Principal Coordinates Analysis
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Given a symmetric matrix, A say, with values representing the associations amongst a set of n units, principal coordinates analysis attempts to find a set of points for the n units in a multidimensional space so that the squared distance between the ith and jth points is given by:

dij = aii + ajj - 2aij

If, for example, A is a similarity matrix then aii and ajj are both equal to 1.0, and so this is equivalent to:

dij = 2 (1 - aij)

Thus similar units are placed close together and dissimilar units are further apart.

The coordinates generated can be arbitrarily located in space as this will not alter the fitted inter-point distances. By convention, the points are centred to have their mean at the origin, and rotated to principal axes, so that the first r dimensions give the best r dimensional fit.

Association Matrix

This specifies the name of the symmetric matrix of associations to be analysed and can be selected from Available Data. You can specify either a similarity or distance matrix.

Add points

Allows you to add new points to the analysis.

Form Similarity Matrix

This forms a similarity matrix for input to Principal Coordinates. You should select this menu if you are starting with raw data variates.

Analysis based on

Specifies whether the analysis is based on a similarity matrix or distance matrix. Whereas similarity matrices can be input directly to the analysis, a dissimilarity (distance) matrix requires preliminary transformation. Selecting dissimilarities will ensure the appropriate transformation is applied to the data.

If B is a distance matrix, so that bij gives the observed distance between the ith and jth units, then the transformation

A = - B * B / 2

will lead to the analysis generating points with inter-point squared distance

dij = aii + ajj - 2aij = 0 + 0 - 2 * (- bij * bij / 2 ) = bij2

Therefore the analysis will give points that generate the supplied distances; the first r dimensions of the solution will give the best r-dimensional fit.

See Also