Geology Reference
In-Depth Information
3.1.4 Delta Test
2
of the associated
In the Delta Test (DT), the conditional expected values
2
ð
y
0
Þ
y
points
x
′
and
x
located within a distance
of each other converge to Var(r) as
δ
δ
approaches zero. In other words, this can expressed as
1
2
2
y
0
x
0
¼
e
ð
y
Þ
jj
x
j
\
d
!
Var r
ðÞ
as
d !
0
ð
3
:
8
Þ
The design of the DT is based on the premise that, for any particular value given
for
, the anticipation in (
3.8
) can be estimated by the sample mean:
δ
X
1
1
2
2
eðÞ
¼
y
j
y
i
ð
3
:
9
Þ
j
I
ðÞ
j
i
;ðÞ2
I
ðÞ
where
I
ðÞ
¼
ð
i
;
j
Þjj
x
j
x
i
j
\
d;
1
i
6
¼
j
M
ð
3
:
10
Þ
which is the set of index pairs (
I, j
) for which the associated points (
x
i
,
x
j
) are
located within distance
δ
of each other. For a given
finite set of data points, the
\
d
number of pairs xi
j
x
i
decreases as
δ
decreases. If one chooses a smaller
δ
eðdÞ
value, it indicates that the sample mean
has a signi
cant sampling error.
Therefore, it is always a wise decision to set a signi
cantly higher value of
δ
in
order to have a better estimate of the sample mean
, as per (
3.8
). However, this
restriction reduces the effectiveness of the sample mean
eðdÞ
eðdÞ
as an estimate of
Var(r). If there is a parametric form of the relationship between
and the respective
δ
sampling mean
eðdÞ
, then one could easily compute
eðdÞ
for the range of
and then
δ
estimate the limit as
0 using regression techniques [
43
], and the parametric
form for the relationship between
ʴ
→
is not apparent in the case of DT. The
Gamma Test handles the same issue discussed in DT, in which the quantities
eðdÞ
and
δ
eðdÞ
and
ʴ
are replaced by analogous items such as
ʳ
and
ʴ
, respectively, in such a way
that
c !
Var r
ðÞ
as
ʴ
→
0.
3.2 Implementation of Entropy Theory
Entropy measurements could be used as an important tool to indicate the quality of
time series used for modeling. Furthermore, entropy statistics are suitable for
decomposition analysis. One could
find several applications of entropy in the
realms of economics and other innovation studies. It can also be used as a quan-
titative measure of the information content of a series of data. This is equivalent to
