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
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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
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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]
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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)