Geology Reference
In-Depth Information
Inference System (ANFIS), and SVMS. Later we describe conjunction models such
as NW, Wavelet-ANFIS (W-ANFIS), and Wavelet-SVM (W-SVM).
4.6.1 Neural Network Autoregressive with Exogenous Inputs
(NNARX) Model
In this topic, a Neural Network Autoregressive with Exogenous Inputs (NNARX)
model is built on the basis of a linear ARX model. The discrete linear ARX model
can be expressed as a simple linear difference equation. A simple linear ARX model
for rainfall-runoff modeling (modi
ed version of (
5.2
)) with one exogenous input
(rainfall) can be expressed as follows:
X
n
a
X
n
b
Q
ð
t
Þ
¼
a
n
Q
ð
t
n
Þþ
b
m
P
ð
t
m
Þþ
e
ð
t
Þ
ð
4
:
47
Þ
n¼1
m¼1
where Q is the runoff, t represents time step, a
n
and b
m
denote the model param-
eters, n
a
and n
b
are the orders associated with the output (Q) and the input (P), and e
(t) is the mapping error. The choice of the appropriate ARX model structure (the
appropriate n
a
and n
b
) is very important;
s
Information Criterion (AIC) and the best input combination selection is selected by
the Gamma Test.
these can be guided with Akaike
'
4.6.2 Fuzzy Inference System
Fuzzy inferences systems are a popular framework, widely used in hydrology,
based on concepts such as fuzzy set theory, fuzzy reasoning, and fuzzy if then rules.
The fuzzy concept was introduced by Lotfi
A. Zadeh (University of California,
Berkeley) in the 1960s. In general, it helps computers to make decisions more like
human beings. One of the major advantages of fuzzy logic is that it can solve
several ambiguities and uncertainties in decision making and it can be applied in
any areas where empirical relationships are not well de
ned or impractical to
model. The basic structure of fuzzy interference systems consists of three con-
ceptual components: (1) a rule base contains selection of fuzzy rules, (2) a database
which de
nes membership functions, and (3) a reasoning mechanism which per-
forms an inference procedure. A defuzzi
cation step is used to get crisp values
which best represent a fuzzy set. Three major fuzzy frameworks are Mamdani fuzzy
models, Sugeno fuzzy models, and Tsukamoto fuzzy models. Mamdani systems go
through six major steps, namely (1)
first determination of set of fuzzy rules, (2)
fuzzi
cation of inputs using suitable membership functions, (3) combining fuzzi
ed
inputs as per the fuzzy rules to a rule strength, (4)
finding the consequence of the
