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Analyzing Stochastic Diffusion Processes
Posterior Inference
Urban Development ProblemResidential houses in Irving, TX
1951 1956
1962 1968
Spatio-temporal Cox ProcessIn a study region D during a period of [0,T], NT events:
Point pattern:
is a Poisson process with inhomogeneous intensity
Specifying the intensity?
are processes forparameters of interest.
where
The cumulative intensityDiscretize the spatio-temporal Cox process in time:
during
The cumulative intensity
for is
We consider models for the cumulative intensity
Spatial point pattern:
Comments
Illustrative growth models (each of which has an explicit solution)
Exponential growth
Gompertz growth
Logistic growth
local growth rate local carrying capacity
Logistic Population Growth
growth rate for region D
carrying capacity for region D
current population at time t
population growth at time t
Model for the aggregate intensity.
Proper ScalingLocal growth model
should scale with the global growth model:
cumulate
cumulate
average
Process Models for the Parametersand initial intensity
are parameter processes which are modeled on log scale as
Hence, giventhe growth curve is fixed. Also, the
μ’s are trend surfaces.
Discretizing Time (Euler Approximation)Back to the original model, the intensity for the spatial point pattern in a time interval:
Difference equation model:
a recursionexplicit transition
Discrete-time Model
Likelihood
Model parameters and latent processes:
stochastic integral
point i in period j
Discretizing SpaceDivide region D into M cells. Rescaling and assuming 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):
Simulated Data Analysis
time 0 year 1
year 2 year 3Initially and successive 5 years
Simulated Data Analysis: Estimation
Simulated Data Analysis: Estimation
Posterior:
Actual:
r K
Simulation: One Step Ahead Prediction
Predicted Actual