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I was reading the Decision Trees user guide of sklearn to understand some of the underlying mathematics behind trees. Everything was fine until I stumbled upon some notation I'm not understanding. This is what appears in the documentation:

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(link to the documentation can be found here)

My problem is I have no idea how to interpret or even read Q_left and Q_right. Is this some set theory specific notation? Any help would be appreciated.

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  • $\begingroup$ The tree is splitting the data to reduce the error, Q is the data, Q left is the data that goes in the left node, Q right is the data that goes in the right node. The first line is saying that Q left is defined by xj being less than or equal to tm, which is the hypothetical best split. Q right, as Wayne said, is everything else, AKA where xj > tm. $\endgroup$
    – doubletrouble
    Commented Sep 24, 2019 at 21:29
  • $\begingroup$ For more information about notation...$Q_{left}$ is defined using set builder notation and $Q_{right}$ is defined using the set difference $\endgroup$
    – user20160
    Commented Sep 24, 2019 at 22:06

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Do you mean you literally don't know what $Q_{left}$ means or how to pronounce it, or do you mean what the right-hand sides mean?

In term of the right-hand sides, the $|$ appears to mean a conditional, so $Q_{left}$ is the set of (x,y) pairs such that $x_j <= t_m$ The backslash in $Q_{right}$ appears to be the set complement: $Q_{right}$ is elements of $Q$ not including the elements of $Q_{left}$.

EDIT: I changed the backslash operator name to the more correct "complement". Wikipedia describes here: https://en.wikipedia.org/wiki/Set_(mathematics)

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  • $\begingroup$ It was about the right hand sides. Most time when you read the notation out loud is quite easy to understand what it means. I didn't know about set substraction notation (even though it was kind of evident based on what Qleft and Qright are. Thanks a lot ! $\endgroup$
    – Juan C
    Commented Sep 24, 2019 at 22:21

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