2  LGCP theory

The LGCP incorporates …

Spatio-temporal? See (camelitti-2013?) to set up the pieces for the spatial-temporal covariance bits. Paula Moraga resources are also very useful.

In the Poisson Point Process the intensity is written as \[ \lambda(s) = \theta(s)b(s) = \exp[\alpha + \beta'x(s) + \gamma + \delta'z(s)] \] In the LGCP we now incorporate a Gaussian Random Field to incorporate spatial correlation between sites \[ \lambda(s) = \theta(s)b(s) = \exp[\alpha + \beta'x(s) + \gamma + \delta'z(s) + w(s)] \] so that the expected value for quadrat \(A\) will be the integral over the quadrat: \[ \Delta(A) = \int_A\lambda(s)ds = \int_A \exp[\alpha + \beta'x(s) + \gamma + \delta'z(s) + w(s)]ds \]

However, this forces to use a covariance structure that grows in size as the area of interest increases in area. Thus, the computational complexity to estimate such covariance structure as \(n\) increases makes this approach inhibitively costly for large regions. Using MCMC to estimate this model is possible, but slow.

Integrated Nested Laplace Approximations have been proposed by (rue-2009?) as a fast way to estimate LGCPs.

The SPDE approach represents the continuous Gaussian Field with a discretely indexed Gaussian Markov Random Field, using a basis function defined on a triangulation of the region of interest.

2.1 Covariance Matrices

From https://www.youtube.com/watch?v=CCuWHaYxVFg&ab_channel=RConsortium: The precision matrix is sparse for Markovian process. Gaussian distributoins with sparse precision matrices are Guassian Markov Random Fields

From https://www.youtube.com/watch?v=Tdb5EPczE9E&ab_channel=TomislavHengl%28OpenGeoHubFoundation%29 and https://www.paulamoraga.com/book-geospatial/sec-geostatisticaldatatheory.html. INLA computes the posterior marginals for the latent Guassian field and the hyperparameters. \[ p(x_i|y) = \int p(x_i|\theta,y)p(\theta|y)d\theta \] ## Mesh

The mesh is is used to project the discrete GRMF to the continuous GF using some spatial weights defined at the vertices of the mesh (did I say that right?). From the Moraga tutorial above, she says the continuous process is estimated using a weighted average of the process at the vertices of the discrete triangulation mesh.

The projection matrix maps the GMRF from the observations to the triangulation nodes. The observation will be the weighted average using the weights and values from the triangulation and projection matrix. \[ S(x_i) \approx \sum^G_{g=1}A_{ig}S_g \]

Simpson 2016