Coupled Time Series Research Articles

Enhancing Time Series Predictability via Structure‐Aware Reservoir Computing

Accurate prediction of the future evolution of observational time series is a paramount challenge in current data‐driven research. While existing techniques struggle to learn useful representations from the temporal correlations, the high dimensionality in spatial domain is always considered as obstacle, leading to the curse of dimensionality and excessive resource consumption. This work designs a novel structure‐aware reservoir computing aiming at enhancing the predictability of coupled time series, by incorporating their historical dynamics as well as structural information. Paralleled reservoir computers with redesigned mixing inputs based on spatial relationships are implemented to cope with the multiple time series, whose core idea originates from the principle of the celebrated Granger causality. Representative numerical simulations and comparisons demonstrate the superior performance of the approach over the traditional ones. This work provides valuable insights into deeply mining both temporal and spatial information to enhance the representation learning of data in various machine learning techniques.

A global evaluation of the associations between long-term dynamics of seed falls and rodents.

One classic system of pulsed resource and animal population is mast seeding and population dynamics of seed-eating rodents in forests. However, we still lack an understanding of the global patterns regarding the contributions of seed falls to rodent outbreaks or population dynamics. We analyzed a global dataset of coupled long-term time series of seed abundances and rodent populations from published literature, including 66 and 89 time series (156 rodent-seed pairs from 37 studies) for rodent and seed abundances, respectively. We found only half of the examined rodent populations showed statistically significant coincidence between rodent outbreak and mast-seeding years. Over all the coupled time series, seed abundance was found to positively correlate with rodent abundance with a one-year lag, and the relative importance of seed abundance was much lower than that of density dependence in affecting rodent population growth rates. We also found the relative importance of seed abundance decreased, but that of rodent density dependence increased with the latitude of study. For the first time, our work provides a global pattern on the associations between seed falls and rodent population dynamics mostly in mid- and high-latitude forests, and highlights the necessity of more long-term studies on this subject in more forest ecosystems.

Reconstructing coupled time series in climate systems using three kinds of machine-learning methods

Abstract. Despite the great success of machine learning, its application in climate dynamics has not been well developed. One concern might be how well the trained neural networks could learn a dynamical system and what will be the potential application of this kind of learning. In this paper, three machine-learning methods are used: reservoir computer (RC), backpropagation-based (BP) artificial neural network, and long short-term memory (LSTM) neural network. It shows that the coupling relations or dynamics among variables in linear or nonlinear systems can be inferred by RC and LSTM, which can be further applied to reconstruct one time series from the other. Specifically, we analyzed the climatic toy models to address two questions: (i) what factors significantly influence machine-learning reconstruction and (ii) how do we select suitable explanatory variables for machine-learning reconstruction. The results reveal that both linear and nonlinear coupling relations between variables do influence the reconstruction quality of machine learning. If there is a strong linear coupling between two variables, then the reconstruction can be bidirectional, and both of these two variables can be an explanatory variable for reconstructing the other. When the linear coupling among variables is absent but with the significant nonlinear coupling, the machine-learning reconstruction between two variables is direction dependent, and it may be only unidirectional. Then the convergent cross mapping (CCM) causality index is proposed to determine which variable can be taken as the reconstructed one and which as the explanatory variable. In a real-world example, the Pearson correlation between the average tropical surface air temperature (TSAT) and the average Northern Hemisphere SAT (NHSAT) is weak (0.08), but the CCM index of NHSAT cross mapped with TSAT is large (0.70). And this indicates that TSAT can be well reconstructed from NHSAT through machine learning. All results shown in this study could provide insights into machine-learning approaches for paleoclimate reconstruction, parameterization scheme, and prediction in related climate research.Highlights: i The coupling dynamics learned by machine learning can be used to reconstruct time series. ii Reconstruction quality is direction dependent and variable dependent for nonlinear systems. iii The CCM index is a potential indicator to choose reconstructed and explanatory variables. iv The tropical average SAT can be well reconstructed from the average Northern Hemisphere SAT.

DISCOVERING CAUSALITIES FROM CARDIOTOCOGRAPHY SIGNALS USING IMPROVED CONVERGENT CROSS MAPPING WITH GAUSSIAN PROCESSES.

Convergent cross mapping (CCM) is designed for causal discovery in coupled time series, where Granger causality may not be applicable because of a separability assumption. However, CCM is not robust to observation noise which limits its applicability on signals that are known to be noisy. Moreover, the parameters for state space reconstruction need to be selected using grid search methods. In this paper, we propose a novel improved version of CCM using Gaussian processes for discovery of causality from noisy time series. Specifically, we adopt the concept of CCM and carry out the key steps using Gaussian processes within a non-parametric Bayesian probabilistic framework in a principled manner. The proposed approach is first validated on simulated data, and then used for understanding the interaction between fetal heart rate and uterine activity in the last two hours before delivery and of interest in obstetrics. Our results indicate that uterine activity affects the fetal heart rate, which agrees with recent clinical studies.

