[R] physical constraint with gam

[Simon Wood]

The spline having a positive value is not the same as a glm coefficient 
having a positive value. When you plot a smooth, say s(x), that is 
equivalent to plotting the line 'beta * x' in a GLM. It is not 
equivalent to plotting 'beta'. The smooths in a gam are (usually) 
subject to `sum-to-zero' identifiability constraints to avoid 
confounding via the intercept, so they are bound to be negative over 
some part of the covariate range. For example, if I have a model y ~ 
s(x) + s(z), I can't estimate the mean level for s(x) and the mean level 
for s(z) as they are completely confounded, and confounded with the 
model intercept term.

I suppose that if you want to interpret the smooths as glm parameters 
varying with the covariate they relate to then you can do, by setting 
the model up as a varying coefficient model, using the `by' argument to 
's'...

gam(snowdepth~s(fsca,by=fsca),data=dat)


this model is `snowdepth_i = f(fsca_i) * fsca_i + e_i' . s(fsca,by=fsca) 
is not confounded with the intercept, so no constraint is needed or 
applied, and you can now interpret the smooth like a local GLM coefficient.

best,
Simon




On 11/05/16 01:30, Dominik Schneider wrote:
> Hi,
> Just getting into using GAM using the mgcv package. I've generated some
> models and extracted the splines for each of the variables and started
> visualizing them. I'm noticing that one of my variables is physically
> unrealistic.
>
> In the example below, my interpretation of the following plot is that the
> y-axis is basically the equivalent of a "parameter" value of a GLM; in GAM
> this value can change as the functional relationship changes between x and
> y. In my case, I am predicting snowdepth based on the fractional snow
> covered area. In no case will snowdepth realistically decrease for a unit
> increase in fsca so my question is: *Is there a way to constrain the spline
> to positive values? *
>
> Thanks
> Dominik
>
> library(mgcv)
> library(dplyr)
> library(ggplot2)
> extract_splines=function(mdl){
>    sterms=predict(mdl,type='terms')
>    datplot=cbind(sterms,mdl$model) %>% tbl_df
>    datplot$intercept=attr(sterms,'constant')
>    datplot$yhat=rowSums(sterms)+attr(sterms,'constant')
>    return(datplot)
> }
> dat=data_frame(snowdepth=runif(100,min =
> 0.001,max=6.7),fsca=runif(100,0.01,.99))
> mdl=gam(snowdepth~s(fsca),data=dat)
> termdF=extract_splines(mdl)
> ggplot(termdF)+
>    geom_line(aes(x=fsca,y=`s(fsca)`))
>
> 	[[alternative HTML version deleted]]
>
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-- 
Simon Wood, School of Mathematics, University of Bristol BS8 1TW UK
+44 (0)117 33 18273     http://www.maths.bris.ac.uk/~sw15190