Spatial data with observations below detection limit (left censored)

[Eamonn Organ]

Hi all,

I have a problem in environmental data where measurements of a spatial process are below an (assumed known) detection limit for some sensors. 

example problems:
ML Stein [1992] - Prediction and inference for truncated spatial data

Oliveira [2005] - Bayesian inference and prediction of Gaussian random fields based on censored data

I'm wondering can this type of data be modelled in INLA. For example taking Steins example. Assume we have a GRF W with some mean and matern covariance. We observed 

Y = f(W) , for W > 0

Y = 0 , for W < 0.

The main extension in this case would be detection limits can vary by device (there are also high quality measurements that always measure exactly, but an arbitrarily small detection limit could be set) .

The closest example I have found is this for time series data:

https://groups.google.com/g/r-inla-discussion-group/c/qsfHH2aAQQo/m/O-HFG9C8CQAJ 

using inla.surv() , I'm wondering can an SPDE spatial model be incorporated with the survival framework?

[Helpdesk (Haavard Rue)]

This determinisic mapping is not possible.

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Håvard Rue
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[Eamonn Organ]

Hi Haavard/All,

Thanks for getting back. 

To clarify, a strict deterministic cut off/mapping is not necessary, I am just interest in a approach where, if I have observations that are say 0, the likelihood is the latent field falls within some (known) low range within a high probability. I am open/flexible with the approach as the alternative is to treat these observations as missing

best regards,

Eamonn

[Helpdesk (Haavard Rue)]

It `not possible' to put constraints on the latent.

it is possible to say that all Y < eps is reported as 0, which is a left
truncated observation. this would imply that the latent is 'low'.

would something like this be useful ?

n <- 10^5
x <- rnorm(n, sd = 0.2)
prec <- 100
s <- sqrt(1/prec)
a <- 0.1
b <- 0.2
y <- a + b * x + s * rnorm(n)

y.left <- 0.1
y[y <= y.left] <- y.left
event <- rep(1, n)
event[y <= y.left] <- 2
Y <- inla.surv(exp(y), event)
r <- inla(Y ~ 1 + x,
data = list(Y = Y, event = event, x = x),
family = "lognormalsurv")
summary(r)

cbind(estimated.values = c(r$summary.fixed$mean, r$summary.hyperpar$mean),
true.values = c(a, b, prec))