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:   <ebohlmanEFxB5s.AEu@netcom.com>
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.
   _________________________________________________________________
   
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   970.05960
   Lawrence, K.Mark; Mahalanabis, Arijit; Mullen, Gary L.; Schmid,
   Wolfgang Ch.
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   [CA] Cohen, S. (ed.) et al., Finite fields and applications.
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   Mullen, Gary L.; Mahalanabis, Arijit; Niederreiter, Harald
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   [CA] Niederreiter, Harald (ed.) et al., Monte Carlo and quasi-Monte
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   Williams, Ronald T.; Prasad, Surendra; Mahalanabis, A.K.; Sibul, Leon
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   [J] IEEE Trans. Acoust. Speech Signal Process. 36, No.4, 425-432
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   Classification: 
          *93E14 Data smoothing (stochastic systems)