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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.

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.

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.

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.

The chromatin conformation capture-related methods such as Hi-C have improved our understanding of nuclear architecture and organization in recent years. However, reconstruction of nuclear architecture from single-cell Hi-C (scHi-C) data is challenging due to limited information of DNA contacts obtained from a single cell. We have previously developed the Recurrence Plot-Based Reconstruction (RPR) method for reconstructing three dimensional (3D) genomic structure from Hi-C data of single haploid cells (1) and diploid cells (2). This algorithm is based on a recurrence plot, a tool of nonlinear time-series analysis for visualizing patterns within a time series (3, 4), and enables the reconstruction of a unique 3D chromosome architecture even from low-coverage DNA contact information. Here we used the RPR method to analyzing published scHi-C data of diploid cells derived from early-stage F1 hybrid embryos (5) as a proof-of-concept for understanding of global developmental changes in chromosomal architecture of early stage embryos. We found that paternal and maternal chromosomes become gradually intermingled from 1 cell to 64 cell stage, and that discrete chromosome territories are largely established between 8 cell and 64 cell stages. We also observed Rabl-like polarization of chromosomes from the 2-to 8-cell stages, but this polarization becomes mostly dissolved by the 64-cell stage. Rabl-like chromosome polarization precedes rod-like extension and parallel alignment of chromosomes, implicating the role of Rabl-like polarization in effective mixing of chromosomes before establishing chromosome territories. We also found cell-to-cell variability in chromatin configuration. A combination of scHi-C and RPR analyses can depict features of the 3D chromatin architecture of individual cells at different developmental stages during early embryogenesis.

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.

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.

The circadian clock is synchronized by environmental cues, mostly by light and temperature. Explaining how the plant circadian clock responds to temperature oscillations is crucial to understanding plant responsiveness to the environment. Here, we found a prevalent temperature-dependent function of the Arabidopsis clock component EARLY FLOWERING 4 (ELF4) in the root clock. Although the clocks in roots are able to run in the absence of shoots, micrografting assays and mathematical analyses show that ELF4 moves from shoots to regulate rhythms in roots. ELF4 movement does not convey photoperiodic information, but trafficking is essential for controlling the period of the root clock in a temperature-dependent manner. Low temperatures favour ELF4 mobility, resulting in a slow-paced root clock, whereas high temperatures decrease movement, leading to a faster clock. Hence, the mobile ELF4 delivers temperature information and establishes a shoot-to-root dialogue that sets the pace of the clock in roots., Research in the Y.H. laboratory is supported by JSPS KAKENHI (Grant Number JP18K11461). The S.J.D. laboratory is funded by the Biotechnology and Biological Sciences Research Council (BB/N018540/1). S.A.K. acknowledges support from the National Institutes of Health (GM067837). The Mas laboratory is funded by the FEDER/Spanish Ministry of Economy and Competitiveness, the Ramon Areces Foundation and the Generalitat de Catalunya (AGAUR). The P.M. laboratory also acknowledges financial support from the CERCA Program, Generalitat de Catalunya and by the Spanish Ministry of Economy and Competitiveness through the Severo Ochoa Program for Centers of Excellence in R&D 2016–2019 (SEV-2015-0533). W.W.C. is a recipient of a Chinese Scholarship Council (CSC) fellowship.

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.

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.

Using a dataset of 150 patients treated with intermittent androgen suppression (IAS) through a fixed treatment schedule, we retrospectively designed a personalized treatment schedule mathematically for each patient. We estimated 100 sets of parameter values for each patient by randomly resampling each patient’s time points to take into account the uncertainty for observations of prostate specific antigen (PSA). Then, we identified 3 types and classified patients accordingly: in type (i), the relapse, namely the divergence of PSA, can be prevented by IAS; in type (ii), the relapse can be delayed by IAS later than by continuous androgen suppression (CAS); in type (iii) IAS was not beneficial and therefore CAS would have been more appropriate in the long run. Moreover, we obtained a treatment schedule of hormone therapy by minimizing the PSA of 3 years later in the worst case scenario among the 100 parameter sets by searching exhaustively all over the possible treatment schedules. If the most frequent type among 100 sets was type (i), the maximal PSA tended to be kept less than 100 ng/ml longer in IAS than in CAS, while there was no statistical difference for the other cases. Thus, mathematically personalized IAS should be studied prospectively.

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.

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.

Short-term prediction for renewable energy outputs up to 6 h is important especially for preparing backup power plants such as thermal power plants and hydro power plants to keep the voltage and the frequency in a power grid constant. However, short-term prediction of solar irradiance for the morning is especially difficult because measurements of solar irradiance before sunrise are zeros and useless for the prediction. Here we propose to use recently derived infinite-dimensional delay coordinates for predicting solar irradiance after sunrise based on a time series before the sunrise. As a result, one can make time series prediction by taking into account the long history of previous changes of solar irradiance. We demonstrate that the examined short-term time series prediction has effectively predictive skills because its prediction errors are smaller by about 85% best compared with the 24 h clear sky index persistence.

We propose a method for generating surrogate data that preserves all the properties of ordinal patterns up to a certain length, such as the numbers of allowed/forbidden ordinal patterns and transition likelihoods from ordinal patterns into others. The null hypothesis is that the details of the underlying dynamics do not matter beyond the refinements of ordinal patterns finer than a predefined length. The proposed surrogate data help construct a test of determinism that is free from the common linearity assumption for a null-hypothesis.