Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation

Conditional Gaussian nonlinear system: A fast preconditioner and a cheap surrogate model for complex nonlinear systems

Developing suitable approximate models for analyzing and simulating complex nonlinear systems is practically important. This paper aims at exploring the skill of a rich class of nonlinear stochastic models, known as the conditional Gaussian nonlinear system (CGNS), as both a cheap surrogate model and a fast preconditioner for facilitating many computationally challenging tasks. The CGNS preserves the underlying physics to a large extent and can reproduce intermittency, extreme events, and other non-Gaussian features in many complex systems arising from practical applications. Three interrelated topics are studied. First, the closed analytic formulas of solving the conditional statistics provide an efficient and accurate data assimilation scheme. It is shown that the data assimilation skill of a suitable CGNS approximate forecast model outweighs that by applying an ensemble method even to the perfect model with strong nonlinearity, where the latter suffers from filter divergence. Second, the CGNS allows the development of a fast algorithm for simultaneously estimating the parameters and the unobserved variables with uncertainty quantification in the presence of only partial observations. Utilizing an appropriate CGNS as a preconditioner significantly reduces the computational cost in accurately estimating the parameters in the original complex system. Finally, the CGNS advances rapid and statistically accurate algorithms for computing the probability density function and sampling the trajectories of the unobserved state variables. These fast algorithms facilitate the development of an efficient and accurate data-driven method for predicting the linear response of the original system with respect to parameter perturbations based on a suitable CGNS preconditioner.

Data-driven stochastic model for cross-interacting processes with different time scales

In this work, we propose a new data-driven method for modeling cross-interacting processes with different time scales represented by time series with different sampling steps. It is a generalization of a nonlinear stochastic model of an evolution operator based on neural networks and designed for the case of time series with a constant sampling step. The proposed model has a more complex structure. First, it describes each process by its own stochastic evolution operator with its own time step. Second, it takes into account possible nonlinear connections within each pair of processes in both directions. These connections are parameterized asymmetrically, depending on which process is faster and which process is slower. They make this model essentially different from the set of independent stochastic models constructed individually for each time scale. All evolution operators and connections are trained and optimized using the Bayesian framework, forming a multi-scale stochastic model. We demonstrate the performance of the model on two examples. The first example is a pair of coupled oscillators, with the couplings in both directions which can be turned on and off. Here, we show that inclusion of the connections into the model allows us to correctly reproduce observable effects related to coupling. The second example is a spatially distributed data generated by a global climate model running in the middle 19th century external conditions. In this case, the multi-scale model allows us to reproduce the coupling between the processes which exists in the observed data but is not captured by the model constructed individually for each process.

A virtual climate library of surface temperature over North America for 1979-2015

The most comprehensive continuous-coverage modern climatic data sets, known as reanalyses, come from combining state-of-the-art numerical weather prediction (NWP) models with diverse available observations. These reanalysis products estimate the path of climate evolution that actually happened, and their use in a probabilistic context-for example, to document trends in extreme events in response to climate change-is, therefore, limited. Free runs of NWP models without data assimilation can in principle be used for the latter purpose, but such simulations are computationally expensive and are prone to systematic biases. Here we produce a high-resolution, 100-member ensemble simulation of surface atmospheric temperature over North America for the 1979-2015 period using a comprehensive spatially extended non-stationary statistical model derived from the data based on the North American Regional Reanalysis. The surrogate climate realizations generated by this model are independent from, yet nearly statistically congruent with reality. This data set provides unique opportunities for the analysis of weather-related risk, with applications in agriculture, energy development, and protection of human life.

Data-adaptive harmonic spectra and multilayer Stuart-Landau models

Harmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure. The corresponding eigenvalues can be grouped per Fourier frequency and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey, furthermore, a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum. The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coefficients stacked by Fourier frequency which can be efficiently modeled-provided the decay of temporal correlations is sufficiently well-resolved-within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators. Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.

Comment on "Nonparametric forecasting of low-dimensional dynamical systems "

The comparison performed in Berry et al. [Phys. Rev. E 91, 032915 (2015)] between the skill in predicting the El Niño-Southern Oscillation climate phenomenon by the prediction method of Berry et al. and the "past-noise" forecasting method of Chekroun et al. [Proc. Natl. Acad. Sci. USA 108, 11766 (2011)] is flawed. Three specific misunderstandings in Berry et al. are pointed out and corrected.

