Geoscience Reference
In-Depth Information
Box 4.2: Maximum Likelihood for Functional Relationship
(MLFR) Method
MLFR (Ripley and Thompson
1987
) works as follows: one is interested in the
linear relation
v
i
¼
ʱ
+
β
·
u
i
where
u
i
and
v
i
are observed as
x
i
(
¼
u
i
+ error)
s
2
(
y
) represent the
variances of
x
i
and
y
i
, respectively, the problem reduces to minimizing the
expression
Q
s
2
(
x
) and
and
y
i
(
¼
v
i
+ error), respectively. If
ʺ
i
ʻ
i
h
i
over
u
i
. First minimizing over
u
i
and
X
x
i
u
i
2
2
ð
Þ
ð
y
i
ʱβu
i
Þ
¼
þ
ʺ
i
ʻ
i
2
introducing the weights
w
i
¼
1/(
ʻ
i
+
β
ʺ
i
), this minimum is reached when
u
i
)
2
]. The weights
w
i
depend on
Q
min
(
ʱ
β
¼
∑
[
w
i
(
y
i
ʱβ
β
,
)
and so does
X
w
i
y
i
βu
i
½
2
ð
Þ
X
w
i
the estimate of
α
that satisfies
ʱ
¼
. The slope
a
is found by
u
i
)
2
]. This, in turn, yields the
weights
w
i
and intercept
a
. The covariance
s
(
a
,
b
) can be calculated from
minimizing
Q
min
(
a
,
β
)
¼
∑
[
w
i
(
y
i
a
β
X
w
i
x
i
w
i
s
2
b
the relation
sa
ðÞ
¼
;
b
ðÞ
. So-called scaled residuals
r
i
can be
q
1
ʻ
i
þb
2
computed by means of:
r
i
¼
. The sum of squares of
these scaled residuals should be approximately equal to number of observa-
tions
2. If estimates of
u
i
are written as
X
i
with
X
i
¼
ð
y
i
ʱ β
u
i
Þ
ʺ
i
w
i
[
ʻ
i
x
i
+
ʺ
i
b
(
y
i
a
)],
a plot of
r
i
against
X
i
should not show any noticeable pattern.
The MLFR method was applied by Agterberg (
2004
) for the purpose of
estimating the age of Paleozoic stage boundaries in the GTS-2004 geologic time
scale (also see Sect.
9.5
).
4.2 Linear Regression
In Box
4.1
, bivariate linear regression was written in the form:
EY
X
¼
β
0
þ β
1
X
with
β
0
¼
ʼ
y
þ ˁ
˃
y
β
1
¼
ˁ
˃
y
˃
x
. In practical applications, the statis-
tical parameters in this expression can be replaced by sample estimates. The Gauss-
Marlov theorem of mathematical statistics (
cf
. Bickel and Doksum
2001
) states
that the least-squares estimate of
E
(
Y
˃
x
ʼ
x
;
j
X
) is unbiased, regardless of the nature of the
frequency distribution of
Y
X
, even if it is not normal. The only condition to be
fulfilled for application is that the residuals for
Y
are random and uncorrelated.
j
4.2.1 Degree of Fit and 95 % - Confidence Belts
A useful expression is: TSS
SSR + RSS meaning that the total sum of squares for
deviations from the mean (TSS) is sum of squares due to regression (SSR) and
¼
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