Correlation, Method of Least Squares, Linear Regression and the General Linear Model - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 115

Uncertainties (1) and (2) can be combined with one another by adding the
variances associated with them. In the preceding example the combined variance
is 0.0116. Suppose now, in the preceding example, that the probability of a (larger)
unit cell is 0.1. It would imply that the intrinsic variance is 0.1, with estimation
variance of 0.16, and combined variance of 0.26. It illustrates that for larger unit
areas and for larger posterior probabilities, relative uncertainty associated with
estimation increases significantly. It is noted that probabilities for groups of adjoin-
ing pixels can be added. The resulting sums can be interpreted as probabilities if
they are less than 1 but must be considered to be expected values if they are greater
than 1.

4.2.3 Elementary Statistics of the Mosaic Model

A small-scale geological map of bedrock in a region is a mosaic on which mineral
deposits are projected as points. As already discussed in Chap. 2 , a simple example
of how one can proceed when information of this type is available is as follows:
Suppose a study area contains one million pixels of which 20 % are underlain by
“favorable” environment A . There are 10 pixels with mineral deposits in this study
area of which 8 are on A . The other 2 are on “unfavorable” A ~ where the ~ symbol
denotes “not”. Therefore, the probability that any pixel contains a deposit is P ( D )

¼

0.000,01. The probability that a pixel on A has a deposit can be written as P ( D j A ) ¼

2.5•10 6 . If a probability
of occurrence map is constructed on the basis of this information, it contains
200,000 pixels with probability 0.000,4, and 800,000 with probability 2.5•10 6 .
The second type of uncertainty is related to precision of the statistics. When
weights-of-evidence modeling is applied, the positive weight for the preceding
example is 0.982 representing the natural log of the ratio P ( A

8/200,000 ¼ 0.000,4; likewise, P ( D j A ~) ¼ 2/800,000 ¼

j D )/ P ( A j D ~)

¼ 0.75/

¼

0.281 2.669, and the negative weight is 1.056 representing the natural log of the

ratio P ( A ~ 0.25/0.719. A useful measure of degree of special

association between a point pattern and a mosaic layer is the contrast C , which is
positive weight minus negative weight ( cf . Chap. 5 ) . The contrast for this example
is 2.04 with approximate standard deviation equal to 1.18. The corresponding
t -value of 1.73 is barely significant at the 95 % level if a one-sided test is used
under the normality assumption. It is interesting to apply other resource estimation
techniques to this simple mosaic model as well.
For example, one can fit the linear model Y

j D )/ P ( A ~ j D ~) ¼

a + b • x where Y is a random
variable assuming the value of 1 at pixels on “ A ” where x

¼

¼ 1, and 0 where x ¼ 0.

37.5•10 6 .
Obviously, this linear regression model exactly reproduces the two probabilities
estimated in the first paragraph of this section. The linear equation also can be used
in logistic regression with Y representing the logit of occurrence instead of the
probability itself ( cf . Sect. 5.2 ). Application of this technique gave a

2.5•10 6

Using the method of least squares, this gives a ¼ and b ¼

¼ 12.90 and

b ¼ 2.773, with variances of 0.50 and 0.624, respectively, and covariance of 0.50.

Conversion of logits into probabilities again reproduces P ( D j A ) ¼ 0.000,4