R help - Dec 2009 - Distance between sets of points in transformed environmental space

Dear friends,

I have several sets of points in a transformed environmental space. Each set 
of points can be represented as a cloud in the environmental space.

This space is spanned by n coordinates, corresponding to the first n PCs of 36 
PCs of some environmental variables (12 monthly minimum temperatures, 12 
monthly maximum temperature, 12 monthly precipitations).

I would like to calculate a "distance" or dissimilarity between each
pair of
sets of points.

Let's label two of those sets as X,Y, where x is in X and y is in Y. We are 
interested in defining a distance between X and Y. I have thought of using the 
following:

1) The Euclidean distance between the centroids of X and Y. Simple and 
effective but does not give much real information on the actual degree of 
overlapping.
2) The median of the all the distances between all pairs of points (x,y). Same 
problem as (1), partially resolved.
3) The proportion of points of X U Y which fall outside the intersection of 
the convex or concave hulls (defined with a smoothing parameter) of X and Y, 
i.e. C(X) intersect C(Y). Very complicated, and does not necessarily lead to

What do you think? Are there any other approaches worth considering?  

Kind Regards
-- 
Corrado Topi

Global Climate Change & Biodiversity Indicators
Area 18,Department of Biology
University of York, York, YO10 5YW, UK
Phone: + 44 (0) 1904 328645, E-mail: ct529 at york.ac.uk

Well, here's another naive post from me (hopefully better than the last
one).

Firstly I'm not sure computing euclidean distance is that simple. I
would assume temperatures and precipitation would need to be
standardised in some way.

I think the notion of how far away something is, and how distinct
location wise something is, are quite different, so maybe two
measures?

For distance per se, I think your first idea is the best.
Plus simple is always good...

For distinctness, given one one of two sets, for each point, you could
just compute the closest point to it. If the closest point is a member
of the same set, we will call that a + point, if the closest point is
a member of the other set, we will call it a - point. In principle the
measure of distinctness would be the sum of the +'s, however there
might need to be some scaling to take into account the number of
points in each set.

There are also a lot of fancy things out there, so someone will
probably come up with a much fancier (and possibly better) idea than
this.

Well, that's just my rant, before I go to bed.


kind regards
-- 
Charlotte Maia
http://sites.google.com/site/maiagx/home

Hi Corrado,

I was thinking about this some more.

Maybe you could use a linear discriminate, i.e. a (hyper)plane that
partitions your points into two sets, such that the misclassification
rate is minimised.

Closeness could be regarded as the number of misclassified points.
Two sets would be distant, if no points are misclassified.

I am assuming there is a standard function in R to do this, no idea
what it is though. Plus this is a reasonably well known technique.

Again the size of the sets needs to be accounted for.
As well as the question, does the distance of set A from B, need to be
the same as the distance of set B from A. Both the nearest neighbour
approach and the discriminant approach, don't necessarily satisfy this
condition.

regards
-- 
Charlotte Maia
http://sites.google.com/site/maiagx/home