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We connect a common conventional value to quantify a recurrence plot with its motifs, which have recently been termed "recurrence triangles." The common practical value we focus on is DET, which is the ratio of the points forming diagonal line segments of length 2 or longer within a recurrence plot. As a topological value, we use different recurrence triangles defined previously. As a measure-theoretic value, we define the typical recurrence triangle frequency dimension, which generally fluctuates around 1 when the underlying dynamics are governed by deterministic chaos. By contrast, the dimension becomes higher than 1 for a purely stochastic system. Additionally, the typical recurrence triangle frequency dimension correlates most precisely with DET among the above quantities. Our results show that (i) the common practice of using DET could be partly theoretically supported using recurrence triangles, and (ii) the variety of recurrence triangles behaves more consistently for identifying the strength of stochasticity for the underlying dynamics. The results in this study should be useful in checking basic properties for modeling a given time series. [ABSTRACT FROM AUTHOR]

Discretizing a nonlinear time series enables us to calculate its statistics fast and rigorously. Before the turn of the century, the approach using partitions was dominant. In the last two decades, discretization via permutations has been developed to a powerful methodology, while recurrence plots have recently begun to be recognized as a method of discretization. In the meantime, horizontal visibility graphs have also been proposed to discretize time series. In this review, we summarize these methods and compare them from the viewpoint of symbolic dynamics, which is the right framework to study the symbolic representation of nonlinear time series and the inverse process: the symbolic reconstruction of dynamical systems. As we will show, symbolic dynamics is currently a very active research field with interesting applications. [ABSTRACT FROM AUTHOR]

Despite a long history of time series analysis/prediction, theoretically few is known on how to predict the maxima better. To predict the maxima of a flow more accurately, we propose to use its local cross sections or plates the flow passes through. First, we provide a theoretical underpinning for the observability using local cross sections. Second, we show that we can improve short-term prediction of local maxima by employing a generalized prediction error, which weighs more for the larger values. The proposed approach is demonstrated by rainfalls, where heavier rains may cause casualties. [ABSTRACT FROM AUTHOR]

Despite the extensive literature related to earthquakes, an effective method to forecast and avoid occasional seismic hazards that cause substantial damage is lacking. The Sun has recently been identified as a potential precursor to earthquakes, although no causal relationship between its activity and the Earth's seismicity has been established. This study was aimed at investigating whether such a relationship exists and whether it can be used to improve earthquake forecasting. The edit distances between earthquake point processes were combined with delay-coordinate distances for sunspot numbers. The comparison of these two indicated the existence of unidirectional causal coupling from solar activity to seismicity on Earth, and a radial basis function regressor showed accuracy improvements in the largest magnitude prediction of next days by 2.6%–17.9% in the odds ratio when sunspot distances were included. [ABSTRACT FROM AUTHOR]

We propose an algorithm to refine the reconstruction of an original time series given a recurrence plot, which is also referred to as a contact map. The refinement process calculates the local distances based on the Jaccard coefficients with the neighbors in the previous resolution for each point and takes their weighted average using local distances. We demonstrate the utility of our method using two examples. [ABSTRACT FROM AUTHOR]

To the best of our knowledge, the method of prediction coordinates is the only forecasting method in nonlinear time series analysis that explicitly uses the stochastic characteristics of a system with dynamical noise. Specifically, it generates multiple predictions to jointly infer the current states and dynamical noises. Recent findings based on hypothesis testing show that weather is nonlinear and stochastic and, therefore, so are renewable energy power outputs. This being the case, in this paper, we apply the method of prediction coordinates to forecast wind power ramps, which are rapid transitions in the wind power output that can deteriorate the quality of the electricity supply. First, the method of prediction coordinates is tested using numerical simulations. Then, we present an example of wind power ramp forecasting with empirical data. The results show that the method of prediction coordinates compares favorably with other methods, validating it as a reliable tool for forecasting transitions in nonlinear stochastic dynamics, particularly in the field of renewable energies. [ABSTRACT FROM AUTHOR]

It has been shown that a permutation can uniquely identify the joint set of an initial condition and a non-autonomous external force realization added to the deterministic system in given time series data. We demonstrate that our results can be applied to time series forecasting as well as the estimation of common external forces. Thus, permutations provide a convenient description for a time series data set generated by non-autonomous dynamical systems. [ABSTRACT FROM AUTHOR]

