Multivariate singular spectrum analysis and the road to phase synchronization

Quasi-critical dynamics in large-scale social systems regulated by sudden events

How do heterogeneous individual behaviors arise in response to sudden events and how do they shape large-scale social dynamics? Based on a five-year naturalistic observation of individual purchasing behaviors, we extract the long-term consumption dynamics of diverse commodities from approximately 2.2 million purchase orders. We subdivide the consumption dynamics into trend, seasonal, and random components and analyze them using a renormalization group. We discover that the coronavirus pandemic, a sudden event acting on the social system, regulates the scaling and criticality of consumption dynamics. On a large time scale, the long-term dynamics of the system, regardless of arising from trend, seasonal, or random individual behaviors, is pushed toward a quasi-critical region between independent (i.e., the consumption behaviors of different commodities are irrelevant) and correlated (i.e., the consumption behaviors of different commodities are interrelated) phases as the pandemic erupts. On a small time scale, short-term consumption dynamics exhibits more diverse responses to the pandemic. While the trend and random behaviors of individuals are driven to quasi-criticality and exhibit scale-invariance as the pandemic breaks out, seasonal behaviors are more robust against regulations. Overall, these discoveries provide insights into how quasi-critical macroscopic dynamics emerges in heterogeneous social systems to enhance system reactivity to sudden events while there may exist specific system components maintaining robustness as a reflection of system stability.

Ocean–Atmosphere Variability in the Northwest Atlantic Ocean during Active Marine Heatwave Years

The Northwest (NW) Atlantic has experienced extreme ecological impacts from Marine Heatwaves (MHWs) within the past decade. This paper focuses on four MHW active years (2012, 2016, 2017, and 2020) and the relationship between Sea Surface Temperature anomalies (SSTA), Sea Surface Salinity anomalies (SSSA), North Atlantic Oscillation (NAO), Geopotential Height anomalies (ZA), and anomalous Jet Stream positions (JSPA). Multichannel singular spectrum analysis (MSSA) reveals the strongest temporal covariances between SSSA and SSTA, and JSPA and SSTA for all years, particularly for 2020 (SSSA–SSTA: 50%, JSPA–SSTA: 51%) indicating that this active MHW year was more atmospherically driven, followed by 2012, which had the second highest temporal covariances (SSSA–SSTA: 47%, JSPA–SSTA: 50%) between these parameters. Spatial correlations for SSSA and SSTA between NAO during MHW active years disrupt the long–term (2010–2020) positive relationship in the NW Atlantic. SSSA and JSPA, and SSSA and SSTA were strongly correlated across the NW Atlantic; 2012 SSSA–JSPA correlations were strong and positive between 56–62°W, and 2016, 2017, and 2020 SSSA–JSPA correlations were mostly strong and negative, with strong positive correlations present near the coastline (70–66°W) or off the NW Atlantic shelf (52–48°W). SSSA–SSTA showed the opposite correlations of similar spatial distributions of SSSA–JSPA for all MHW active years. This indicates strong relationships between JSPA, SSSA, and SSTA during MHWs. Understanding the temporal and spatial interplay between these parameters will aid in better monitoring and prediction of MHWs.

Inversion of Groundwater Storage Variations Considering Lag Effect in Beijing Plain, from RadarSat-2 with SBAS-InSAR Technology

