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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]

The effects of changes in weather variables, including precipitation dependence on the days-of-the-week, have known applications in weather predictions. However, the use of these effects to improve weather forecasting has not been determined. Here we investigate if conditioning past data somehow by considering the days-of-the-week helps us to obtain the better short-term time series prediction for precipitation. Especially, we demonstrate that short-term time series prediction of precipitation up to 2 h ahead can be improved using the data points of the days whose differences from the current day are multiples of 14. For short-term predictions, we employ infinite-dimensional delay coordinates (Hirata et al., Sci. Rep. 5, 15736, 2015) to reconstruct the underlying dynamics. Although the results demonstrate that the two-week periodicity seems to exist in the weather at Tokyo, and thus some anthropogenic activities could influence weather, the mechanism of the influence remains unclear. [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]

The identification of recurrences at various timescales in extreme event-like time series is challenging because of the rare occurrence of events which are separated by large temporal gaps. Most of the existing time series analysis techniques cannot be used to analyze an extreme event-like time series in its unaltered form. The study of the system dynamics by reconstruction of the phase space using the standard delay embedding method is not directly applicable to event-like time series as it assumes a Euclidean notion of distance between states in the phase space. The edit distance method is a novel approach that uses the point-process nature of events. We propose a modification of edit distance to analyze the dynamics of extreme event-like time series by incorporating a nonlinear function which takes into account the sparse distribution of extreme events and utilizes the physical significance of their temporal pattern. We apply the modified edit distance method to event-like data generated from point process as well as flood event series constructed from discharge data of the Mississippi River in the USA and compute their recurrence plots. From the recurrence analysis, we are able to quantify the deterministic properties of extreme event-like data. We also show that there is a significant serial dependency in the flood time series by using the random shuffle surrogate method. [ABSTRACT FROM AUTHOR]

Records for observing dynamics are usually complied by a form of time series. However, time series can be a challenging type of dataset for deep neural networks to learn. In deep neural networks, pairs of inputs and outputs are usually fed for constructive mapping. Such inputs are typically prepared as static images in successful applications. And so, here we propose two methods to prepare such inputs for learning the dynamical properties behind time series. In the first method, we simply array a time series in the shape of a rectangle as an image. In the second method, we convert a time series into a distance matrix using delay coordinates, or an unthresholded recurrence plot. We demonstrate that the second method performs well in inferring a slow driving force from observations of a forced system within which there are symmetry and almost invariant subsets. [ABSTRACT FROM AUTHOR]

The identification of recurrences at various time scales in extreme event-like time series is challenging because of the rare occurrence of events which are separated by large temporal gaps. Most of the existing time series analysis techniques cannot be used to analyse extreme event-like time series in its unaltered form. The study of the system dynamics by reconstruction of the phase space using the standard delay embedding method is not directly applicable to event-like time series as it assumes a Euclidean notion of distance between states in the phase space. The edit distance method is a novel approach that uses the point-process nature of events. We propose a modification of edit distance to analyze the dynamics of extreme event-like time series by incorporating a nonlinear function which takes into account the sparse distribution of extreme events and utilizes the physical significance of their temporal pattern. We apply the modified edit distance method to event-like data generated from point process as well as flood event series constructed from discharge data of the Mississippi River in USA, and compute their recurrence plots. From the recurrence analysis, we are able to quantify the deterministic properties of extreme event-like data. We also show that there is a significant serial dependency in the flood time series by using the random shuffle surrogate method. [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]

