Predicting observed and hidden extreme events in complex nonlinear dynamical systems with partial observations and short training time series.

Extreme events appear in many complex nonlinear dynamical systems. Predicting extreme events has important scientific significance and large societal impacts. In this paper, a new mathematical framework of building suitable nonlinear approximate models is developed, which aims at predicting both the observed and hidden extreme events in complex nonlinear dynamical systems for short-, medium-, and long-range forecasting using only short and partially observed training time series. Different from many ad hoc data-driven regression models, these new nonlinear models take into account physically motivated processes and physics constraints. They also allow efficient and accurate algorithms for parameter estimation, data assimilation, and prediction. Cheap stochastic parameterizations, judicious linear feedback control, and suitable noise inflation strategies are incorporated into the new nonlinear modeling framework, which provide accurate predictions of both the observed and hidden extreme events as well as the strongly non-Gaussian statistics in various highly intermittent nonlinear dyad and triad models, including the Lorenz 63 model. Then, a stochastic mode reduction strategy is applied to a 21-dimensional nonlinear paradigm model for topographic mean flow interaction. The resulting five-dimensional physics-constrained nonlinear approximate model is able to accurately predict extreme events and the regime switching between zonally blocked and unblocked flow patterns. Finally, incorporating judicious linear stochastic processes into a simple nonlinear approximate model succeeds in learning certain complicated nonlinear effects of a six-dimensional low-order Charney-DeVore model with strong chaotic and regime switching behavior. The simple nonlinear approximate model then allows accurate online state estimation and the short- and medium-range forecasting of extreme events.

Using machine learning to predict extreme events in complex systems.

Extreme events and the related anomalous statistics are ubiquitously observed in many natural systems, and the development of efficient methods to understand and accurately predict such representative features remains a grand challenge. Here, we investigate the skill of deep learning strategies in the prediction of extreme events in complex turbulent dynamical systems. Deep neural networks have been successfully applied to many imaging processing problems involving big data, and have recently shown potential for the study of dynamical systems. We propose to use a densely connected mixed-scale network model to capture the extreme events appearing in a truncated Korteweg-de Vries (tKdV) statistical framework, which creates anomalous skewed distributions consistent with recent laboratory experiments for shallow water waves across an abrupt depth change, where a remarkable statistical phase transition is generated by varying the inverse temperature parameter in the corresponding Gibbs invariant measures. The neural network is trained using data without knowing the explicit model dynamics, and the training data are only drawn from the near-Gaussian regime of the tKdV model solutions without the occurrence of large extreme values. A relative entropy loss function, together with empirical partition functions, is proposed for measuring the accuracy of the network output where the dominant structures in the turbulent field are emphasized. The optimized network is shown to gain uniformly high skill in accurately predicting the solutions in a wide variety of statistical regimes, including highly skewed extreme events. The technique is promising to be further applied to other complicated high-dimensional systems.

Linear and nonlinear statistical response theories with prototype applications to sensitivity analysis and statistical control of complex turbulent dynamical systems.

Statistical response theory provides an effective tool for the analysis and statistical prediction of high-dimensional complex turbulent systems involving a large number of unresolved unstable modes, for example, in climate change science. Recently, the linear and nonlinear response theories have shown promising developments in overcoming the curse-of-dimensionality in uncertain quantification and statistical control of turbulent systems by identifying the most sensitive response directions. We offer an extensive illustration of using the statistical response theory for a wide variety of challenging problems under a hierarchy of prototype models ranging from simple solvable equations to anisotropic geophysical turbulence. Directly applying the linear response operator for statistical responses is shown to only have limited skill for small perturbation ranges. For stronger nonlinearity and perturbations, a nonlinear reduced-order statistical model reduction strategy guaranteeing model fidelity and sensitivity provides a systematic framework to recover the multiscale variability in leading order statistics. The linear response operator is applied in the training phase for the optimal nonlinear model responses requiring only the unperturbed equilibrium statistics. The statistical response theory is further applied to the statistical control of inherently high-dimensional systems. The statistical response in the mean offers an efficient way to recover the control forcing from the statistical energy equation without the need to run the expensive model. Among all the testing examples, the statistical response strategy displays uniform robust skill in various dynamical regimes with distinct statistical features. Further applications of the statistical response theory include the prediction of extreme events and intermittency in turbulent passive transport and a rigorous saturation bound governing the total statistical growth from initial and external uncertainties.

