Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

5.3. Biplots 93ButandSubstituting in (5.3.6) givesX ′ X =(ULA ′ ) ′ (ULA ′ )= AL(U ′ U)LA ′= AL 2 A ′ ,(X ′ X) −1 = AL −2 A ′ .δ 2 hi =(n − 1)(g h − g i ) ′ L(A ′ A)L −2 (A ′ A)L(g h − g i )=(n − 1)(g h − g i ) ′ LL −2 L(g h − g i )(as the columns of A are orthonormal),=(n − 1)(g h − g i ) ′ (g h − g i ),as required.An adaptation to the straightforward factorization given above for α =0improves the interpretation of the plot still further. If we multiply the g iby (n − 1) 1/2 and correspondingly divide the h j by (n − 1) 1/2 , then thedistances between the modified g i are equal (not just proportional) to theMahalanobis distance and, if m = 2 < p, then the Euclidean distancebetween gh ∗ and g∗ i gives an easily visualized approximation to the Mahalanobisdistance between x h and x i . Furthermore, the lengths h ′ j h j areequal to variances of the variables. This adaptation was noted by Gabriel(1971), and is used in the examples below.A further interesting property of the biplot when α = 0 is that measurescan be written down of how well the plot approximates(a) the column-centred data matrix X;(b) the covariance matrix S;(c) the matrix of Mahalanobis distances between each pair of observations.These measures are, respectively, (Gabriel 1971)/ r∑(a) (l 1 + l 2 ) l k ;k=1/ r∑(b) (l1 2 + l2)2 lk 2;k=1/ r∑(c) (l1 0 + l2)0 lk 0 =2/r.k=1Because l 1 ≥ l 2 ≥ ··· ≥ l r , these measures imply that the biplot gives abetter approximation to the variances and covariances than to the (Mahalanobis)distances between observations. This is in contrast to principalcoordinate plots, which concentrate on giving as good a fit as possible to