The long-term over-exploitation of groundwater has not only caused the compaction of aquifer thickness and surface deformation but has also further aggravated the loss of groundwater storage (GWS) in Beijing plain. The South-to-North Water Diversion Project (SNWDP) furnishes a new source of water for Beijing. By reviewing related studies, it was found that there are few studies on the realization of GWS estimation based on InSAR technology considering the lag effect. Therefore, in this study, firstly, the long-time series deformation characteristics of Beijing plain were obtained from 46 RadarSat-2 images using small baseline subset interferometric synthetic aperture radar (SBAS-InSAR) technology. Secondly, the seasonal components of surface deformation and hydraulic head change were extracted by means of multichannel singular spectrum analysis (MSSA), verifying the separation accuracy by means of Monto Carlo-SSA (MC-SSA). Finally, for the hydrodynamic delay (aquifer water supply/drainage) of the complex aquifer system, we introduced the time lag cross-correlation (TLCC) approach to correct the hysteresis response of seasonal deformation relative to the variation of the aquifer system head, so as to realize the estimation of aquifer storage properties and GWS loss, even unrecoverable GWS (UGWS). The results showed that the average annual variation of total GWS (TGWS) in Beijing plain was −6.702 × 107 m3, of which the depletion volume of UGWS was −6.168 × 107 m3, accounting for 92.03% of the TGWS. On a temporal scale, the depletion of UGWS lagged behind the total head change, with about one year of lag time. On a spatial scale, in contrast to the north of Beijing plain, the depletion of UGWS in the south only recovered briefly after 2015 and then continued to decline. This further indicated that the process of the decline of middle-deep confined head and long-term GWS loss caused by over-exploitation of groundwater was irreversible. These findings are of great significance to optimize the allocation of groundwater resources, reduce the harm of land subsidence and protect groundwater resources.

Harmonic cross-correlation decomposition for multivariate time series

We introduce harmonic cross-correlation decomposition (HCD) as a tool to detect and visualize features in the frequency structure of multivariate time series. HCD decomposes multivariate time series into spatiotemporal harmonic modes with the leading modes representing dominant oscillatory patterns in the data. HCD is closely related to data-adaptive harmonic decomposition (DAHD) [Chekroun and Kondrashov, Chaos 27, 093110 (2017)10.1063/1.4989400] in that it performs an eigendecomposition of a grand matrix containing lagged cross-correlations. As for DAHD, each HCD mode is uniquely associated with a Fourier frequency, which allows for the definition of multidimensional power and phase spectra. Unlike in DAHD, however, HCD does not exhibit a systematic dependency on the ordering of the channels within the grand matrix. Further, HCD phase spectra can be related to the phase relations in the data in an intuitive way. We compare HCD with DAHD and multivariate singular spectrum analysis, a third related correlation-based decomposition, and we give illustrative applications to a simple traveling wave, as well as to simulations of three coupled Stuart-Landau oscillators and to human EEG recordings.

Tipping points induced by parameter drift in an excitable ocean model

Numerous systems in the climate sciences and elsewhere are excitable, exhibiting coexistence of and transitions between a basic and an excited state. We examine the role of tipping between two such states in an excitable low-order ocean model. Ensemble simulations are used to obtain the model's pullback attractor (PBA) and its properties, as a function of a forcing parameter [Formula: see text] and of the steepness [Formula: see text] of a climatological drift in the forcing. The tipping time [Formula: see text] is defined as the time at which the transition to relaxation oscillations (ROs) arises: at constant forcing this occurs at [Formula: see text]. As the steepness [Formula: see text] decreases, [Formula: see text] is delayed and the corresponding forcing amplitude decreases, while remaining always above [Formula: see text]. With periodic perturbations, that amplitude depends solely on [Formula: see text] over a significant range of parameters: this provides an example of rate-induced tipping in an excitable system. Nonlinear resonance occurs for periods comparable to the RO time scale. Coexisting PBAs and total independence from initial states are found for subsets of parameter space. In the broader context of climate dynamics, the parameter drift herein stands for the role of anthropogenic forcing.

