Formulation of Knorr-Held (KH) models

[pav]

There seem to be two ways of using INLA to estimate
Knorr-Held spacetime models. One is with Elias Krainski's
inla.knmodels function. The other, as exemplified (eg) in
Blangiardo+Cameletti (B+C), is to use INLA's group feature.
In this connection, I have some questions: apologies in
advance if these are dumb, but on the other hand, they
should be easy to answer!

I. In the group (B+C) specification, does it matter which
(space or time) index is the first f-argument and which is
specified in the "group" option, as long as you get the
other settings to match up correctly? If it *does* matter,
which way is correct?  (I've seen both).


II. Everything I've read about KH models suggests that for
Types 2--4 there's an issue of rank deficiency that needs
to be addressed. Elias Krainski does this explicitly in
knmodels; B+C, as far as I can see, do not. Doesn't this mean
that there's a potential identification problem with B+C-type
specifications? If not, is there a simple explanation as to why not?


III. Does it make sense to estimate a KH Type-I spacetime
model when we have family="gaussian"? In that case wouldn't
we have 2 essentially identical iid effects so that only one
is identified? (Note that in the examples I've seen,
including B+C themselves, the family is Poisson, so this issue
doesn't arise for them).

[Finn Lindgren]

1. In principle/theory, it doesn’t matter which of space and time is the main part and which is the group part of the specification, but in practice, space is almost always better to have as the main part (in particular for more general space-time models; K-H models are a somewhat odd special case)

2. Yes, the K-H models are meant to have constraints imposed on them to handle the rank deficiency; the theory is based on that, so implementation that don’t handle the rank deficiency actually don’t implement the K-H models, but rather the rank-deficient “proto-models” involved in the K-H model definitions. These models _can_ be ok as priors if the data itself is sufficiently informative, but it can easily lead to rank-deficient posterior distributions. Conclusion; Elias implementation should be preferred if one specifically wants the original K-H models.

Now, I don’t use those models myself, since the complex rank-deficiency makes the models extremely hard to I interpret (I prefer having proper priors, either separable with matern*ar1 or ar2, or nonseparable, like in INLAspacetime), but yes, if Type 1 models are simply completely independent iid variables, then combining that with additive Gaussian noise requires additional information to get an identifiable model.

Finn

On 4 Sep 2023, at 19:01, pav <vit...@osu.edu> wrote:



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