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| {{Cleanup|date=July 2010}}
| | I’m Nereida from Tarrel doing my final year engineering in Computing and Information Science. I did my schooling, secured 79% and hope to find someone with same interests in Cycling.<br>xunjie 時間の残りの半分大変疲れたクレジットカードあればと言った物語のデモですが、 |
| | | ファッショントレンドをリード、 |
| In [[statistics]] and in [[probability theory]], '''distance correlation''' is a measure of [[statistical dependence]] between two [[random variable]]s or two [[random vector]]s of arbitrary, not necessarily equal [[Euclidean vector|dimension]]. An important property is that this measure of dependence is zero if and only if the [[multivariate random variable|random variables]] are [[statistically independent]]. This measure is derived from a number of other quantities that are used in its specification, specifically: '''distance variance''', '''distance standard deviation''' and '''distance covariance'''. These take the same roles as the ordinary [[Moment (mathematics)|moment]]s with corresponding names in the specification of the [[Pearson product-moment correlation coefficient]].
| | 豊富な経験を蓄積してきた、 [http://www.schochauer.ch/_js/r/e/mall/watch/gaga/ �����ߥ�� �rӋ ��ǥ��`��] 最も便利なものでいっぱい3.Oversizedバッグ。 |
| | | 見て小冊子「コピー」と同様の本を保持している。 |
| These distance-based measures can be put into an indirect relationship to the ordinary moments by an [[#Alternative formulation: Brownian covariance|alternative formulation]] (described below) using ideas related to [[Brownian motion]], and this has led to the use of names such as '''Brownian covariance''' and '''Brownian distance covariance'''.
| | 市場適応性と能力に投資家がリスクに抵抗することを確認するために効果的に製品の多様化。 [http://bigapplefilmfestival.com/css/shop/Duvetica.php ���`�� �����ȥ�å�] トップ5のグローバルベビー用品製造工場全体の強度へのアクセス。 |
| | | 国内市場の消費のための20パーセント304万トンとなりました。 |
| [[Image:Distance Correlation Examples.svg|thumb|400px|right|Several sets of (''x'', ''y'') points, with the Distance correlation coefficient of ''x'' and ''y'' for each set. Compare to the graph on [[correlation]]]]
| | 子どもたちのシングルブレストデザイン大きな範囲、[http://www.tstcom.com/Backup/images/jp/shop/toryburch/ �ȥ�`�Щ`�� ؔ�� ����] この日当たりの良いシーズンを書いてお店に快適な印刷上のパリッとした白い線のためだけのリゾート2014シリーズで出版のオーストラリアブランドツィンマーマン、 |
| | | 私たちはしっかりと技術革新は、 |
| ==Background==
| | 国内外でのビジネスコミュニティと手をつないで行く、 |
| | | 間違いなくザラ自然看板になるのウィンドウを変更すると、 [http://www.tobler-verlag.ch/flash/ja/li/top/gaga/ �����ߥ�� �rӋ ���] |
| The classical measure of dependence, the [[Pearson product-moment correlation coefficient|Pearson correlation coefficient]],<ref>Pearson (1895)</ref> is mainly sensitive to a linear relationship between two variables. Distance correlation was introduced in 2005 by [[Gabor J Szekely]] in several lectures to address this deficiency of Pearson’s [[correlation]], namely that it can easily be zero for dependent variables. Correlation = 0 (uncorrelatedness) does not imply independence while distance correlation = 0 does imply independence. The first results on distance correlation were published in 2007 and 2009.<ref name=SR2007>Székely, Rizzo and Bakirov (2007)</ref><ref name=SR2009>Székely & Rizzo (2009)</ref> It was proved that distance covariance is the same as the Brownian covariance.<ref name=SR2009/> These measures are examples of [[energy distance]]s.
