Space-time Modelling Using Differential Equations

1. Space-time Modelling Using Differential Equations Alan E. Gelfand, ISDS, Duke University (with J. Duan and G. Puggioni)

2. The Contribution • Space-time data collection is increasing • This talk is entirely about modelling for such data in the case of geo-coded locations • Modelling the data directly or using random effects to explain the data • Using process models to provide process realizations • Using classes of parametric functions to provide realizations • Using classes of differential equations to provide realizations

3. Two applications • Two examples working with differential equations • A realization of a space-time point pattern driven by a random space-time intensity – the application is to urban development measured through housing construction • Spatio-temporal data collection - soil moisture measurements in time and space assumed to be a realization of a hydrological model – work in progress

4. Important Points • A differential equation in time at every spatial location, i.e., parameters indexed by location • The parameters in the differential equation vary spatially as realizations of a spatial process • ALTERNATIVELY, the differential equation is a stochastic differential equation (SDE), e.g., a spatial Ornstein-Uhlenbeck process (Brix and Diggle) • OR, the rate parameter in the differential equation is assumed to change over time. It can be modelled as a realization of a spatio-temporal process • OR, the rate parameter can be modelled using a SDE, yielding an SDE embedded within the differential equation

5. Spatio-temporal Point Process Models • Overview: Urban Development & Spatio-temporal Point Processes. • Differential Equation Models for Cumulative Intensity • Model Fitting & Inference • Data Examples: • Simulated data • Urban development data for Irving, TX

6. Urban Development Problem • Residential houses in Irving, TX 1951 1956 1962 1968

7. Our objectives • Space-time point pattern of urban development using a spatio-temporal Cox process model. • Work with housing development (available at high resolution) surrogate for population growth (not available at high resolution) • Use population models, expect housing dynamics to be similar to population dynamics • Differential equation models for surfaces. Insight into interpretable mechanisms of growth • Bayesian inference and prediction: • Discretizing time and space(replace integral by sum) • Kernel convolution approximation( to handle large sample size)

8. Spatio-temporal Cox Process In a study region D during a period of [0,T], NT events: Point pattern: where is a Poisson process with inhomogeneous intensity Specifying the intensity? are processes for parameters of interest.

9. The cumulative intensity Discretize the spatio-temporal Cox process in time: Spatial point pattern: during The cumulative intensity for is We consider models for the cumulative intensity

10. Comments

11. Three Growth Models • Exponential growth • Gompertz growth • Logistic growth local growth rate local carrying capacity

12. Logistic Population Growth population growth at time t current population at time t average growth rate for region D carrying capacity for region D Model for the aggregate intensity.

13. Proper Scaling Local growth model should scale with the global growth model: cumulate cumulate average

14. Process Models for the Parameters and initial intensity are parameter processes which are modeled on log scale as • Hence, given the growth curve is fixed. Also, the μ’s are trend surfaces.

15. Diffusion Model (SDE) for Growth Rate • Can not add scaled spatial Brownian motion to the logistic diff eqn. Instead, a time-varying growth rate at each location Let • Spatial Ornstein-Uhlenbeck (OU) process model: • where Wt (s) is spatial Brownian motion and, again • It induces a stationary process with separable space-time covariance:

16. Discretizing Time Back to the original model, the intensity for the spatial point pattern in a time interval: • Difference equation model: explicit transition a recursion

17. Discrete-time Model • Model parameters and latent processes: • Likelihood point i in period j stochastic integral

18. Discretizing Space Divide region D into M cells. Assume homogeneous intensity in each cell. We obtain (with r(m), k(m) average growth rate and cumulative carrying capacity): with induced transition The joint likelihood (product Poisson):

19. Parametrizing the spatial processes • Growth rate, carry capacity and initial intensity for each cell: M is very large (2500 in our example) !

20. Kernel Convolution Approximation • A dimension reduction approach (2500—>100) • Kernel Convolution (Xia & Gelfand, 2006): study region D centroid of block l block l covering region

21. Kernel Convolution Approximation • Approximate centroid of cell m area of block l kernel function centroid of block l • Let where

22. Recursive Calculation • Calculate random effects: Sequentially, use

23. Simulated Data Analysis time 0 year 1 year 2 year 3 Initially and successive 5 years

24. Simulated Data Analysis: Estimation

25. Simulated Data Analysis: Estimation r K Posterior: Actual:

26. Simulation: One Step Ahead Prediction Predicted Actual

27. New Houses in Irving, TX: 1952—1957 We use 1951 –1966 to fit our model, leaving 1967 and 1968 out for prediction and validation.

28. Data Analysis for Irving, TX centroid of block l Divide region D into M cells

29. Data Analysis for Irving, TX with points at t0 r K

30. Prediction: New Houses in1967 & 1968 One Step Ahead 1967 Two Step Ahead 1968

31. Future work on this project • Intensity/type of development – marked point pattern • Underlying determinants of development, e.g., zoning, roads, time-varying? - add as covariates in rates, carrying capacities • Holes, e.g., lakes, parks, externalities – force zero growth • Fit the SDE within the DE • A model where the observations drive the intensity, e.g., a self-exciting process

40. Soil moisture loss, i.e., transpiration and drainage as a function of soil moisture

42. A simulated data set

43. Results

45. First Differences

46. Wilting point and field capacity

47. Wilting point and field capacity