What is the official name of a specific type of combination algorithm

Question

Say that I have the following set of variables:

[A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z]

The values represent a list of variables from a dataset. Each variable has a certain level of correlation with the target variable $. The correlation increases or decreases depending on the sort of combination you use against your target $. For example, BCJX might have a higher correlation with $ than OQTVW. I'm going to test each possible combination against the training algorithm and output it all, with their accuracy score, in a concise CSV file. But I don't know the name of the algorithm that might combine these variables in every possible way.

In other words, I want to find the combination with the highest correlation value and the smallest dimension.

Answer 1

You are probably looking for subsets of a set. If a set has $n$ elements, it has $2^n$ subsets when counting also the empty set (there is a clear correspondence to all bit strings of length $n$). So if in your case $n=26$, there are about 67 million subsets of variables to check which might be unfeasible.

Answer 2

This seems reminiscent of design of experiments problem. (link) Rather than check every possible subset, it would be much more feasible do a sample which emphasizes maximizing change. The whole subject was pioneered by R.A. Fisher, who was hired to improve the yield of farmer's crops. Since each experiment required considerable time and land to execute, he wanted to make the most of every single experiment.

Generally, in real world examples, a large part of results are caused by just pairwise interactions. So, if for every two events (x,y), you have an experiment where both x and y happen, then you are likely to have very strong coverage of how the system behaves. With the lack of information provided, I am not sure there a limit to how many variables you can observe or not, but a common reseasonable event would be including a given variable A, as well as excluding A.

To say clearly, simply create a set of subsets $S$, such that for any two variables $A,B$, there exist at least one subset in $S$ containing both $A$ and $B$, a subset containing neither $A$ nor $B$, a subset containing $A$ but not $B$, and $B$ not $A$. While the power set grows exponentially, pairwise coverage is much, much, much smaller, and typically captures the essence of functions whose are not random and uncorrelated. Of course, increasing things to include all 3-way interactions or higher is also viable, and increases tyro coverage, but in many cases this is not necessary.