Question

In MATLAB, Implement a hybrid clustering algorithm which combines hierarchical clustering and k-means clustering.

Answer 1

Proof:-

Let ξ ∼ Np(0, I). Since X d= Σ 1 2 l ξ + µl , we have

Y j,l (X) = X 0 (Σ −1 j − Σ −1 l )X + 2X 0 (Σ −1 l µl − Σ −1 j µj ) + µ 0 jΣ −1 j µj − µ 0 lΣ −1 l µl

d= (Σ 1 2 l ξ + µl ) 0 (Σ −1 j − Σ −1 l )(Σ 1 2 l ξ + µl ) + 2(Σ 1 2 l ξ + µl ) 0 (Σ −1 l µl − Σ −1 j µj ) + µ 0 jΣ −1 j µj − µ 0 lΣ −1 l µl

= ξ 0 (Σj|l − I)ξ + 2ξ 0Σ 1 2 l Σ −1 j (µl − µj ) + (µl − µj ) 0 (Σ −1 j )(µl − µj ) (3)

where Σj|l = Σ 1 2 l Σ −1 j Σ 1 2 l . Let the spectral decomposition of Σj|l be given by Σj|l = Γj|lΛj|lΓ 0 j|l , where Λj|l is a diagonal matrix containing the eigenvalues λ1, λ2, . . . λp of Σj|l , and Γj|l is an orthogonal matrix containing the eigenvectors γ1 , γ2 , . . . , γp of Σj|l . Since Z ≡ Γj|l 0 ξ ∼ Np(0, I) as well, we get from (3) that

Y j,l (X) d= ξ 0 (Γj|lΛj|lΓ 0 j|l − Γj|lΓ 0 j|l )ξ + 2ξ 0 (Γj|lΛj|lΓ 0 j|lΣ − 1 2 1 )(µl − µj ) + (µl − µj ) 0 (Σ − 1 2 1 Γj|lΛj|lΓ 0 j|lΣ − 1 2 l )(µl − µj )

= (Γ 0 j|l ξ) 0 (Λj|l − I)(Γ 0 j|l ξ) + 2(Γ 0 j|l ξ) 0 (Λj|lΓj|lΣ − 1 2 l )(µl − µj ) + (µl − µj ) 0 (Σ − 1 2 l Γj|lΛj|lΓ 0 j|lΣ − 1 2 l )(µl − µj )

d= X p i=1 (λi − 1)Zi 2 + 2λi δiZi + λi δ 2 i , (4)

where δi , i = 1, 2, . . . , p are as in the statement of the theorem. We can simplify (4) further based on the values of λi : If λi > 1: (λi − 1)Z 2 i + 2λi δiZi + λi δ 2 i = (√ λi − 1Zi + λi δi/ √ λi − 1)2 − λi δ 2 i /(λi − 1), while for λi < 1: (λi − 1)Zi 2 + 2λi δiZi + λi δ 2 i = −( √ 1 − λiZi − λi δi/ √ 1 − λi ) 2 − λi δ 2 i /(λi − 1). In both cases, (λi − 1)Zi 2 + 2λi δiZi + λi δ 2 i is distributed as a (λi − 1)χ 2 l ,λ2 i δ 2 i /(λi −1)2 -random variable shifted by −λi δ 2 i /(λi − 1). When λi = 1, (λi − 1)Z 2 i + 2λi δiZi + λi δ 2 i = 2δiZi + δ 2 i . The theorem follows from some further minor rearrangement of terms.

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