# Using similarity measures as distances

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:

1. 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?
2. 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)?
• The simplest thing you can do is use a proper distance to begin with, e.g. simply the Euclidean distance in the attribute space. In this case, metric MDS turns out to be the same as simply performing a PCA on the original data. The reason is that if your data are already living in a space, you don't have to use MDS to embed them into a space. The question is of course whether Euclidean distance makes sense with respect to your binary attributes. – A. Donda Jul 22 '15 at 13:57
• There exist many ways to convert a similarity into a distance. Actually, you may use any way or invent your own if it will really make mathematical and substantial sense for you. – ttnphns Jul 22 '15 at 14:35
• If the pairwise similarity matrix is PSD then the similarity is also a metric As far as I know word "metric" is reserved for distances (dissimilarities). If a specific similarity measure matrix is always PSD the similarity could be then called "euclidean" because it spreads euclidean space and can be converted by the cosine law into the corresponding euclidean distance. As for metric distances - not every metric distance is euclidean distance. – ttnphns Jul 22 '15 at 14:39