Distance matrix
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In mathematics, a distance matrix is a matrix (two-dimensional array) containing the distances, taken pairwise, of a set of points. It is therefore a symmetric N×N matrix containing non-negative reals as elements, given N points in Euclidean space. The number of pairs of points N×(N-1)/2 is the number of independent elements in the distance matrix. Distance matrices are closely related to adjacency matrices, with the difference that the latter only provides the information which vertices are connected but does not tell about costs or distances between the vertices.
For example, suppose this data is to be analyzed. Where pixel euclidean distance is the distance metric.
The distance matrix would be:
| a | b | c | d | e | f | |
|---|---|---|---|---|---|---|
| a | 0 | 184 | 222 | 177 | 216 | 231 |
| b | 184 | 0 | 45 | 123 | 128 | 200 |
| c | 222 | 45 | 0 | 129 | 121 | 203 |
| d | 177 | 123 | 129 | 0 | 46 | 83 |
| e | 216 | 128 | 121 | 46 | 0 | 83 |
| f | 231 | 200 | 203 | 83 | 83 | 0 |
This data can then be viewed in graphic form. In this image, black denotes a distance of 0 and white is maximal distance.
In bioinformatics, distances matrices are used to represent protein structures in a coordinate-independent manner, as well as the pairwise distances between two sequences in sequence space. They are used in structural and sequential alignment, and for the determination of protein structures from NMR or X-ray crystallography.
Sometimes it is more convenient to express data as a similarity matrix.
See also
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