Geoscience Reference
In-Depth Information
Box 4.1 (continued)
frequency distribution of the random variable ( Z 2 j
z 1 ) which denotes the
probability of Z 2 ¼
z 2 when it is given that Z 1 has assumed the value z 1 . The
method of moments yields: E ( Z 2 j
2 ( Z 2 j
2 . Transforming
z 1 )
¼ ˁ
z 1 ;
˃
z 1 )
¼
1
ˁ
back to X and Y : EY x ¼ ʼ y þ ˁ ˃ y
2 Y x ¼ ˃
y 1
2
ð
x
ʼ x
Þ;
and
˃
ð
ˁ
Þ
.
˃ x
The expected value of Y for a given value X
¼
x , is commonly written in
the form EY x ¼ β 0 þ β 1 x with
β 0 ¼ ʼ y þ ˁ ˃ y
˃ x ʼ x ; β 1 ¼ ˁ ˃ y
˃ x . It denotes
the linear regression of the dependent variable Y on the independent or
explanatory variable X . The frequency distribution of the correlation coeffi-
cient ( r ) was derived by Fisher ( 1915 ) who also showed that ½ log e 1 þ r
is
1 r
log e 1 þˁ
ˁ
2 n 1
normally distributed with, approximately, mean of
1 ˁ þ
and
½
ð
Þ
1
variance of
3 .
n
There are many situations in which one wishes to determine the relationship
between two random variables that are both subject to uncertainty. For example,
suppose that a set of rock or ore samples is chemically analyzed in a new
laboratory and it is necessary to determine whether or not the newly produced
results are unbiased by having the same analyses done in a laboratory known to
produce unbiased results. This problemwassolvedbyRipleyandThompson
( 1987 ) who developed a maximum likelihood fitting method for (linear)
functional relationship (MLFR) that can be applied in other situations as well.
Their method generalizes the original “major axis” method (Agterberg 1974 ).
The major axis differs from the linear regression line in that it assumes that both
variables ( X and Y ) are subject to uncertainty. The two methods have in common
that the best-fitting straight line passes through the point with x and y equal to their
sample means. It is readily shown that the estimator of slope of the major axis (
β
)
2 P x 0 y 0
P x 0 2
where x 0 and y 0 represent deviations of x and y from
satisfies tan 2
β ¼
y 0 2
their sample means. A disadvantage of the major axis is that it is not independent
of scale (unit distances along X -and Y -axes). For this reason, the data for x and
y are often standardized before the major axis is constructed. This result is known
as the reduced major axis that passes though the origin with slope equal to
β ¼
no . If more than two random variables are analyzed simulta-
neously, the correlation coefficients form a correlation matrix that can be
subjected to principal component analysis. This is a multivariate extension of
constructing the reduced major axis of two variables. If principal component
analysis is applied to the variance-covariance matrix instead of to the correlation
matrix of more than two variables the result is equivalent to a multivariate
extension of estimation of the major axis for two variables. Principal component
analysis and a modification of it called factor analysis are well known multivariate
methods with useful geoscience applications (see, e.g., Davis 2003 ).
s ðÞ
sðÞ
arctan
Search WWH ::




Custom Search