Incorporation of Structural Health Monitoring Coupled Data in Dynamic Extreme Stress Prediction of Steel Bridges Using Dynamic Coupled Linear Models

In this article, an approach for using structural health monitoring coupled extreme stress data in dynamic extreme stress prediction of steel bridges is presented, where the coupled extreme stress data means the extreme stress data with dynamicity, randomness, and trend. Firstly, the modeling processes about dynamic coupled linear models (DCLM) are provided based on a supposed coupled time series; furthermore, the dynamic probabilistic recursion processes about DCLM are given with Bayes method; secondly, the monitoring dynamic coupled extreme stress data is taken as a time series, historical monitoring coupled extreme stress data‐based DCLM and the corresponding Bayesian probabilistic recursion processes are given for predicting bridge extreme stresses; furthermore, the monitoring mechanism is provided for monitoring the prediction precision of DCLM; finally, the monitoring coupled extreme stress data of a steel bridge is used to illustrate the proposed approach which can provide the foundations for bridge reliability prediction and assessment.

Detecting Causality using Deep Gaussian Processes.

Convergent cross mapping (CCM) is a state space reconstruction (SSR)-based method designed for causal discovery in coupled time series, where Granger causality may not be applicable due to a separability assumption. However, CCM requires a large number of observations and is not robust to observation noise which limits its applicability. Moreover, in CCM and its variants, the SSR step is mostly implemented with delay embedding where the parameters for reconstruction usually need to be selected using grid search-based methods. In this paper, we propose a Bayesian version of CCM using deep Gaussian processes (DGPs), which are naturally connected with deep neural networks. In particular, we adopt the framework of SSR-based causal discovery and carry out the key steps using DGPs within a non-parametric Bayesian probabilistic framework in a principled manner. The proposed approach is first validated on simulated data and then tested on data used in obstetrics for monitoring the well-being of fetuses, i.e., fetal heart rate (FHR) and uterine activity (UA) signals in the last two hours before delivery. Our results indicate that UA affects the FHR, which agrees with recent clinical studies.

Coupled rainfall and water vapour stable isotope time series reveal tropical atmospheric processes on multiple timescales

AbstractHigh‐frequency stable isotope data are useful for validating atmospheric moisture circulation models and provide improved understanding of the mechanisms controlling isotopic compositions in tropical rainfall. Here, we present a near‐continuous 6‐month record of O‐ and H‐isotope compositions in both water vapour and daily rainfall from Northeast Australia measured by laser spectroscopy. The data set spans both wet and dry seasons to help address a significant data and knowledge gap in the southern hemisphere tropics. We interpret the isotopic records for water vapour and rainfall in the context of contemporaneous meteorological observations. Surface air moisture provided near‐continuous tracking of the links between isotopic variations and meteorological events on local to regional spatial scales. Power spectrum analysis of the isotopic variation showed a range of significant periodicities, from hourly to monthly scales, and cross‐wavelet analysis identified significant regions of common power for hourly averaged water vapour isotopic composition and relative humidity, wind direction, and solar radiation. Relative humidity had the greatest subdiurnal influence on isotopic composition. On longer timescales (weeks to months), isotope variability was strongly correlated with both wind direction and relative humidity. The high‐frequency records showed diurnal isotopic variations in O‐ and H‐isotope compositions due to local dew formation and, for deuterium excess, as a result of evapotranspiration. Several significant negative isotope anomalies on a daily scale were associated with the activity of regional mesoscale convective systems and the occurrence of two tropical cyclones. Calculated air parcel back trajectories identified the predominant moisture transport paths from the Southwest Pacific Ocean, whereas moisture transport from northerly directions occurred mainly during the wet season monsoonal airflow. Water vapour isotope compositions reflected the same meteorological events as recorded in rainfall isotopes but provided much more detailed and continuous information on atmospheric moisture cycling than the intermittent isotopic record provided by rainfall. Improved global coverage of stable isotope data for atmospheric water vapour is likely to improve simulations of future changes to climate drivers of the hydrological cycle.

Can Community Structure Causally Determine Dynamics of Constituent Species? A Test Using a Host-Parasite Community.

