Biplots

The previous two Subsections have shown that, in principal component analysis, both the descriptor-axes and object-vectors can be plotted in the reduced space. This led Jolicoeur & Mosimann (1960), and subsequently Gabriel (1971, 1982), to plot these projections together in the same diagram, called biplot.

Scalings Two types of biplots may be used to represent PCA results (Gabriel, 1982; ter in PCA Braak, 1994): distance biplots are made of the juxtaposition of matrices U (eigenvectors scaled to lengths 1) and F (eq. 9.4, where each principal component k is scaled to variance = Xk), whereas correlation biplots use matrix UA1/2 for descriptors (each eigenvector k is scaled to length JXk) and a matrix G = FA-1/2 for objects whose columns have unit variances. Matrices F and U, or G and UA1/2, can be used together in biplots because the products of the eigenvectors with the object score matrices reconstruct the original (centred) matrix Y perfectly: FU' = Y and G(UA1/2)' = Y. Actually, the eigenvectors and object score vectors may be multiplied by any constant without changing the interpretation of a biplot.

Distance • Distance biplot (Fig. 9.3a) — The main features of a distance biplot are the biplot following: (1) Distances among objects in the biplot are approximations of their

Euclidean distances in multidimensional space. (2) Projecting an object at right angle on a descriptor approximates the position of the object along that descriptor. (3) Since descriptors have lengths 1 in the full-dimensional space (eq. 9.7), the length of the projection of a descriptor in reduced space indicates how much it contributes to the formation of that space. (4) The angles among descriptor vectors are meaningless.

Correlation • Correlation biplot (Fig. 9.3b) — The main features of a correlation biplot are the biplot following: (1) Distances among objects in the biplot are not approximations of their

Euclidean distances in multidimensional space. (2) Projecting an object at right angle on a descriptor approximates the position of the object along that descriptor. (3) Since descriptors have lengths Sj in full-dimensional space (eq. 9.9), the length of the projection of a descriptor in reduced space is an approximation of its standard deviation. (4) The angles between descriptors in the biplot reflect their correlations. When the relationships among objects are important for interpretation, this type of biplot is inadequate; use a distance biplot in this case.

For the numerical example, the positions of the objects in the correlation biplot are computed as follows: