Bayesian Analysis

Modeling space-time data using stochastic differential equations

Jason A. Duan, Alan E. Gelfand, and C. F. Sirmans

Full-text: Open access

Abstract

This paper demonstrates the use and value of stochastic differential equations for modeling space-time data in two common settings. The first consists of point-referenced or geostatistical data where observations are collected at fixed locations and times. The second considers random point pattern data where the emergence of locations and times is random. For both cases, we employ stochastic differential equations to describe a latent process within a hierarchical model for the data. The intent is to view this latent process mechanistically and endow it with appropriate simple features and interpretable parameters. A motivating problem for the second setting is to model urban development through observed locations and times of new home construction; this gives rise to a space-time point pattern. We show that a spatio-temporal Cox process whose intensity is driven by a stochastic logistic equation is a viable mechanistic model that affords meaningful interpretation for the results of statistical inference. Other applications of stochastic logistic differential equations with space-time varying parameters include modeling population growth and product diffusion, which motivate our first, point-referenced data application. We propose a method to discretize both time and space in order to fit the model. We demonstrate the inference for the geostatistical model through a simulated dataset. Then, we fit the Cox process model to a real dataset taken from the greater Dallas metropolitan area.

Article information

Source
Bayesian Anal., Volume 4, Number 4 (2009), 733-758.

Dates
First available in Project Euclid: 22 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ba/1340369822

Digital Object Identifier
doi:10.1214/09-BA427

Mathematical Reviews number (MathSciNet)
MR2570086

Zentralblatt MATH identifier
1330.86023

Keywords
geostatistical data point pattern hierarchical model stochastic logistic equation Markov chain Monte Carlo urban development

Citation

Duan, Jason A.; Gelfand, Alan E.; Sirmans, C. F. Modeling space-time data using stochastic differential equations. Bayesian Anal. 4 (2009), no. 4, 733--758. doi:10.1214/09-BA427. https://projecteuclid.org/euclid.ba/1340369822


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