Structures of communities have been widely studied with the assumption that they not only are a useful bookkeeping tool but also can causally influence dynamics of the populations from which they emerge. However, convincing tests of this assumption have remained elusive because generally the only way to alter a community property is by manipulating its constituent populations, thereby preventing independent measurements of effects on those populations. There is a growing body of evidence that methods like convergent cross-mapping (CCM) can be used to make inferences about causal interactions using state space reconstructions of coupled time series, a method that relies on only observational data. Here we show that CCM can be used to test the causal effects of community properties using a well-studied Slovakian rodent-ectoparasite community. CCM identified causal drivers across the organizational scales of this community, including evidence that host dynamics were influenced by the degree to which the community at large was connected and clustered. Our findings add to the growing literature on the importance of community structures in disease dynamics and argue for a broader use of causal inference in the analysis of community dynamics.

Partially ordered permutation complexity of coupled time series

A new information-theoretic complexity measure of coupling among multiple time series is proposed based on ordinal patterns. We regard ordinal patterns extracted from time series as total orders and exploit their algebraic properties to introduce a multiplication among them. The result of the multiplication is in general a partial order. A Kullback–Leibler divergence for the partial orders gives rise to a complexity measure of coupling in the sense that it vanishes in two extreme cases: when time series are uncoupled and when they are identical. The performance of the proposed complexity measure is tested on two model multivariate dynamical systems and its advantage over the existing complexity measures is discussed.

Local detrended cross-correlation analysis for non-stationary time series

We propose a method called local detrended cross-correlation analysis (LDCCA) to quantify temporal power-law cross-correlation characteristics of coupled time series at local samples. The proposed method is validated with uncoupled Gaussian white noises, coupled ARFIMA processes and Hénon maps. As an example, electrical probe technologies are employed to detect the flow structure information of gas–liquid churn flows in a vertical pipe, and temporal cross-correlation characteristics of flow interfacial structures are investigated using the proposed LDCCA. The results show that the proposed LDCCA can provide beneficial insights to local dynamic evolution behaviors of the flow interfacial structures.

Persistent homology of time-dependent functional networks constructed from coupled time series.

We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example consists of biological data in the form of functional magnetic resonance imaging data that were acquired from human subjects during a simple motor-learning task in which subjects were monitored for three days during a five-day period. With these examples, we demonstrate that (1) using persistent homology to study functional networks provides fascinating insights into their properties and (2) the position of the features in a filtration can sometimes play a more vital role than persistence in the interpretation of topological features, even though conventionally the latter is used to distinguish between signal and noise. We find that persistent homology can detect differences in synchronization patterns in our data sets over time, giving insight both on changes in community structure in the networks and on increased synchronization between brain regions that form loops in a functional network during motor learning. For the motor-learning data, persistence landscapes also reveal that on average the majority of changes in the network loops take place on the second of the three days of the learning process.

Characterization of flow pattern transitions for horizontal liquid–liquid pipe flows by using multi-scale distribution entropy in coupled 3D phase space

Abstract Horizontal oil–water two-phase flows often exist in many industrial processes. Uncovering the dynamic mechanism of the flow pattern transition is of great significance for modeling the flow parameters. In this study we propose a method called multi-scale distribution entropy (MSDE) in a coupled 3D phase space, and use it to characterize the flow pattern transitions in horizontal oil–water two-phase flows. Firstly, the proposed MSDE is validated with Lorenz system and ARFIMA processes. Interestingly, it is found that the MSDE is dramatically associated with the cross-correlations of the coupled time series. Then, through conducting the experiment of horizontal oil–water two-phase flows, the upstream and downstream flow information is collected using a conductance cross-correlation velocity probe. The coupled cross-correlated signals are investigated using the MSDE method, and the results indicate that the MSDE is an effective tool uncovering the complex dynamic behaviors of flow pattern transitions.

Asymmetric Dynamic Attunement of Speech and Gestures in the Construction of Children's Understanding.

As children learn they use their speech to express words and their hands to gesture. This study investigates the interplay between real-time gestures and speech as children construct cognitive understanding during a hands-on science task. 12 children (M = 6, F = 6) from Kindergarten (n = 5) and first grade (n = 7) participated in this study. Each verbal utterance and gesture during the task were coded, on a complexity scale derived from dynamic skill theory. To explore the interplay between speech and gestures, we applied a cross recurrence quantification analysis (CRQA) to the two coupled time series of the skill levels of verbalizations and gestures. The analysis focused on (1) the temporal relation between gestures and speech, (2) the relative strength and direction of the interaction between gestures and speech, (3) the relative strength and direction between gestures and speech for different levels of understanding, and (4) relations between CRQA measures and other child characteristics. The results show that older and younger children differ in the (temporal) asymmetry in the gestures–speech interaction. For younger children, the balance leans more toward gestures leading speech in time, while the balance leans more toward speech leading gestures for older children. Secondly, at the group level, speech attracts gestures in a more dynamically stable fashion than vice versa, and this asymmetry in gestures and speech extends to lower and higher understanding levels. Yet, for older children, the mutual coupling between gestures and speech is more dynamically stable regarding the higher understanding levels. Gestures and speech are more synchronized in time as children are older. A higher score on schools’ language tests is related to speech attracting gestures more rigidly and more asymmetry between gestures and speech, only for the less difficult understanding levels. A higher score on math or past science tasks is related to less asymmetry between gestures and speech. The picture that emerges from our analyses suggests that the relation between gestures, speech and cognition is more complex than previously thought. We suggest that temporal differences and asymmetry in influence between gestures and speech arise from simultaneous coordination of synergies.