Navigating differential structures in complex networks

Structural changes in a network representation of a system, due to different experimental conditions, different connectivity across layers, or to its time evolution, can provide insight on its organization, function, and on how it responds to external perturbations. The deeper understanding of how gene networks cope with diseases and treatments is maybe the most incisive demonstration of the gains obtained through this differential network analysis point of view, which led to an explosion of new numeric techniques in the last decade. However, where to focus one's attention, or how to navigate through the differential structures in the context of large networks, can be overwhelming even for a few experimental conditions. In this paper, we propose a theory and a methodological implementation for the characterization of shared "structural roles" of nodes simultaneously within and between networks. Inspired by recent methodological advances in chaotic phase synchronization analysis, we show how the information about the shared structures of a set of networks can be split and organized in an automatic fashion, in scenarios with very different (i) community sizes, (ii) total number of communities, and (iii) even for a large number of 100 networks compared using numerical benchmarks generated by a stochastic block model. Then, we investigate how the network size, number of networks, and mean size of communities influence the method performance in a series of Monte Carlo experiments. To illustrate its potential use in a more challenging scenario with real-world data, we show evidence that the method can still split and organize the structural information of a set of four gene coexpression networks obtained from two cell types × two treatments (interferon-β stimulated or control). Aside from its potential use as for automatic feature extraction and preprocessing tool, we discuss that another strength of the method is its "story-telling"-like characterization of the information encoded in a set of networks, which can be used to pinpoint unexpected shared structure, leading to further investigations and providing new insights. Finally, the method is flexible to address different research-field-specific questions, by not restricting what scientific-meaningful characteristic (or relevant feature) of a node shall be used.

Cardiac and Respiratory Self-Gating in Radial MRI Using an Adapted Singular Spectrum Analysis (SSA-FARY)

Cardiac Magnetic Resonance Imaging (MRI) is time-consuming and error-prone. To ease the patient's burden and to increase the efficiency and robustness of cardiac exams, interest in methods based on continuous steady-state acquisition and self-gating has been growing in recent years. Self-gating methods extract the cardiac and respiratory signals from the measurement data and then retrospectively sort the data into cardiac and respiratory phases. Repeated breathholds and synchronization with the heart beat using some external device as required in conventional MRI are then not necessary. In this work, we introduce a novel self-gating method for radially acquired data based on a dimensionality reduction technique for time-series analysis (SSA-FARY). Building on Singular Spectrum Analysis, a zero-padded, time-delayed embedding of the auto-calibration data is analyzed using Principle Component Analysis. We demonstrate the basic functionality of SSA-FARY using numerical simulations and apply it to in-vivo cardiac radial single-slice bSSFP and Simultaneous Multi-Slice radiofrequency-spoiled gradient-echo measurements, as well as to Stack-of-Stars bSSFP measurements. SSA-FARY reliably detects the cardiac and respiratory motion and separates it from noise. We utilize the generated signals for high-dimensional image reconstruction using parallel imaging and compressed sensing with in-plane wavelet and (spatio-)temporal total-variation regularization.

Estimation of temporal covariances in pathogen dynamics using Bayesian multivariate autoregressive models

It is well recognised that animal and plant pathogens form complex ecological communities of interacting organisms within their hosts, and there is growing interest in the health implications of such pathogen interactions. Although community ecology approaches have been used to identify pathogen interactions at the within-host scale, methodologies enabling robust identification of interactions from population-scale data such as that available from health authorities are lacking. To address this gap, we developed a statistical framework that jointly identifies interactions between multiple viruses from contemporaneous non-stationary infection time series. Our conceptual approach is derived from a Bayesian multivariate disease mapping framework. Importantly, our approach captures within- and between-year dependencies in infection risk while controlling for confounding factors such as seasonality, demographics and infection frequencies, allowing genuine pathogen interactions to be distinguished from simple correlations. We validated our framework using a broad range of synthetic data. We then applied it to diagnostic data available for five respiratory viruses co-circulating in a major urban population between 2005 and 2013: adenovirus, human coronavirus, human metapneumovirus, influenza B virus and respiratory syncytial virus. We found positive and negative covariances indicative of epidemiological interactions among specific virus pairs. This statistical framework enables a community ecology perspective to be applied to infectious disease epidemiology with important utility for public health planning and preparedness.

