Geology Reference
In-Depth Information
2M
X
M
1
2
c
M
k
ðÞ
¼
y
Ni
;ðÞ
y
i
ð
1
k
p
Þ
ð
3
:
4
Þ
i¼1
where y
Ni
;ðÞ
is the corresponding y-value for the kth nearest neighbor of Xi
i
in (
3.3
).
In order to compute
Γ
, a least squares regression line is constructed for the p points
ð
d
M
k
ðÞ; c
M
k
ðÞ
Þ
:
c
¼
A
d þ C
ð
3
:
5
Þ
The intercept on the vertical axis
d
¼
0 is the
Γ
value (Fig.
3.1
), and, as can be
shown:
c
M
ðÞ!
Var
ðÞ
in probability as
d
M
ðÞ!
0
ð
3
:
6
Þ
Calculating the regression line gradient can also provide helpful information on
the complexity of the system under investigation. A formal mathematical justifi-
-
cation of the method can be found in Evans and Jones [
22
].
The graphical output of this regression line (
3.5
) provides very useful infor-
mation. First, it is remarkable that the vertical intercept
Γ
of the y (or Gamma) axis
offers an estimate of the best MSE achievable utilizing a modeling technique for
unknown smooth functions of continuous variables [
22
]. Second, the gradient offers
an indication of a model
'
s complexity (a steeper gradient indicates a model of
greater complexity).
The Gamma Test is a non-parametric method and the results apply regardless of
the particular techniques used to build subsequently a model of f. We can stan-
dardize the result by considering another term, V
ratio
, which returns a scale invariant
noise estimate between zero and one. The V
ratio
is be de
ned as
C
V
ratio
¼
ð
3
:
7
Þ
r
2
ðÞ
y
Fig. 3.1 The regression plot
for the gamma test
