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Natural and social systems exhibit complex behavior reflecting their rich dynamics, whose governing laws are not fully known. This study develops a unified data-driven approach to estimate predictability of such systems when several independent realizations of the system's evolution are available. If the underlying dynamics are quasi-linear, the signal associated with the variable external factors, or forcings, can be estimated as the ensemble mean; this estimation can be optimized by filtering out the part of the variability with a low ensemble-mean-signal-to-residual-noise ratio. The dynamics of the residual internal variability is then encapsulated in an optimal, in a Bayesian sense, linear stochastic model able to predict the observed behavior. This model's self-forecast covariance matrices define a basis of patterns (directions) associated with the maximum forecast skill. Projecting the observed evolution onto these patterns produces the corresponding component time series. These ideas are illustrated by applying the proposed analysis technique to (1) ensemble data of regional sea-surface temperature evolution in the tropical Pacific generated by a state-of-the-art climate model and (2) consumer-spending records across multiple regions of the Russian Federation. These examples map out a range of possible solutions—from a solution characterized by a low-dimensional forced signal and a rich spectrum of predictable internal modes (1)—to the one in which the forced signal is extremely complex, but the number of predictable internal modes is limited (2). In each case, the proposed decompositions offer clues into the underlying dynamical processes, underscoring the usefulness of the proposed framework. [ABSTRACT FROM AUTHOR]

We suggest a new method for building data-driven dynamical models from observed multidimensional time series. The method is based on a recurrent neural network with specific structure, which allows for the joint reconstruction of both a low-dimensional embedding for dynamical components in the data and an evolution operator. The key link of the method is a Bayesian optimization of both model structure and the hypothesis about the data generating law, which is needed for constructing the cost function for model learning. First, the performance of the method is successfully tested in the situation when a signal from a low-dimensional dynamical system is hidden in noisy multidimensional observations. Second, the method is used for building the data-driven model of the low frequency variability (LFV) in the quasigeostrophic model of the Earth's midlatitude atmosphere—a high-dimensional chaotic system. It is demonstrated that the key regimes of the atmospheric LFV are reproduced correctly in data simulations by means of the obtained model. [ABSTRACT FROM AUTHOR]

We present a detailed description of a new approach for the extraction of principal nonlinear dynamical modes (NDMs) from high-dimensional data. The method of NDMs allows the joint reconstruction of hidden scalar time series underlying the observational variability together with a transformation mapping these time series to the physical space. Special Bayesian prior restrictions on the solution properties provide an efficient recognition of spatial patterns evolving in time and characterized by clearly separated time scales. In particular, we focus on adaptive properties of the NDMs and demonstrate for model examples of different complexities that, depending on the data properties, the obtained NDMs may have either substantially nonlinear or linear structures. It is shown that even linear NDMs give us more information about the internal system dynamics than the traditional empirical orthogonal function decomposition. The performance of the method is demonstrated on two examples. First, this approach is successfully tested on a low-dimensional problem to decode a chaotic signal from nonlinearly entangled time series with noise. Then, it is applied to the analysis of 250-year preindustrial control run of the INMCM4.0 global climate model. There, a set of principal modes of different nonlinearities is found capturing the internal model variability on the time scales from annual to multidecadal. [ABSTRACT FROM AUTHOR]