Re: modifying mahalanobis distance
- From: "Ray Koopman" <koopman@xxxxxx>
- Date: 22 Oct 2006 23:03:24 -0700
shaobo hou wrote:
Well, what I am trying to achieve is that in addition to computing the Mahalanobis distance of a data point to a Gaussian, the data point also has some uncertainty (also Gaussian) associated with it, as represented by the matrix C. The greater the uncertainty about the data point, the greater the distance should be.
Another idea I have considered is to compute the integral of the product of the Gaussian and the uncertainty about the data point, but this involves matrix inversion, which will make it too slow for my uses.
Does this describe your situation? You have a population of objects
whose locations in some p-dimensional space have a multivariate normal
distribution with mean vector m and covariance matrix K. The measured
location of any object contains error. That is, every measurement y
can be written as x + e, where x is the true location of an object,
and e is the measurement error. For each measurement, the sampling
distribution of the error is p-variate normal with mean vector 0
and covariance matrix C.
If that's what you have then the sampling distribution of y for a
randomly chosen object is p-variate normal with mean vector m and
covariance matrix K+C, and your post suggests that you want to talk
about (y-m)'K^{-1}(y-m) = (x-m)'K^{-1}(x-m) + 2(x-m)'K^{-1}e +
e'K^{-1}e. The first term on the right is the Mahalanobis distance
of x from m. The expected value of the second term is 0. The third
term is always positive, with expected value trace(CK^{-1}).
.
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