Geoscience Reference
In-Depth Information
variable could be the grade at an unsampled location denoted
Z
(
u
) where
u
represents a location coordinates vector. A set
of random variables is called a random function (RF). The
set of grades over a stationary geologic population
A
is a
random function {
Z
(
u
),
u
∈
A
}.
2.2
Probability Distributions
Probabilities are closely associated to proportions. A prob-
ability of 0.8 or 80 % assigned to an event means that the
proportion of times it will occur, in similar circumstances,
is 0.8 or 8/10 or 80 %. The
similar circumstances
relates to
our decision of stationarity. In some cases we calculate the
probabilities directly through proportions. For example, the
probability for a mineral grade within a particular geologic
unit to be less than a particular threshold could be calculated
by counting the number of samples below the threshold and
dividing by the total number of data.
There are many cases, however, when probabilities can-
not be calculated from proportions. This is particularly true
for conditional probabilities, that is, probability values given
certain a set of data events. Consider the probability that a
mineral grade be less than a particular threshold given one
measurement 50 m away that is twice the threshold and an-
other measurement 75 m away that is just below the thresh-
old. In such cases, we do not have multiple replications to
calculate an experimental proportion. We must rely on prob-
abilistic models and well established probability laws.
Probability distributions are characterized as parametric
or non-parametric. A parametric distribution model has a
closed analytical expression for the probability, and is com-
pletely determined by a finite number of parameters, as for
example the Gaussian distribution model with parameters
mean (m) and standard deviation (s) that control the center
and spread of the distribution, respectively.
It is common to consider probability distributions that
relate to one continuous or categorical variable at a time.
Such distributions are called univariate distributions. Two
examples: (1) the probability for a continuous variable to be
less than a particular threshold, or (2) the probability for a
particular lithology to prevail at a certain location. When we
consider probability distributions of more than one variable
at a time, then we call them multivariate distributions. The
distribution of two variables is a bivariate distribution. For
example, the probability of one grade being less than one
threshold and a second grade being less than another thresh-
old is a bivariate probability.
There are a large number of references for probability
and basic statistics. Some general statistical ones and also
some related to spatial data include Borradaile (
2003
); Davis
(
1986
); Koch and Link (
1986
); Ripley (
1987
); and Rohatgi
and Ehsanes Saleh (
2000
).
Fig. 2.1
Cumulative distribution of 2,993 data values. The cumulative
frequency or probability is the probability to be less than the threshold
value
2.2.1
Univariate Distributions
The cumulative distribution function (CDF) is the universal
way to express a state of incomplete knowledge for a con-
tinuous variable. Consider an RV denoted by
Z
. The CDF
F(z)
is defined as:
F z
( )
= ≤∈
Prob Z
{
z
}
[ 0,1]
The lowercase
z
denotes a threshold.
Prob
{
·
} denotes a
probability or proportion. An example CDF is shown on
Fig.
2.1
; the z-variable is between 2 and 35 and is most prob-
ably between 20 and 30.
A cumulative histogram is an experimental CDF based on
the data. It is useful to see all of the data values on one plot
and sometimes can be used to isolate statistical populations.
Cumulative histograms do not depend on a bin width, and
can be created at the resolution of the data.
An important challenge is to determine how representa-
tive each sample is of the actual mineralization. This issue
is discussed in more detail in Chap. 5. It is also important to
determine whether the distribution of all samples adequate-
ly represents the actual grade distribution in the deposit, or
whether certain weighting should be applied.
The interval probability of
Z
occurring in an interval from
a
to
b
(where
b
>
a
) is the difference in the CDF values eval-
uated at values
b
and
a
:
Prob Z
{
∈=
[ , ]}
a b
F b
( )
−
F a
( )
The probability density function (PDF) is the derivative of
the CDF, if it is differentiable. Applying the fundamental
theorem of calculus, the CDF can be obtained by integrating
the PDF:
