knitr::opts_chunk$set( collapse = TRUE, comment = "#>", dpi=300, out.width="50%" ) library(ggplot2)
We first generate a mixture of bivariate normal distributions. The distributions differ only by their x- and y-displacement, that is, by their mean values. There are two predictors grp1
and grp2
which predict the differences in means. grp1
predicts differences in the first dimension and grp2
predicts differences in the second dimension. Without focus parameter, both predictors are needed to distinguish all four groups. If one of the two means is chosen as a focus parameter, only one of the two predictors is important.
library(semtree) set.seed(123) N <- 1000 grp1 <- factor(sample(x = c(0,1), size=N, replace=TRUE)) grp2 <- factor(sample(x = c(0,1), size=N, replace=TRUE)) noise <- factor(sample(x = c(0,1),size=N, replace=TRUE)) Sigma <- matrix(byrow=TRUE, nrow=2,c(2,0.2, 0.2,1)) obs <- MASS::mvrnorm(N,mu=c(0,0), Sigma=Sigma) obs[,1] <- obs[,1] + ifelse(grp1==1,3,0) obs[,2] <- obs[,2] + ifelse(grp2==1,3,0) df.biv <- data.frame(obs, grp1, grp2, noise) names(df.biv)[1:2] <- paste0("x",1:2) manifests<-c("x1","x2")
The following code specifies a bivariate Gaussian model with five parameters:
model.biv <- mxModel("Bivariate_Model", type="RAM", manifestVars = manifests, latentVars = c(), mxPath(from="x1",to=c("x1","x2"), free=c(TRUE,TRUE), value=c(1.0,.2) , arrows=2, label=c("VAR_x1","COV_x1_x2") ), mxPath(from="x2",to=c("x2"), free=c(TRUE), value=c(1.0) , arrows=2, label=c("VAR_x2") ), mxPath(from="one",to=c("x1","x2"), label=c("mu1","mu2"), free=TRUE, value=0, arrows=1), mxData(df.biv, type = "raw") ); result <- mxRun(model.biv) summary(result)
This is how the data look in a 2D space:
df.biv.pred <- data.frame(df.biv, leaf=factor(as.numeric(df.biv$grp2)*2+as.numeric(df.biv$grp1))) ggplot(data = df.biv.pred, aes(x=x1, y=x2, group=leaf))+ geom_density_2d(aes(colour=leaf))+ viridis::scale_color_viridis(discrete=TRUE)+ theme_classic()
Now, we choose the mean of the second dimension mu2
as focus parameter. We expect that only predictor grp2
. This is what we see in a single tree.
fp <- "mu2" # predicted by grp2 #fp <- "mu1" # predicted by grp1 tree.biv <- semtree(model.biv, data=df.biv, constraints = list(focus.parameters=fp))
plot(tree.biv)
Now, we are repeating the same analysis in a forest.
forest <- semforest(model.biv, data=df.biv, constraints = list(focus.parameters=fp), control=semforest.control(num.trees=10, control=semtree.control(method="score",alpha=1)))
By default, we see that individual trees are fully grown (without a p-value threshold). The first split is according to grp2
because it best explains the group differences. Subsequent splits are according to grp1
even though the chi2 values are close to zero. They only appear because there is no p-value-based stopping criterion.
plot(forest$forest[[1]])
Now, let us investigate the permutation-based variable importance:
vim <- varimp(forest, method="permutationFocus") plot(vim, main="Variable Importance")
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