Principal component analysis biplots - Understanding Biplots - page 90

Ta b l e 3 . 4 Six artificial variables constructed from the first two variables of Table 1.1
by first centring them, then adding four zero dummy variables and finally applying a
random orthogonal rotation to the six-variable data set, rounding the final values to four
decimal places.

Va r 1 Va r 2 Va r 3 Va r 4 Va r 5 Va r 6

a 1 . 2547 0 . 9841 − 0 . 4140 0 . 8834 − 1 . 3810 1 . 2630

b 1 . 0845 0 . 9866 − 0 . 4420 0 . 8032 − 1 . 3329 0 . 9533

c 0 . 4454 0 . 9602 − 0 . 5251 0 . 4914 − 1 . 1157 − 0 . 1733

d 0 . 5392 0 . 9874 − 0 . 5273 0 . 5439 − 1 . 1713 − 0 . 0316

e 0 . 6137 0 . 6357 − 0 . 2981 0 . 4770 − 0 . 8335 0 . 4607

f 1 . 1539 1 . 1747 − 0 . 5476 0 . 8909 − 1 . 5461 0 . 8873

g 0 . 7635 0 . 7546 − 0 . 3483 0 . 5829 − 0 . 9998 0 . 6101

h 0 . 4340 0 . 7760 − 0 . 4128 0 . 4324 − 0 . 9236 − 0 . 0063

i 0 . 6617 0 . 5026 − 0 . 2082 0 . 4611 − 0 . 7116 0 . 6827

j − 0 . 4129 − 0 . 2927 0 . 1170 − 0 . 2817 0 . 4226 − 0 . 4473

k − 0 . 0250 − 0 . 5455 0 . 3337 − 0 . 1707 0 . 5659 0 . 5100

m0 . 1351 0 . 0754 − 0 . 0257 0 . 0862 − 0 . 1175 0 . 1671

n − 0 . 7239 − 1 . 0570 0 . 5417 − 0 . 6521 1 . 2976 − 0 . 2308

p 0 . 3298 0 . 4899 − 0 . 2520 0 . 2995 − 0 . 5998 0 . 0967

q − 0 . 3411 0 . 1725 − 0 . 1598 − 0 . 1121 − 0 . 0751 − 0 . 7912

1 . 0023 1 . 8891

− 1 . 0134

1 . 0268

− 2 . 2324 − 0 . 1135

r

− . − . . − . . − .

s 1 4953 2 2401 1 1540 1 3635 2 7385 0 4191

t − 1 . 1912 − 1 . 2507 0 . 5889 − 0 . 9308 1 . 6351 − 0 . 8772

u − 1 . 2055 − 0 . 9300 0 . 3882 − 0 . 8443 1 . 3110 − 1 . 2292

v − 1 . 4743 − 1 . 5245 0 . 7143 − 1 . 1452 1 . 9996 − 1 . 1095

w − 1 . 5486 − 2 . 5484 1 . 3365 − 1 . 4786 3 . 0700 − 0 . 2016

original variables SPR and RGF . Attempts to find these involve factor rotation methods
and are not discussed here (see Lawley and Maxwell, 1971).
We remark that readers who try to verify the above results by calling PCAbipl with
the data given in Table 3.4 will find that, due to the rounding, the quality will not exactly
equal unity nor will the adequacies be exactly as given in Table 3.5. It is better to use
the true input matrix we obtained with the following function:

function (X = aircraft.data[,2:3], new.orthog.rot = FALSE)
{ X.new <- cbind(scale(X, scale = FALSE), matrix(0,

nrow = nrow(X), ncol = 4))
if (new.orthog.rot == TRUE)
random.ortmat <- svd(matrix(rnorm(36), nrow = 6))$v
else set.seed(3015)
random.ortmat <- svd(matrix(rnorm(36), nrow = 6))$v
X.rot <- X.new %*% random.ortmat
dimnames(X.rot)[[2]] <- paste("V", 1:6, sep = "")
X.rot

}

Thus, although adequacy is a popular measure, it has its limitations. Also, the over-
all quality, depending on the trace operator, is a little crude. Can we do better? In the
definition of overall quality of fit we used the trace of the squared residuals. However,
calculating the 'variance accounted for' ratio individually for each diagonal element