Statistical dynamical model to predict extreme events and anomalous features in shallow water waves with abrupt depth change.

Understanding and predicting extreme events and their anomalous statistics in complex nonlinear systems are a grand challenge in climate, material, and neuroscience as well as for engineering design. Recent laboratory experiments in weakly turbulent shallow water reveal a remarkable transition from Gaussian to anomalous behavior as surface waves cross an abrupt depth change (ADC). Downstream of the ADC, probability density functions of surface displacement exhibit strong positive skewness accompanied by an elevated level of extreme events. Here, we develop a statistical dynamical model to explain and quantitatively predict the above anomalous statistical behavior as experimental control parameters are varied. The first step is to use incoming and outgoing truncated Korteweg-de Vries (TKdV) equations matched in time at the ADC. The TKdV equation is a Hamiltonian system, which induces incoming and outgoing statistical Gibbs invariant measures. The statistical matching of the known nearly Gaussian incoming Gibbs state at the ADC completely determines the predicted anomalous outgoing Gibbs state, which can be calculated by a simple sampling algorithm verified by direct numerical simulations, and successfully captures key features of the experiment. There is even an analytic formula for the anomalous outgoing skewness. The strategy here should be useful for predicting extreme anomalous statistical behavior in other dispersive media.

Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification.

A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction-diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker-Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems.

Complex multiscale systems are ubiquitous in many areas. This research expository article discusses the development and applications of a recent information-theoretic framework as well as novel reduced-order nonlinear modeling strategies for understanding and predicting complex multiscale systems. The topics include the basic mathematical properties and qualitative features of complex multiscale systems, statistical prediction and uncertainty quantification, state estimation or data assimilation, and coping with the inevitable model errors in approximating such complex systems. Here, the information-theoretic framework is applied to rigorously quantify the model fidelity, model sensitivity and information barriers arising from different approximation strategies. It also succeeds in assessing the skill of filtering and predicting complex dynamical systems and overcomes the shortcomings in traditional path-wise measurements such as the failure in measuring extreme events. In addition, information theory is incorporated into a systematic data-driven nonlinear stochastic modeling framework that allows effective predictions of nonlinear intermittent time series. Finally, new efficient reduced-order nonlinear modeling strategies combined with information theory for model calibration provide skillful predictions of intermittent extreme events in spatially-extended complex dynamical systems. The contents here include the general mathematical theories, effective numerical procedures, instructive qualitative models, and concrete models from climate, atmosphere and ocean science.

Beating the curse of dimension with accurate statistics for the Fokker-Planck equation in complex turbulent systems.

Solving the Fokker-Planck equation for high-dimensional complex dynamical systems is an important issue. Recently, the authors developed efficient statistically accurate algorithms for solving the Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures, which contain many strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy with a small number of samples [Formula: see text], where a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious Gaussian kernel density estimation in the remaining low-dimensional subspace. In this article, two effective strategies are developed and incorporated into these algorithms. The first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions, which allows an extremely efficient parallel computation due to the small size of each individual block. The second strategy exploits statistical symmetry for a further reduction of [Formula: see text] The resulting algorithms can efficiently solve the Fokker-Planck equation with strongly non-Gaussian PDFs in much higher dimensions even with orders in the millions and thus beat the curse of dimension. The algorithms are applied to a [Formula: see text]-dimensional stochastic coupled FitzHugh-Nagumo model for excitable media. An accurate recovery of both the transient and equilibrium non-Gaussian PDFs requires only [Formula: see text] samples! In addition, the block decomposition facilitates the algorithms to efficiently capture the distinct non-Gaussian features at different locations in a [Formula: see text]-dimensional two-layer inhomogeneous Lorenz 96 model, using only [Formula: see text] samples.

Effective control of complex turbulent dynamical systems through statistical functionals.

Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among complex systems in science and engineering, including climate, material, and neural science. Control of these complex systems is a grand challenge, for example, in mitigating the effects of climate change or safe design of technology with fully developed shear turbulence. Control of flows in the transition to turbulence, where there is a small dimension of instabilities about a basic mean state, is an important and successful discipline. In complex turbulent dynamical systems, it is impossible to track and control the large dimension of instabilities, which strongly interact and exchange energy, and new control strategies are needed. The goal of this paper is to propose an effective statistical control strategy for complex turbulent dynamical systems based on a recent statistical energy principle and statistical linear response theory. We illustrate the potential practical efficiency and verify this effective statistical control strategy on the 40D Lorenz 1996 model in forcing regimes with various types of fully turbulent dynamics with nearly one-half of the phase space unstable.

Simple stochastic dynamical models capturing the statistical diversity of El Niño Southern Oscillation.

The El Niño Southern Oscillation (ENSO) has significant impact on global climate and seasonal prediction. A simple modeling framework is developed here that automatically captures the statistical diversity of ENSO. First, a stochastic parameterization of the wind bursts including both westerly and easterly winds is coupled to a simple ocean-atmosphere model that is otherwise deterministic, linear, and stable. Second, a simple nonlinear zonal advection with no ad hoc parameterization of the background sea-surface temperature (SST) gradient and a mean easterly trade wind anomaly representing the multidecadal acceleration of the trade wind are both incorporated into the coupled model that enables anomalous warm SST in the central Pacific. Then a three-state stochastic Markov jump process is used to drive the wind burst activity that depends on the strength of the western Pacific warm pool in a simple and effective fashion. It allows the coupled model to simulate the quasi-regular moderate traditional El Niño, the super El Niño, and the central Pacific (CP) El Niño as well as the La Niña with realistic features. In addition to the anomalous SST, the Walker circulation anomalies at different ENSO phases all resemble those in nature. In particular, the coupled model succeeds in reproducing the observed episode during the 1990s, where a series of 5-y CP El Niños is followed by a super El Niño and then a La Niña. Importantly, both the variance and the non-Gaussian statistical features in different Niño regions spanning from the western to the eastern Pacific are captured by the coupled model.

State estimation and prediction using clustered particle filters.

Particle filtering is an essential tool to improve uncertain model predictions by incorporating noisy observational data from complex systems including non-Gaussian features. A class of particle filters, clustered particle filters, is introduced for high-dimensional nonlinear systems, which uses relatively few particles compared with the standard particle filter. The clustered particle filter captures non-Gaussian features of the true signal, which are typical in complex nonlinear dynamical systems such as geophysical systems. The method is also robust in the difficult regime of high-quality sparse and infrequent observations. The key features of the clustered particle filtering are coarse-grained localization through the clustering of the state variables and particle adjustment to stabilize the method; each observation affects only neighbor state variables through clustering and particles are adjusted to prevent particle collapse due to high-quality observations. The clustered particle filter is tested for the 40-dimensional Lorenz 96 model with several dynamical regimes including strongly non-Gaussian statistics. The clustered particle filter shows robust skill in both achieving accurate filter results and capturing non-Gaussian statistics of the true signal. It is further extended to multiscale data assimilation, which provides the large-scale estimation by combining a cheap reduced-order forecast model and mixed observations of the large- and small-scale variables. This approach enables the use of a larger number of particles due to the computational savings in the forecast model. The multiscale clustered particle filter is tested for one-dimensional dispersive wave turbulence using a forecast model with model errors.

Simple dynamical models capturing the key features of the Central Pacific El Niño.

