Both the estimated maximal Lyapunov exponent and the noise limit of the resultant sequence of fluctuating CLs suggest chaotic dynamics. Apparently chaotic heart rate variability (HRV) seen in sinus rhythm can be produced by stochastic modulation of the SAN. The identification of HRV data as chaotic by use of time series measures such as a positive maximal Lyapunov exponent or positive noise limit requires both caution and a quantitative, predictive mechanistic model that is fully deterministic.

Fractal, entropic and chaotic approaches to complex physiological time series analysis: a critical appraisal

A wide variety of methods based on fractal, entropic or chaotic approaches have been applied to the analysis of complex physiological time series. In this paper, we show that fractal and entropy measures are poor indicators of nonlinearity for gait data and heart rate variability data. In contrast, the noise titration method based on Volterra autoregressive modeling represents the most reliable currently available method for testing nonlinear determinism and chaotic dynamics in the presence of measurement noise and dynamic noise.