Global Positioning System Reference
In-Depth Information
Rule 2: if
x
is
A
2
and
y
is
B
2
; then
f
2
=
p
2
x
+
q
2
y
+
r
2
where the symbols
A
and
B
denote the fuzzy sets defined for membership functions of
x
and
y
in the premise parts. The symbols
p
,
q
and
r
denote the consequent parameters of the
output functions
f
(Takagi and Sugeno, 1985; Jang, 1993; Yılmaz, 2010). The Gaussian
function is usually used as input membership function μ
i
(
x
) (see Equation 11) with the
maximum value equal to 1 and the minimum value equal to 0:
⎡
2
⎤
⎛
⎞
xb
−
()
⎢
⎥
μ
x
=
exp
−
⎜
i
(11)
⎟
i
⎢
a
⎥
⎝
⎠
i
⎣
⎦
where
a
i
,
b
i
are the premise parameters that define the gaussian-shape according to their
changing values. Yılmaz and Arslan (2008) apply various membership functions and
investigate the effect of the each function on the approximation accuracy of the data set.
In the associated ANFIS architecture of Figure 9, the functions of the layers can be explained
as such that in
Layer 1
, inputs are divided subspaces using selected membership function, in
Layer 2
, firing strength of a rule is calculated by multiplying incoming signals, in
Layer 3
, the
firing strengths are normalised and in
Layer 4
, the consequent parameters (
p
i
,
q
i
, r
i
) are
determined and finally in
Layer 5
, the final output is obtained by summing of all incoming
signals.
Using the designed architecture, in the running steps of the ANFIS, basically, it takes the
initial fuzzy system and tunes it by means of a hybrid technique combining gradient descent
back-propagation and mean least-squares optimization algorithms (see Yılmaz and Arslan,
2008). At each epoch, an error measure, usually defined as the sum of the squared difference
between actual and desired output, is reduced. Training stops when either the predefined
epoch number or error rate is obtained. The gradient descent algorithm is mainly
implemented to tune the non-linear premise parameters while the basic function of the
mean least-squares is to optimize or adjust the linear consequent parameters (Jang, 1993;
Takagi and Sugeno, 1985).
After determination of the local geoid model using either of the methods, the success of the
method can be assessed using various statistical measures such as the coefficient of
determination, R
2
, and the root mean square error, RMSE, of geoidal heights at the reference
benchmarks:
j
( )
2
ˆ
∑
AA
−
i
2
i
=
1
R
=−
1
(12)
j
(
)
2
∑
AA
−
i
i
=
1
j
(
)
2
ˆ
∑
AA
−
i
i
i
=
1
RMSE
=
(13)
j
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