Geoscience Reference
In-Depth Information
between established usages in dif erent parts of the world (Streckeisen 1974,
1976), discriminant analysis is based solely on mathematical constraints for
the classii cation of objects. Furthermore, discriminant analysis assumes
normality for the measured values within a class, which is probably not
a valid assumption for the Streckeisen classii cation. Since normality is
important for the success of the method, the user must be careful to ensure
that this condition is actually met, especially when analyzing compositional
(closed) data (see also Section 9.5).
Discriminant analysis was i rst introduced by Sir Ronald A. Fisher (1936)
to discriminate between two or more populations of l owering plant. h ei rst
step involves determining the discriminant function
Y
i
that best separate two
groups of objects described by the normally-distributed variables
X
i
h e parameters
a
i
are determined to maximize the distance between the
multivariate means of the individual groups. In other words, we determine
a
i
such that the ratio of the distances between the means of the groups to
the distances between group members is high. In the second step, having
determined the discriminant function from a training set of objects, new
objects can be assigned to one group or the other. Using the Streckeisen
diagram, a rock sample is assigned to an established rock type such as granite
on the basis of the percentages of Q, A, P and F.
As an example we i rst create a synthetic data set of granite rock samples,
described by two variables,
x
1
and
x
2
. h ese two variables could represent the
percentages of two chemical elements expressed as oxides (in weight percent).
Let us assume that we know from preliminary studies that these rock samples
come from three dif erent granites that were formed at dif erent times during
three separate magmatic events. Apart from natural inhomogeneities within
a granite intrusion, we can assume that the measured values from the granite
samples are normally distributed. In this example we will i rst determine
the discriminant functions separating the three groups (or types of granite).
We then use the discriminant functions to assign rock samples (which were
collected during a subsequent i eld campaign) to one of the three types of
granite.
We i rst clear the workspace.
clear
We then reset the random number generator.
rng(0)