The Central Pacific El Niño (CP El Niño) has been frequently observed in recent decades. The phenomenon is characterized by an anomalous warm sea surface temperature (SST) confined to the central Pacific and has different teleconnections from the traditional El Niño. Here, simple models are developed and shown to capture the key mechanisms of the CP El Niño. The starting model involves coupled atmosphere-ocean processes that are deterministic, linear, and stable. Then, systematic strategies are developed for incorporating several major mechanisms of the CP El Niño into the coupled system. First, simple nonlinear zonal advection with no ad hoc parameterization of the background SST gradient is introduced that creates coupled nonlinear advective modes of the SST. Secondly, due to the recent multidecadal strengthening of the easterly trade wind, a stochastic parameterization of the wind bursts including a mean easterly trade wind anomaly is coupled to the simple atmosphere-ocean processes. Effective stochastic noise in the wind burst model facilitates the intermittent occurrence of the CP El Niño with realistic amplitude and duration. In addition to the anomalous warm SST in the central Pacific, other major features of the CP El Niño such as the rising branch of the anomalous Walker circulation being shifted to the central Pacific and the eastern Pacific cooling with a shallow thermocline are all captured by this simple coupled model. Importantly, the coupled model succeeds in simulating a series of CP El Niño that lasts for 5 y, which resembles the two CP El Niño episodes during 1990-1995 and 2002-2006.

Simple stochastic model for El Niño with westerly wind bursts.

Atmospheric wind bursts in the tropics play a key role in the dynamics of the El Niño Southern Oscillation (ENSO). A simple modeling framework is proposed that summarizes this relationship and captures major features of the observational record while remaining physically consistent and amenable to detailed analysis. Within this simple framework, wind burst activity evolves according to a stochastic two-state Markov switching-diffusion process that depends on the strength of the western Pacific warm pool, and is coupled to simple ocean-atmosphere processes that are otherwise deterministic, stable, and linear. A simple model with this parameterization and no additional nonlinearities reproduces a realistic ENSO cycle with intermittent El Niño and La Niña events of varying intensity and strength as well as realistic buildup and shutdown of wind burst activity in the western Pacific. The wind burst activity has a direct causal effect on the ENSO variability: in particular, it intermittently triggers regular El Niño or La Niña events, super El Niño events, or no events at all, which enables the model to capture observed ENSO statistics such as the probability density function and power spectrum of eastern Pacific sea surface temperatures. The present framework provides further theoretical and practical insight on the relationship between wind burst activity and the ENSO.

Concrete ensemble Kalman filters with rigorous catastrophic filter divergence.

The ensemble Kalman filter and ensemble square root filters are data assimilation methods used to combine high-dimensional, nonlinear dynamical models with observed data. Ensemble methods are indispensable tools in science and engineering and have enjoyed great success in geophysical sciences, because they allow for computationally cheap low-ensemble-state approximation for extremely high-dimensional turbulent forecast models. From a theoretical perspective, the dynamical properties of these methods are poorly understood. One of the central mysteries is the numerical phenomenon known as catastrophic filter divergence, whereby ensemble-state estimates explode to machine infinity, despite the true state remaining in a bounded region. In this article we provide a breakthrough insight into the phenomenon, by introducing a simple and natural forecast model that transparently exhibits catastrophic filter divergence under all ensemble methods and a large set of initializations. For this model, catastrophic filter divergence is not an artifact of numerical instability, but rather a true dynamical property of the filter. The divergence is not only validated numerically but also proven rigorously. The model cleanly illustrates mechanisms that give rise to catastrophic divergence and confirms intuitive accounts of the phenomena given in past literature.

Statistical energy conservation principle for inhomogeneous turbulent dynamical systems.

