从mgcv :: gam拟合中获得预测，该预测包含矩阵“ by”变量到平滑

Question

I just discovered that mgcv::s() permits one to supply a matrix to its by argument, permitting one to smooth a continuous variable with separate smooths for each of a combination of variables (and their interactions if so desired). However, I'm having trouble getting sensible predictions from such models, for example:

library(mgcv) #for gam
library(ggplot2) #for plotting

#Generate some fake data
set.seed(1) #for replicability of this example
myData = expand.grid(
    var1 = c(-1,1)
    , var2 = c(-1,1)
    , z = -10:10
)
myData$y = rnorm(nrow(myData)) + (myData$z^2 + myData$z*4) * myData$var1 + 
                                 (3*myData$z^2 + myData$z) * myData$var2 
    #note additive effects of var1 and var2

#plot the data
ggplot(
    data = myData
    , mapping = aes(
        x = z
        , y = y
        , colour = factor(var1)
        , linetype = factor(var2)
    )
)+
geom_line(
    alpha = .5
)

#reformat to matrices
zMat = matrix(rep(myData$z,times=2),ncol=2)
xMat = matrix(c(myData$var1,myData$var2),ncol=2)

#get the fit
fit = gam(
    formula = myData$y ~ s(zMat,by=xMat,k=5)
)

#get the predictions and plot them
predicted = myData
predicted$value = predict(fit)
ggplot(
    data = predicted
    , mapping = aes(
        x = z
        , y = value
        , colour = factor(var1)
        , linetype = factor(var2)
    )
)+
geom_line(
    alpha = .5
)

Yields this plot of the input data:

And this obviously awry plot of the predicted values:

Whereas replacing the gam fit above with:

fit = gam(
    formula = y ~ s(z,by=var1,k=5) + s(z,by=var2,k=5)
    , data = myData
)

but otherwise running the same code yields this reasonable plot of predicted values:

What am I doing wrong here?

Answer 1

The use of vector-valued inputs to mgcv smooths is taken up here . It seems to me that you are misunderstanding these model types.

Your first formula

myData$y ~ s(zMat,by=xMat,k=5)

fits the model

y ~ f(z)*x_1 + f(z)*x_2

That is, mgcv estimates a single smooth function f(). This function is evaluated at each covariate, with the weightings supplied to the by argument.

Your second formula

y ~ s(z,by=var1,k=5) + s(z,by=var2,k=5)

fits the model

y ~ f_1(z)*x_1 +f_2(z)*x_2

where f_1() and f_2() are two different smooth functions. Your data model is essentially the second formula, so it is not surprising that it gives a more sensible looking fit.

The first formula is useful when you want an additive model where a single function is evaluated on each variable, with given weightings.