Issue with zero-mean Matérn field in Log-Gaussian Cox Process using inlabru

[Daniele Poggio]

Dear colleagues,

I am using inlabru (a wrapper for INLA) to build a Log-Gaussian Cox Process (LGCP) model for spatial observations.

I aim to describe the observations with a simple model composed of:

• a time-dependent intercept and
• a zero-mean Matérn field,
  fixed for all time points considered.

The model is defined as:

log(lambda(s,t)) = Intercept_t + w(s)  

Here:

• s is the spatial location,
• t is the time,
• Intercept_t​ is the time-dependent intercept, and
• w(s) is a zero-mean Matérn field.

Following the tutorial 2D LGCP in inlabru, I defined a mesh for the study region:

  The blue line in the image represents the boundary of the study area.  

  I compute the model using the following R code:

# Define the Matérn model
matern <- inla.spde2.pcmatern(
  mesh,
  # P(\sigma > \sigma_0) = p_{\sigma}
  prior.sigma = c(0.1, 0.01),
  # P(\rho < \rho_0) = p_{\rho}
  prior.range = c(50, 0.5)
)

# Define the component model
cmp <- geometry + season ~ Intercept(season, model = "factor_full") + W(geometry, model = matern) - 1

# Set bru options
bru_options_set(
  bru_verbose = TRUE,
  verbose = TRUE,
  bru_max_iter = 1,
  control.inla = list(int.strategy = "eb")
)

# Fit the LGCP model
fit <- lgcp(cmp,
  data = points,
  samplers = search_area_polygon,
  domain = list(
    geometry = mesh,
    season = seq_len(5)
  )
)

Here, search_area_polygon is an sfc_POLYGON object that defines the boundaries of the study region.

When I observe the mean value of the Matérn field, I notice the following issue:

• Inside the study area (where the observations are located), the field is strongly positive.
• Outside the study area (where there are no observations), the field is negative.

I would like the Gaussian field to have a zero-mean, but it does not appear to behave as such. I would expect these deviations to be "absorbed" by the intercept component.

Here is an image of the Matérn field for reference:

Can anyone suggest whether there is an issue with my approach or provide advice on how to resolve this?

Thank you in advance!

Best regards,

Daniele Poggio

[Helpdesk (Haavard Rue)]

you can add constr=T in inla.spde2.pcmatern to impose a (weighted-)sum-to-zero
constraint on the spde model, hence you can separate the 'intercept' from the
spde

On Mon, 2024-11-18 at 07:36 -0800, Daniele Poggio wrote:
> Dear colleagues,
> I am using inlabru (a wrapper for INLA) to build a Log-Gaussian Cox Process
> (LGCP) model for spatial observations.
> I aim to describe the observations with a simple model composed of:

>  * a time-dependent intercept and
>  * a zero-mean Matérn field,

>    fixed for all time points considered.
> The model is defined as:
> log(lambda(s,t)) = Intercept_t + w(s)  
> Here:

>  * s is the spatial location,
>  * t is the time,
>  * Intercept_t​ is the time-dependent intercept, and
>  * w(s) is a zero-mean Matérn field.

> Following the tutorial 2D LGCP in inlabru, I defined a mesh for the study
> region:

>  * Inside the study area (where the observations are located), the field is
> strongly positive.
>  * Outside the study area (where there are no observations), the field is

> negative.
> I would like the Gaussian field to have a zero-mean, but it does not appear to
> behave as such. I would expect these deviations to be "absorbed" by the
> intercept component.
> Here is an image of the Matérn field for reference:

> GF.png
> Can anyone suggest whether there is an issue with my approach or provide
> advice on how to resolve this?
> Thank you in advance!
> Best regards,
> Daniele Poggio

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--
Håvard Rue
he...@r-inla.org

[Finn Lindgren]

One thing to note:

Even if one used constr=TRUE, that would impose the constraint to the whole mesh, which wouldn’t prevent the effect you’re seeing, as it would still be confounded with the intercepts on the observed subdomain; one would need to apply the integrate-to-zero constraint only on the study domain. This is also possible; the extraconstr.int can be used to supply a constraint against an indicator function defined on the mesh nodes; 1 inside and 0 for vertices outside the study domain. That way, it can more fully ignore the outside parts. The format is list(A=matrix(…, 1, fm_dof(mesh)), e=1), I think, where the … are the 0/1 values. fm_is_inside or fm_is_within is the function that can help evaluate those.

Finn

On 18 Nov 2024, at 18:28, Daniele Poggio <poggiod...@gmail.com> wrote:



Dear colleagues,

I am using inlabru (a wrapper for INLA) to build a Log-Gaussian Cox Process (LGCP) model for spatial observations.

I aim to describe the observations with a simple model composed of:

• a time-dependent intercept and
• a zero-mean Matérn field,
  fixed for all time points considered.

The model is defined as:

log(lambda(s,t)) = Intercept_t + w(s)  

Here:

• s is the spatial location,
• t is the time,
• Intercept_t​ is the time-dependent intercept, and
• w(s) is a zero-mean Matérn field.

Following the tutorial 2D LGCP in inlabru, I defined a mesh for the study region:

<mesh.png>

<GF.png>

Can anyone suggest whether there is an issue with my approach or provide advice on how to resolve this?

Thank you in advance!

Best regards,

Daniele Poggio

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<mesh.png>

<GF.png>