Geoscience Reference
In-Depth Information
Fig. 2.5
A sketch of a lognormal distribution
a normal CDF, as n increases towards infinity.
The corollary
of this is that the product of a great number of independent,
identically distributed RV′s tends to be normally distributed.
The theoretical justification of the normal distribution is of
little practical importance; however, we commonly observe
that the distribution of grades becomes more symmetric
and normal-like as the volume of investigation becomes
large—the randomness of the grades is averaged and the re-
sults tend to a normal distribution.
Fig. 2.4
A sketch of a normal or Gaussian distribution
Analytical Theory of Probabilities
(1812), and is now called
the Theorem of de Moivre-Laplace. Laplace used the normal
distribution in the analysis of errors of experiments. The im-
portant method of least squares optimization was introduced
by Legendre in 1806. Gauss, who claimed to have used the
method since 1794, justified it rigorously in 1809 by assum-
ing a normal distribution of the errors.
The Gaussian distribution is fully characterized by its two
parameters, the mean and the variance. The standard normal
PDF has a mean of zero and a standard deviation of one.
The CDF of the Gaussian distribution has no closed form
analytical expression, but the standard normal CDF is well
tabulated in literature. The Gaussian distribution has a char-
acteristic symmetric bell shaped curve about its mean; thus
the mean and median are the same, see Fig.
2.4
.
The lognormal distribution is important because of its
history in spatial statistics and geostatistics. Many earth
science variables are non-negative and positively skewed.
The lognormal distribution is a simple distribution that
can be used to model non-negative variables with positive
skewness. A positive random variable is said to be log-
normally distributed if
X
=
ln(Y)
is normally distributed
(Fig.
2.5
). There are many grade distributions that are ap-
proximately lognormal. These distributions are also charac-
terized by two parameters, a mean and a variance, although
three-parameter lognormal distributions have been used in
mining, see for example Sichel (
1952
). Lognormal distribu-
tions can be characterized by either their arithmetic or their
logarithmic parameters.
The Central Limit theorem (see for example Lapin
1983
)
states that
the sum of a great number of independent equally
distributed (not necessarily Gaussian) standardized random
variables (RV) tends to be normally distributed, i.e. if n RV's
Z
i
have the same CDF and zero means, the RV tends toward
2.2.2
Parametric and Non-parametric
Distributions
A parametric distribution model has an analytical expression
for either the PDF or the CDF, as for the Gaussian density
function and the Lognormal distribution. Parametric distribu-
tions sometimes relate to an underlying theory, as does the nor-
mal distribution to the Central Limit Theorem. There are many
parametric distributions that are used in different settings in-
cluding the lognormal, uniform, triangular, and exponential
distributions. Modern geostatistics makes extensive use of the
Gaussian distribution because of its mathematical tractability.
The lognormal distribution is important as well, but mostly
from an historical perspective. In general, however, modern
geostatistics is not overly concerned with other parametric
distributions because data from any distribution can be trans-
formed to any other distribution including the Gaussian one if
needed. Adopting a parametric distribution for the data values
may be the only option in presence of very sparse data; a non-
parametric distribution is used when there are sufficient data.
There is no general theory for earth science related vari-
ables that would predict the parametric form for probabil-
ity distributions. Nevertheless, certain distribution shapes
are commonly observed. There are statistical tests to judge
whether a set of data values follow a particular parametric
distribution. But these tests are of little value in resource es-
timation because they require that the data values all be inde-
pendent one from another, which is not the case in practice.
