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