1 Data

Part of the iris dataset

X <- iris[1:5,1:4] %>% as.matrix
X

#>   Sepal.Length Sepal.Width Petal.Length Petal.Width
#> 1          5.1         3.5          1.4         0.2
#> 2          4.9         3.0          1.4         0.2
#> 3          4.7         3.2          1.3         0.2
#> 4          4.6         3.1          1.5         0.2
#> 5          5.0         3.6          1.4         0.2

2 Centering

\[ \mathbf{H} = \mathbf{I} - \frac{1}{n} \mathbf{1}\mathbf{1}^T , \] \[ \mathbf{X}_c = \mathbf{H}\mathbf{X} \] \[ \mathbf{H}\mathbf{X}_c = \mathbf{X}_c \]

H <- diag(nrow(X)) - matrix(1/nrow(X),nrow=nrow(X),ncol=nrow(X))
Xc <- H%*%X
colMeans(Xc)

#>  Sepal.Length   Sepal.Width  Petal.Length   Petal.Width 
#>  2.220446e-16  5.551115e-18  1.620969e-17 -8.326673e-18

H%*%Xc-Xc

#>       Sepal.Length   Sepal.Width  Petal.Length  Petal.Width
#> [1,] -2.220446e-16  5.551115e-17 -1.521006e-17 8.326673e-18
#> [2,] -2.289835e-16  0.000000e+00 -1.521006e-17 8.326673e-18
#> [3,] -2.498002e-16 -1.387779e-17 -2.775558e-17 8.326673e-18
#> [4,] -2.220446e-16 -2.775558e-17 -1.387779e-17 8.326673e-18
#> [5,] -2.220446e-16  0.000000e+00 -1.756928e-17 8.326673e-18

3 Gram matrix

Gram matrix: \(\mathbf{G}=\mathbf{X}\mathbf{X}^T\) Here we work on centered data so \(\mathbf{G}=\mathbf{X}_c\mathbf{X}_c^T\)

G <- Xc%*%t(Xc)

4 Squared Distance matrix

\[ \mathbf{D}_X = \mathbf{N} - 2\mathbf{X}\mathbf{X}^T + \mathbf{N}^T , \]

N <- matrix(diag(G),nrow(Xc),nrow(Xc))
N

#>       [,1]  [,2]  [,3]  [,4]  [,5]
#> [1,] 0.106 0.106 0.106 0.106 0.106
#> [2,] 0.080 0.080 0.080 0.080 0.080
#> [3,] 0.042 0.042 0.042 0.042 0.042
#> [4,] 0.110 0.110 0.110 0.110 0.110
#> [5,] 0.122 0.122 0.122 0.122 0.122

dist2 <- N-2*G+t(N)
dist2

#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.00 0.29 0.26 0.42 0.02
#> [2,] 0.29 0.00 0.09 0.11 0.37
#> [3,] 0.26 0.09 0.00 0.06 0.26
#> [4,] 0.42 0.11 0.06 0.00 0.42
#> [5,] 0.02 0.37 0.26 0.42 0.00

dist(Xc)^2

#>      1    2    3    4
#> 2 0.29               
#> 3 0.26 0.09          
#> 4 0.42 0.11 0.06     
#> 5 0.02 0.37 0.26 0.42

5 Link Gram Matrix and Squared Distance Matrix

\[\mathbf{G} =\mathbf{X}_c\mathbf{X}_c^T=-\frac{1}{2}\mathbf{H}\mathbf{D}_X\mathbf{H}\]

#>        [,1]   [,2]   [,3]   [,4]   [,5]
#> [1,]  0.106 -0.052 -0.056 -0.102  0.104
#> [2,] -0.052  0.080  0.016  0.040 -0.084
#> [3,] -0.056  0.016  0.042  0.046 -0.048
#> [4,] -0.102  0.040  0.046  0.110 -0.094
#> [5,]  0.104 -0.084 -0.048 -0.094  0.122

-1/2*H%*%dist2%*%H

#>        [,1]   [,2]   [,3]   [,4]   [,5]
#> [1,]  0.106 -0.052 -0.056 -0.102  0.104
#> [2,] -0.052  0.080  0.016  0.040 -0.084
#> [3,] -0.056  0.016  0.042  0.046 -0.048
#> [4,] -0.102  0.040  0.046  0.110 -0.094
#> [5,]  0.104 -0.084 -0.048 -0.094  0.122

-1/2*H%*%dist2%*%H - G

#>               [,1]          [,2]          [,3]          [,4]          [,5]
#> [1,] -8.326673e-17 -5.551115e-17  6.938894e-18 -4.163336e-17 -6.938894e-17
#> [2,] -8.326673e-17  0.000000e+00  3.122502e-17  5.551115e-17 -4.163336e-17
#> [3,] -1.387779e-17  3.469447e-17  7.632783e-17  9.714451e-17  0.000000e+00
#> [4,]  1.387779e-17  4.857226e-17  9.020562e-17  1.249001e-16  1.387779e-17
#> [5,] -8.326673e-17 -4.163336e-17 -1.387779e-17  4.163336e-17 -4.163336e-17

6 Show that N cancels out when multiplying with H

\[ \mathbf{H N H} = \mathbf{0}\]

\[ \mathbf{H N}^T \mathbf{H} = \mathbf{0}\]

N%*%H

#>               [,1]          [,2]          [,3]         [,4]          [,5]
#> [1,]  9.048318e-19  9.048318e-19 -6.034062e-18 9.048318e-19 -3.619327e-18
#> [2,] -8.520962e-18 -1.582068e-18  5.356826e-18 1.887379e-18  6.328271e-18
#> [3,] -4.662937e-19  3.003153e-18  3.003153e-18 1.268430e-18  1.865175e-18
#> [4,]  8.604228e-19  8.604228e-19  8.604228e-19 8.604228e-19 -3.441691e-18
#> [5,]  7.271961e-19  7.271961e-19  7.271961e-19 7.271961e-19 -2.908784e-18

H%*%t(N)

#>               [,1]          [,2]          [,3]          [,4]          [,5]
#> [1,]  9.048318e-19 -8.520962e-18 -4.662937e-19  8.604228e-19  7.271961e-19
#> [2,]  9.048318e-19 -1.582068e-18  3.003153e-18  8.604228e-19  7.271961e-19
#> [3,] -6.034062e-18  5.356826e-18  3.003153e-18  8.604228e-19  7.271961e-19
#> [4,]  9.048318e-19  1.887379e-18  1.268430e-18  8.604228e-19  7.271961e-19
#> [5,] -3.619327e-18  6.328271e-18  1.865175e-18 -3.441691e-18 -2.908784e-18

Session info

#> [1] "2024-10-03 06:25:01 UTC"

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MDS: Link Squared Distance Matrix and Gram Matrix in Practice

Lieven Clement

statOmics, Ghent University (https://statomics.github.io)