Although there are various models of epidemic diseases, there are a few individual-based models that can guide susceptible individuals on how they should behave in a pandemic without its appropriate treatment. Such a model would be ideal for the current coronavirus disease 2019 (COVID-19) pandemic. Thus, here, we propose a topological model of an epidemic disease, which can take into account various types of interventions through a time-dependent contact network. Based on this model, we show that there is a maximum allowed number of persons one can see each day for each person so that we can suppress the epidemic spread. Reducing the number of persons to see for the hub persons is a key countermeasure for the current COVID-19 pandemic. [ABSTRACT FROM AUTHOR]

In this paper, we propose to use linear programming methods or a more specialized method, namely, the Hungarian method, for speeding up the exact calculation of an edit distance for marked point processes [Y. Hirata and K. Aihara, Chaos 25, 123117 (2015)]. The key observation is that the problem of calculating the edit distance reduces to a matching problem on a bipartite graph. Our preliminary numerical results show that the proposed implementations are faster than the conventional ones by a factor of 10–1000. [ABSTRACT FROM AUTHOR]

We present a model-free forecast algorithm that dynamically combines multiple forecasts using multivariate time series data. The underlying principle is based on the fact that forecast performance depends on the position in the state space. This property is exploited to weight multiple forecasts via a local loss function. Specifically, additional weights are assigned to appropriate forecasts depending on their positions in a state space reconstructed via delay coordinates. The function evaluates the forecast error discounted by the distance in the space, whereas most existing methods discount the error in relation to time. In addition, forecasts are selected with the function for each time step based on the existing multiview embedding approach; by contrast, the original multiview embedding selects the reconstructions in advance before starting the forecast. The proposed prediction method has the smallest mean squared error among conventional ensemble methods for the Rössler and the Lorenz'96I models. The results of comparison of the proposed method with conventional machine-learning methods using a flood forecast example indicate that the proposed method yields superior accuracy. We also demonstrate that the proposed method might even correctly forecast the maximum water level of rivers without any prior knowledge. [ABSTRACT FROM AUTHOR]

The identification of directional couplings (or drive-response relationships) in the analysis of interacting nonlinear systems is an important piece of information to understand their dynamics. This task is especially challenging when the analyst’s knowledge of the systems reduces virtually to time series of observations. Spurred by the success of Granger causality in econometrics, the study of cause-effect relationships (not to be confounded with statistical correlations) was extended to other fields, thus favoring the introduction of further tools such as transfer entropy. Currently, the research on old and new causality tools along with their pitfalls and applications in ever more general situations is going through a time of much activity. In this paper, we re-examine the method of the joint distance distribution to detect directional couplings between two multivariate flows. This method is based on the forced Takens theorem, and, more specifically, it exploits the existence of a continuous mapping from the reconstructed attractor of the response system to the reconstructed attractor of the driving system, an approach that is increasingly drawing the attention of the data analysts. The numerical results with Lorenz and Rössler oscillators in three different interaction networks (including hidden common drivers) are quite satisfactory, except when phase synchronization sets in. They also show that the method of the joint distance distribution outperforms the lowest dimensional transfer entropy in the cases considered. The robustness of the results to the sampling interval, time series length, observational noise, and metric is analyzed too. [ABSTRACT FROM AUTHOR]

We review our recent work on chaos in neurons and its application to neural networks from perspective of chaos engineering. Especially, we analyze a dataset of a squid giant axon by newly combining our previous work of identifying Devaney's chaos with surrogate data analysis, and show that an axon can behave chaotically. Based on this knowledge, we use a chaotic neuron model to investigate possible information processing in the brain. [ABSTRACT FROM AUTHOR]

A new parameter estimation method for nonlinear systems from time series data is proposed. For the purpose of unbiased estimation, we employ the idea of bootstrap method on regression problems. Our method can be applied into even short and noisy data and is expected to give us a robust estimation. Some benchmarks of estimating chaotic models show its practical applicability. We also try to apply this method to analysis for intermittent hormonal therapy for prostate cancer by using a mathematical model and real clinical data. [ABSTRACT FROM AUTHOR]

