- View SourceDear all out there,

I have got 2 (maybe very basic) questions, the first one about the

requirement of stationary mean in semivariance analysis, the 2. about the

correlation dimension of fractal analysis.

1.

before performing a semivariance analysis, any secular trend must be

removed from the data in order to meet the requirement of stationarity of

the mean. Question: how do you decide at which scale fluctuations are

actually trends ? Stationarity of the empirical data means regional mean =

ยต(x) = const. over x, but over which range of x the regional mean has to

be taken ?

As an example, I add (see zipped attachment <examp1.doc>) the spatial

distribution of count rates over a monazite (a radioactive mineral)

containing beach of Brazil. The dots represent the sampling locations. The

picture has been produced using 'naive' kriging with Surfer Software, i.e.

using the default settings for the variogram and assuming isotropy, just

in order to grossly visualize the distribution. I would say that there is

an obvious trend, represented by the maximum between easting 300 and 450

m; but is the maximum at ca. 160 m also part of the trend ? Doesn't the

big maximum in fact consist of 3 maxima at ca. 340, 370 and 410 m,

respectively, which should be modelled by the trend surface ? (Apart from

the problem, how to model such a trend structure.)

2.

It seems to me that the correlation dimension is quite a useful tool to

assess the topologic structure of the spatial distribution of a variable;

or could be if used properly. I use this kind of fractal dimension because

it is (as I think) the easiest to calculate: D :<=> AM(n(r)) ~ r^D, where

the left hand side denotes the number of points within distance r from a

fixed point x, averaged over all points x. D is then easily calculated by

log regression.

Now the question: there is always an 'edge effect' to D produced by the

fact that the sampling area is inevitably limited in space. An infinite

complete regular quadratic sampling grid, e.g., has D = 2, but the same

grid with finite extension has D < 2, because the points at the border

have naturally less neighbours than points within the grid and therefore

contribute to AM(n(r)) by lowering D and thus inferring a fractal

patchiness of the structure which is clearly an artefact. For this reason,

D depends heavily on the overall size (extent) of the grid (number of

sampling points), regardless of its structure, which makes this quantity

somewhat questionable, I think. Does somebody know how to deal with this

problem ?

Thank you very much & regards, PB

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Peter Bossew

Georg Sigl-Gasse 13/11

A-1090 Vienna, Austria

ph. +43-1-3177627

e-mail: p.bossew@...