# Re: [MORPHMET] Mahalanobis distance in cluster analysis of shape variables

```I can not speak directly to why it is frequently used in GM cluster analysis
but I would like to mention how I look at Mahalanobis distance based on its
calculation.```
```
Mahalanobis distance is not a pure distance metric like Euclidian or Manhattan
distance, as you have stated it is ‘standardized’.  What doe that really mean?
It sounds supeficially good.

One way of computing it is to rotate the k-landmark data set to simplest form
treating the landmarks as factors.  This way would consider all landmarks to
have a common covariance structure in XY or XYZ in three dimensions.  That is a
already a streetch, since not all landmarks can be assumed to have the same
about their centroid and rotated to coincide, which has eliminated a dgeree of
freedom of variability that can have consequences.

Furthermore not all species landmarks can be expected to have the same
covariance structure, which is an assumption made in the ordinary Mahalanobis
distance application to strut analysis between populations or species.  The
assumption of similar data structure of course applies to the null hypothesis
where there is no difference.  The typical statistical test explodes when the
null hypothesis is falsified so just when you want the Mahalanobis distance
metric to be accurate it starts misbehaving.

After rotation to simplest axes one does an 1 df F-test between each of the
landmarks.  These tests are all independent so they can be summed together to
produce a k df F-test which is Mahalonobis D squared.    So Mahalonobis D is
the square root of the sum of independent F-tests, but those F-tests are based
on all sorts of assumptions about the variance of the landmarks.  I immagine on
could modify calculation of D by limiting the sum over the top 95 or 99%
variance components of the principal components.

Many times applications of analytical techniques are judged by whether they
‘work’ or not.   If a clustering method works for you, use it(?).  I am of the
opinion that I use statistics to convince myself rather than the audience.   A
confluence on many arguments is used to make a case.

Joe

-·.  .· ·.  .><((((º>·.  .· ·.  .><((((º>·.  .· ·.  .><((((º> .··.· >=-
=º}}}}}><
Joseph G. Kunkel, Research Professor
UNE Biddeford ME 04005
http://www.bio.umass.edu/biology/kunkel/

> On Jan 30, 2016, at 7:11 AM, Elahep <ellie.parv...@gmail.com> wrote:
>
>
> Hello all,
>
>
>
> I have seen in many GM articles people use Mahalanobis distance for cluster
> analysis. What is the advantage of using Mahalanobis distance over Euclidian
> distance as similarity measure in cluster analysis of shape variables?
>
> As far as I know Mahalanobis distance is the standardized form of Euclidean
> between variables and weights all variables equally.
>
> Why this distance measure is frequently used in GM cluster analysis??
>
>
>
>
> Elahe
>
>
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