Geoscience Reference
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a
z
-value (
fractile of normal distribution in standard form) by changing its sign if
it belongs to the younger stage A. The value of
z
was transformed into a probability
P
¼
3,
t
e
+ 3]. The frequency corresponding to
3 is equal to 0.999 of which the natural logarithm is equal to
¼
ʦ
(
z
) for values on the interval [
t
e
0.001. Consequently,
values outside the interval [
t
e
3] yield probabilities which are approximately
1 (or 0 for the log-likelihood function) and these were not used for further analysis.
Thus a natural window is provided screening out dates that are not in the vicinity of
the age of chronostratigraphic boundary that is to be estimated. Most probabilities
are greater than 0.5. Only inconsistent dates (asterisks in Table
3.1
) give probabil-
ities less than 0.5. The value of the log-likelihood function for
t
e
is the sum of the
logs of the probabilities as illustrated for
4.6 in Table
3.1
.
Log-likelihood values for Run No. 1 are shown in Table
3.2
with
t
e
ranging from
3 to 7 in steps of 0.1. The largest log-likelihood value is reached for
t
e
¼
t
e
¼
5.6 and this
value was selected as the rst approximation of the true age of the stage boundary.
Ten values before and after 5.6 were used to t a parabola that is shown in Fig.
3.5
.
The tted parabola is more or less independent of the number of values (
¼
21) used
to t it and of width of neighborhood (
2). However, the neighborhood should not
be made too wide because of random uctuations (local minima and maxima near
t
e
¼
¼
3 or 7; see e.g. Table
3.2
). Such edge effects should be avoided. They are due to
the fact that the initial range of simulated time was arbitrarily set equal to 10 in the
computer simulation experiment. The peak of the best-tting parabola provides the
second approximation (
s) of
the corresponding normal distribution can be used to estimate the 95 % condence
interval
m
¼
m) of the estimated age. The standard deviation (
¼
1.96
s
also shown in Fig.
3.5
.
The sum of squares
E
2
for
L
a
, using inconsistent dates only, also is tabulated in
Table
3.1
as a function of
t
e
. The rst approximation of its minimum value is 5.3.
The corresponding parabola is shown in Fig.
3.5
. Te mean value resulting from
L
a
is
about 0.3 less than the mean based on
L
and its standard deviation is nearly the
same. It is fortuitous that the mean based on
L
a
is closer to the population mean
(
5) than that based on
L
. On the average, the
L
method gives better results (see
results for 50 runs summarized at the end of this section).
Means and standard deviations obtained for the rst ten runs of the computer
simulation experiment are listed in Table
3.3
. If they could be estimated, results
obtained for
L
a
are close to those for
L
. The estimated standard deviations tend to be
slightly smaller or much greater. It can be seen from the results for Run No. 7 shown
in Fig.
3.6
that the large standard deviations are due to a break-down of the
L
a
method if there are no inconsistent dates. Results obtained by means of the method
of scoring (see, e.g., Rao
1973
, pp. 366-374) also are shown in Table
3.3
.In
application of this method, the following procedure was followed. As before, the
log-likelihood was calculated for 0.1 increments in
t
e
and the largest of these values
was used as the initial guess. Suppose that this value is written as
y
. Then two other
values
x
and
z
were calculated representing log-likelihood values close to
y
at very
small distances
10
4
and 10
4
along the
t
e
-axis. The quantities
D
1
¼
x
)·10
4
0.5(
z
2
y
+
z
)·10
8
were used to obtain a second approximation of the mean
by subtracting
D
1
/
D
2
from the initial guess. The procedure was repeated until the
and
D
2
¼
(
x
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