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I had a discussion with my colleague some time ago and he stated that a correlation between the predictor variable and the outcome variable is not a good measure to perform feature selection, especially when it comes to tree-based classification/regression methods. His justification for that opinion was that the certain predictor $A$ might not be correlated to the outcome $Y$ at all, but after a split by some other predictor $B$ (remember we are talking about tree-based methods), predictor $A$ may reveal a great predictive power. Just as a note - let's say it does not necessarily to be correlation, what he actually meant is any pre-training (filter approach) measure that scores one predictor variable at a time.

For that time, I agreed with him because it seems quite intuitive, but when I think about this now, I can't really create an example to prove this statement. When I think about it or try to create an example, I get an impression that if $B$ is able to split the $A$ in a way that it becomes greatly correlated with $Y$ (within the subsets defined by $B$), then $B$ itself is in fact very highly correlated with $Y$, so there's probably no need to use $A$ anyway. On the other hand, the statement must be somehow true, because it is basically how the tree-based methods work. So maybe there is a possibility in correlation improvement, but only in a small amount and both of the variables have to be meaningfully correlated to $Y$ anyway (resulting in a fact that feature elimination due to low correlation would still make sense)?

So my main question is: is the statement true, and if it is, could you provide an example?

I have one side question as well: is it possible to combine two predictor variables, that are very weakly correlated to the outcome variable, in a way that the engineered third predictor (e.g. $C = A*B$ as a silly example) will be very strongly correlated to the outcome variable?

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Here's an example that should be in everyone's toolbox regarding regression/decision trees, as it succinctly makes two very important points.

X shaped data, with the arm of the X controlled by a factor

The vertical axis shows a response $y$. There are two predictor variables, $x_1$ ranges from $-1$ to $1$ continuously. The relationship between $y$ and $x_1$ is constructed so that no matter how you partition the $x_1$ axis, the response always averages out to zero. In particular:

> cor(df$x_2, df$y)
[1] -0.001792121

Or, for all intents and purposes, zero.

The other variable $x_2$ allows you to distinguish the "arm" of the $X$ shaped data.

This data has two very interesting features:

  • The true or correct decision tree for this data splits on $x_2$ first. This allows it to distinguish the arms of the $X$ shape, and immediately breaks the zero correlation structure between $x_1$ and $y$ for all sub-splits.

  • A greedy algorithm will generally not find the optimal fist split (you could get very lucky, but probably it will find some noisy split on $x_1$)! The first split on $x_2$ does not lead to a reduction in squared error, it is only important because it lets later splits focus on the association between $x_1$ and $y$.

Here is the R code I used to make this plot, so you can experiment yourself

df <- data.frame(
  x_1 = c(seq(-1, 1, length.out=250), seq(1, -1, length.out=250)),
  x_2 = rep(c(1, -1), each=250)
)
df$y <- df$x_1*df$x_2 + rnorm(500, 0, .05)

library(ggplot2)
ggplot(data=df) + geom_point(aes(x=x_1, y=y, color=factor(x_2)))
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    $\begingroup$ Good answer, just a couple of small additions: This is also closely related to the so-called XOR problem. A range of solutions has been suggested for this, including explicitly adding interaction variables in the tree, looking two steps ahead (instead of just one) in the greedy search, or trying to grow globally optimal trees (e.g., through evolutionary algorithms). However, these do not come "for free" as these require either more a-priori knowledge or computational effort. $\endgroup$ – Achim Zeileis Sep 1 '16 at 0:26
  • $\begingroup$ Well, this basically answers my question. It seems the statement was right after all. Thanks for the great example. $\endgroup$ – Matek Sep 1 '16 at 8:33
  • $\begingroup$ @AchimZeileis This XOR problem is actually very interesting. Would you be able to provide some names/papers about algorithms trying to tackle this problem? Especially regarding decision trees? $\endgroup$ – Matek Sep 7 '16 at 22:49
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    $\begingroup$ Our R package evtree provides evolutionary learning of globally optimal trees (CRAN.R-project.org/package=evtree). The corresponding JSS paper (dx.doi.org/10.18637/jss.v061.i01) also has some references to related work. $\endgroup$ – Achim Zeileis Sep 8 '16 at 9:34

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