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I need to check if a numeric variable $y$ depends on a set of other numeric variables $(x_1, x_2, ..., x_n)$. I do not know anything about the form of possible dependency (I do not even know if it exists). So, my goal is to check if there is something.

The first thing that comes to mind is to perform a linear regression and to check if there is a statistically significant relation. However, it might be that the relation is not linear.

Moreover, the number of dimensions in my case is larger (about 100). So, I am afraid that it can cause some problems.

I herd that there is a method to check if there is a dependency by checking if close $y_i$ mean closer $\vec x_i$. So, it is something like that. For each $y_i$ I search the closest $y_j$ and then how far from each other are the corresponding $\vec x_i$ and $\vec x_j$. But how does exactly it work? For example, how to define a cutoff for deciding if there is a dependency.

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A common approach is to use a tree method. Tree methods can model all kinds of relationships, not only linear relationships. In R one can fit a random forest and then request the variable importance (http://www.inside-r.org/packages/cran/randomforest/docs/importance).

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