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| ==Definitions==
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| ===Distance covariance===
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| Let us start with the definition of the '''sample distance covariance'''. Let (''X''<sub>''k''</sub>, ''Y''<sub>''k''</sub>), ''k''= 1, 2, ..., ''n'' be a [[statistical sample]] from a pair of real valued or vector valued random variables (''X'', ''Y''). First, compute all pairwise [[Euclidean distance|distances]]
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| :<math>
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| \begin{align}
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| a_{j, k} &= \|X_j-X_k\|, \qquad j, k =1,2,\ldots,n,
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| \\ b_{j, k} &= \|Y_j-Y_k\|, \qquad j, k=1,2,\ldots,n,
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| \end{align}
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| </math>
| |
| | |
| where || ⋅ || denotes [[Euclidean norm]]. That is, compute the ''n'' by ''n'' distance matrices (''a''<sub>''j'', ''k''</sub>) and (''b''<sub>''j'', ''k''</sub>). Then take all doubly centered distances
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| :<math>
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| A_{j, k} := a_{j, k}-\overline{a}_{j.}-\overline{a}_{.k} + \overline{a}_{..}, \qquad
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| B_{j, k} := b_{j, k} - \overline{b}_{j.} -\overline{b}_{.k} + \overline{b}_{..},
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| </math>
| |
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| where <math>\textstyle \overline{a}_{j.}</math> is the {{math|''j''}}-th row mean, <math>\textstyle \overline{a}_{.k}</math> is the{{math|''k''}}-th column mean, and <math>\textstyle \overline{a}_{..}</math> is the grand mean of the distance matrix of the ''X'' sample. The notation is similar for the ''b'' values. (In the matrices of centered distances (''A''<sub>''j'', ''k''</sub>) and (''B''<sub>''j'',''k''</sub>) all rows and all columns sum to zero.) The squared '''sample distance covariance''' is simply the arithmetic average of the products ''A''<sub>''j'', ''k''</sub>''B''<sub>''j'', ''k''</sub>:
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| :<math>
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| \operatorname{dCov}^2_n(X,Y) := \frac{1}{n^2}\sum_{j, k = 1}^n A_{j, k}\,B_{j, k}.
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| </math>
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| The statistic ''T''<sub>''n''</sub> = ''n'' dCov<sup>2</sup><sub>''n''</sub>(''X'', ''Y'') determines a consistent multivariate test of independence of random vectors in arbitrary dimensions. For an implementation see ''dcov.test'' function in the ''energy'' package for [[R (programming language)|R]].<ref name=energy>[http://cran.us.r-project.org/web/packages/energy/index.html energy package for R]</ref>
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| The population value of '''distance covariance''' can be defined along the same lines. Let ''X'' be a random variable that takes values in a ''p''-dimensional Euclidean space with probability distribution {{math| μ}} and let ''Y'' be a random variable that takes values in a ''q''-dimensional Euclidean space with probability distribution {{math| ν}}, and suppose that ''X'' and ''Y'' have finite expectations. Write
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| :<math>a_\mu(x):= \operatorname{E}[\|X-x\|], \quad D(\mu) := \operatorname{E}[a_\mu(X)], \quad d_\mu(x, x') := \|x-x'\|-a_\mu(x)-a_\mu(x')+D(\mu).
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| </math>
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| Finally, define the population value of squared distance covariance of ''X'' and ''Y'' as
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| :<math>\operatorname{dCov}^2(X, Y) := \operatorname{E}\big[d_\mu(X,X')d_\nu(Y,Y')\big].</math>
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| One can show that this is equivalent to the following definition:
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| :<math>
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| \begin{align}
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| \operatorname{dCov}^2(X,Y) & := \operatorname{E}[\|X-X'\|\,\|Y-Y'\|] + \operatorname{E}[\|X-X'\|]\,\operatorname{E}[\|Y-Y'\|] \\
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| &\qquad - \operatorname{E}[\|X-X'\|\,\|Y-Y''\|] - \operatorname{E}[\|X-X''\|\,\|Y-Y'\|]
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| \\
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| & = \operatorname{E}[\|X-X'\|\,\|Y-Y'\|] + \operatorname{E}[\|X-X'\|]\,\operatorname{E}[\|Y-Y'\|] \\
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| &\qquad - 2\operatorname{E}[\|X-X'\|\,\|Y-Y''\|],
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| \end{align}
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| </math>
| |
| | |
| where '''''E''''' denotes expected value, and <math>\textstyle (X, Y),</math> <math>\textstyle (X', Y'),</math> and <math>\textstyle (X'',Y'')</math> are independent and identically distributed. Distance covariance can be expressed in terms of Pearson’s covariance,
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| '''cov''', as follows:
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| :<math>\operatorname{dCov}^2(X,Y) = \operatorname{cov}(\|X-X'\|,\|Y-Y'\|) - 2\operatorname{cov}(\|X-X'\|,\|Y-Y''\|).
