Geology Reference
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ef
ciently the Gamma statistic (of the best mean squared error (MSE) on an output
that can be attained by any smooth model) directly from the unseen data. The GT is
often seen as an improved version of the Delta Technique (DT) [ 60 ]. The
description of DT is given later after a detailed description of GT.
3.1.1 Background on Gamma Statistic, V-Ratio, and M-Test
The GT estimates the minimum MSE which can be achieved when modeling the
unseen data using any continuous nonlinear models. The GT was
rst reported by
Koncar [ 46 ] and Agalbj
rn et al. [ 2 ], and later enhanced and discussed in detail by
many researchers [ 20 , 21 , 42 , 77 , 78 ].
Only a brief introduction to the GT is given here and the interested reader should
consult the aforementioned papers for further details. The basic idea is quite distinct
from the earlier attempts with nonlinear analysis. Suppose we have a set of data
observations of the form
ö
f
ð
x i ;
y i
Þ;
1
i
M
g
ð 3 : 1 Þ
R m are vectors con
where the input vectors xi i Є
ned to some closed bounded set
R m and, without loss of generality, the corresponding outputs yi i Є
C
R are scalars.
Є
The vectors x contain predicatively useful factors in
uencing the output y. The only
assumption made is that the underlying relationship of the system is of the fol-
lowing form:
y
¼
f
ð
x 1 ... x m
Þ þ
r
ð 3 : 2 Þ
where f is a smooth function and r is a random variable which represents noise.
Without loss of generality, it can be assumed that the mean of r
is distribution is
zero (since any constant bias can be subsumed into the unknown function f) and that
the variance of the noise Var(r) is bounded. The domain of a possible model is now
restricted to the class of smooth functions which have bounded
'
first partial deriv-
atives. The Gamma statistic
Γ
is an estimate of the model
'
is output variance which
cannot be accounted for by a smooth data model.
The GT is based on Ni ; k
½
;
which are the kth 1
ð
k p
Þ
nearest neighbors
x Ni ; ½ 1
ð
k p
Þ
for each vector
x i 1
ð
i M
Þ
. Speci
cally, the Gamma Test is
derived from the Delta function of the input vectors:
M X
M
i¼1 x Ni ;ðÞ x i
1
2
d M k
ðÞ ¼
ð
1
k
p
Þ
ð 3 : 3 Þ
where
denotes Euclidean distance, and the corresponding Gamma function of
the output values:
j
j
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