Distance Covariance/Covariation

Sample Distance Covariance Formula

Given a statistical sample {(𝑋₁, 𝑌₁), …, (𝑋_𝑛, 𝑌_𝑛)} compute their respective doubly-centered distance matrices 𝐴 and 𝐵:

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Let (𝑋_𝑖, 𝑌_𝑖), 𝑖 = 1, 2, …, 𝑛 be a statistical sample from a pair of real-valued or vector-valued random variables (𝑋, 𝑌)

First, compute the 𝑛x𝑛 distance matrices 𝑎 and 𝑏

• 𝑎_𝑖𝑗 = ||𝑋_𝑖 - 𝑋_𝑗||
• 𝑏_𝑖𝑗 = ||𝑌_𝑖 - 𝑌_𝑗||

where:

• ||⋅|| denotes norm

Next, compute the doubly-centered distance matrices 𝐴 and 𝐵:

• 𝐴_𝑖𝑗 = 𝑎_𝑖𝑗 - 𝑎̅_𝑖· - 𝑎̅_·𝑗 + 𝑎̅
• 𝐵_𝑖𝑗 = 𝑏_𝑖𝑗 - 𝑏̅_𝑖· - 𝑏̅_·𝑗 + 𝑏̅

where:

• 𝑎̅_𝑖· is the 𝑖^th row mean of 𝑎
• 𝑎̅_·𝑗 is the 𝑗^th column mean of 𝑎
• 𝑎̅ is the grand mean of 𝑎
• the notation is similar for the b values

The sample distance covariance (a scalar) is simply the arithmetic average of the products 𝐴_𝑖𝑗𝐵_𝑖𝑗:

• 𝑑𝐶𝑜𝑣²(𝑋,𝑌) = (1/𝑛²) 𝛴_{1≤𝑖≤𝑛}𝛴_{1≤𝑗≤𝑛}[𝐴_𝑖𝑗𝐵_𝑖𝑗]

Subpages

• Distance Covariance／Covariation - Java Code
• Distance Covariance／Covariation - Properties