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1. Nonlinear Dynamics for the Ising Model

   https://doi.org/10.1007/s00220-024-05129-w  

   Caputo, Pietro; Sinclair, Alistair (November 2024, Communications in Mathematical Physics)

   Abstract We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pairwise interactions, and captures a number of important nonlinear models from various fields, including chemical reaction networks, Boltzmann’s model of an ideal gas, recombination in population genetics and genetic algorithms. In the context of spin systems, it is a natural generalization of linear dynamics based on Markov chains, such as Glauber dynamics and block dynamics, which are by now well understood. However, the inherent nonlinearity makes the dynamics much harder to analyze, and rigorous quantitative results so far are limited to processes which converge to essentially trivial stationary distributions that are product measures. In this paper we provide the first quantitative convergence analysis for natural nonlinear dynamics in a combinatorial setting where the stationary distribution contains non-trivial correlations, namely spin systems at high temperatures. We prove that nonlinear versions of both the Glauber dynamics and the block dynamics converge to the Gibbs distribution of the Ising model (with given external fields) in times$$O(n\log n)$$ $O(nlogn)$and$$O(\log n)$$ $O(logn)$respectively, wherenis the size of the underlying graph (number of spins). Given the lack of general analytical methods for such nonlinear systems, our analysis is unconventional, and combines tools such as information percolation (due in the linear setting to Lubetzky and Sly), a novel coupling of the Ising model with Erdős-Rényi random graphs, and non-traditional branching processes augmented by a “fragmentation” process. Our results extend immediately to any spin system with a finite number of spins and bounded interactions. 

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2. A Nonstationary Standardized Precipitation Index (NSPI) Using Bayesian Splines

   https://doi.org/10.1175/JAMC-D-21-0244.1  

   Stagge, James H.; Sung, Kyungmin (July 2022, Journal of Applied Meteorology and Climatology)

   Abstract The standardized precipitation index (SPI) measures meteorological drought relative to historical climatology by normalizing accumulated precipitation. Longer record lengths improve parameter estimates, but these longer records may include signals of anthropogenic climate change and multidecadal natural climate fluctuations. Historically, climate nonstationarity has either been ignored or incorporated into the SPI using a quasi-stationary reference period, such as the WMO 30-yr period. This study introduces and evaluates a novel nonstationary SPI model based on Bayesian splines, designed to both improve parameter estimates for stationary climates and to explicitly incorporate nonstationarity. Using synthetically generated precipitation, this study directly compares the proposed Bayesian SPI model with existing SPI approaches based on maximum likelihood estimation for stationary and nonstationary climates. The proposed model not only reproduced the performance of existing SPI models but improved upon them in several key areas: reducing parameter uncertainty and noise, simultaneously modeling the likelihood of zero and positive precipitation, and capturing nonlinear trends and seasonal shifts across all parameters. Further, the fully Bayesian approach ensures all parameters have uncertainty estimates, including zero precipitation likelihood. The study notes that the zero precipitation parameter is too sensitive and could be improved in future iterations. The study concludes with an application of the proposed Bayesian nonstationary SPI model for nine gauges across a range of hydroclimate zones in the United States. Results of this experiment show that the model is stable and reproduces nonstationary patterns identified in prior studies, while also indicating new findings, particularly for the shape and zero precipitation parameters. Significance StatementWe typically measure how bad a drought is by comparing it with the historical record. With long-term changes in climate or other factors, however, a typical drought today may not have been typical in the recent past. The purpose of this study is to build a model that measures drought relative to a changing climate. Our results confirm that the model is accurate and captures previously noted climate change patterns—a drier western United States, a wetter eastern United States, earlier summer weather, and more extreme wet seasons. This is significant because this model can improve drought measurement and identify recent changes in drought. 

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3. Causal Discovery and Forecasting in Nonstationary Environments with State-Space Models

   Biwei Huang, Kun Zhang (January 2019, Proceedings of the 36th International Conference on Machine Learning)

   In many scientific fields, such as economics and neuroscience, we are often faced with nonstationary time series, and concerned with both finding causal relations and forecasting the values of variables of interest, both of which are particularly challenging in such nonstationary environments. In this paper, we study causal discovery and forecasting for nonstationary time series. By exploiting a particular type of state-space model to represent the processes, we show that nonstationarity helps to identify the causal structure, and that forecasting naturally benefits from learned causal knowledge. Specifically, we allow changes in both causal strengths and noise variances in the nonlinear state-space models, which, interestingly, renders both the causal structure and model parameters identifiable. Given the causal model, we treat forecasting as a problem in Bayesian inference in the causal model, which exploits the time-varying property of the data and adapts to new observations in a principled manner. Experimental results on synthetic and real-world data sets demonstrate the efficacy of the proposed methods. 

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4. Spatio-Temporal Expanding Distance Asymptotic Framework for Locally Stationary Processes

   https://doi.org/10.1007/s13171-020-00213-4  

   Chu, Tingjin; Liu, Jialuo; Zhu, Jun; Wang, Haonan (January 2020, Sankhya A)

   Spatio-temporal data indexed by sampling locations and sampling time points are encountered in many scientific disciplines such as climatology, environ- mental sciences, and public health. Here, we propose a novel spatio-temporal expanding distance (STED) asymptotic framework for studying the proper- ties of statistical inference for nonstationary spatio-temporal models. In particular, to model spatio-temporal dependence, we develop a new class of locally stationary spatio-temporal covariance functions. The STED asymp- totic framework has a fixed spatio-temporal domain for spatio-temporal pro- cesses that are globally nonstationary in a rescaled fixed domain and locally stationary in a distance expanding domain. The utility of STED is illus- trated by establishing the asymptotic properties of the maximum likelihood estimation for a general class of spatio-temporal covariance functions. A simulation study suggests sound finite-sample properties and the method is applied to a sea-surface temperature dataset. 

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5. Functional central limit theorem via nonstationary projective conditions

   Florence Merlevede and Magda Peligrad (April 2023, Progress in probability)

   Radosław Adamczak, Nathael Gozlan (Ed.)

   In this paper we survey some recent progress on the Gaussian approximation for nonstationary dependent structures via martingale methods. First we present general theorems involving projective conditions for triangular arrays of random variables and then present various applications for rho-mixing and alpha-dependent triangular arrays, stationary sequences in a random time scenery, application to the quenched FCLT, application to linear statistics with alpha-dependent innovations, application to functions of a triangular stationary Markov chain. 

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