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Space-Time Modeling Using Differential Equations: Insights into Urban Development

This presentation by Alan E. Gelfand, J. Duan, and G. Puggioni explores advanced modeling techniques for increasing space-time data collection, particularly focusing on geo-coded locations. It emphasizes the use of differential equations and random effects models to interpret spatial data such as urban development and soil moisture measurements. Two case studies demonstrate applications in housing construction and hydrological models. Key points include parameter variability, stochastic differential equations, and Bayesian inference techniques for insightful prediction and understanding of growth mechanisms.

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Space-Time Modeling Using Differential Equations: Insights into Urban Development

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  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

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

  33. A simulated data set

  34. Results

  35. First Differences

  36. Wilting point and field capacity

  37. Wilting point and field capacity

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