GNSS in Practical Determination of Regional Heights - Global Navigation Satellite Systems: Signal,Theory and Applications - page 141

usually results in an insufficient or rough approximation of the surface, unnecessarily use of
a higher degree function may produce an over fitted surface that may reveal unrealistic and
optimistic values at the test points. Another critical phase of determining polynomial surface
is selecting the significant parameters and hence ignoring the insignificant ones in the model
that this decision also bases on statistical criteria. After calculating the polynomials with
least squares adjustment, the statistical significance of the model parameters can be
analyzed using F-test with the null hypothesis H o : X i = 0 and the alternative hypothesis
H 1 : X i ≠ 0 (Draper and Smith, 1998). The F-statistic is used to verify the null hypothesis and
computed as a function of observations (Dermanis and Rossikopoulos, 1991):

T
i

− 1

XQ X

XXi

F = (10)

i i

2

ˆ

t σ

ˆ σ is a-posteriori variance, t is the number of tested parameters. The null hypothesis
is accepted if

2

where

F α is obtained from the standard statistical tables for a
confidence level α and degrees of freedom r that means the tested parameters are
insignificant and deleted from the model. If the contrary is true and

FF α

≤ , where

tr tr

FF > is fulfilled, then
the parameters remain in the model. After clarifying the optimal form of a polynomial
model with significance tests of parameters, the performance of the calculated model is
tested empirically, considering the geoid residuals at the benchmarks of the network. The
tests are repeated with the polynomials in varying orders and hence an appropriate order of
polynomial is determined for the data depending on the comparisons of test results.

tr

3.1.2.2 Adaptive network based fuzzy inference system

ANFIS is an artificial intelligence inspired soft computing method that is first purposed in
the late 1960's depending on fuzzy logic and fuzzy set theory introduced by Zadeh (1965).
After that this method was used in various disciplines for controlling the systems and
modelling non-stationary phenomena, and recently applied in geoid determination, as well
(see e.g. Ayan et al, 2005; Yılmaz and Arslan, 2008). The computation algorithm of the
method mainly bases on feed-forward adaptive networks and fuzzy inference systems. A
fuzzy inference system is typically designed by defining linguistic input and output
variables and an inference rule base. Initially, the resulting system is just an approximation
for an adequate model. Hence, its premise and consequent parameters are tuned based on
the given data in order to optimize the system performance and this process bases on a
supervised learning algorithm (Jang, 1993).

In computations with ANFIS, depending on the fuzzy rule structures, there are different
neural-fuzzy systems such as Mamdani, Tsukamoto and Takagi-Sugeno (Jang, 1993). Tung
and Quek (2009) can be referred for a review on implementation of different neural-fuzzy
systems. In Figure 9 a two input, two-fuzzy ruled, one output type 3 fuzzy model is
illustrated. In this example Takagi-Sugeno's fuzzy if-then rules are used and the output of
each rule is a linear combination of input variables plus a constant term, and the final output
is a weighted average of each rule's output.

In the associate fuzzy reasoning in the figure and corresponding equivalent ANFIS structure:

Rule 1: if x is A 1 and y is B 1 ; then f 1 = p 1 x + q 1 y + r 1