yii y2i y3i

Object x1

y4i yii y2i y3i

Object x1

Figure 7.2 On the left, representation of five objects in an A-space with two descriptors. The thickness of the lines that join the objects is proportional to their degree of resemblance with regard to the two descriptors, i.e. their proximity in the A-space. On the right, a similar representation of four descriptors in an I-space consisting of two objects.

possible to obtain an ordination of objects in low-dimension space using either the R method of principal component analysis or the Q method of principal coordinate analysis. Interestingly, these two analyses lead to the same ordination of the objects when the principal coordinates are computed using coefficient Di (Section 7.4), although the results of Q and R analyses are not always reducible to each other.

Following the terminology of Williams & Dale (1965), the space of descriptors A-space (attributes) is called "A-space". In this space, the objects may be represented along axes which correspond to the descriptors. For example, sites may be positioned in an A-space whose axes are the abundances of various species. A low-dimension space obtained using the ordination methods described in Chapter 9, in which the objects are represented, is also an A-space. Symmetrically, the space of reference in which the descriptors are positioned relative to axes corresponding to objects (or individuals) is I-space called "I-space". For example, species could be positioned in an I-space whose axes would be the different sites. An I-space representation is more artificial than an Aspace plot because, in I-space, values on the Xj axes are yj values (Table 2.1) and are therefore states of the observed descriptors. Figure 7.2 illustrates the A- and I-spaces.

The number of dimensions that can be represented on paper is limited to two or eventually three. Hence, one generally imagines distances between objects or descriptors as embedded in a 2- or 3-dimensional space. Section 7.4 will show that such models can be extended to a large number of dimensions. Distances and similarities computed in the present chapter will, in most instances, be based on measurements made in high-dimensional spaces.

In addition to the methods described in the present and following chapters, there exist approaches allowing the analysis of the whole data box instead of subsets of it, as was the case in the six modes described above. Examples are found in Williams & Stephenson (1973), Williams et al. (1982), Cailliez & Pages (1976), Marcotorchino & Michaud (1979), Kroonenberg (1983: three-way principal component analysis ), and Carlier & Kroonenberg (1996: three-way correspondence analysis).

Euclidean The A- and I-spaces are called metric or Euclidean because the reference axes are space quantitative and metric. Even though it might be interesting to plot objects in A-space, or descriptors in I-space, in order to detect their clustering structure, the clustering methods described in Chapter 8 are free from the restrictions inherent to metric spaces. As a matter of fact, clustering may be achieved without reference to any particular space, metric or not. This makes the two dimensions of figures, drawn on paper, available for illustrating other aspects of the clustering, e.g. the tree of the successive partitions of the set of objects.

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