Understanding the complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering shear turbulence as well as climate atmosphere ocean science is a grand challenge of contemporary science with important societal impact. In such inhomogeneous turbulent dynamical systems there is a large dimensional phase space with a large dimension of unstable directions where a large-scale ensemble mean and the turbulent fluctuations exchange energy and strongly influence each other. These complex features strongly impact practical prediction and uncertainty quantification. A systematic energy conservation principle is developed here in a Theorem that precisely accounts for the statistical energy exchange between the mean flow and the related turbulent fluctuations. This statistical energy is a sum of the energy in the mean and the trace of the covariance of the fluctuating turbulence. This result applies to general inhomogeneous turbulent dynamical systems including the above applications. The Theorem involves an assessment of statistical symmetries for the nonlinear interactions and a self-contained treatment is presented below. Corollary 1 and Corollary 2 illustrate the power of the method with general closed differential equalities for the statistical energy in time either exactly or with upper and lower bounds, provided that the negative symmetric dissipation matrix is diagonal in a suitable basis. Implications of the energy principle for low-order closure modeling and automatic estimates for the single point variance are discussed below.

Blended particle filters for large-dimensional chaotic dynamical systems.

A major challenge in contemporary data science is the development of statistically accurate particle filters to capture non-Gaussian features in large-dimensional chaotic dynamical systems. Blended particle filters that capture non-Gaussian features in an adaptively evolving low-dimensional subspace through particles interacting with evolving Gaussian statistics on the remaining portion of phase space are introduced here. These blended particle filters are constructed in this paper through a mathematical formalism involving conditional Gaussian mixtures combined with statistically nonlinear forecast models compatible with this structure developed recently with high skill for uncertainty quantification. Stringent test cases for filtering involving the 40-dimensional Lorenz 96 model with a 5-dimensional adaptive subspace for nonlinear blended filtering in various turbulent regimes with at least nine positive Lyapunov exponents are used here. These cases demonstrate the high skill of the blended particle filter algorithms in capturing both highly non-Gaussian dynamical features as well as crucial nonlinear statistics for accurate filtering in extreme filtering regimes with sparse infrequent high-quality observations. The formalism developed here is also useful for multiscale filtering of turbulent systems and a simple application is sketched below.

Conceptual dynamical models for turbulence.

Understanding the complexity of anisotropic turbulent processes in engineering and environmental fluid flows is a formidable challenge with practical significance because energy often flows intermittently from the smaller scales to impact the largest scales in these flows. Conceptual dynamical models for anisotropic turbulence are introduced and developed here which, despite their simplicity, capture key features of vastly more complicated turbulent systems. These conceptual models involve a large-scale mean flow and turbulent fluctuations on a variety of spatial scales with energy-conserving wave-mean-flow interactions as well as stochastic forcing of the fluctuations. Numerical experiments with a six-dimensional conceptual dynamical model confirm that these models capture key statistical features of vastly more complex anisotropic turbulent systems in a qualitative fashion. These features include chaotic statistical behavior of the mean flow with a sub-Gaussian probability distribution function (pdf) for its fluctuations whereas the turbulent fluctuations have decreasing energy and correlation times at smaller scales, with nearly Gaussian pdfs for the large-scale fluctuations and fat-tailed non-Gaussian pdfs for the smaller-scale fluctuations. This last feature is a manifestation of intermittency of the small-scale fluctuations where turbulent modes with small variance have relatively frequent extreme events which directly impact the mean flow. The dynamical models introduced here potentially provide a useful test bed for algorithms for prediction, uncertainty quantification, and data assimilation for anisotropic turbulent systems.

Statistically accurate low-order models for uncertainty quantification in turbulent dynamical systems.

