Discrete nonhomogeneous and nonstationary logistic and Markov regression models for spatiotemporal data with unresolved external influences

Abstract

Dynamical systems with different characteristic behavior at multiple scales can be modeled with hybrid methods combining a discrete model (e.g., corresponding to the microscale) triggered by a continuous mechanism and vice versa. A data-driven black-box-type framework is proposed, where the discrete model is parametrized with adaptive regression techniques and the output of the continuous counterpart (e.g., output of partial differential equations) is coupled to the discrete system of interest in the form of a fixed exogenous time series of external factors. Data availability represents a significant issue for this type of coupled discrete-continuous model, and it is shown that missing information/observations can be incorporated in the model via a nonstationary and nonhomogeneous formulation. An unbiased estimator for the discrete model dynamics in presence of unobserved external impacts is derived and used to construct a data-based nonstationary and nonhomogeneous parameter estimator based on an appropriately regularized spatiotemporal clustering algorithm. One-step and long-term predictions are considered, and a new Bayesian approach to discrete data assimilation of hidden information is proposed. To illustrate our method, we apply it to synthetic data sets and compare it with standard techniques of the machine-learning community (such as maximum-likelihood estimation, artificial neural networks and support vector machines).