Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition

Revealing trends and persistent cycles of non-autonomous systems with autonomous operator-theoretic techniques

An important problem in modern applied science is to characterize the behavior of systems with complex internal dynamics subjected to external forcings. Many existing approaches rely on ensembles to generate information from the external forcings, making them unsuitable to study natural systems where only a single realization is observed. A prominent example is climate dynamics, where an objective identification of signals in the observational record attributable to natural variability and climate change is crucial for making climate projections for the coming decades. Here, we show that operator-theoretic techniques previously developed to identify slowly decorrelating observables of autonomous dynamical systems provide a powerful means for identifying nonlinear trends and persistent cycles of non-autonomous systems using data from a single trajectory of the system. We apply our framework to real-world examples from climate dynamics: Variability of sea surface temperature over the industrial era and the mid-Pleistocene transition of Quaternary glaciation cycles.

Estimating predictability of a dynamical system from multiple samples of its evolution

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.

Revealing recurrent regimes of mid-latitude atmospheric variability using novel machine learning method

The low-frequency variability of the extratropical atmosphere involves hemispheric-scale recurring, often persistent, states known as teleconnection patterns or regimes, which can have a profound impact on predictability on intra-seasonal and longer timescales. However, reliable data-driven identification and dynamical representation of such states are still challenging problems in modeling the dynamics of the atmosphere. We present a new method, which allows us both to detect recurring regimes of atmospheric variability and to obtain dynamical variables serving as an embedding for these regimes. The method combines two approaches from nonlinear data analysis: partitioning a network of recurrent states with studying its properties by the recurrence quantification analysis and the kernel principal component analysis. We apply the method to study teleconnection patterns in a quasi-geostrophical model of atmospheric circulation over the extratropical hemisphere as well as to reanalysis data of geopotential height anomalies in the mid-latitudes of the Northern Hemisphere atmosphere in the winter seasons from 1981 to the present. It is shown that the detected regimes as well as the obtained set of dynamical variables explain large-scale weather patterns, which are associated, in particular, with severe winters over Eurasia and North America. The method presented opens prospects for improving empirical modeling and long-term forecasting of large-scale atmospheric circulation regimes.

Contributions to surface air temperature trends estimated from climate time series: Medium-term causalities

Contributions of various natural and anthropogenic factors to trends of surface air temperatures at different latitudes of the Northern and Southern hemispheres on various temporal horizons are estimated from climate data since the 19th century in empirical autoregressive models. Along with anthropogenic forcing, we assess the impact of several natural climate modes including Atlantic Multidecadal Oscillation, El-Nino/Southern Oscillation, Interdecadal Pacific Oscillation, Pacific Decadal Oscillation, and Antarctic Oscillation. On relatively short intervals of the length of two or three decades, contributions of climate variability modes are considerable and comparable to the contributions of greenhouse gases and even exceed the latter. On longer intervals of about half a century and greater, the contributions of greenhouse gases dominate at all latitudinal belts including polar, middle, and tropical ones.

Applying interval stability concept to empirical model of middle Pleistocene transition

Interval stability is a novel method for the study of complex dynamical systems, allowing for the estimation of their stability to strong perturbations. This method describes how large perturbation should be to disrupt the stable dynamical regime of the system (attractor). In our work, interval stability is used for the first time to study the properties of a real natural system: to analyze the stability of the earth's climate system during the last 2.6×10⁶ years. The main abrupt shift in global climate during this period is the middle Pleistocene transition (MPT), which occurred about 1×10⁶ years ago as a change of the periodicity of glacial cycles from 41 to 100 kyr. On the basis of the empirical nonlinear stochastic model proposed in our recent work, we demonstrate that the global climate stability to any perturbations decreases throughout the Pleistocene period (including the MPT), enhancing its response to fast (with a millennial scale or less) internal disturbances.

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.

On Time Scales of Intrinsic Oscillations in the Climate System

Proxy temperature data records featuring local time series, regional averages from areas all around the globe, as well as global averages, are analyzed using the Slow Feature Analysis (SFA) method. As explained in the paper, SFA is much more effective than the traditional Fourier analysis in identifying slow-varying (low-frequency) signals in data sets of a limited length. We find the existence of a striking gap from ~1000 to about ~20,000 years, which separates intrinsic climatic oscillations with periods ranging from ~60 years to ~1000 years, from the longer time-scale periodicities (20,000 year+) involving external forcing associated with Milankovitch cycles. The absence of natural oscillations with periods within the gap is consistent with cumulative evidence based on past data analyses, as well as with earlier theoretical and modeling studies.

Bayesian framework for simulation of dynamical systems from multidimensional data using recurrent neural network

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.