Priors on AR(1) in spatio temporal model

[TheCorinna1994]

Dear all,

I have a spatio-temporal model for different rain scenarios.

I fit Rain_mean with a gamma distribution

Rain_max with a bgev distribution

and Dry_spells with a negative binomial distribution

the formula is given by

e.g. Rain_mean~ -1+Intercept+Latitude+Longitude+Elevation+f(field, model=spde.nonstat, group=field.group, control.group=list(model="ar1"))

the parameter estimate for the temporal correlation coefficient is for each scenario in the interval 0,93-0,99 which as far as I know is too high and indicates a non-stationary process

Would you add pc.priors in order to avoid overfitting?

Or would you change to a RW1 process? (I tried but it does not work, maybe I need priors too but I do not know which one to implement)

Can you maybe give me an advice on how to solve that problem or do you think these high temporal correlations can be argued somehow?

Very much looking forward to your answer.

Best,

Corinna

[Finn Lindgren]

Hi,

1) high temporal correlation in the model described is expected when
you don't have anything else in the model to capture persistent
spatial structure; if your analysis domain is large enough,
long+lat+elev won't be enough for that.
The spatio-temporal process is then forced to deal with that, and the
only way to do that is to have a large temporal correlation. Adding a
purely spatial component may solve this problem (but also induces some
confounding between the components; whether this is a problem or not
depends on the amount of data etc).
2) A high AR(1) coefficient is interpretable as non-stationarity in
name only, and depends on how the limit is taken; in the inla "ar1"
model, the process model is always stationary, and rho-->1 leads to a
temporally _constant_ model, in the form of a single random variable
with variance given by the model.

Note that this is different from the intrinsic-stationary rw1 model,
that by default has a sum-to-zero constraint, and scale.model=TRUE
rescales the average variance. The effect of imposing the constraint
results in a truly nonstationary model. However, without the
constraint, the model is intrinsic-stationary (the first order
differences are stationary). Some literature calls that a
non-stationary model because of the Brownian motion anchored at B(0)=0
is a truly non-stationary process, but when used as a random walk, the
nonstationarity concept isn't really the right way to think about it,
as it mostly behaves a a stationary process, just with infinite
correlation range and large overall variance.
When the issue leading to a large AR coefficient is that there are
unmodeled spatially persistent effects in the data, then applying a
random walk doesn't solve problem at all, as it's not constructed to
be persistent! (and when constrained to sum to zero ight even make
things worse).

One thing to check is what happens if you remove the time aspect of
the model entirely, and just have a spatial component. If that
component has spatial structure, you likely need it in the model even
when you add time.

In some case, one can get away with the joint space-time component
capturing everything, but not always.

Finn

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Finn Lindgren
email: finn.l...@gmail.com

[TheCorinna1994]

Dear Finn,

what did work better now was the following implementation:

e.g. Rain_mean~ -1+Intercept+Latitude+Longitude+Elevation+f(field_only, model=spde.nonstat)+f(field, model=spde.nonstat, group=field.group, control.group=list(model="ar1"))

About the confounding: I do not know how to see if this problem exists. Do you have any idea?

For the paper I would write the model the following way: 

\eta(s_i,t_k)=\alpha+\gamma_1 Longitude+\gamma_2 Latitude+\gamma_3 Elevation+ v(s_i) + z(s_i,t_k) with  v(s_i)   the purely spatial term and

         z(s_i,t_k)=az(s_i,t_{k-1})+w(s_i,t_k)  with w(s_i,t_k) again a spatial field.

Is this correct?

I highly appreciate your help!

Thanks.

Corinna