Disease-causing microorganisms, including viruses, bacteria, protozoa and fungi, form complex communities within animals and plants. These microorganisms can coexist harmoniously or even beneficially, or they may competitively interact for host resources. Well-studied examples include interactions between viruses and bacteria in the respiratory tract. Whilst ecological studies have revealed that some pathogens do interact within their hosts, identifying interactions from available population scale data from health authorities is challenging. This is exacerbated by a lack of large-scale data describing the infection patterns of multiple pathogens within single populations over long time frames. Furthermore, methods for evaluating whether infection frequencies of different pathogens fluctuate together or not over time cannot readily account for alternative explanations. For example, human pathogens may have related seasonal patterns depending on the age groups they infect and the weather conditions they survive in, and not because they are interacting. We developed a robust statistical framework to identify pathogen-pathogen interactions from population scale diagnostic data. This framework serves as a crucial step in identifying such important interactions and will guide new studies to elucidate their underpinning mechanisms. This will have important consequences for public health preparedness and the design of effective disease control interventions.

Faint phase synchronization detection through structured orthomax rotations in singular spectrum analysis

Multivariate singular spectrum analysis (M-SSA), with a structured varimax rotation, is a method that allows a deep characterization of phase synchronization (PS) phenomena in an almost automatic fashion. It has been increasingly used in the study of PS in networks of nonlinear, real-world, and numeric systems. This paper investigates the impact of the other recently developed structured orthomax rotations on the M-SSA ability to characterize PS. The results show that by using the structured quartimax rotation, a very faint and intermittent PS regime can be detected, in contrast with the structured varimax (which demands a stronger, more consolidated PS regime). This is due to the fact that the different rotations do not have the same efficiency in achieving a simple structure of the M-SSA eigenvectors. Nevertheless, for well-established PS regimes, the same robustness of the original M-SSA approach against high levels of additive Gaussian noise was found for the structured quartimax and biquartimax rotations. However, for all approaches we found an overshoot of the qualitative range for the PS onset due to noise.

Automatic Grouping in Singular Spectrum Analysis

Singular spectrum analysis (SSA) is a non-parametric forecasting and filtering method that has many applications in a variety of fields such as signal processing, economics and time series analysis. One of the four steps of the SSA, which is called the grouping step, plays a pivotal role in the SSA because reconstruction and forecasting of results are directly affected by the outputs of this step. Usually, the grouping step of SSA is time consuming as the interpretable components are manually selected. An alternative more optimized approach is to apply automatic grouping methods. In this paper, a new dissimilarity measure between two components of a time series that is based on various matrix norms is first proposed. Then, using the new dissimilarity matrices, the capabilities of different hierarchical clustering linkages are compared to identify appropriate groups in the SSA grouping step. The performance of the proposed approach is assessed using the corrected Rand index as validation criterion and utilizing various real-world and simulated time series.

Synchronization of world economic activity

Common dynamical properties of business cycle fluctuations are studied in a sample of more than 100 countries that represent economic regions from all around the world. We apply the methodology of multivariate singular spectrum analysis (M-SSA) to identify oscillatory modes and to detect whether these modes are shared by clusters of phase- and frequency-locked oscillators. An extension of the M-SSA approach is introduced to help analyze structural changes in the cluster configuration of synchronization. With this novel technique, we are able to identify a common mode of business cycle activity across our sample, and thus point to the existence of a world business cycle. Superimposed on this mode, we further identify several major events that have markedly influenced the landscape of world economic activity in the postwar era.

Introduction to focus issue: Synchronization in large networks and continuous media-data, models, and supermodels

The synchronization of loosely coupled chaotic systems has increasingly found applications to large networks of differential equations and to models of continuous media. These applications are at the core of the present Focus Issue. Synchronization between a system and its model, based on limited observations, gives a new perspective on data assimilation. Synchronization among different models of the same system defines a supermodel that can achieve partial consensus among models that otherwise disagree in several respects. Finally, novel methods of time series analysis permit a better description of synchronization in a system that is only observed partially and for a relatively short time. This Focus Issue discusses synchronization in extended systems or in components thereof, with particular attention to data assimilation, supermodeling, and their applications to various areas, from climate modeling to macroeconomics.

