Macky Macky - 25 days ago 12
R Question

Multi Dimensional Scaling

I'm trying to apply MDS to a distance matrix based on disagreements (namely it is the "voting" dataset in the "HSAUR" package.) I'm trying to reduce it to 2 dimensions and plot without the cmdscale() function, but can't get the same result when I try to do it myself. Here is the code;

library(HSAUR)

n <- 15

deltaD = voting
deltaDstar = deltaD^2

I = matrix(0,n,n)
diag(I) <- 1

J = matrix(1,n,n)

H = I-n^-1*J

Q = -0.5*H%*%deltaDstar%*%H

reseigen = eigen(Q)
lambda = reseigen$values
E = reseigen$vectors
Lambda = matrix(0,n,n)
diag(Lambda) <- lambda

Yhat = E[,1:2]%*%Lambda[1:2,1:2]^1/2
Yhat

x1 <- Yhat[,1]
x2 <- Yhat[,2]

plot(x1, x2, type = "n", xlim=c(-10,5), ylim=c(-6,8), xlab = "Coordinate 1",
ylab = "Coordinate 2", asp=1)

text(x1, x2, rownames(deltaD), cex = 0.6)


I'm following the standart textbook notation. This is the data matrix Yhat I get:

[,1] [,2]
[1,] -102.227945 0.1306901
[2,] -93.369153 46.4283081
[3,] 62.778582 -1.6069442
[4,] 30.708488 39.6033985
[5,] -59.614466 -17.4749816
[6,] -41.422942 -21.1382075
[7,] -94.185208 -5.0311437
[8,] 63.513501 -1.3529431
[9,] 72.856275 -0.3352204
[10,] 49.323040 -0.1241045
[11,] 54.595017 -4.7480531
[12,] 67.283718 -4.3435477
[13,] -53.094269 -28.0575071
[14,] 2.341299 -2.5952789
[15,] 40.514064 0.6455349


Compared to the one from cmdscale():

[,1] [,2]
Hunt(R) -9.1640883 0.02161894
Sandman(R) -8.3699537 7.68023459
Howard(D) 5.6277025 -0.26582292
Thompson(D) 2.7528216 6.55124865
Freylinghuysen(R) -5.3440596 -2.89073549
Forsythe(R) -3.7133046 -3.49671135
Widnall(R) -8.4431079 -0.83225871
Roe(D) 5.6935834 -0.22380571
Heltoski(D) 6.5311040 -0.05545261
Rodino(D) 4.4214984 -0.02052953
Minish(D) 4.8940977 -0.78542948
Rinaldo(R) 6.0315595 -0.71851563
Maraziti(R) -4.7595652 -4.64131141
Daniels(D) 0.2098827 -0.42931460
Patten(D) 3.6318295 0.10678526


They seem correlated but I don't understand what causes the different results. I would be glad to get a correction on the code. Thanks a lot in advance.

Answer

It's just about operator precedence: you need to change the line:

Yhat = E[,1:2]%*%Lambda[1:2,1:2]^1/2 # it's computing half of the dominant eigenvalues matrix

to

Yhat = E[,1:2]%*%Lambda[1:2,1:2]^(1/2) # take square-root of the dominant  eigenvalues matrix

and then you get exactly same results as cmdscale:

Yhat
            [,1]        [,2]
 [1,] -9.1640883  0.02161894
 [2,] -8.3699537  7.68023459
 [3,]  5.6277025 -0.26582292
 [4,]  2.7528216  6.55124865
 [5,] -5.3440596 -2.89073549
 [6,] -3.7133046 -3.49671135
 [7,] -8.4431079 -0.83225871
 [8,]  5.6935834 -0.22380571
 [9,]  6.5311040 -0.05545261
[10,]  4.4214984 -0.02052953
[11,]  4.8940977 -0.78542948
[12,]  6.0315595 -0.71851563
[13,] -4.7595652 -4.64131141
[14,]  0.2098827 -0.42931460
[15,]  3.6318295  0.10678526
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