[R] coupled statistical models

W. C. Thacker Carlisle.Thacker at noaa.gov
Fri Jan 30 20:11:37 CET 2004

Can someone point me to the appropriate functions for 
fitting multiple statistical models that are coupled 
to each other.

The data are measurements of salinity s and temperature t 
at stations id and pressures at p as well as surface elevations h 
at stations id.  The problem is, for any new station, to estimate 
s at all p, given t at all p and given h.

If h is ignored, the for each p, there would be an 
independent model like rlm(s~t+I(t^2)).  No problem.  

(I have been using a loop, but I think they can be computed 
simultaneously using lm(s~as.factor(p)/(t+I(t^2))-1).
However, rlm(s~as.factor(p)/(t+I(t^2))-1) does not converge!)

The h data couple the models.  Surface elevation reflects the
water's specific volume at each depth (pressure).  The volume vp 
(p to indicate pressure) depends on s,t, and p, and h is essentially
a sum over the volume at each p.  When the individual models are
computed, coefficients should be chosen so that this is satisfied.
How to do this in R?

Without the coupling the least-squares formulation would seek 
coefficients a_p and b_p to minimize:
   sum_p(sum_p((s_id,p - a_p t_id,p + b_k)^2)).
Minimization at each p is independent.
With coupling, there would be an additional term:
where vp is a function of a_p t_id,p + b_k and of t_id.

The functions for vp are nonlinear, but if necessary with some work 
it should be possible to get a linear function that is good enough.

Also, not all stations have values at all pressures.




William Carlisle Thacker                            
Atlantic Oceanographic and Meteorological Laboratory
4301 Rickenbacker Causeway, Miami, Florida 33149 USA
Office: (305) 361-4323           Fax: (305) 361-4392

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