Transcript Space-time Modelling Using Stochastic Partial Differential

Space-time Modelling Using
Differential Equations
Alan E. Gelfand, ISDS, Duke
University
(with J. Duan and G. Puggioni)
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
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
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
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
Urban Development Problem
Residential houses in Irving, TX
1951
1956
1962
1968
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)
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.
The cumulative intensity
Discretize the spatio-temporal Cox process in time:
Spatial point pattern:
The cumulative intensity for
during
is
We consider models for the cumulative intensity
Comments
Three Growth Models
Exponential growth
Gompertz growth
Logistic growth
local growth rate
local carrying capacity
Logistic Population Growth
population growth
at time t
average growth rate
for region D
current population at
time t
carrying capacity for
region D
Model for the aggregate intensity.
Proper Scaling
Local growth model should scale with the global growth model:
cumulate
cumulate
average
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.
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:
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
Discrete-time Model
Model parameters and latent processes:
Likelihood
point i in period j
stochastic integral
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):
Parametrizing the spatial processes
Growth rate, carry capacity and initial intensity for each cell:
M is very large (2500 in our example) !
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
Kernel Convolution Approximation
Approximate
centroid of cell m
area of block l
Let
where
kernel function
centroid of block l
Recursive Calculation
Calculate random effects:
Sequentially, use
Simulated Data Analysis
time 0
year 2
year 1
year 3
Initially and successive 5 years
Simulated Data Analysis: Estimation
Simulated Data Analysis: Estimation
r
Posterior:
Actual:
K
Simulation: One Step Ahead Prediction
Predicted
Actual
New Houses in Irving, TX: 1952—1957
We use 1951 –1966 to fit our model, leaving 1967
and 1968 out for prediction and validation.
Data Analysis for Irving, TX
centroid of
block l
Divide region
D into M cells
Data Analysis for Irving, TX
with points at t0
r
K
Prediction: New Houses in1967 & 1968
One Step Ahead
1967
Two Step Ahead
1968
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
Soil moisture loss, i.e., transpiration and
drainage as a function of soil moisture
A simulated data set
Results
First Differences
Wilting point and field capacity
Wilting point and field capacity