Quantifiers for randomness of chaotic pseudo-random number generators

From Continuous-Time Chaotic Systems to Pseudo Random Number Generators: Analysis and Generalized Methodology

The use of chaotic systems in electronics, such as Pseudo-Random Number Generators (PRNGs), is very appealing. Among them, continuous-time ones are used less because, in addition to having strong temporal correlations, they require further computations to obtain the discrete solutions. Here, the time step and discretization method selection are first studied by conducting a detailed analysis of their effect on the systems’ statistical and chaotic behavior. We employ an approach based on interpreting the time step as a parameter of the new “maps”. From our analysis, it follows that to use them as PRNGs, two actions should be achieved (i) to keep the chaotic oscillation and (ii) to destroy the inner and temporal correlations. We then propose a simple methodology to achieve chaos-based PRNGs with good statistical characteristics and high throughput, which can be applied to any continuous-time chaotic system. We analyze the generated sequences by means of quantifiers based on information theory (permutation entropy, permutation complexity, and causal entropy × complexity plane). We show that the proposed PRNG generates sequences that successfully pass Marsaglia Diehard and NIST (National Institute of Standards and Technology) tests. Finally, we show that its hardware implementation requires very few resources.

Quantifying interdependence using the missing joint ordinal patterns

In this paper, we develop the concept of forbidden/missing ordinal patterns into the forbidden/missing joint ordinal patterns and propose the ratio of the number of missing joint ordinal patterns (RMJPs) as a sign of interdependence. RMJP in a surrogate analysis can be used to differentiate the forbidden joint ordinal patterns from the missing joint ordinal patterns due to small sample effects. We first apply RMJP to the simulated time series: a two-component autoregressive fractionally integrated moving average process, the Hénon map, and the Rössler system using active control and discuss the effect of the length of the time series, embedding dimension, and noise contamination. RMJP has been proven to be capable of measuring the interdependence in the numerical simulation. Then, RMJP is further used on the electroencephalogram (EEG) time series for empirical analysis to explore the interdependence of brain waves. With results by RMJP obtained from a widely used open dataset of the sleep EEG time series from healthy subjects, we find that RMJP can be used to quantify the brain wave interdependence under different sleep/wake stages, reveal the overall sleep architecture, and indicate a higher level of interdependence as sleep gets deeper. The findings are consistent with existing knowledge in sleep medicine. The proposed RMJP method has shown its validity and applicability and may assist automatic sleep quantification or bring insight into the understanding of the brain activity during sleep. Furthermore, RMJP can be used on sleep EEG under various pathological conditions and in large-scale sleep studies, helping to investigate the mechanisms of the sleep process and neuron synchronization.

Analysis of ischaemic crisis using the informational causal entropy-complexity plane

In the present work, an ischaemic process, mainly focused on the reperfusion stage, is studied using the informational causal entropy-complexity plane. Ischaemic wall behavior under this condition was analyzed through wall thickness and ventricular pressure variations, acquired during an obstructive flow maneuver performed on left coronary arteries of surgically instrumented animals. Basically, the induction of ischaemia depends on the temporary occlusion of left circumflex coronary artery (which supplies blood to the posterior left ventricular wall) that lasts for a few seconds. Normal perfusion of the wall was then reestablished while the anterior ventricular wall remained adequately perfused during the entire maneuver. The obtained results showed that system dynamics could be effectively described by entropy-complexity loops, in both abnormally and well perfused walls. These results could contribute to making an objective indicator of the recovery heart tissues after an ischaemic process, in a way to quantify the restoration of myocardial behavior after the supply of oxygen to the ventricular wall was suppressed for a brief period.

Bandt-Pompe symbolization dynamics for time series with tied values: A data-driven approach

In 2002, Bandt and Pompe [Phys. Rev. Lett. 88, 174102 (2002)] introduced a successfully symbolic encoding scheme based on the ordinal relation between the amplitude of neighboring values of a given data sequence, from which the permutation entropy can be evaluated. Equalities in the analyzed sequence, for example, repeated equal values, deserve special attention and treatment as was shown recently by Zunino and co-workers [Phys. Lett. A 381, 1883 (2017)]. A significant number of equal values can give rise to false conclusions regarding the underlying temporal structures in practical contexts. In the present contribution, we review the different existing methodologies for treating time series with tied values by classifying them according to their different strategies. In addition, a novel data-driven imputation is presented that proves to outperform the existing methodologies and avoid the false conclusions pointed by Zunino and co-workers.