The value of a broad temporal and spatial perspective in understanding dynamics of kelp forest ecosystems

Many ecological processes play out over longer time scales and larger spatial scales than can be studied in a traditional 2–4-year grant cycle. Uncertainties in future funding hinder efforts to implement comprehensive research programs that integrate coupled time series observations of physical variables and ecological responses, manipulative experiments and synthetic analyses over the long term. Such research is essential for advancing our understanding of ecological responses associated with climate change, and the physical and biological processes that control them. This need is perhaps greatest for ecosystems that display highly dynamic and spatially complex patterns that are difficult to explain with short-term, small-scale studies. Such is the case for kelp forest ecosystems, which often show tremendous spatial and temporal variability in resource supply, consumer control and physical disturbance across spatial scales of metres to hundreds of kilometres and temporal scales of hours to decades. Here we present four examples from the Santa Barbara Coastal Long-term Ecological Research project that demonstrate the value of a broad temporal and spatial perspective in understanding the causes and ecological consequences of short-term local dynamics of giant kelp forests of California, USA.

Dimensional reduction of conditional algebraic multi-information via transcripts

Symbolic representation is a standard and powerful technique in time series analysis. In an ordinal symbolic representation the symbols are the so-called ordinal patterns, which can be identified with permutations. Transcripts exploit the fact that permutations build a group, the transcript of a pair of permutations being the product of the second permutation times the inverse of the first one. This particular setting can be easily generalized to any representation which elements belong to an algebraic group. The dimensional reduction of conditional multi-information via transcripts, proved in this paper, perfectly shows the potential of such algebraic symbolic representations. Specifically, given N+M group-valued random variables, the multi-information of N variables conditioned on the other M variables can also be calculated as a multi-information of N transcripts conditioned on M-1 transcripts, under some restrictions. Such a dimensional reduction can be crucial when estimating a conditional multi-information from short time series. Applications include two popular ordinal indicators of the information flow in coupled time series, namely, symbolic transfer entropy and momentary sorting information transfer. As a by-product of the above results, two new information directionality indicators based on ordinal transcripts are proposed, the simplest one being an (unconditioned) mutual information.

Permutation complexity of coupled time series

Permutation complexity of coupled time series

Application of joint permutations for predicting coupled time series

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.

Coupling between time series: A network view

Recently, the visibility graph has been introduced as a novel method for analyzing time series, which maps a time series to a complex network. In this paper we introduce a new algorithm of visibility, “cross-visibility”, which reveals the conjugation of two coupled time series. The correspondence between the two time series is mapped to a network, “the cross-visibility graph”, to demonstrate the correlation between them. We have applied the algorithm to several correlated and uncorrelated time series, generated by the linear stationary ARFIMA process, in order to better understand the results of the cross-visibility of empirical series. The comparison between the degree distribution of coupled and uncoupled (shuffled) series' networks demonstrates the emergence of super nodes (extremely high-degree nodes) in the uncoupled ones. Furthermore, we have applied the algorithm to real-world data from the financial trades of two companies and oil, and observed significant small-scale coupling in their dynamics.

Information directionality in coupled time series using transcripts

In ordinal symbolic dynamics, transcripts describe the algebraic relationship between ordinal patterns. Using the concept of transcript, we exploit the mathematical structure of the group of permutations to derive properties and relations among information measures of the symbolic representations of time series. These theoretical results are then applied for the assessment of coupling directionality in dynamical systems, where suitable coupling directionality measures are introduced depending only on transcripts. These measures improve the reliability of the information flow estimates and reduce to well-established coupling directionality quantifiers when some general conditions are satisfied. Furthermore, by generalizing the definition of transcript to ordinal patterns of different lengths, several of the commonly used information directionality measures can be encompassed within the same framework.

Scale dependence of the directional relationships between coupled time series

Using the cross-correlation of the wavelet transformation, we propose a general method of studying the scale dependence of the direction of coupling for coupled time series. The method is first demonstrated by applying it to coupled van der Pol forced oscillators and coupled nonlinear stochastic equations. We then apply the method to the analysis of the log-return time series of the stock values of the IBM and General Electric (GE) companies. Our analysis indicates that, on average, IBM stocks react earlier to possible common sector price movements than those of GE.