Distance Variance/Variation

Sample Distance Variance

Calculate the doubly-centered distance matrix (𝐴) of a statistical sample {𝑋₁, …, 𝑋_𝑛}:

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

First, compute the 𝑛x𝑛 distance matrix (𝑎)

• 𝑎_𝑖𝑗 = ||𝑋_𝑖 - 𝑋_𝑗||

where:

• ||⋅|| denotes Euclidean distance/Euclidean norm

Next, compute the doubly-centered distance matrix (𝐴)

• 𝐴_𝑖𝑗 = 𝑎_𝑖𝑗 - 𝑎̅_𝑖· - 𝑎̅_·𝑗 + 𝑎̅

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

All rows and all columns of 𝐴 sum to zero

The sample distance variance (a scalar) is simply the arithmetic average of the products 𝐴_𝑖𝑗𝐴_𝑖𝑗:

• 𝑑𝑉𝑎𝑟(𝑋) = (1/𝑛²) 𝛴_{1≤𝑖≤𝑛}𝛴_{1≤𝑖≤𝑛}[𝐴_𝑖𝑗𝐴_𝑖𝑗]

Subpages

• Distance Variance／Variation - Properties