Date: Jan 18, 2013 12:10 PM
Author: Herman Rubin
Subject: Re: Mahalanobis_distance and Gaussian distribution
On 2013-01-17, David Jones <email@example.com> wrote:
> "Rich Ulrich" wrote in message
> On Thu, 17 Jan 2013 11:21:24 -0000, "David Jones"
> [snip, a bunch]
>>The Mahalanobis distances may be dimensionless with respect to the units of
>>the underlying observations but that does not men that they are immediately
>>comparable across different sources of data. Even of the number of
>>dimensions is the same you still need to look at context. For example, if
>>used in some formal testing procedure, the power of such tests can be
>>different. Consider two different set of observations on the underlying
>>quantity, one with rather more random observation error than the other.
>>For different dimensions, consider the case where the dimensions are much
>>more different, say 2 and 100. Then a typical value of Mahalanobis
>>for a point from the second population would be 100, but this would be very
>>unusual value for a point from the first population. In fact the sets of
>>values of distances for the two populations would hardly overlap. If this
>>meaningful for whatever way you intend to use the distances then OK. But
>>many uses are of the kind where you are looking for datapoints that are
>>unusual with respect to an initial distribution ... the Mahalanobis
>>is not (without some transformation) directly usable in a comparison
>>sets of data with different dimensions, as exemplified in the case above
>>where a value of 100 is unusual for one population but not the other.
> I'm asking myself -- to judge which is more of an outlier, Why
> can't we consider the "p-value" of each of these two
> chisquared distributions with different df's?
> I'm not saying that this is a good idea. -- I *suspect* that there
> is something shaky about it, or I might have heard of it being
> done before, and it doesn't seem familiar. Or, is that just
> because the circumstances are too rare in my reading?
Other than as a component of risk in evaluating a testing procedure or
similar, the p-value is a relic of the alchemical past before it was
realized that one must consider the risks and benefits in the various
states of nature,, and weight them according to the user's weights.
The Mahalanobis distance is a measure of the separation of distributions,
with a precise meaning if the distributions are normal, but so what?
It is what is going to be done with it which counts.
At least, a Bayesian approach to comparing distances, which would
(I believe) involve the appropriate non-central chi-squared
distributions, would be the way to compare if that were the problem.
> The answer here is that you can always compare the p-values of
> test-statistics to see how unusual the observed statistics are compared to
> their corresponding distributions. But that doesn't tell you anything about
> a comparison of the worth of those test statistics. For example, given the
> assumption of a normal distribution with unknown mean you can use either the
> usual t-statistic to test for a given mean, or one based the number of
> values on either side of that mean. The p-values for the test statistics
> would be valid and comparable in a sense. Yet you know that one test is more
> powerful than the other. (Here the question is not just comparing two
> statistics derived from the same data, but two statistics derived from
> different data, but this doesn't add much extra to the question.)
> Thus in the Mahalanobis distance case, there really needs to be some
> consideration of power. If one thinks of plotting "probability of
> rejection" against some representative quantification of departure from the
> mean, it seems that this is most naturally the covariance-matrix-weighted
> quadratic form based on the difference of the means of null and alternative.
> These x-axis variables are therefore different for two Mahalanobis distances
> which are derived from different basic variables or from variables with
> different measurement errors contributing to the covariance matrix and
> certainly from observation vectors of different lengths. Of course, one
> could start from a univariate measure of "size of departure" and map this
> through some model of how this affects the means of the "alternative" models
> in each case: then one would be able to compare the powers of the tests on
> the same scale, but this would clearly depend on how the changes in means
> are modelled for each space. Of course, if there really is just one
> underlying variable controlling the change being sought by a test, it would
> be possible to devise some optimal test statistic aligned to the given
> direction, which would argue for not using the Mahalanobis distance.
> It is not clear what the OP's context for asking actually was, as there are
> a number of possibilities even for a single population case:
> (i) comparison of Mahalanobis distances based on different subspaces
> (ii) comparison of p-values for Mahalanobis distances based on different
> (iii) choice of which of several Mahalanobis distances based on different
> subspaces to use for single test statistics
> (iv) possible benefits of combining subspaces to create a better test
> There is also the possibility that there are multiple populations, in which
> case there may be a clustering problem, and the apparent separation of
> populations may be important.
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
firstname.lastname@example.org Phone: (765)494-6054 FAX: (765)494-0558