Subject: What is the Mahalanobid distance? From: jmack@p3.net Date: 1997/08/27 Message-Id: <872714138.1214@dejanews.com> Newsgroups: sci.math,sci.image.processing,sci.physics [More Headers] Can some one provide me with and explain the formula for the Mahalanobois distance? How is this distance different from the Euclidean distance? I've seen that the Mahalanobois distance has been used in some image processing techniques, some have used it for image segmentation/max. liklihood method. Thanks John Mack jmack@p3.net Subject: Re: What is the Mahalanobid distance? From: ebohlman@netcom.com (Eric Bohlman) Date: 1997/09/03 Message-Id: Newsgroups: sci.math,sci.image.processing,sci.physics [More Headers] In matrix notation, the Euclidean distance between vectors x and y is (x-y)'I(x-y) where ' denotes transpose and I is the identity matrix (ones on the diagonal, zeroes elswhere. The Mahalanobis distance is (x-y)'S^-1(x-y) where S is the covariance matrix of the set from which the vectors are drawn and ^-1 denotes inverse. Euclidean distance weights each component of the vectors equally. Mahalanobis distance gives lower weights to components that are strongly correlated with each other. It also standardizes the components so that differences in scale between the components don't affect the distances. Another measure is standardized Euclidean distance, (x-y)'V^-1(x-y), where V is the matrix whose diagonal entries are the variances of the components and non-diagonal entries are zero (i.e. S with the non-diagonal entries zeroed). This standardizes the components (it's the same thing as dividing each component by its standard deviation and then taking the Euclidean distance) but does not penalize for correlation. _________________________________________________________________ _________________________________________________________________ 970.05960 Lawrence, K.Mark; Mahalanabis, Arijit; Mullen, Gary L.; Schmid, Wolfgang Ch. Construction of digital \$(t, m, s)\$-nets from linear codes. (English) [CA] Cohen, S. (ed.) et al., Finite fields and applications. Proceedings of the 3rd international conference, Glasgow, UK, July 11--14, 1995. Cambridge: Cambridge University Press, Lond. Math. Soc. Lect. Note Ser. 233, 189-208 (1996). [ISSN 0076-0552] [ISBN 0-521-56736-X] _________________________________________________________________ 838.65004 Mullen, Gary L.; Mahalanabis, Arijit; Niederreiter, Harald Tables of \$(t,m,s)\$-net and \$(t,s)\$-sequence parameters. (English) [CA] Niederreiter, Harald (ed.) et al., Monte Carlo and quasi-Monte Carlo methods in scientific computing. Proceedings of a conference at the University of Nevada, Las Vegas, Nevada, USA, June 23-25, 1994. Berlin: Springer-Verlag, Lect. Notes Stat., Springer-Verlag. 106, 58-86 (1995). [ISBN 0-387-94577-6/pbk] _________________________________________________________________ 850.93873 Williams, Ronald T.; Prasad, Surendra; Mahalanabis, A.K.; Sibul, Leon H. An improved spatial smoothing technique for bearing estimation in a multipath environment. (English) [J] IEEE Trans. Acoust. Speech Signal Process. 36, No.4, 425-432 (1988). [ISSN 0096-3518] Classification: *93E14 Data smoothing (stochastic systems)