Environmental Engineering Reference
In-Depth Information
dent given an output description so that:
river level forecasting as described below. See
Qin (2005) and Qin and Lawry (2005) for discus-
sions on the application of LID3 to many different
classification and prediction problems.
Y
n
¼
mF
1
; ...;
F
n
j
F
y
mF
i
j
F
y
ð
8
:
5
Þ
i
¼
1
In addition, Randon and Lawry (2002) proposed a
Semi-Naive Bayes model where the above inde-
pendence assumption was weakened to allow for
groups of dependent variables with independence
assumed between each group.
Fuzzy Naive and Semi-Naive Bayes have been
applied to a wide range of application areas includ-
ing vision, medical classification and time series
prediction (see Randon 2004). In the flood man-
agement domain Fuzzy Naive Bayes has been
applied to the prediction of sea levels in the North
Sea (Randon et al. 2008), and flow prediction for
the Bird Creek catchment, as described
below. Also, inhydrological applications ofweath-
er radar, a Fuzzy Naive Bayes classifier has been
applied to classify precipitation and non-precipi-
tation echoes (Rico-Ramirez and Cluckie 2008).
The Fuzzy Bayesian approach
The Fuzzy Bayesian approach was proposed in
Randon and Lawry (2002) and Randon (2004) as
a Label Semantics based method for combining
Bayesian learning algorithms with fuzzy labels.
For classification problems the standard Bayesian
approach determines the conditional probability
of input variables x
1
,
...
, x
n
given each class C
i
.
Bayes' theorem is then applied to determine the
probability of each class given instantiations of
x
1
,
...
, x
n
:
px
1
; ...;
ð
x
n
j
C
i
Þ
P
ð
C
i
Þ
PC
i
j
x
1
; ...;
x
n
ð
Þ ¼
P
i
px
1
; ...;
ð
8
:
3
Þ
ð
x
n
j
C
i
Þ
P
ð
C
i
Þ
Here P(C
i
) is the proportion of examples in the
training data that are of class C
i
.
The Fuzzy Bayesian approach applies Bayes'
theorem to mass functions instead of standard
probability distributions. A set of labels LA
i
is
defined for each variable x
i
and (for continuous
valued prediction problems) labels LA
y
are defined
for the output variable y. A multi-dimensional
mass function (referred to as a mass relation) is
then evaluated from the training data conditional
on each set of description labels for y; this
is denoted mF
1
; ...;
Classification of Weather Radar Images
The quantitative use of radar-based precipitation
estimations in hydrological modelling for flood
forecasting has been limited due to different
sources of uncertainty in the rainfall estimation
process. The factors that affect radar rainfall esti-
mations are well known and have been discussed
by several authors (Battan 1973; Austin 1987;
Doviak and Zrnic 1993; Collier 1996). These in-
clude factors such as radar calibration, signal
attenuation, clutter and anomalous propagation,
variation of the Vertical Reflectivity of Precipita-
tion (VPR), range effects, Z-R relationships, vari-
ation of the drop size distribution, vertical air
motions, beam overshooting the shallow precipi-
tation and sampling issues among others.
The VPR is an important source of uncertainty
in the estimation of precipitation using radars.
The variation is largely due to factors such as the
growth or evaporation of precipitation, the ther-
modynamic phase of the hydrometeors, ormelting
and wind effects. As the range increases from the
where F
i
LA
i
for
i
¼
1,
...
, n and F
y
LA
y
. Bayes' theorem is then
applied to obtain amass function on the labels for y
given label sets describing the input variables:
F
n
j
F
y
mF
y
¼
mF
1
; ...;
F
n
j
F
y
mF
y
ð
8
:
4
Þ
mF
y
j
F
1
; ...;
F
n
P
F
y
mF
1
; ...;
F
n
j
F
y
Here m(F
y
) is the mass function for y derived from
the training data by aggregating m
y
(F
y
) across all
values of y in the database.
Randon (2004) investigated the use of a Naive
Bayes algorithmin this context where all the input
variables are assumed to be conditionally indepen-