We previously proposed, on theoretical grounds, that the cerebellum must regulate the dimensionality of its neuronal activity during motor learning and control to cope with the low firing frequency of inferior olive neurons, which form one of two major inputs to the cerebellar cortex. Such dimensionality regulation is possible via modulation of electrical coupling through the gap junctions between inferior olive neurons by inhibitory GABAergic synapses. In addition, we previously showed in simulations that intermediate coupling strengths induce chaotic firing of inferior olive neurons and increase their information carrying capacity. However, there is no in vivo experimental data supporting these two theoretical predictions. Here, we computed the levels of synchrony, dimensionality, and chaos of the inferior olive code by analyzing in vivo recordings of Purkinje cell complex spike activity in three different coupling conditions: carbenoxolone (gap junctions blocker), control, and picrotoxin (GABA-A receptor antagonist). To examine the effect of electrical coupling on dimensionality and chaotic dynamics, we first determined the physiological range of effective coupling strengths between inferior olive neurons in the three conditions using a combination of a biophysical network model of the inferior olive and a novel Bayesian model averaging approach. We found that effective coupling co-varied with synchrony and was inversely related to the dimensionality of inferior olive firing dynamics, as measured via a principal component analysis of the spike trains in each condition. Furthermore, for both the model and the data, we found an inverted U-shaped relationship between coupling strengths and complexity entropy, a measure of chaos for spiking neural data. These results are consistent with our hypothesis according to which electrical coupling regulates the dimensionality and the complexity in the inferior olive neurons in order to optimize both motor learning and control of high dimensional motor systems by the cerebellum. Author summary: Computational theory suggests that the cerebellum must decrease the dimensionality of its neuronal activity to learn and control high dimensional motor systems effectively, while being constrained by the low firing frequency of inferior olive neurons, one of the two major source of input signals to the cerebellum. We previously proposed that the cerebellum adaptively controls the dimensionality of inferior olive firing by adjusting the level of synchrony and that such control is made possible by modulating the electrical coupling strength between inferior olive neurons. Here, we developed a novel method that uses a biophysical model of the inferior olive to accurately estimate the effective coupling strengths between inferior olive neurons from in vivo recordings of spike activity in three different coupling conditions. We found that high coupling strengths induce synchronous firing and decrease the dimensionality of inferior olive firing dynamics. In contrast, intermediate coupling strengths lead to chaotic firing and increase the dimensionality of the firing dynamics. Thus, electrical coupling is a feasible mechanism to control dimensionality and chaotic firing of inferior olive neurons. In sum, our results provide insights into possible mechanisms underlying cerebellar function and, in general, a biologically plausible framework to control the dimensionality of neural coding. [ABSTRACT FROM AUTHOR]

Hepatic epithelioid hemangioendothelioma is a rare neoplasm with a variable malignant potential and a high risk of recurrence. No general treatment guidelines have been established. Fortunately, we were able to minimize immunosuppressant after liver transplantation because of a full HLA‐matched case. There was no recurrence 1 year after treatment. [ABSTRACT FROM AUTHOR]

• We discuss practical scheduling methods of hormone therapy for prostate cancer. • We infer a schedule of hormone therapy based on a mathematical model. • We compare three types of scheduling methods and the current clinical protocol. • The sparse search can find good schedules without failure with not so much costs. • The model predictive control method is promising for its reasonability and usability. Hormone therapy is one of the popular therapeutic methods for prostate cancer. Intermittent androgen suppression (IAS) is the method which stops and resumes hormone therapy repeatedly. The efficacy of IAS differs depending on patients; both the cases have been reported where the relapse of cancer happened and did not happen, for the patients who had undergone IAS. For the patients who cannot avoid the relapse of cancer by IAS, we should delay the relapse of cancer as later as possible. Here we compared some practical methods of determining when to stop and restart hormone therapy for IAS using an existing mathematical model of prostate cancer. The method we suggest is to determine the ratio of on-treatment period and off-treatment period sparsely for each cycle, namely the "sparse search." We also compared the performance of the sparse search with the exhaustive search and the model predictive control. We found that the sparse search can find a good treatment schedule without failure, and the computational cost is not so high compared to the exhaustive method. In addition, we focus on the model predictive control (MPC) method which has been applied to the scheduling of IAS in some existing studies. The MPC is computationary efficient, although it does not always find an optimal schedule in the numerical experiments here. We believe that the MPC method might be also promising because of its reasonable computational costs and its possibility of expanding of the model. [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]