We propose a method for achieving an optimal protocol of intermittent androgen suppression for the treatment of prostate cancer. Since the model that reproduces the dynamical behavior of the surrogate tumor marker, prostate specific antigen, is piecewise linear, we can obtain an analytical solution for the model. Based on this, we derive conditions for either stopping or delaying recurrent disease. The solution also provides a design principle for the most favorable schedule of treatment that minimizes the rate of expansion of the malignant cell population. [ABSTRACT FROM AUTHOR]

We propose a widely applicable scheme of coupling that models competitions among dynamical systems for fixed amounts of resources. Two oscillators coupled in this way synchronize in antiphase. Three oscillators coupled circularly show a number of oscillation modes such as rotation and partially in-phase synchronization. Intriguingly, simple oscillators in the model also produce complex behavior such as spontaneous switching among different modes. The dynamics reproduces well the spatiotemporal oscillatory behavior of a true slime mold Physarum, which is capable of computational optimization. [ABSTRACT FROM AUTHOR]

In this paper we demonstrate how to construct signals (time series) of continuous-time dynamical systems that exhibit a given symbolic dynamics. This is achieved without construction of the ordinary differential equations that generate the flow. This construction is of theoretical interest and is useful as a source of dynamical data that can be used to test various data analysis algorithms. [ABSTRACT FROM AUTHOR]

I propose a method for reconstructing multi-dimensional dynamical noise inspired by the embedding theorem of Muldoon et al. [Dyn. Stab. Syst. 13, 175 (1998)] by regarding multiple predictions as different observables. Then, applying the embedding theorem by Stark et al. [J. Nonlinear Sci. 13, 519 (2003)] for a forced system, I produce time series forecast by supplying the reconstructed past dynamical noise as auxiliary information. I demonstrate the proposed method on toy models driven by auto-regressive models or independent Gaussian noise. [ABSTRACT FROM AUTHOR]

Describing a time series parsimoniously is the first step to study the underlying dynamics. For a time-discrete system, a generating partition provides a compact description such that a time series and a symbolic sequence are one-to-one. But, for a time-continuous system, such a compact description does not have a solid basis. Here, we propose to describe a time-continuous time series using a local cross section and the times when the orbit crosses the local cross section. We show that if such a series of crossing times and some past observations are given, we can predict the system's dynamics with fine accuracy. This reconstructability neither depends strongly on the size nor the placement of the local cross section if we have a sufficiently long database. We demonstrate the proposed method using the Lorenz model as well as the actual measurement of wind speed. [ABSTRACT FROM AUTHOR]

In this work, we introduce a model for predicting multivariate time series data. This model was obtained by partitioning the state space with joint permutations. We review the theoretical framework of the previous works, show a simple extension to multivariate data, and compare its performance to the previous model obtained by permutations for predicting scalar time series data. [ABSTRACT FROM AUTHOR]

We propose an algorithm based on cross-entropy to determine parameters of a piecewise linear model, which describes intermittent androgen suppression therapy for prostate cancer. By comparing with clinical data, the parameter estimation for the switched system shows good fitting accuracy and efficiency. We further optimize switching time points for the piecewise linear model to obtain a feasible therapeutic schedule. The simulation results of therapeutic effect are superior to those of previous strategy. [ABSTRACT FROM AUTHOR]

We experimentally study strange nonchaotic attractors (SNAs) and chaotic attractors by using a nonlinear integrated circuit driven by a quasiperiodic input signal. An SNA is a geometrically strange attractor for which typical orbits have nonpositive Lyapunov exponents. It is a difficult problem to distinguish between SNAs and chaotic attractors experimentally. If a system has an SNA as a unique attractor, the system produces an identical response to a repeated quasiperiodic signal, regardless of the initial conditions, after a certain transient time. Such reproducibility of response outputs is called consistency. On the other hand, if the attractor is chaotic, the consistency is low owing to the sensitive dependence on initial conditions. In this paper, we analyze the experimental data for distinguishing between SNAs and chaotic attractors on the basis of the consistency. [ABSTRACT FROM AUTHOR]

We introduce a low-dimensional description for a high-dimensional system, which is a piecewise affine model whose state space is divided by permutations. We show that the proposed model tends to predict wind speeds and photovoltaic outputs for the time scales from seconds to 100 s better than by global affine models. In addition, computations using the piecewise affine model are much faster than those of usual nonlinear models such as radial basis function models. [ABSTRACT FROM AUTHOR]