Distance correlation: Definition & Examples

Distance correlation is a nonnegative measure of dependence built from distances between observations. It can reveal relationships that Pearson correlation misses, such as a response to the magnitude of a signed market move. For a quant researcher, it is a diagnostic for dependence beyond linear co-movement; it does not supply a trading direction or a hedge ratio.

For Euclidean random vectors $X$ and $Y$ with finite first moments, meaning $\mathbb{E}[\lVert X\rVert]<\infty$ and $\mathbb{E}[\lVert Y\rVert]<\infty$ , population distance correlation satisfies

0\le\mathcal{R}(X,Y)\le1, \qquad \mathcal{R}(X,Y)=0 \iff X\text{ and }Y\text{ are independent}.

This independence characterisation is a population result. The standard definition assigns zero when either population distance variance vanishes, as happens for an almost surely constant variable. Such a variable is independent of every other variable. For nonconstant variables, the normalisation has a positive denominator. These conventions are part of the definition in Székely, Rizzo and Bakirov (2007), §2.2.

Compare distances, then centre them

Take

n

matched observations

(x_i,y_i)

. Here the distances are ordinary Euclidean distances, not squared distances:

a_{ij}=\lVert x_i-x_j\rVert, \qquad b_{ij}=\lVert y_i-y_j\rVert.

Double-centre each distance matrix: subtract its row mean and column mean, then add its overall mean. Thus

A_{ij}=a_{ij}-\bar a_{i\cdot}-\bar a_{\cdot j}+\bar a_{\cdot\cdot}, \qquad B_{ij}=b_{ij}-\bar b_{i\cdot}-\bar b_{\cdot j}+\bar b_{\cdot\cdot}.

Every mean here includes all entries along its indicated dimension, including the original zero diagonal. The centred diagonal generally becomes nonzero.

The biased V-statistic version uses

\begin{aligned} \mathcal{V}_n^2(X,Y)&=\frac{1}{n^2}\sum_{i,j=1}^{n}A_{ij}B_{ij},\\ \mathcal{V}_n^2(X)&=\frac{1}{n^2}\sum_{i,j=1}^{n}A_{ij}^{2}, \qquad \mathcal{V}_n^2(Y)=\frac{1}{n^2}\sum_{i,j=1}^{n}B_{ij}^{2}. \end{aligned}

The first quantity is squared distance covariance; the other two are squared distance variances. For a positive denominator,

\mathcal{R}_n^2(X,Y)= \frac{\mathcal{V}_n^2(X,Y)} {\sqrt{\mathcal{V}_n^2(X)\mathcal{V}_n^2(Y)}}, \qquad \mathcal{R}_n(X,Y)=\sqrt{\mathcal{R}_n^2(X,Y)}.

If either observed series is constant, set

\mathcal{R}_n=0

under this convention. The final square root matters: reporting

\mathcal{R}_n^2

as distance correlation changes the coefficient. The authors' calculation reference gives these exact normalisations.

Three pairs that cancel linearly

Use the original synthetic observations

Observation	$X$	$Y=X^2$
1	−1	1
2	0	0
3	1	1

The sample means are $\bar x=0$ and $\bar y=2/3$ . The centred Pearson cross-products sum to $-1/3+0+1/3=0$ , so Pearson correlation is zero. Yet the supplied rule determines $Y$ exactly from $X$ .

For the $X$ distances, the first row mean is $1$ and the overall mean is $8/9$ . Consequently $A_{11}=0-1-1+8/9=-10/9$ . Applying the same centring to every entry gives

\mathcal{V}_n^2(X,Y)=\frac8{81},\qquad \mathcal{V}_n^2(X)=\frac{40}{81},\qquad \mathcal{V}_n^2(Y)=\frac{16}{81}.

Therefore

\mathcal{R}_n^2= \frac{8/81}{\sqrt{(40/81)(16/81)}} =\frac1{\sqrt{10}}, \qquad \mathcal{R}_n=10^{-1/4}\approx0.56234.

Three synthetic pairs are (−1,1), (0,0) and (1,1). The figure shows double-centred distance matrices multiplied by nine. Pearson correlation is zero, squared distance covariance is 8/81, and biased distance correlation is 0.56234.

Original synthetic three-observation example. Matrix entries are multiplied by nine for legibility; the calculation uses their unscaled values.

A deterministic nonlinear rule need not produce distance correlation one. Observations 1 and 3 are farthest apart in $X$ but identical in $Y$ : the two distance patterns are not proportional. Nor is the coefficient signed. Negating every $y_i$ leaves all pairwise distances, and therefore distance correlation, unchanged.

Keep the coefficient separate from the test

A positive sample coefficient alone does not establish population dependence. The biased statistic can be positive under independence and has positive finite-sample bias under nondegenerate independent sampling. It is not a p-value, a probability of dependence, or evidence of causation. The three pairs above expose the arithmetic; they do not constitute a causal experiment.

For i.i.d. paired observations under the independence null, the

Y

labels are exchangeable relative to

X

. A permutation test compares the observed statistic with statistics after reassignment of those labels; the authors' test reference distinguishes the coefficient from its calibrated test. Ordinary unrestricted shuffling need not be valid for serially dependent financial observations. The resampling scheme must respect the dependence structure and justify its own null calibration.

Also record the estimator. U-centred distance matrices produce an unbiased estimator of squared distance covariance for

n>3

under the paper's sampling assumptions. That estimate can be negative; this does not make the population squared distance covariance negative. It is a different estimator, unavailable for this

n=3

example, and cannot simply replace the numerator while retaining all interpretations of the nonnegative V-statistic. See Székely and Rizzo (2014), §3.1.

For the linear baseline and its limitations, Covariance and Correlation develops the population identities and an uncorrelated-but-dependent example. Its full lesson requires membership, with a public preview.

Sources

Székely, Rizzo and Bakirov (2007), Measuring and Testing Dependence by Correlation of Distances, §2.2, Definitions 2–5, and the population properties in §2 — finite-first-moment framework, zero if and only if independence, and sample conventions.
Székely and Rizzo (2014), Partial Distance Correlation with Methods for Dissimilarities, §3.1, Definition 2 and Proposition 1 — U-centring and unbiased squared distance covariance for $n>3$ .
Rizzo and Székely, energy: distance covariance and correlation, Details — double-centring and the distance-variance normalisation.
Rizzo and Székely, energy: distance covariance and correlation tests, Details and Value — permutation inference, test statistics and p-values.

Compare distances, then centre them

Three pairs that cancel linearly

Keep the coefficient separate from the test

Sources

Related terms

Correlation

Independence

P-value