Cailliez

The Cailliez correction is a transformation used in multidimensional scaling (MDS) to convert a dissimilarity matrix that is not Euclidean into a Euclidean one by adding a constant to all pairwise dissimilarities. It addresses the common problem that non-Euclidean distances produce negative eigenvalues in the Gram matrix used by classical MDS, which prevents a faithful Euclidean embedding.

The procedure involves determining the smallest nonnegative constant c such that the adjusted dissimilarities d_ij' = d_ij + c yield an Euclidean distance matrix. Once c is found, classical MDS is applied to the adjusted distances, producing coordinates in a low-dimensional space that best preserve the modified dissimilarities. In practice, the value of c is chosen so that the double-centered matrix derived from the adjusted distances is positive semidefinite, ensuring a valid Euclidean representation.

The Cailliez correction is one of several techniques for handling non-Euclidean dissimilarities. It is often compared with other methods, such as the Lingoes correction, which offer alternative ways to modify distances to achieve Euclidean embeddability while attempting to preserve the relative structure of the data.

Named after Cailliez, the method is widely cited in statistical and data-analytic applications that use multidimensional scaling, including fields such as psychology, ecology, and geography, where researchers frequently encounter non-Euclidean dissimilarities and seek a meaningful Euclidean representation for visualization and interpretation.