Prediction of Occurrence of Discrete Events - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 162

Suppose that X 0 is a row vector {1, X 1 , X 2 ,

, X p } with values of p explanatory
variables while Y is a binary variable that can only assume the values 1 or 0. The
explanatory variables can be binary like the variables used for WofE earlier in this
chapter but they can assume any other value on the real line. The event in a cell or at
a location is given by the following two equations:

...

1 X

e ʲ 0 þʲ 1 X 1 þ...þʲ p X p

PY ¼ ¼ ˀ ðÞ ¼ X

e ʲ 0 þʲ 1 X 1 þ...þʲ p X p

1

0 X

¼

PY ¼ 1 ˀ ðÞ X

This represents a non-linear model; the unknown coefficients then can be
estimated using the scoring method of maximum likelihood with Newton-Raphson
iteration (Agterberg 1992 ). In applications to presence or absence of deposits, after
reaching convergence, the sum of all estimates of the n probabilities ˀ j , which can

be written as S , should be equal to the total number of known mineral deposits, N .
Weighted Logistic Regression (WLR) is a variant of logistic regression. It was
originally developed for maps with mosaic patterns consisting of numerous small
polygons with yes-no data for different map layers. Polygons with the same
characteristic features belong to the same “unique condition”. The unique condi-
tions can be regarded as separate observations to be weighted according to the areas
they occupy on the map of the study area. Because the observations have different
weights according to the areas occupied by the unique conditions, WLR differs
slightly from ordinary logistic regression.
Logistic regression is used in many branches of science. Cox ( 1966 ) has pro-
vided a detailed account of the logistic qualitative response model, its multivariate
extension employing several explanatory variables, and its relation to discriminant
analysis. An elementary introduction to the method with illustrative examples is
provided by Hosmer and Lemeshow ( 1989 ). There is a close connection between
logistic regression and linear discriminant analysis ( cf . Agterberg 1974 ). Recently,
discriminant analysis was used by Grunsky et al. ( 2013 ) in a study of lake sediment
geochemistry of the Melville Peninsula. It provides a basis for distinguishing
between different map units which are assigned probabilities of occurrence on the
basis of the lake sediment geochemistry.
Chung ( 1978 ) published LOGIST, a computer program for logistic regression.
Pregibon ( 1981 ) used the approach to estimate frequencies, which are independent
binomial responses, thus expanding the method to deal with multiple qualitative
responses. Agterberg ( 1989c ) published LOGDIA, which is a generalization of
LOGIST in that frequencies of more than one discrete event could be estimated and
logistic regression diagnostics were provided. The further extension (LOGPOL
program) to make the method applicable to observations for polygons with different
areas is described in Agterberg ( 1992 ).