physical constraint with gam

Hi again,
I'm looking for some clarification on 2 things.
1. On that last note, I realize that s(x1,x2) would be the other 
obvious interaction to compare with - and I see that you recommend 
te(x1,x2) if they are not on the same scale.

2. If s(x1,by=x1) gives you a "parameter" value similar to a GLM when 
you plot s(x1):x1, why does my function above return the same yhat as 
predict(mdl,type='response') ? Shouldn't each of the terms need to be 
multiplied by the variable value before applying 
rowSums()+attr(sterms,'constant') ??

Thanks again
Dominik

On Wed, May 11, 2016 at 10:11 AM, Dominik Schneider 
<Dominik.Schneider at colorado.edu 
<mailto:Dominik.Schneider at colorado.edu>> wrote:

    Hi Simon, Thanks for this explanation.
    To make sure I understand, another way of explaining the y axis in
    my original example is that it is the contribution to snowdepth
    relative to the other variables (the example only had fsca, but my
    actual case has a couple others). i.e. a negative s(fsca) of -0.5
    simply means snowdepth 0.5 units below the intercept+s(x_i), where
    s(x_i) could also be negative in the case where total snowdepth is
    less than the intercept value.

    The use of by=fsca is really useful for interpreting the marginal
    impact of the different variables. With my actual data, the term
    s(fsca):fsca is never negative, which is much more intuitive. Is
    it appropriate to compare magnitudes of e.g. s(x2):x2 / mean(x2)
    and s(x2):x2 / mean(x2)  where mean(x_i) are the mean of the
    actual data?

    Lastly, how would these two differ: s(x1,by=x2); or
    s(x1,by=x1)*s(x2,by=x2) since interactions are surely present and
    i'm not sure if a linear combination is enough.

    Thanks!
    Dominik


    On Wed, May 11, 2016 at 3:11 AM, Simon Wood <simon.wood at bath.edu
    <mailto:simon.wood at bath.edu>> wrote:

        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)`))

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