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topics). Given such a dicotomy of approaches to the topic, we should not be making a choice here but rather attempting to describe both views. I reverted your recent edits so as not to lose this other viewpoint, but the article certainly needs work. Finding some reliable sources for each viewpoint would be a great start.
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From a mathematical perspective, if it's not symmetric, it's not really a distance matrix. Intuitively, distance implies the distance from a to b should be equivalent to the distance from b to a. If this doesn't hold, I don't think it should be called a distance matrix. This property is not exclusive
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Can you think of applications that use the term loosely? In my experience, distance implies some structure in the matrix that can then be used by algorithms for clustering or tree building. If a matrix doesn't actually have any of this structure, I don't think the term distance should be used. Maybe
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This article is a bit of a mess. The opening sentence clearly talks about distance as a metric, but the applications in bioinformatics and related fields use distance with a looser non-metric meaning (and while they even talk about metrics, these are not the mathematically defined terms). In these
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It makes the claim that the matrix need not be symmetric and that it need not be hollow. The rules 2 and 3 under formalization directly contradict this. They state precisely in no uncertain terms that the matrix must be both symmetric and hollow. It also seems to want to allow for complex valued
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I believe that you made the wrong choice here. The "Formalization" section was only added recently, and, as you point out, contradicted material that was already on the page for a long while. The application areas seem to use this concept with a non-metric distance in mind (look at the See also
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We currently have the applications of a "distance matrix" spread in sections "Bioinformatics", "Data Mining and
Machine Learning", "Information retrieval", "Chemistry", and "Other Applications". Some of these about specific definitions of distances, some talk about specific algorithm run on a
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I believe we should talk keep field-specific definition of distances inside the "field" sections, but move the algorithms out to a more general section. Right now we are basically mentioning clustering algorithms twice, once as phylogeny, once as data-mining.
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with the merger. Adjacency matrices are simply a way to tell whether two vertices are connected or not, and by how many paths. Distance matrices give the distances between two vertices, not whether they are connected or not.
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metrics which is not in line with the standard definition of a metric. As it stands it is a complete mess. The article should at least be self consistent even if it is inaccurate.--
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applications, distances can be negative and don't have to be symmetric. I could help fixing up the mathematical side of this topic, but the other applications are beyond my ken.
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While searching for something more explicit to give you I stumbled upon the shortest path problem in networks which is a clear example. I've amended the article accordingly.
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To be fair, I removed it, but after reading your comment, I agree that covering both points of view is the best way forward, at least until we uncover more sources.
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Too bad no one of you added any information about adjacency matrices to the article - at least that should have been done before removing the tags... --
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in the math definition non-negativity is too restrictive because transformations can usually resolve this, but I think asymmetry should at least hold.
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with the merger. However, these topics are somewhat related and there should be a sentence or two describing their relationship. --
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Unlike a
Euclidean distance matrix, the matrix does not need to be symmetric—that is, the values xi,j do not necessarily equal xj,i.
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The section "Comparison with
Euclidean distance matrix" contradicts the "Formalization" section completely
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Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
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Groenen, P. (1997) Modern Multidimensional Scaling. Theory and Applications. Springer.
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Knowledge. If you would like to participate, please visit the project page, where you can join
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Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979) Multivariate
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These topic are closely related. Could we merged them in some way?
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Basically everything here are definitions of distance functions.
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My current preferred outline to rewrite into would look like:
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