Many thanks to the following for answers to the posted spatial
questions.

Further contributions are still welcome !


Eric Grist

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Eric--

I'm not sure, but I think random field theory is what you are looking
for.

The following will keep you up at nights.

Adler, Robert J. (1981) The geometry of random field. New York : J.
Wiley,
c1981.

Good luck,
Wil

-----William Irwin-------------------Laboratory for Affective
Neuroscience-----
Department of Psychology                       lab:
608-262-4443
University  of Wisconsin                       fax:
608-265-2875
1202 West Johnson Street                  http: psyphz.psych.wisc.edu
Madison, Wisconsin 53706                  [log in to unmask]

----------------------------------------------------------------------

It seems that your first 2 questions might both be approached via an
extended form of kriging. Originally this was developed for the case of
just a single realisation of spatially discrete observations with the
semi-variogram being estimated by taking the average of squared
differences of observations for those pairs of observations whose
distance apart in space is within some reasonable interval. This leads
to a plot of the semi-variogram against distance which characterises
spatial dependence. (This is related to a spatial correlation function.)
In your case this could be extended to construct semi-variograms based
on pairs a given distance apart in space and 0 time-steps apart in time
(combining across all time frames), then 1 time-step apart, 2 time-steps
apart, etc.. Various types of plots might be tried of these functions,
but you could start with superimposed variograms for lags of 0,1,2, ....
time-steps. The advantage of the above approach is that it would not
rely on having a fixed configuration of sampling points, since this is
handled by the averaging within spatial-distance cells. However, if the
problem is closer to being one of a fixed configuration with many
missing observations, you might prefer to modify an approach based
directly on estimating correlations. If these are unfamiliar to you,
Kriging and Semi-variograms have made it into a few statistical texts. I
believe they are covered on Cressie's book, whose tiltle I can't
presently remember.

David Jones, Institute of Hydrology.
--------------------------------------------------------------------

Eric
The only thing which comes to mind (and it might be completely off-beam)
is Gower's Procrustes Analysis (and Generalised Procrustes Analysis for
comparing more than two configurations of points). I've not looked at
any of the original literature (IIRC there's a reference to Gower 1975
in the Genstat manual; Krzanowski's Principals of Multivariate Analysis,
OUP, 1988 may also cover it), just used the procedures available in
Genstat for it.
It uses rigid rotations to match multi-dimensional configurations of
points (blast - they've got to be *the same* points, just described by
different coordinate systems. We used to use it to match scores for sets
of food samples on a variety of different attributes given by different
food tasters - the idea was that the tasters may use the individual
terms differently - may in fact use different terms entirely - but the
relative configuration of the samples should be the same - sample 1, 1A
and 1B should be closer than samples 2, 3 and 4, for example) Maybe
there's something of use in that ramble...

Duncan

Duncan Hedderley
Statistics Research & Consulting Centre
Massey University/Te Kunenga ki Purehuroa
Private Bag 11-222
Palmerston North
New Zealand/Aotearoa
------------------------------------------------------------------

Eric,
I'm not sure I'm in a position to do you a lot of good, but I'm sending
some of this on to a person who (may) do this for a living.  A
statistician, yet!

My quesiton is, what are you going to deduce/determine from the
measurements?  Such as a 2-D profile of Temperature, for ex.?

RA Fisher basically asked himself (in 1925), if I make a measurement,
what will I learn?  Then he follwed a specific dataset through to the
conclusion (or worked backwards), and discovered that orthogonal arrays
gave him nice results.  Voilla!  DoE!

If I am to develop a general area function for Temp, or whaever, I'd
guess that I would ideally want orthogonal measurements.  Failing that
ideal, I'd guess that a real set of measurmeents, with their own factor
configuration, would result in some kind of partial confounding.  But,
in your case, this may not be a significnat issue, as you are not
interested in adjusting the Temp., but in describing it.  Confounding
would only get significnat when you wish to estimate Temp at locations
away form the sampling location.  The further away, of course, the worse
off you are.

Now I'm rambling.  So what are you trying to accomplish/predict from
your measurments?

Jay
--
Jay Warner
Principal Scientist
Warner Consulting, Inc.
4444 North Green Bay Road
Racine, WI 53404-1216
USA

Ph: (414) 634-9100
FAX: (414) 681-1133
email:  [log in to unmask]
web: http://www.a2q.com

Power to the data!

--------------------------------------------------------------------
The original questions were:


(1) Suppose I have a FLAT finite 2 dimensional region and take S samples
from different (point) sites at t different times for t=1,2,3..n

BOTH the location AND number of samples taken at each time step t
will in general be different (ie change in time).

What statistic(s) would best give a measure of how the spatial
arrangment of the sampling regime changes at each time step in:-

(a) a relative sense (the points may be translated)
(b) an absolute sense (relative to some fixed reference frame)?

For example, the samples might be the temperatures of a tank of fluid
at different surface locations respectively (a) not taking into account
the sampling locations relative to the tank bottom, (b)taking into
account sampling locations relative to the tank bottom.

(2) Question (1) again but now with the samples taken 3 dimensionally
ie not confined to the fluid surface.

(3) Suppose I have a continuous finite 2 dimensional surface within
fixed boundaries which changes in shape at times t=1,2,3,..n (for
example the membrane of a drum). What statistic(s) would be
be used to measure how much the overall shape changes at each time step?




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