A framework for low-order predictive statistical modeling and uncertainty quantification in turbulent dynamical systems is developed here. These reduced-order, modified quasilinear Gaussian (ROMQG) algorithms apply to turbulent dynamical systems in which there is significant linear instability or linear nonnormal dynamics in the unperturbed system and energy-conserving nonlinear interactions that transfer energy from the unstable modes to the stable modes where dissipation occurs, resulting in a statistical steady state; such turbulent dynamical systems are ubiquitous in geophysical and engineering turbulence. The ROMQG method involves constructing a low-order, nonlinear, dynamical system for the mean and covariance statistics in the reduced subspace that has the unperturbed statistics as a stable fixed point and optimally incorporates the indirect effect of non-Gaussian third-order statistics for the unperturbed system in a systematic calibration stage. This calibration procedure is achieved through information involving only the mean and covariance statistics for the unperturbed equilibrium. The performance of the ROMQG algorithm is assessed on two stringent test cases: the 40-mode Lorenz 96 model mimicking midlatitude atmospheric turbulence and two-layer baroclinic models for high-latitude ocean turbulence with over 125,000 degrees of freedom. In the Lorenz 96 model, the ROMQG algorithm with just a single mode captures the transient response to random or deterministic forcing. For the baroclinic ocean turbulence models, the inexpensive ROMQG algorithm with 252 modes, less than 0.2% of the total, captures the nonlinear response of the energy, the heat flux, and even the one-dimensional energy and heat flux spectra.

Efficient stochastic superparameterization for geophysical turbulence.

Efficient computation of geophysical turbulence, such as occurs in the atmosphere and ocean, is a formidable challenge for the following reasons: the complex combination of waves, jets, and vortices; significant energetic backscatter from unresolved small scales to resolved large scales; a lack of dynamical scale separation between large and small scales; and small-scale instabilities, conditional on the large scales, which do not saturate. Nevertheless, efficient methods are needed to allow large ensemble simulations of sufficient size to provide meaningful quantifications of uncertainty in future predictions and past reanalyses through data assimilation and filtering. Here, a class of efficient stochastic superparameterization algorithms is introduced. In contrast to conventional superparameterization, the method here (i) does not require the simulation of nonlinear eddy dynamics on periodic embedded domains, (ii) includes a better representation of unresolved small-scale instabilities, and (iii) allows efficient representation of a much wider range of unresolved scales. The simplest algorithm implemented here radically improves efficiency by representing small-scale eddies at and below the limit of computational resolution by a suitable one-dimensional stochastic model of random-direction plane waves. In contrast to heterogeneous multiscale methods, the methods developed here do not require strong scale separation or conditional equilibration of local statistics. The simplest algorithm introduced here shows excellent performance on a difficult test suite of prototype problems for geophysical turbulence with waves, jets, and vortices, with a speedup of several orders of magnitude compared with direct simulation.

Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum and intermittency.

This paper motivates, develops and reviews elementary models for turbulent tracers with a background mean gradient which, despite their simplicity, have complex statistical features mimicking crucial aspects of laboratory experiments and atmospheric observations. These statistical features include exact formulas for tracer eddy diffusivity which is non-local in space and time, exact formulas and simple numerics for the tracer variance spectrum in a statistical steady state, and the transition to intermittent scalar probability density functions with fat exponential tails as certain variances of the advecting mean velocity are increased while satisfying important physical constraints. The recent use of such simple models with complex statistics as unambiguous test models for central contemporary issues in both climate change science and the real-time filtering of turbulent tracers from sparse noisy observations is highlighted throughout the paper.

Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability.

Many processes in science and engineering develop multiscale temporal and spatial patterns, with complex underlying dynamics and time-dependent external forcings. Because of the importance in understanding and predicting these phenomena, extracting the salient modes of variability empirically from incomplete observations is a problem of wide contemporary interest. Here, we present a technique for analyzing high-dimensional, complex time series that exploits the geometrical relationships between the observed data points to recover features characteristic of strongly nonlinear dynamics (such as intermittency and rare events), which are not accessible to classical singular spectrum analysis. The method employs Laplacian eigenmaps, evaluated after suitable time-lagged embedding, to produce a reduced representation of the observed samples, where standard tools of matrix algebra can be used to perform truncated singular-value decomposition despite the nonlinear geometrical structure of the dataset. We illustrate the utility of the technique in capturing intermittent modes associated with the Kuroshio current in the North Pacific sector of a general circulation model and dimensional reduction of a low-order atmospheric model featuring chaotic intermittent regime transitions, where classical singular spectrum analysis is already known to fail dramatically.