Observability and synchronization of neuron models

Observability is the property that enables recovering the state of a dynamical system from a reduced number of measured variables. In high-dimensional systems, it is therefore important to make sure that the variable recorded to perform the analysis conveys good observability of the system dynamics. The observability of a network of neuron models depends nontrivially on the observability of the node dynamics and on the topology of the network. The aim of this paper is twofold. First, to perform a study of observability using four well-known neuron models by computing three different observability coefficients. This not only clarifies observability properties of the models but also shows the limitations of applicability of each type of coefficients in the context of such models. Second, to study the emergence of phase synchronization in networks composed of neuron models. This is done performing multivariate singular spectrum analysis which, to the best of the authors' knowledge, has not been used in the context of networks of neuron models. It is shown that it is possible to detect phase synchronization: (i) without having to measure all the state variables, but only one (that provides greatest observability) from each node and (ii) without having to estimate the phase.

Enhancing multivariate singular spectrum analysis for phase synchronization: The role of observability

Multivariate singular spectrum analysis (M-SSA) was recently adapted to study systems of coupled oscillators. It does not require an a priori definition for phase nor detailed knowledge of the individual oscillators, but it uses all the variables of each system. This aspect could be restrictive for practical applications, since usually just a few (sometimes only one) variables are measured. Based on dynamical systems and observability theories, we first show how to apply the M-SSA with only one variable and show the conditions to achieve good performance. Next, we provide numerical evidence that this single-variable approach enhances the explanatory power compared to the original M-SSA when computed with all the system variables. This could have important practical implications, as pointed out using benchmark oscillators.

Matrix formulation and singular-value decomposition algorithm for structured varimax rotation in multivariate singular spectrum analysis

Groth and Ghil [Phys. Rev. E 84, 036206 (2011)PLEEE81539-375510.1103/PhysRevE.84.036206] developed a modified varimax rotation aimed at enhancing the ability of the multivariate singular spectrum analysis (M-SSA) to characterize phase synchronization in systems of coupled chaotic oscillators. Due to the special structure of the M-SSA eigenvectors, the modification proposed by Groth and Ghil imposes a constraint in the rotation of blocks of components associated with the different subsystems. Accordingly, here we call it a structured varimax rotation (SVR). The SVR was presented as successive pairwise rotations of the eigenvectors. The aim of this paper is threefold. First, we develop a closed matrix formulation for the entire family of structured orthomax rotation criteria, for which the SVR is a special case. Second, this matrix approach is used to enable the use of known singular value algorithms for fast computation, allowing a simultaneous rotation of the M-SSA eigenvectors (a Python code is provided in the Appendix). This could be critical in the characterization of phase synchronization phenomena in large real systems of coupled oscillators. Furthermore, the closed algebraic matrix formulation could be used in theoretical studies of the (modified) M-SSA approach. Third, we illustrate the use of the proposed singular value algorithm for the SVR in the context of the two benchmark examples of Groth and Ghil: the Rössler system in the chaotic (i) phase-coherent and (ii) funnel regimes. Comparison with the results obtained with Kaiser's original (unstructured) varimax rotation (UVR) reveals that both SVR and UVR give the same result for the phase-coherent scenario, but for the more complex behavior (ii) only the SVR improves on the M-SSA.

Principal nonlinear dynamical modes of climate variability

We suggest a new nonlinear expansion of space-distributed observational time series. The expansion allows constructing principal nonlinear manifolds holding essential part of observed variability. It yields low-dimensional hidden time series interpreted as internal modes driving observed multivariate dynamics as well as their mapping to a geographic grid. Bayesian optimality is used for selecting relevant structure of nonlinear transformation, including both the number of principal modes and degree of nonlinearity. Furthermore, the optimal characteristic time scale of the reconstructed modes is also found. The technique is applied to monthly sea surface temperature (SST) time series having a duration of 33 years and covering the globe. Three dominant nonlinear modes were extracted from the time series: the first efficiently separates the annual cycle, the second is responsible for ENSO variability, and combinations of the second and the third modes explain substantial parts of Pacific and Atlantic dynamics. A relation of the obtained modes to decadal natural climate variability including current hiatus in global warming is exhibited and discussed.