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| </math> | |
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| This identity shows that the distance covariance is not the same as the covariance of distances, cov(||''X''-''X' ''||, ||''Y''-''Y' '' ||). This can be zero even if ''X'' and ''Y'' are not independent.
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| Alternately, the squared distance covariance can be defined as the weighted {{math|''L''<sub>2</sub>}} norm of the distance between the joint [[Characteristic function (probability theory)|characteristic function]] of the random variables and the product of their marginal characteristic functions:<ref name=SR2009a>Székely & Rizzo (2009) Theorem 7, (3.7), p. 1249.</ref>
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| <math>
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| \operatorname{dCov}^2(X,Y)= \frac{1}{c_p c_q}\int_{\mathbb{R}^{p+q}} \frac{\left| \phi_{X,Y}(s, t) - \phi_X(s)\phi_Y(t) \right|^2}{|s|_p^{1+p} |t|_q^{1+q}} dt\,ds
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| </math>
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| where ''ϕ''<sub>''X'', ''Y''</sub>(''s'', ''t''), {{nowrap|''ϕ''<sub>''X''</sub>(''s''),}} and {{nowrap|''ϕ''<sub>''Y''</sub>(''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''<sub>''p''</sub>, ''c''<sub>''q''</sub> 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|author= Székely, G. J. and 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<sup>isX</sup>'' and ''e<sup>itY</sup>'' are cyclic representations of ''X'' and ''Y'' with different periods given by ''s'' and ''t'', and the expression {{nowrap|''ϕ''<sub>''X'', ''Y''</sub>(''s'', ''t'') - ''ϕ''<sub>''X''</sub>(''s'') ''ϕ''<sub>''Y''</sub>(''t'')}} in the numerator of the characteristic function definition of distance covariance is simply the classical covariance of ''e<sup>isX</sup>'' and ''e<sup>itY</sup>''. The characteristic function definition clearly shows that
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| dCov<sup>2</sup>(''X'', ''Y'') = 0 if and only if ''X'' and ''Y'' are independent.
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| ===Distance variance===
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| The ''distance variance'' is a special case of distance covariance when the two variables are identical.
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| The population value of distance variance is the square root of
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| :<math>
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| \operatorname{dVar}^2(X) := \operatorname{E}[\|X-X'\|^2] + \operatorname{E}^2[\|X-X'\|] - 2\operatorname{E}[\|X-X'\|\,\|X-X''\|],
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| </math>
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| where <math>\operatorname{E}</math> denotes the expected value, <math>X'</math> is an independent and identically distributed copy of <math>X</math> and <math>X''</math> is independent of <math>X</math> and <math>X'</math> and has the same distribution as <math>X</math> and <math>X'</math>.
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| The ''sample distance variance'' is the square root of
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| :<math>
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| \operatorname{dVar}^2_n(X) := \operatorname{dCov}^2_n(X,X) = \tfrac{1}{n^2}\sum_{k,\ell}A_{k,\ell}^2,
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| </math>
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| which is a relative of [[Corrado Gini]]’s [[mean difference]] introduced in 1912 (but Gini did not work with centered distances).
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| ===Distance standard deviation===
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| The ''distance standard deviation'' is the square root of the ''distance variance''.
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| ===Distance correlation===
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| The ''distance correlation'' <ref name=SR2007/><ref name=SR2009/> of two random variables is obtained by dividing their ''distance covariance'' by the product of their ''distance standard deviations''. The distance correlation is
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| :<math> | |
| \operatorname{dCor}(X,Y) = \frac{\operatorname{dCov}(X,Y)}{\sqrt{\operatorname{dVar}(X)\,\operatorname{dVar}(Y)}},
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| </math>
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| and the ''sample distance correlation'' is defined by substituting the sample distance covariance and distance variances for the population coefficients above.