Complexity of Simple, Switched and Skipped Chaotic Maps in Finite Precision

In this paper we investigate the degradation of the statistic properties of chaotic maps as consequence of their implementation in a digital media such as Digital Signal Processors (DSP), Field Programmable Gate Arrays (FPGA) or Application-Specific Integrated Circuits (ASIC). In these systems, binary floating- and fixed-point are the numerical representations available. Fixed-point representation is preferred over floating-point when speed, low power and/or small circuit area are necessary. Then, in this paper we compare the degradation of fixed-point binary precision version of chaotic maps with the one obtained by using floating point 754-IEEE standard, to evaluate the feasibility of their FPGA implementation. The specific period that every fixed-point precision produces was investigated in previous reports. Statistical characteristics are also relevant, it has been recently shown that it is convenient to describe the statistical characteristic using both, causal and non-causal quantifiers. In this paper we complement the period analysis by characterizing the behavior of these maps from an statistical point of view using cuantifiers from information theory. Here, rather than reproducing an exact replica of the real system, the aim is to meet certain conditions related to the statistics of systems.

Diagnosing the Dynamics of Observed and Simulated Ecosystem Gross Primary Productivity with Time Causal Information Theory Quantifiers

Data analysis and model-data comparisons in the environmental sciences require diagnostic measures that quantify time series dynamics and structure, and are robust to noise in observational data. This paper investigates the temporal dynamics of environmental time series using measures quantifying their information content and complexity. The measures are used to classify natural processes on one hand, and to compare models with observations on the other. The present analysis focuses on the global carbon cycle as an area of research in which model-data integration and comparisons are key to improving our understanding of natural phenomena. We investigate the dynamics of observed and simulated time series of Gross Primary Productivity (GPP), a key variable in terrestrial ecosystems that quantifies ecosystem carbon uptake. However, the dynamics, patterns and magnitudes of GPP time series, both observed and simulated, vary substantially on different temporal and spatial scales. We demonstrate here that information content and complexity, or Information Theory Quantifiers (ITQ) for short, serve as robust and efficient data-analytical and model benchmarking tools for evaluating the temporal structure and dynamical properties of simulated or observed time series at various spatial scales. At continental scale, we compare GPP time series simulated with two models and an observations-based product. This analysis reveals qualitative differences between model evaluation based on ITQ compared to traditional model performance metrics, indicating that good model performance in terms of absolute or relative error does not imply that the dynamics of the observations is captured well. Furthermore, we show, using an ensemble of site-scale measurements obtained from the FLUXNET archive in the Mediterranean, that model-data or model-model mismatches as indicated by ITQ can be attributed to and interpreted as differences in the temporal structure of the respective ecological time series. At global scale, our understanding of C fluxes relies on the use of consistently applied land models. Here, we use ITQ to evaluate model structure: The measures are largely insensitive to climatic scenarios, land use and atmospheric gas concentrations used to drive them, but clearly separate the structure of 13 different land models taken from the CMIP5 archive and an observations-based product. In conclusion, diagnostic measures of this kind provide data-analytical tools that distinguish different types of natural processes based solely on their dynamics, and are thus highly suitable for environmental science applications such as model structural diagnostics.

Multiscale recurrence analysis of spatio-temporal data

The description and analysis of spatio-temporal dynamics is a crucial task in many scientific disciplines. In this work, we propose a method which uses the mapogram as a similarity measure between spatially distributed data instances at different time points. The resulting similarity values of the pairwise comparison are used to construct a recurrence plot in order to benefit from established tools of recurrence quantification analysis and recurrence network analysis. In contrast to other recurrence tools for this purpose, the mapogram approach allows the specific focus on different spatial scales that can be used in a multi-scale analysis of spatio-temporal dynamics. We illustrate this approach by application on mixed dynamics, such as traveling parallel wave fronts with additive noise, as well as more complicate examples, pseudo-random numbers and coupled map lattices with a semi-logistic mapping rule. Especially the complicate examples show the usefulness of the multi-scale consideration in order to take spatial pattern of different scales and with different rhythms into account. So, this mapogram approach promises new insights in problems of climatology, ecology, or medicine.