Acta Montanistica Slovaca (Dec 2001)
The indicators related to a regionalized variable
Abstract
In many areas, one has to deal not only with quantitative variables but also very often with categorical variables. Then a common problem consists of estimating the probability for each category to prevail at any particular location. In the seventies, in order to make prediction for future selective mining, geostatistics had to deal with the problem of estimating reserves above the cut-off grades. In 1982 at the 17th APCOM symposium in Colorado, Andre Journel presented the earliest concepts of Indicator approach to estimation of spatial distribution. Using an indicator approach these probabilities can by established. Since then practitioners have used techniques based on kriging of indicators. The indicators above a threshold define the ore at this cut-off or, in a non-mining context, define the geometric set of values above the cut-off. In the last decade of the past century the interest in geometric problems has increased, in particular for simulation. For this reason, the development of techniques for simulating random function Z(x) from indicators I[Z(x)<z] at different thresholds z was started. The concept of indicator transforms is one of simplest and (possibly) most elegant in modern geostatistics. In many such cases several thresholds and indicators corresponding to different random sets related to the variable under study. These sets depend on each other and their mutual arrangement is an important structural characteristic of the variable. The indicator approach is as follows. Select discriminator cut-off value, which should not be confused with an economic cut-off or some critical level of toxicity. All samples with measurements above this value are coded as 1. All samples below selected cut-off value are coded as 0. This new measurement is the indicator yes/no, presence/absence, etc. Each pair of samples that goes into the experimental semi-variogram will be {0,0} if both samples are bellow the cut-off value, giving a difference of zero or {1,1} if both samples are above the cut-off, also giving a difference of zero or finally {1,0} or {0,1} if one sample is above and the other is below. The calculated graph will be the average of these differences and represents the predictability of being above or below cut-off. Resulted kriged map of indicator values is interpreted as the probability that unknown value is above the cut-off value. The paper deals with theory of mathematical background and application of indicator approach to estimation of probability unknown values above selected cut-off.