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| For easy computation of sample distance correlation see the ''dcor'' function in the ''energy'' package for [[R (programming language)|R]].<ref name=energy />
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| ==Properties==
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| ===Distance correlation===
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| (i) <math>0\leq\operatorname{dCor}_n(X,Y)\leq1</math> and <math>0\leq\operatorname{dCor}(X,Y)\leq1</math>.
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| (ii) <math>\operatorname{dCor}(X,Y) = 0</math> if and only if <math>X</math> and <math>Y</math> are independent.
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| (iii) <math>\operatorname{dCor}_n(X,Y) = 1</math> implies that dimensions of the linear spaces spanned by <math>X</math> and <math>Y</math> samples respectively are almost surely equal and if we assume that these subspaces are equal, then in this subspace <math>Y = A + b\,\mathbf{C}X</math> for some vector <math>A</math>, scalar <math>b</math>, and [[orthonormal matrix]] <math>\mathbf{C}</math>.
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| ===Distance covariance===
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| (i) <math>\operatorname{dCov}(X,Y)\geq0</math> and <math>\operatorname{dCov}_n(X,Y)\geq0</math>.
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| (ii) <math>\operatorname{dCov}^2(a_1 + b_1\,\mathbf{C}_1\,X, a_2 + b_2\,\mathbf{C}_2\,Y) = |b_1\,b_2|\operatorname{dCov}^2(X,Y)</math>
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| for all constant vectors <math>a_1, a_2</math>, scalars <math>b_1, b_2</math>, and orthonormal matrices <math>\mathbf{C}_1, \mathbf{C}_2</math>.
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| (iii) If the random vectors <math>(X_1, Y_1)</math> and <math>(X_2, Y_2)</math> are independent then
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| :<math>
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| \operatorname{dCov}(X_1 + X_2, Y_1 + Y_2) \leq \operatorname{dCov}(X_1, Y_1) + \operatorname{dCov}(X_2, Y_2).
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| </math>
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| Equality holds if and only if <math>X_1</math> and <math>Y_1</math> are both constants, or <math>X_2</math> and <math>Y_2</math> are both constants, or <math>X_1, X_2, Y_1, Y_2</math> are mutually independent.
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| (iv) <math>\operatorname{dCov}(X,Y) = 0</math> if and only if <math>X</math> and <math>Y</math> are independent.
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| This last property is the most important effect of working with centered distances.
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| The statistic <math>\operatorname{dCov}^2_n(X,Y)</math> is a biased estimator of <math>\operatorname{dCov}^2(X,Y)</math>. Under independence of X and Y <ref>Székely and Rizzo (2009), Rejoinder</ref>
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| :<math>
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| \operatorname{E}[\operatorname{dCov}^2_n(X,Y)] = \frac{n-1}{n^2} \left\{(n-2) \operatorname{dCov}^2(X,Y)+ \operatorname{E}[\|X-X'\|]\,\operatorname{E}[\|Y-Y'\|] \right\} = \frac{n-1}{n^2}\operatorname {E}[\|X-X'\|]\,\operatorname{E}[\|Y-Y'\|].
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| </math>
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| ===Distance variance===
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| (i) <math>\operatorname{dVar}(X) = 0</math> if and only if <math>X = \operatorname{E}[X]</math> almost surely.
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| (ii) <math>\operatorname{dVar}_n(X) = 0</math> if and only if every sample observation is identical.
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| (iii) <math>\operatorname{dVar}(A + b\,\mathbf{C}\,X) = |b|\operatorname{dVar}(X)</math> for all constant vectors <math>A</math>, scalars <math>b</math>, and orthonormal matrices <math>\mathbf{C}</math>.
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| (iv) If <math>X</math> and <math>Y</math> are independent then <math>\operatorname{dVar}(X + Y)\leq\operatorname{dVar}(X) + \operatorname{dVar}(Y)</math>.
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| Equality holds in (iv) if and only if one of the random variables <math>X</math> or <math>Y</math> is a constant.
