I have a matrix where each row corresponds to an observation with binary attributes, and I am interested in performing multidimensional scaling using
cmdscale on this data. I am looking into binary distance measures but I am having some trouble on how to correctly define the distance matrix that is needed as input:
From what I understand, a similarity measure needs to satisfy three properties (boundary conditions, symmetry, identity/indiscernability). If the pairwise similarity matrix is PSD then the similarity is also a metric. In the case of dissimilarities, they must satisfy non-negativity, symmetry, identity/indiscernability. If the dissimilarity meets the triangle inequality, it is also a distance measure (and a metric).
- How can I transform (and under what conditions) a similarity measure into a distance measure? Would it be correct to do this with $d = 1 - s$ if the similarity measure is also a metric?
I am interested in analyzing the symmetry / asymmetry properties of several binary similarity measures (i.e. see how the MDS output behaves with measures that take positive matches and negative matches into account; or only positive matches).
- Are the symmetry / asymmetry properties of a similarity measure preserved if I convert them to dissimilarities (or distances)?