Distance correlation: Difference between revisions

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where ''ϕ''_{''X'', ''Y''}(''s'', ''t''), {{nowrap|''ϕ''_''X''(''s''),}} and {{nowrap|''ϕ''_''Y''(''t'')}} are the [[Characteristic function (probability theory)|characteristic functions]] of {{nowrap|(''X'', ''Y''),}} ''X'', and ''Y'', respectively, ''p'', ''q'' denote the Euclidean dimension of ''X'' and ''Y'', and thus of ''s'' and ''t'', and ''c''_''p'', ''c''_''q'' are constants. The weight function <math>({c_p c_q}{|s|_p^{1+p} |t|_q^{1+q}})^{-1}</math> is chosen to produce a scale equivariant and rotation invariant measure that doesn't go to zero for dependent variables.<ref name=SR2009a/><ref>{{cite journal|author1=Székely, G. J. |author2=Rizzo, M. L.|title=On the uniqueness of distance covariance|journal= Statistics & Probability Letters | year=2012| volume=82 |issue=12 | pages=2278–2282 | doi=10.1016/j.spl.2012.08.007}}</ref> One interpretation<ref name=neustats2012>{{cite web|url=http://www.neustats.com/neu-da-documentation/how-distance-correlation-works/|title=How distance correlation works|accessdate=2012-12-13}}</ref> of the characteristic function definition is that the variables ''e^isX'' and ''e^itY'' are cyclic representations of ''X'' and ''Y'' with different periods given by ''s'' and ''t'', and the expression {{nowrap|''ϕ''_{''X'', ''Y''}(''s'', ''t'') − ''ϕ''_''X''(''s'') ''ϕ''_''Y''(''t'')}} in the numerator of the characteristic function definition of distance covariance is simply the classical covariance of ''e^isX'' and ''e^itY''. The characteristic function definition clearly shows that

dCov²(''X'', ''Y'') = 0 if and only if ''X'' and ''Y'' are independent.