Reply to "Comment on 'Nonparametric forecasting of low-dimensional dynamical systems' "

In this Reply we provide additional results which allow a better comparison of the diffusion forecast and the "past-noise" forecasting (PNF) approach for the El Niño index. We remark on some qualitative differences between the diffusion forecast and PNF, and we suggest an alternative use of the diffusion forecast for the purposes of forecasting the probabilities of extreme events.

Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics

Many physical systems are described by nonlinear differential equations that are too complicated to solve in full. A natural way to proceed is to divide the variables into those that are of direct interest and those that are not, formulate solvable approximate equations for the variables of greater interest, and use data and statistical methods to account for the impact of the other variables. In the present paper we consider time-dependent problems and introduce a fully discrete solution method, which simplifies both the analysis of the data and the numerical algorithms. The resulting time series are identified by a NARMAX (nonlinear autoregression moving average with exogenous input) representation familiar from engineering practice. The connections with the Mori-Zwanzig formalism of statistical physics are discussed, as well as an application to the Lorenz 96 system.

Nonparametric forecasting of low-dimensional dynamical systems

This paper presents a nonparametric modeling approach for forecasting stochastic dynamical systems on low-dimensional manifolds. The key idea is to represent the discrete shift maps on a smooth basis which can be obtained by the diffusion maps algorithm. In the limit of large data, this approach converges to a Galerkin projection of the semigroup solution to the underlying dynamics on a basis adapted to the invariant measure. This approach allows one to quantify uncertainties (in fact, evolve the probability distribution) for nontrivial dynamical systems with equation-free modeling. We verify our approach on various examples, ranging from an inhomogeneous anisotropic stochastic differential equation on a torus, the chaotic Lorenz three-dimensional model, and the Niño-3.4 data set which is used as a proxy of the El Niño Southern Oscillation.

Multiscale atmospheric dynamics: cross-frequency phase-amplitude coupling in the air temperature

Interactions between dynamics on different temporal scales of about a century long record of data of the daily mean surface air temperature from various European locations have been detected using a form of the conditional mutual information, statistically tested using the Fourier-transform and multifractal surrogate data methods. An information transfer from larger to smaller time scales has been observed as the influence of the phase of slow oscillatory phenomena with the periods around 6-11 yr on the amplitudes of the variability characterized by the smaller temporal scales from a few months to 4-5 yr. The overall effect of the slow oscillations on the interannual temperature variability within the range 1-2 ° C has been observed in large areas of Europe.

Very early warning of next El Niño

The most important driver of climate variability is the El Niño Southern Oscillation, which can trigger disasters in various parts of the globe. Despite its importance, conventional forecasting is still limited to 6 mo ahead. Recently, we developed an approach based on network analysis, which allows projection of an El Niño event about 1 y ahead. Here we show that our method correctly predicted the absence of El Niño events in 2012 and 2013 and now announce that our approach indicated (in September 2013 already) the return of El Niño in late 2014 with a 3-in-4 likelihood. We also discuss the relevance of the next El Niño to the question of global warming and the present hiatus in the global mean surface temperature.

Improved El Nino forecasting by cooperativity detection

Although anomalous episodic warming of the eastern equatorial Pacific, dubbed El Niño by Peruvian fishermen, has major (and occasionally devastating) impacts around the globe, robust forecasting is still limited to about 6 mo ahead. A significant extension of the prewarning time would be instrumental for avoiding some of the worst damages such as harvest failures in developing countries. Here we introduce a unique avenue toward El Niño prediction based on network methods, inspecting emerging teleconnections. Our approach starts from the evidence that a large-scale cooperative mode--linking the El Niño basin (equatorial Pacific corridor) and the rest of the ocean--builds up in the calendar year before the warming event. On this basis, we can develop an efficient 12-mo forecasting scheme, i.e., achieve some doubling of the early-warning period. Our method is based on high-quality observational data available since 1950 and yields hit rates above 0.5, whereas false-alarm rates are below 0.1.