inst/Open_issues.md

In marjoleinF/gamtree: GAM-based Recursive Partitioning

To-do list

• Add support for fitting GAMs using bam() instead of gam() (e.g., create bam_fit function, or include additional argument type = c("gam", "bam") in gam_fit() function). Using bam() also allows for specifying the coef argument (initial values for model coefficients), which could be passed from the model in the current node.

• Issue: The estimated smooths from the tree and the full GAM have similar, but not the same coefficients. Is that problematic? See section below.

• Testing: Evaluate performance using real and simulated data.

• Allow for having an intercept-only model in the terminal nodes

• Develop nicer summary and print methods

Issue: Different coefficient estimates

library("gamtree")
gt1 <- gamtree(Pn ~ s(PAR) | Species, data = eco, verbose = FALSE, 
               cluster = eco$specimen)
gt2 <- gamtree(Pn ~ s(PAR) | noise + s(cluster_id, bs="re") | Species,
               data = eco, verbose = FALSE, cluster = eco$specimen)

The estimated coefficients from the final tree use the predictions based on global effects from the full GAM from the second-to-last iteration:

coef(gt1)

##   (Intercept)   s(PAR).1  s(PAR).2  s(PAR).3  s(PAR).4  s(PAR).5   s(PAR).6
## 2    3.700746  -4.229937 -8.925765 -6.419708 -3.123013  4.133930 9.13784734
## 4    4.710368  -3.992768 -9.397897 -7.868900 -3.059983 -1.058191 5.48472474
## 5    5.285132 -10.109006 23.118991 14.836167 41.452806 55.693627 0.02877788
##     s(PAR).7   s(PAR).8 s(PAR).9
## 2   1.107162   6.000138 12.78004
## 4   2.962870   7.508345 11.30423
## 5 -11.010620 -49.776075 44.36823

coef(gt1$tree)

##   (Intercept)   s(PAR).1   s(PAR).2   s(PAR).3   s(PAR).4  s(PAR).5  s(PAR).6
## 2    3.680405  -4.450707  -9.273774  -6.890283 -2.9187555 -3.301982  9.802856
## 4    4.875050   3.700263 -12.796646 -11.071199 -0.6819658 -2.496246  7.689810
## 5    5.196168 -10.773631  18.830462  10.236554 21.7896631  5.485991 29.524685
##       s(PAR).7   s(PAR).8 s(PAR).9
## 2 -0.007723617   6.302115 12.59880
## 4 -7.066515961  12.101445 15.78517
## 5 51.559591044 -44.310344 41.96981

The ‘severity’ of the differences between the estimated coefficients is difficult to judge. If we look at the plotted partial effects of the smooth terms, we see very similar, but not identical effects.

The partial effects as estimated in the tree nodes:

plot(gt2, which = "nodes", gamplot_ctrl = list(residuals = TRUE, ylim = c(-3.8, 4.2)))

## Warning in plot.gamtree(gt2, which = "nodes", gamplot_ctrl = list(residuals =
## TRUE, : Argument which should specify one of 'tree', 'terms' or 'both'

The partial effects as estimated in the full model:

par(mfrow = c(2, 2))
plot(gt2$gamm, residuals = TRUE, ylim = c(-3.8, 4.2))

We can also compare the predicted values:

newdat <- eco
newdat$x <- newdat$PAR
preds <- data.frame(gam = predict(gt2$gamm),
                    tree = predict(gt2$tree, newdata = newdat, 
                                   type = "response"))
cor(preds)

##            gam      tree
## gam  1.0000000 0.9981619
## tree 0.9981619 1.0000000

colMeans(preds)

##      gam     tree 
## 4.130796 4.134305

sapply(preds, var)

##      gam     tree 
## 1.748725 1.747762

sapply(preds, max)

##      gam     tree 
## 7.501499 7.274412

sapply(preds, min)

##      gam     tree 
## 1.135808 1.255259

cols <- c(rep("white", times = 2), "yellow", "orange", "white", 
          "purple", "blue") 

The predicted values are very similar, but not identical.

The differences are likely due to the different ways of estimating a smooth with the by argument specified (which is used for estimating the full GAM), and estimating separate smooths in each subgroup (which is done in estimating the partition with local GAMs). The scale and smoothing parameters differ between these two approaches:

Smoothing and scale parameters for the full GAM:

gt2$gamm$sp

## s(PAR):.tree2 s(PAR):.tree4 s(PAR):.tree5 s(cluster_id) 
##  2.772777e-03  3.548033e-03  1.357165e-04  6.748856e+01

gt2$gamm$scale

## [1] TRUE

For the GAMs in the terminal nodes, we obtain different values for the smoothing parameters:

gt2$tree[[2]]$node$info$object$sp

##      s(PAR) 
## 0.003401764

gt2$tree[[4]]$node$info$object$sp

##      s(PAR) 
## 0.001136981

gt2$tree[[5]]$node$info$object$sp

##       s(PAR) 
## 0.0001236358

This is probably due to a separate scale parameter being estimated in each node, instead of a single global one in the full GAM:

gt2$tree[[2]]$node$info$object$scale

## [1] TRUE

gt2$tree[[4]]$node$info$object$scale

## [1] TRUE

gt2$tree[[5]]$node$info$object$scale

## [1] TRUE

marjoleinF/gamtree documentation built on Sept. 17, 2021, 5:46 a.m.