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Hodgeman County Study, part 6 of 12


Allocation by Extension

The use that regionalized classification makes of discriminant analysis is to a certain degree analogous to the use of crossvalidation in kriging and different from the classical use of discriminant analysis. Classical use of discriminant analysis employs the training set for calibration and then proceeds to classify vectorial measurements without assigments.

In regionalized classification the interest is in the calculation of the probabilities p(i)(z) for the same realizations in the training set already classified by the prior cluster analysis. Once one has all the probabilities p(i)(z), reallocation of the sites to the group with the highest probability is a trivial endeavor. This reallocation offers an opportunity to check results and compare methods in case the user wants to consider more than one type of cluster analysis or discriminant analysis. Results from cluster and discriminant analysis should be comparable only with minor variations that one should employ to select methods and parameters yielding the most consistent results.

The final step in regionalized classification is the mapping of groups, which one can accomplish by arbitrarily assigning colors or black and white patterns to the groups. As arbitrary as the color or pattern selection may be, it always helps to select a combination of alternatives that maximizes contrast to the eye, which customarily requires some experimentation by trial and error.

Most mapping procedures require a collection of values regularly spaced at short intervals, which is rarely the case of training sets. In addition, interpolation presumes that the variable is continuous. In those circumstances Harff and Davis (1990) recommend mapping the group probabilities p(i)(z) by treating them as regionalized variables and then use the allocation rule to produce the discontinuous group map. Remember that z is a shorthand for z(x), where x is the location of the site. Then the probabilities are actually regionalized variables p(z(x)) or p(x) that depend on location. One does group allocation node by node performing a grid-to-grid operation in which one assigns each node to the group with the highest probability.

Considering that each site must be fully sampled to allow for the cluster and discriminant analysis, there is no advantage on using cokriging for the estimations. Although the probabilities sum to a constant, use of alr transformation is not feasible due to the numerous zeros both in the numerator resulting in a null argument for the logarithm, or in the denominator producing unacceptable ratios.

Ordinary or universal kriging, the default geostatistical options of choice, suffer from the inability to restrict the estimates to an interval—0-1 in this instance—let alone to force the probabilities to sum to one. Estimates outside the 0-1 interval, however, are rare and never far away from the interval. A common solution to force a vector in a coregionalization to add to one is to rescale the values. Considering that such correction does not change the ranking of the membership probabilities, the allocation is insensitive to the rescaling.

Alternative strategies involving the interpolation of the coregionalization itself or of the generalized distances instead of the membership probabilities are deceptive choices. Although the probabilities end up honoring all constraints, the use of estimates instead of true values results in unaccounted propagation of errors.

Allocation in regionalized classification remains open to improvements.

Algorithm 3

This is a procedure for the regionalized classification of fully sampled coregionalizations involving p attributes.
  1. Assign the sites to one and only one of g groups either by using Algorithm 1 or any other cluster analysis procedure, or on the basis of external information
  2. For each site calculate the group probabilities either by Algorithm 2 or any other discriminant analysis procedure deemed appropriate.
  3. If the mapping procedure does not require a regular grid of values, go to step 4. Otherwise, use some form of kriging to produce grids of estimated values for every group probability.
  4. For every site or node, assign the site or node to the group with the largest probability.
  5. Prepare a map showing the group assignment. This is the regionalized classification of the area under study, conditional to the values and attributes in the sampling.
  6. Prepare a map of the highest probability per node. This is the probability that the site in the regionalized classification indeed belongs to the group in the regionalized classification.
  7. Prepare a censored map eliminating nodes or sites likely to be misclassified, which are more likely to occur when the highest probability is only marginally larger than the second highest probability. Previous Page--Discriminant Analysis || Next Page--The Dakota Aquifer Program Study
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    Updated Sept. 16, 1996.
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