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| ==Generalization==
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| Distance covariance can be generalized to include powers of Euclidean distance. Define
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| :<math> | |
| \begin{align}
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| \operatorname{dCov}^2(X, Y; \alpha) &:= \operatorname{E}[\|X-X'\|^\alpha\,\|Y-Y'\|^\alpha] + \operatorname{E}[\|X-X'\|^\alpha]\,\operatorname{E}[\|Y-Y'\|^\alpha]\\
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| &\qquad - 2\operatorname{E}[\|X-X'\|^\alpha\,\|Y-Y''\|^\alpha].
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| \end{align}
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| </math>
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| Then for every <math>0<\alpha<2</math>, <math>X</math> and <math>Y</math> are independent if and only if <math>\operatorname{dCov}^2(X, Y; \alpha) = 0</math>. It is important to note that this characterization does not hold for exponent <math>\alpha=2</math>; in this case for bivariate <math>(X, Y)</math>, <math>\operatorname{dCor}(X, Y; \alpha=2)</math> is a deterministic function of the Pearson correlation.<ref name=SR2007>Székely & Rizzo (2007) Theorem 7, p. 2785.</ref> If <math>a_{k,\ell}</math> and <math>b_{k,\ell}</math> are <math>\alpha</math> powers of the corresponding distances, <math>0<\alpha\leq2</math>, then <math>\alpha</math> sample distance covariance can be defined as the nonnegative number for which
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| :<math>
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| \operatorname{dCov}^2_n(X, Y; \alpha):= \frac{1}{n^2}\sum_{k,\ell}A_{k,\ell}\,B_{k,\ell}.
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| </math>
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| One can extend <math>\operatorname{dCov}</math> to [[metric space|metric-space]]-valued [[random variables]] <math>X</math> and <math>Y</math>: If <math>X</math> has law <math>\mu</math> in a metric space with metric <math>d</math>, then define <math>a_\mu(x):= \operatorname{E}[d(X, x)]</math>, <math>D(\mu) := \operatorname{E}[a_\mu(X)]</math>, and (provided <math>a_\mu</math> is finite, i.e., <math>X</math> has finite first moment), <math>d_\mu(x, x') := d(x, x')-a_\mu(x)-a_\mu(x')+D(\mu)</math>. Then if <math>Y</math> has law <math>\nu</math> (in a possibly different metric space with finite first moment), define
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| :<math>
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| \operatorname{dCov}^2(X, Y) := \operatorname{E}\big[d_\mu(X,X')d_\nu(Y,Y')\big].
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| </math>
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| This is non-negative for all such <math>X, Y</math> iff both metric spaces have negative type.<ref name=Lyonsdcov>Lyons, R. (2011) "Distance covariance in metric spaces". {{arXiv|1106.5758}}</ref>
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| Here, a metric space <math>(M, d)</math> has negative type
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| if <math>(M, d^{1/2})</math> is [[isometry|isometric]] to a subset of a [[Hilbert space]].<ref>Klebanov, L. B. (2005) ''N-distances and their Applications'', Karolinum Press,
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| Charles University, Prague.</ref>
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| If both metric spaces have strong negative type, then <math>\operatorname{dCov}^2(X, Y)= 0</math> iff <math>X, Y</math> are independent.<ref name=Lyonsdcov/>
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| ==Alternative formulation: Brownian covariance==
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| Brownian covariance is motivated by generalization of the notion of covariance to stochastic processes. The square of the covariance of random variables X and Y can be written in the following form:
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| :<math>
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| \operatorname{cov}(X,Y)^2 = \operatorname{E}\left[
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| \big(X - \operatorname{E}(X)\big)
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| \big(X^\mathrm{'} - \operatorname{E}(X^\mathrm{'})\big)
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| \big(Y - \operatorname{E}(Y)\big)
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| \big(Y^\mathrm{'} - \operatorname{E}(Y^\mathrm{'})\big)
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| \right]
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| </math>
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| where E denotes the [[expected value]] and the prime denotes independent and identically distributed copies. We need the following generalization of this formula. If U(s), V(t) are arbitrary random processes defined for all real s and t then define the U-centered version of X by
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| :<math>
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| X_U := U(X) - \operatorname{E}_X\left[ U(X) \mid \left \{ U(t) \right \} \right]
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| </math>
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| whenever the subtracted conditional expected value exists and denote by Y<sub>V</sub> the V-centered version of Y.<ref name=SR2009/><ref>Bickel & Xu (2009)</ref><ref>
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| Kosorok (2009)</ref> The (U,V) covariance of (X,Y) is defined as the nonnegative number whose square is
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| :<math>
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| \operatorname{cov}_{U,V}^2(X,Y) := \operatorname{E}\left[X_U X_U^\mathrm{'} Y_V Y_V^\mathrm{'}\right]
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| </math>
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| whenever the right-hand side is nonnegative and finite. The most important example is when U and V are two-sided independent [[Brownian motion]]s /[[Wiener process]]es with expectation zero and covariance
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| |s| + |t| - |s-t| = 2 min(s,t). (This is twice the covariance of the standard Wiener process; here the factor 2 simplifies the computations.) In this case the (U,V) covariance is called '''Brownian covariance''' and is denoted by
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| :<math>
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| \operatorname{cov}_W(X,Y).
