# Transforming the distance value from a center, to a probability value

Let $c_i$ be the center of a micro-cluster (i.e. we have many centers representing some fragments of clusters). Let $c_1$ be the center which is the closest to a new data-point $x$, such that $d_{x,c_1} = distance(x, c_1)$.

(1) Is it possible to express d by a probability $d_{x,c_1}$ by a probability $p_{x,c_1}$ which tends to 1 as $d_{x,c_1}$ is higher, and to 0 as $d_{x,c_1}$ is smaller ? I think it is more intuitive to manipulate a probability instead of a distance.

(2) Same question as (1), however this time I suppose that $p_{x,c_1} = 0$ if $d_{x,c_1} < radius$.

• If we take "probability" merely in the sense of some value in the interval $[0,1]$ then the answer is of course, in many simple ways (e.g., rescale an inverse tangent in #1) and for a full account you would want to ask this question on the math site. But what is the intended interpretation of the "probability"? – whuber Apr 22 '13 at 14:53
• @whuber What I want is: the more $x$ is far from its closest center c1, the more likely it is to different from c1. So with some probability $p_{x, c1}$ (proportional to distance(x,c1)) we want to "say that x is different from c1" to behaves accordingly. – shn Apr 22 '13 at 15:35
• That's fine--but given that there are infinitely many ways to do this, and they can differ enormously among themselves, what constraints or guidance can you provide concerning which to choose from? What justification do you have that will allow any such transformation to be interpreted as a probability rather than as some otherwise meaningless number between zero and one? – whuber Apr 22 '13 at 16:28
• @whuber Because I'll do inside a loop, something like: if( uniform_random([0,1]) < $P_{x,c1}$ ) then "create a a new center at x"; else "e.g. assign x to c1" – shn Apr 23 '13 at 16:30
• That's just an algorithm: it does not justify calling the results "probabilities," even if random values were used in the algorithm itself. – whuber Apr 23 '13 at 16:32

Yes, you can do so. You need to live with a lot of assumptions, and most likely this will not live up to your expectations.

But in essence, the function that you are looking for is the "cumulative density function" (cdf) of the distribution of the distances. IIRC you can assume these to be rescaled-beta distributed, at least some question here on stats.SE said so...

See e.g.

Mahalanobis distance distribution of multivariate normally distributed points

Distribution of an observation-level Mahalanobis distance

• If I estimate the distribution of the distances (previously observed) using some method (I don't know which), is it possible to have the cdf which will reflect the probability I'm looking for ? – shn Apr 23 '13 at 16:33
• Exactly what "distribution of the distances" do you mean? If it's the distribution to the nearest center, assuming a uniform distribution of possible points within some region, it's almost surely not a rescaled beta (or anything else with a simple description), so I'm wondering whether you meant some other set of distances. – whuber Apr 23 '13 at 16:34
• Well, he seems to be interested in the distribution of distances from the center. Probably assuming a multivariate normal distribution., and mahalanobis distances. I've added a link to a related question. – Anony-Mousse Apr 24 '13 at 8:57