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Home Multivariate Data Basic Knowledge Distance and Similarity Measures  
See also: Cluster Analysis, Distance Matrix  
Distance and Similarity MeasuresDistances between objects in multidimensional space form the basis for many multivariate methods of data analysis. Using a different method for calculating the distances may influence the results of a method considerably. Similarity of objects and distances between them are closely related and are often confused. While the term distance is used more precisely in a mathematical sense, the particular meaning of the term similarity often depends on the circumstances and its field of application. In general, the distance d_{ij }between any two points in ndimensional space may be calculated by the equation given by Minkowski: with k being the index of the coordinates, and p determining the type of distance. There are three special cases of the Minkowski distance:
The Mahalanobis distance is related to the Euclidean distance, and results in the same values for uncorrelated and standardized data. It can easily be calculated by including the inverse covariance matrix into the distance computations: Another distance measure, which is rather a measure for the similarity between two objects, has been proposed by Jaccard (it is also called Tanimoto coefficient): , with (x.y) being the inner product of the two vectors x, and y. Note that the Jaccard coefficient equals 1.0 for objects with zero distance. Furthermore, the Tanimoto coefficient can be appied to binary data, as well: T = N_{xy} / (N_{x} + N_{y}  N_{xy})
with N_{x}, N_{y}.... number of 1bits in the vectors x and y, and


Home Multivariate Data Basic Knowledge Distance and Similarity Measures 