Dakota Aquifer Program--Geologic Framework
Hodgeman County Study, part 5 of 12
In practice the answer to the question of how to group vectorial
observations is never straightforward.
First there is the problem of deciding how many groups, which to a
large extent one can settle in terms of external information or rules
based on the total sum of errors that one can obtain running cluster
A second and more difficult problem is the group assignment for
observations that are not like any of the typical observations in
a group and at the same time poorly resemble more than one group.
Cluster analysis assignment for such problematic observations is
unstable and varies depending on the measure of similarity and the
If the quality of group assignments is a concern, cluster analysis
does not offer easy answers. That is the realm of discriminant
analysis, which decides the assignment of any vector
based on allocation probabilities.
Discriminant analysis, however, requires a training classification,
which in regionalized classification is the one generally produced
by cluster analysis.
The basis of discriminant analysis for assigning observations to
one of a given number of groups is the minimization of the total
misallocation cost. The procedure assumes that it is possible to
partition the attribute space in as many mutually exclusive and
exhaustive regions as there
are groups. A site is said to belong to group
when the vector
of attributes associated with
the site falls in .
Let be a vectorial random function
with probability density function
and let be a realization of
for short. Let
be the proportional share of
observations in the group
whose probability density function is
. Then the probability
that the site characterized
by belongs to the
The proof directly follows from Bayes' Theorem. If one considers
that can be regarded as the a
priori probability of sampling group ,
then is the posterior
probability that belongs to group
The optimality in terms of the misallocation cost is assured by
assigning to the group with the
highest probability (McLachlan, 1992, p. 7).
The kernel of discriminant analysis is the calculation of the
probabilities in Theorem 1. As is always the case in statistics,
there are non-parametric and parametric methods. By far,
multivariate normal methods prevail among the latter and overall.
Let be a realization of a
vectorial random function at site
for short. Let be the vectorial
mean and the covariance matrix
of group . Then the Mahalanobis
distance from to the centroid of
group is the squared weighted distance
Let be a normal vectorial random
function with heteroscedastic normal group distributions. If
is the a priori probability
f group and
Mahalanobis distance in Definition 3, then the probability
that the site characterized
by belongs to the
From Theorem 1
If the probability distribution for the group is normal,
where is the dimension of the attribute space. Considering that in addition (McLachlan, 1992, p. 5), then
Cancellation of the terms proves the theorem.
Let be a normal vectorial
random function with homoscedastic normal group distributions.
If is the a priori probability
of group and
Mahalanobis distance in Definition 3 computed using the common
covariance matrix , then
the probability that the
site characterized by belongs to
the group is
From Theorem 2, considering that all covariance matrices are the same
Proof follows by the cancellation of the covariance matrix
determinant term and consolidation of exponents after the trivial
Use of the model in Theorem 3 results in hyperplanes for the
boundaries of the group regions ,
thus the term of linear discriminant analysis, while quadratic
discriminant analysis deals with different covariance matrices
and second order surfaces arising from the heteroscedastic model.
Whatever the model of discriminant analysis used for regionalized
classification, the model requires a training set to determine
the number of groups and for the assignment of realizations to
the different groups in order to have some data for estimating the
centroid and the covariance matrix of such groups.
This is a procedure for the calculation of group probabilities
for vectorial measurements. The method employs a normal model of
- Run Algorithm 1 using the whole sampling of the coregionalization.
- Break the sampling into groups based on the total sum of errors, some external criteria, or both.
- Use the vectorial measurements to estimate all group centroids, covariance matrices, and make a decision about the a priori group probabilities. The alternatives are:
- If the sampling properly represents the true group sizes, the best estimate of the a priori probabilities are the relative proportions of the sites per group
- If the proportions of sites per group are regarded as not being indicative of the a priori group probabilities and the user believes that he or she has some better external information to assess the probabilities, the user may employ such external assessment.
- If the proportions of sites per group are regarded as not being indicative of the a priori group probabilities and the user lacks ways to assess them, they can be made equal to , in which case they are ignored in the calculations.
- Compare the group covariance matrices and decide whether they are sufficiently similar to assume homoscedasticity.
- Calculate the Mahalanobis distance in Definition 3 for measurement . Use the average of all group covariances if the assumption is that the covariance is homoscedastic.
- Compute for measurement the probability of belonging to each of the groups
The choice of must be
consistent with the decision of homoscedasticity made in step 4. The
term is the
Mahalanobis distance in Definition 3; and
is the estimate of the covariance
matrix for group ;
- , the discriminant analysis is linear with unknown or equal a priori group probabilities;
- , the discriminant analysis is linear with known and different a priori group probabilities;
- , the discriminant analysis is quadratic with unknown or equal a priori group probabilities;
- , the discriminant analysis is quadratic with known and different a priori group probabilities.
- Go back to step 7 until assigning group probabilities for all
sites in the training set.
Major statistical computer packages such as IMSL(1987) and SAS (1990) have implementations of Algorithm 2 among several other procedures.
Statisticians concur that quadratic discriminant analysis indeed
provides superior results if the group covariances are considerably
different and the group sizes are large. Quadratic discriminant
analysis, however, is more sensitive to deviations from
multinormality and assignment errors in the training set
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Start of Hodgeman County Study
Kansas Geological Survey, Dakota Aquifer Program
Updated Sept. 16, 1996.
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