Since the two columns of the matrix of component scores are the coordinates of the five objects with respect to the principal axes, they can be used to plot the objects with respect to principal axes I and II (Fig. 9. 2c). It is easy to verify (Fig. 9.2d) that, in this two-descriptor example, the objects are positioned by the principal components in the same way as in the original system of descriptor-axes. Principal component analysis has simply rotated the axes by 26° 34' in such a way that the new axes correspond to the two main components of variability. When there are more than two descriptors, as it is usually the case in ecology, principal component analysis still only performs a rotation of the system of descriptor-axes, but now in multidimensional space. In that case, principal components I and II define the plane allowing the representation of the largest amount of variance. The objects are projected on this plane in such a way as to preserve, as much as possible, the relative Euclidean distances they have in the multidimensional space of the original descriptors.
The relative positions of the objects in the rotated p-dimensional space of principal components are the same as in the p-dimensional space of the original descriptors Euclidean (Fig. 9.2d). This means that the Euclidean distances among objects (D1, eq. 7.34) have distance been preserved through the rotation of axes. This important property of principal component analysis is noted in Table 9.1.
The quality of the representation in a reduced Euclidean space with m dimensions only (m < p) may be assessed by the ratio:
( m ^
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