Modeling Space-Time Data Using Stochastic Differential Equations

This paper demonstrates the use and value of stochastic difierential equations for modeling space-time data in two common settings. The flrst consists of point-referenced or geostatistical data where observations are collected at flxed locations and times. The second considers random point pattern data where the emergence of locations and times is random. For both cases, we employ stochas- tic difierential 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 afiords mean- ingful interpretation for the results of statistical inference. Other applications of stochastic logistic difierential equations with space-time varying parameters in- clude modeling population growth and product difiusion, which motivate our flrst, point-referenced data application. We propose a method to discretize both time and space in order to flt the model. We demonstrate the inference for the geosta- tistical model through a simulated dataset. Then, we flt the Cox process model to a real dataset taken from the greater Dallas metropolitan area.

[1]  N. Shephard,et al.  Likelihood INference for Discretely Observed Non-linear Diffusions , 2001 .

[2]  Y. Ogata.  Space-Time Point-Process Models for Earthquake Occurrences , 1998 .

[3]  P. Brown,et al.  Blur‐generated non‐separable space–time models , 2000 .

[4]  Peter Breeze,et al.  Point Processes and Their Statistical Inference , 1991 .

[5]  Alan E. Gelfand,et al.  Spatial process modelling for univariate and multivariate dynamic spatial data , 2005 .

[6]  Sw. Banerjee,et al.  Hierarchical Modeling and Analysis for Spatial Data , 2003 .

[7]  T. Gneiting.  Nonseparable, Stationary Covariance Functions for Space–Time Data , 2002 .

[8]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

[9]  Leonhard Held,et al.  Gaussian Markov Random Fields: Theory and Applications , 2005 .

[10]  J. Møller,et al.  Log Gaussian Cox Processes , 1998 .

[11]  M. Stein.  Space–Time Covariance Functions , 2005 .

[12]  R. Bass,et al.  Infinite dimensional stochastic differential equations of Ornstein-Uhlenbeck type , 2005, math/0503165.

[13]  J. Møller,et al.  Statistical Inference and Simulation for Spatial Point Processes , 2003 .

[14]  伊藤 清.  Foundations of stochastic differential equations in infinite dimensional spaces , 1984 .

[15]  Haotian Hang,et al.  Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics , 2004 .

[16]  Osnat Stramer,et al.  On Bayesian Analysis of Nonlinear Continuous-Time Autoregression Models , 2004 .

[17]  P. Diggle,et al.  Spatiotemporal prediction for log‐Gaussian Cox processes , 2001 .

[18]  Darren J. Wilkinson,et al.  Bayesian inference for nonlinear multivariate diffusion models observed with error , 2008, Comput. Stat. Data Anal..

[19]  H. Schurz.  MODELING, ANALYSIS AND DISCRETIZATION OF STOCHASTIC LOGISTIC EQUATIONS , 2007 .

[20]  N. Cressie,et al.  Classes of nonseparable, spatio-temporal stationary covariance functions , 1999 .

[21]  V. Mahajan,et al.  Innovation Diffusion Models of New Product Acceptance. , 1987 .

[22]  S. Sathananthan,et al.  Stability analysis of a stochastic logistic model , 2003 .

[23]  Richard H. Jones,et al.  Models for Continuous Stationary Space-Time Processes , 1997 .

[24]  Anders Brix,et al.  Space‐time Multi Type Log Gaussian Cox Processes with a View to Modelling Weeds , 2001 .

[25]  Peter J. Diggle,et al.  Spatio-Temporal Point Processes: Methods and Applications , 2005 .

[26]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[27]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[28]  Robert Haining,et al.  Statistics for spatial data: by Noel Cressie, 1991, John Wiley & Sons, New York, 900 p., ISBN 0-471-84336-9, US $89.95 , 1993 .