able to make stronger statements about dependence than the Pearson $R$, the Kendall $\tau$, or the Spearman $\rho$, even when the two variates have different dimension.

For doubly centered distance matrices $A$ and $B$ of random variables or vectors $X$ and $Y$, the distance covariance is defined: $$\mathscr{V}^2_n(X,Y) = \frac{1}{n}\sum^n_{k,\ell=1}A_{k\ell}B_{k\ell}
    \label{eq:dcov}$$ From this distance covariance, we can compute the distance correlation similarly to the computation of the Pearson correlation: $$\mathscr{R}^2(X,Y) = \frac{\mathscr{V}^2_n(X,Y)}{\sqrt{\mathscr{V}^2_n(X)}\sqrt{\mathscr{V}^2_n(Y)}}
    \label{eq:dcorr}$$

Distance Correlation Computation
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To compute the distance correlation, one must first be able to compute the distance covariance defined by (\[eq:dcov\]). First, the pairwise distances between samples of the response need to be computed and placed into a matrix. Similarly, the pairwise distances between samples of the current predictor are placed into another matrix. The distance measure can be any that satisfies the triangle inequality, but Euclidean distance is typical. Since the distances are pairwise in the number of samples and involve only a single dimension at a time, the computational complexity of this step is $O(N^2)$.

The matrices are then doubly centered by removing the row mean from each row and the column mean from each column, then adding the grand mean back in. These 