Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media

Nonlinear dynamical stochastic models are ubiquitous in different areas.Their statistical properties are often of great interest,but are also very challenging to compute.Many excitable media models belong to such types of complex systems with large state dimensions and the associated covariance matrices have localized structures.In this article,a mathematical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed.Rigorous mathematical analysis shows that the local effect from the diffusion results in an exponential decay of the components in the covariance matrix as a function of the distance while the global effect due to the mean field interaction synchronizes different components and contributes to a global covariance.The analysis is based on a comparison with an appropriate linear surrogate model,of which the covariance propagation can be computed explicitly.Two important applications of these theoretical results are discussed.They are the spatial averaging strategy for efficiently sampling the covariance matrix and the localization technique in data assimilation.Test examples of a linear model and a stochastically coupled Fitz Hugh-Nagumo model for excitable media are adopted to validate the theoretical results.The latter is also used for a systematical study of the spatial averaging strategy in efficiently sampling the covariance matrix in different dynamical regimes.

A case study on the shareholder network effect of stock market data:An SARMA approach

One of the key research problems in financial markets is the investigation of inter-stock dependence.A good understanding in this regard is crucial for portfolio optimization.To this end,various econometric models have been proposed.Most of them assume that the random noise associated with each subject is independent.However,dependence might still exist within this random noise.Ignoring this valuable information might lead to biased estimations and inaccurate predictions.In this article,we study a spatial autoregressive moving average model with exogenous covariates.Spatial dependence from both response and random noise is considered simultaneously.A quasi-maximum likelihood estimator is developed,and the estimated parameters are shown to be consistent and asymptotically normal.We then conduct an extensive analysis of the proposed method by applying it to the Chinese stock market data.