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of measurements should be used instead original values if the variable of interest is
part of a closed number system. They advocate use of logits even when histograms
for individual variables are constructed or when standard deviations are being
calculated. Suppose such measurements are written as P i ( i
, n ). For
trace elements the logit transformation log e { P i /(1- P i )} is equivalent to a logarith-
mic transformation log e P i because P i then is very small. It is well-known that logits
are nearly equal to probits ( cf . Sect. 2.3.1 ) except for the smallest and largest values
in a sample. The relationship between logits and probits is approximately linear for
the range 0.05
¼
1, 2,
...
P i 0.95 as shown in Fig. 12.8 . In Sect. 12.8 , measurements will
be replaced by their probits (instead of logits) because this transformation is helpful
in 2-D cell composition modeling, especially for extrapolation of statistics from
smaller to larger cells and vice versa.
12.3 Non-linear Process Modeling
A question to which new answers are being sought is: Where does the randomness in
Nature come from? Nonlinear process modeling is providing new clues to answers.
Deterministic models can produce chaotic results. A famous early example was
described in Poincar ´ 's ( 1899 ) study of the three-body problem in mechanics. This
author observed that in some systems it may happen that small differences in the
initial conditions produce very great ones in the final phenomena, and “
prediction
becomes impossible.” Nevertheless, the randomness created by non-linear processes
obeys its own specific laws. Many scientists including Turcotte ( 1997 ) have pointed
out that, in chaos theory, otherwise deterministic Earth process models can contain
terms that generate purely random responses. Examples include the logistic equation
and the van der Pol equation with solutions that contain unstable fixed points or
bifurcations. Multifractals, which are spatially intertwined fractals and were antic-
ipated by Hans de Wijs (in 1948), provide a novel way of approach to problem-
solving in situations where the attributes display strongly positively skewed fre-
quency distributions. The standard geostatistical model used in ordinary kriging
assumes a semivariogram with both range and nugget effect. The range extends to
distances at which results from other deterministic or random processes begin to
overshadow local variability. The nugget effect often is the result of relatively wide
sampling between points that hides short-distance variability. The multifractal
semivariogram shows sharp decrease toward zero near the origin. Local singularity
mapping uses this short-distance variability to delineate places with relatively strong
enrichment or depletion of element concentration on geochemical maps and in other
applications (Cheng and Agterberg 2009 ). This method provides a new approach for
mineral exploration and regional environmental assessment ( cf . Sect. 11.5 ). These
nonlinear developments are closely related to statistical theory of extreme events
(Beirlant et al. 2005 ).
...
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