Correlation, Method of Least Squares, Linear Regression and the General Linear Model - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 116

2.5•10 6 . The preceding four methods (probability calculus,
weights-of-evidence, linear least squares, and logistic regression) all produce the
same estimates of the probabilities (uncertainty type 1). However, they produce
slightly different answers for the variances of these probabilities (uncertainty type 2).
Some remarks on other applications pertaining to the mosaic model are as
follows. This model was used by Bernknopf et al. ( 2007 ) for different rock units
with probabilities of occurrence for mineral deposits of different types. Probabili-
ties and expected values were modified according to relative amount of exposure of
each rock unit by these authors. In the context of weights of evidence modeling,
Carranza ( 2009 ) asked the question of what would be the optimum pixel size. For
the mosaic model, the answer to this question is simply that pixels should be
sufficiently small to allow precise estimates of relative areas of rock units on the
map. Further size decrease does not affect estimation results when mineral deposits
are modeled as points, because of the dichotomous nature of every rock unit
represented by a mosaic model.

and P(D j A ~) ¼

4.3 General Model of Least Squares

The linear model discussed in Sect. 4.2 can be generalized by including p additional
explanatory variables X 1 , X 2 , β p X p .

As before Y is a random variable with the same variance as the random variable with
zero mean that distorts the deterministic component

, X p , so that: E ( Y

j X ) ¼ β 0 + β 1 X 1 + β 2 X 2 + ... +

...

β 0 + β 1 X 1 + β 2 X 2 + ... + β p X p .

It is convenient to use matrix algebra and write:

2
4

3
5

2
4

3
5

2
4

3
5

2
4

3
5

1 X 11 X 21
1 X 12 X 22
:

:: X p 1

β 1
β 2
:
:
β p

Y 1
Y 2
:
:
Y n

E 1
E 2
:
:
E n

:: X p 2

:: : : :

¼ þ

: : :
: : :
1 X 1 n X 2 n

:: :
:: :
::

X pn

Box 4.4: Multiple Regression
The preceding matrix equation also can be written as Y ¼ X β + E.

β ¼ X 0 X 1 X 0 Y. The

Best estimates of the coefficients satisfy:

XX 0 X 1 X 0 Y. In regional mineral
resource appraisal, it can be convenient to define a matrix D

X β ¼

Y

estimated values are ¼

X(X 0 X) 1 X 0

¼

Y

so that ¼ DY. The 95-% confidence interval of any individual

q

X 0
k

X 0 X 1 X k

Y k

estimated value is t 0 : 975 s . If all estimated

values are considered simultaneously, the wider belt

(continued)