[R-sig-Geo] Temporal marked point process with time-varying covariates

[Mayeul KAUFFMANN]

Dear David,
Thanks a lot for your prompt and detailed answer!

You wrote:
"I do not think that the ETAS model will have any relevance for your
situation. The spatial ETAS model is slightly peculiar [...]"
What I find very interesting in an epidemic-type model like the spatial ETAS
model is the analogy that is possible to make between earthquakes and violent
events: earthquakes are likely to be followed by aftershocks shortly after and
in the vicinity. Similarly, violence fuels violence locally: a victory by an
armed group may trigger a retaliatory action by another armed group, soon after
and in the same area. For the other etas_gif function, "spatial coordinates of
the events are not taken into account" whereas the non-etas models do not take
into account the history of the point process. Can I mimic this time-space
dependence among events without the spatial ETAS model, or (sorry to insist),
can I use the spatial ETAS model for my purpose?

You also wrote:
"What sort of functional relationship are you proposing for your data? Can you
write an equation for your ground intensity function?" and "There is no reason
why other covariates cannot be part of the defined ground intensity function, or
the mark distribution."
OK, great. I am trying to do this with R but I'm not sure how to program it. In
all pages of the PtProcess package manual where the "data" parameter is
mentioned, it is a dataframe containing data about the (potentially marked)
point process. I need to take into account the value of covariates even in
places where there are no events. In the earthquake field, for example, one
might want to use the type of soil as a covariate: in this case, how to have one
of the gif be dependent on the covariates? You seem to imply that it should not
be too difficult: "given your specification of your conditional intensity as a
function of the covariates, you should be able to mimic the required structure,
and write your own ground intensity function."
For instance, starting from the conditional intensity function given (again) in
help(etas_spatial):
lambda(t,x,y|Ht) = mu + A SUM_{i: ti < t} exp{beta(Mi-Mo)} g(x-xi, y-yi) f(t-ti)
then I might consider trying to introduce the covariates here (is it what you
suggested?), in a simplistic way:
lambda(t,x,y|Ht) = h(a1.Z1 + a2.Z2 + ...) + mu + A SUM_{i: ti < t}
exp{beta(Mi-Mo)} g(x-xi, y-yi) f(t-ti) (equation 1)
where my h(.) function would be a monotonous function of a linear combination of
the Z1, Z2,... covariates, those covariates being measured at (t, x, y). (I
might also consider a multiplicative influence of h() on lambda instead of the
above additive one).
Then, if the above makes sense, I have to modify the following part of the
etas_normal0() R function:
ci <- mu + A * sum(exp(beta * Mi) * omori * spatial)
by adding a term that would be dependent on (xj, yj, tj), namely that will
search for the values of the covariates at (xj, yj, tj). 
(By the way: the i index in your source code is j index [hence previous events]
in equations 4 to 8 of Ogata(2006) paper, and your xj is simply x in the same
paper [hence current point-time], am I correct?)
Finally, I will have to add the a1, a2... coefficients of the linear model into
the "params" vector.
Last not least, there is no room for the covariates themselves in the function
parameters (if I am correct this is true for all functions of the package??). Do
I have to make my function search for those values globally (outside of the
scope of the function) ???

I hope I did not write too many inaccuracies above, and I thank you again for
your time!
I have some colleagues who work on environmental spatial statistics and I agree
with your assessment in your JSS paper that this package may be used in many
fields.
Best regards,
Mayeul 




_____________________________________________________
Dr. Mayeul KAUFFMANN, Conflict Specialist
European Commission, Joint Research Centre (JRC)
Institute for the Protection and Security of the Citizen (IPSC)
Global Security and Crisis Management - ISFEREA
Via E. Fermi 2749 - I-21027 Ispra (VA), ITALY
Phone: (+39) 033278 5071
http://isferea.jrc.ec.europa.eu/Staff/Pages/Kauffmann-Mayeul.aspx

(Office: building 48c, 1st floor, room 123. TP: 483)