Friend Reza:

Check the value difference metric...

http://www.sciencedirect.com/science/article/pii/S0167865514002049

you can use formula that compute similarities and differences for nominal attributes (Ref: Data mining concepts and techniques, page 392) and after that use it for distance.

We compute the probability P(A) = 20/40, P(B) = 3/40, P(AB) = 7/40, P(O) = 10/40. Then the distance between value B and some other value is 1 - P(B).

We assume they are true nominal attributes (that is, they cannot be meaninfully ordered in any way). However, we may be able to construct some kind of taxonomy or ontology and then apply some similarity metric. Maybe a taxonomy can be applied to the blood groups?

A more simple approach is to calculate the modal value, which is equivalent to the average for numeric values.

Regards, David N.

Well let me second the idea of Henk Smid, (gravity and spatial interaction) but don't be too quick to assume that the social science methods are not grounded in various physical analogies and indeed derivations from statistical mechanics, entropy, information theory tools are common. The main idea is that objects which are in closer spatial proximity might be expected to be interactive; this idea has had a huge influence in regional science, transportation, social-migration etc. It is an interesting idea to adapt this framework to include measurement of similarity / difference.

There is a bit more about this in the attached review papers.

Article Isard's Contributions to Spatial Interaction Modeling

Article Entropy‐Based Spatial Interaction Models for Trip Distribution

Dr. Koczkodaj: I'm having a bit of a hard time to see how this is relevant.

Let's take all the parts: an impossibility result (non-existence) such as Arrow is grounded on some axioms (in this case, properties of individual valuations / theory of value, choices, etc) and the result in lay persons terms, as far as I can tell, is to show some difficulty in assembling a social welfare function (an aggregation of individual preferences) with certain reasonable properties. The only way to convert an impossibility proof would seem to me to alter the axioms. And, these are already well known and established. So do you see something here that I am missing? Maybe the idea that distance metrics obey some "similar" axioms (reflexivity, transitivity, symmetry, ..)? Do you  have a way to generalize this?

The question began with a way to compare observations and I think it is basically a measurement issue for multivariate objects (with some possible non-scalar properties). I admit to going off on a bit of a tangent myself with the fact that there are spatial interaction measures that relate to similarity. That one, at the very least is underpinned by Tobler's first law. (See articles by my colleagues for background.)

Article Tobler's First Law and Spatial Analysis

Article Tobler's First Law of Geography: A Big Idea for a Small World?

Dear Reza

The first thing that you sould do is to mapp the nominal variable into a new set of binary variables for each level (you can see Payle, Data Preparation for data mining). Then, there are several distance metrics for binary vectors such as jaccard index, tanimoto distance, simple matching among others.

If you want to do exploration, you can assign discrete values to the nomanl levels and do PCA analysis bearing in mind that they are not regular continuous variables.

Another approach is to do PCA over the continuous variables and use the nominal ones for labeling points, and search for pattern in the PCA space.

Take into account thta if you mix continuous and binary variables for clustering, binary variables will ussually rule the clustering organization. In this case you can do clusering over the continuous variables and another clustering with only the binary ones (that represent the nominal ones) and compare the clusters. If you do hieararchical clusering, you can compare the dendrograms through cophenetic distance.

Hope this helps

Try discrete metric: http://en.wikipedia.org/wiki/Discrete_space

Given subject, its distance to other is 0 (zero) if they have the same blod group or the distance is 1 if they have differente blod group.

http://en.wikipedia.org/wiki/Discrete_space

If you have other atributs, you may have a binary vector where each component is given by a discrete metric. Then the distance is given by this vector norm.

I suggest you read the article: http://www.researchgate.net/publication/278403191_Numerical_Coding_of_Nominal_Data

Perhaps there you will find the answer.

Article Numerical Coding of Nominal Data

To give an answer, you have to tell us what do you want to do after you have the distances.

Dear Prof. Orloci

First of all, I'm so appreciative of your interest.

Reaching to data relation is fundamental knowledge in machine learning. When data are explained by numerical attributes, relations are clear. But in nominal attributes relations and computation of new items expressed by this attributes are vague. All and all, better description of data with nominal attributes help us to gain more valuable knowledge from collect of data.

Best regards