Analyzing Stochastic Diffusion Processes

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

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):

Analyzing Stochastic Diffusion ProcessesSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Spatio-temporal Cox ProcessThe cumulative intensityCommentsIllustrative growth models (each of which has an explicit solution)Process Models for the ParametersSlide Number 31Discretizing Time (Euler Approximation)Discrete-time ModelDiscretizing Space