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| </math>
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| There is a surprising coincidence: The Brownian covariance is the same as the distance covariance:
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| :<math>
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| \operatorname{cov}_{\mathrm{W}}(X, Y) = \operatorname{dCov}(X, Y),
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| </math>
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| and thus '''Brownian correlation''' is the same as distance correlation.
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| On the other hand, if we replace the Brownian motion with the deterministic identity function ''id'' then Cov<sub>id</sub>(X,Y) is simply the absolute value of the classical Pearson [[covariance]],
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| :<math>
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| \operatorname{cov}_{\mathrm{id}}(X,Y) = \left\vert\operatorname{cov}(X,Y)\right\vert.
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| </math>
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| ==See also==
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| * [[RV coefficient]]
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| * For a related third-order statistic, see [[Skewness#Distance skewness|Distance skewness]].
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *Bickel, P.J. and Xu, Y. (2009) "Discussion of: Brownian distance covariance", ''Annals of Applied Statistics'', 3 (4), 1266–1269. {{doi|10.1214/09-AOAS312A}} [http://arxiv4.library.cornell.edu/PS_cache/arxiv/pdf/0912/0912.3295v2.pdf Free access to article]
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| *Gini, C. (1912). Variabilità e Mutabilità. Bologna: Tipografia di Paolo Cuppini.
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| *Pearson, K. (1895). "Note on regression and inheritance in the case of two parents", ''[[Proceedings of the Royal Society]]'', 58, 240–242
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| *Pearson, K. (1920). "Notes on the history of correlation", ''[[Biometrika]]'', 13, 25–45.
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| * Székely, G. J. Rizzo, M. L. and Bakirov, N. K. (2007). "Measuring and testing independence by correlation of distances", ''[[Annals of Statistics]]'', 35/6, 2769–2794. {{doi| 10.1214/009053607000000505}} [http://personal.bgsu.edu/~mrizzo/energy/AOS0283-reprint.pdf Reprint]
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| * Székely, G. J. and Rizzo, M. L. (2009). "Brownian distance covariance", ''Annals of Applied Statistics'', 3/4, 1233–1303. {{doi| 10.1214/09-AOAS312}} [http://personal.bgsu.edu/~mrizzo/energy/AOAS312.pdf Reprint]
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| *Kosorok, M. R. (2009) "Discussion of: Brownian Distance Covariance", ''Annals of Applied Statistics'', 3/4, 1270–1278. {{doi|10.1214/09-AOAS312B}} [http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.0822v1.pdf Free access to article]
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| ==External links==
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| *[http://personal.bgsu.edu/~mrizzo/energy.htm E-statistics (energy statistics)]
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| {{DEFAULTSORT:Distance Correlation}}
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| [[Category:Statistical dependence]]
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| [[Category:Statistical distance measures]]
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| [[Category:Theory of probability distributions]]
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| [[Category:Multivariate statistics]]
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| [[Category:Covariance and correlation]]
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