Prediction of Occurrence of Discrete Events - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 163

Box 5.4: Newton-Raphson Iteration

The logistic regression model can be written in the form: Logit
ˀ ( X ) ¼ ʲ 0 + ʲ 1 X 1 + ... + ʲ p X p . Suppose that Y with elements Y j is a column
vector consisting of n ones and zeros denoting presence or absence in n very
small unit cells (e.g. n 10 6 ) in a study area with N unique conditions.

Suppose further that the weights w i with

¼

∑ w i ¼ n represent numbers of unit

cells for the i -th unique condition. The ( N N ) diagonal matric V with

X Y j 1

where

Y j

Y j

non-zero elements V ii ¼ w i : is an estimated

value of ˀ j ( X ). If the maximum likelihood method is used for estimation, a

column vector of scores S for differences between observed and estimated
values of Y is made to converge until X 0 S ¼ 0. Newton-Raphson iteration

( t )+{X T V( t )X)} 1 X T V( t ){X results in successive estimates: ( t +1) ¼ β ( t )

β β

+ V 1 ( t )S( t )}, t At the beginning of the process an arbitrary vector

of coefficients (e.g., with all coefficients set equal to 0) is used. After conver-
gence, the estimated logits are converted into probabilities. The LOGPOL
program, which is based on Newton-Raphson iteration with weights w i , was
incorporated in the Spatial Data Modeller (SDM, Sawatzky et al. 2009 ).

¼ 1, 2, ...

5.2.1 Meguma Terrane Gold Deposits Example

Weights of evidence (WofE) modeling and weighted logistic regression (WLR) are
different types of application of the loglinear model ( cf . Agterberg 1992 ). In WLR,
the patterns are not necessarily conditionally independent as in WofE. WLR can
also be used in situations where the explanatory variables have many classes or are
continuous. In the Meguma Terrane gold deposits example, the map patterns that
were selected are approximately conditionally independent. The conditional inde-
pendence (CI) hypothesis can be tested in various ways. It was already shown by
means of the Kolmogorov-Smirnov test (Fig. 5.10 ) that CI is approximately satis-
fied for the seven map layers used for Fig. 5.9 . One simple way for testing CI is to
compare the sum S of all posterior probabilities with the total number of deposits
( N ). For Fig. 5.9 , the total number of deposits predicted by the posterior probabil-
ities is S

68 representing the actual number of gold deposits.
If all patterns would have been conditionally independent, their predicted total
would have been 68 as well. Minor violations of the CI hypothesis account for
WofE overestimating the total number of gold occurrences (

¼ 75.2 exceeding N ¼

68) by about 10 %.
Bonham-Carter ( 1994 ) introduced the so-called omnibus test stating that CI is
approximately satisfied if over-estimation is less than 15 %. The hypothesis S

¼

N
can be tested statistically (Agterberg and Cheng 2002 ). For a systematic compar-
ison of various CI testing methods, see Thiart et al. ( 2006 ).
Figure 5.18 (top) shows results of WLR applied to the data set previously used
for the WofE result shown in Fig. 5.9 . Figure 5.18 (bottom) is the corresponding
t -value map. The t -values are the posterior probabilities divided by their standard

¼