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

5.3. Biplots 91columns, L is an (r × r) diagonal matrix with elements l 1/21 ≥ l 1/22 ≥···≥lr1/2 ,andr is the rank of X. Now define L α ,for0≤ α ≤ 1, as the diagonalmatrix whose elements are l α/21 ,l α/22 , ··· ,lrα/2 with a similar definition forL 1−α , and let G = UL α , H ′ = L 1−α A ′ . ThenGH ′ = UL α L 1−α A ′ = ULA ′ = X,and the (i, j)th element of X can be writtenx ij = g ′ ih j , (5.3.2)where g i ′,i=1, 2,...,n and h′ j ,j=1, 2,...,p are the rows of G and H,respectively. Both the g i and h j have r elements, and if X has rank 2, allcould be plotted as points in two-dimensional space. In the more generalcase, where r>2, it was noted in Section 3.5 that (5.3.1) can be writtenx ij =r∑k=1u ik l 1/2ka jk (5.3.3)which is often well approximated bym∑m˜x ij = u ik l 1/2ka jk , with m 2, but thegraphical representation is then less clear. Gabriel (1981) referred to theextension to m ≥ 3asabimodel, reserving the term ‘biplot’ for the casewhere m = 2. However, nine years later Gabriel adopted the more commonusage of ‘biplot’ for any value of m (see Gabriel and Odoroff (1990), whichgives several examples of biplots including one with m = 3). Bartkowiakand Szustalewicz (1996) discuss how to display biplots in three dimensions.In the description of biplots above there is an element of non-uniqueness,as the scalar α which occurs in the definition of G and H can take any valuebetween zero and one and still lead to a factorization of the form (5.3.2).Two particular values of α, namely α =0andα = 1, provide especiallyuseful interpretations for the biplot.