Communications in Applied Mathematics and Computational Science

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

Jana de Wiljes, Lars Putzig, and Illia Horenko

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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).

Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 9, Number 1 (2014), 1-46.

Dates
Received: 29 November 2012
Revised: 22 October 2013
Accepted: 15 January 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.camcos/1513732104

Digital Object Identifier
doi:10.2140/camcos.2014.9.1

Mathematical Reviews number (MathSciNet)
MR3212866

Zentralblatt MATH identifier
1314.62010

Subjects
Primary: 62-07: Data analysis 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 62M05: Markov processes: estimation 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 65C60: Computational problems in statistics
Secondary: 62M02: Markov processes: hypothesis testing 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 62M30: Spatial processes 62M45: Neural nets and related approaches 62H11: Directional data; spatial statistics

Keywords
nonstationary nonhomogeneous discrete spatiotemporal time-series analysis Markov regression logistic data assimilation

Citation

de Wiljes, Jana; Putzig, Lars; Horenko, Illia. Discrete nonhomogeneous and nonstationary logistic and Markov regression models for spatiotemporal data with unresolved external influences. Commun. Appl. Math. Comput. Sci. 9 (2014), no. 1, 1--46. doi:10.2140/camcos.2014.9.1. https://projecteuclid.org/euclid.camcos/1513732104


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