It should be noted that some systems give also probability for the classification.
However, this probability is based on distance and is affected by the variables
normalization and thus may be biased. Hence, the current chapter displays a method
which deals only with the binary classification, assuming that a true WQE will
require additional human investigation.
It should be noted that the classification process must take into consideration the
number of centroids, for example, by normalizing the summation of the distance by
the number of centroids (see step 2 in what follows). Examples for the classification
process as used in this paper include:
• Step1: Calculate the RBF for a given point.
• Step 2: Divide the result of step 1 by the number of centroids used for this
calculation.
• Step 3: If the result of step 2 is above a threshold (henceforth high RBF level
(HRL)), classify as true and classify others as false.
The numerical example given in the data set analysis section of this chapter will
illustrate this process.
Equation (2) gives an implicit form of this function. The term DET refers to
detection process as described above.
Equation 2: RBF with classification
h x
ð Þ ¼ Det
X N
m¼1
w m e
Àγ x m Àμ m
½
2
ð
Þ
(
)
ð2Þ
Once again, using a matrix the general notation yields Eq. 3.
Equation 3: General form of a RBF where the squared matrix is denoted by Φ
e
Àγ x 1 Àμ 1
½
2
e
Àγ x 1 Àμ 2
½
2
. . . e
Àγ x 1 Àμ N
½
2
e
Àγ x 2 Àμ 1
½
2
e
Àγ x 2 Àμ 2
½
2
. . . e
Àγ x 2 Àμ N
½
2
⋮
⋮
. . .
⋮
e
Àγ x K Àμ 1
½
2
e
Àγ x K Àμ 2
½
2
. . . e
Àγ x K Àμ N
½
2
2
6
6
6
6
4
3
7
7
7
7
5
w 1
w 2
⋮
w N
2
6
6
6
4
3
7
7
7
5
ð3Þ
The equation set above can be written in short as Φw ¼ y where Φ is the term in
the first squared parenthesis and w is the correspondent weights vector.
3 RBF Parameters Selection
As it was set out earlier, the values of W(s) and HRL should be defined or initialized
in some manner. We turn now into the last stage of the methodology. This is the
stage that involves setting the values of these parameters. Let us assume that the set
of known centroids (which will be called the training set) is based on known and
Using Radial Basis Function for Water Quality Events Detection
147
However, this probability is based on distance and is affected by the variables
normalization and thus may be biased. Hence, the current chapter displays a method
which deals only with the binary classification, assuming that a true WQE will
require additional human investigation.
It should be noted that the classification process must take into consideration the
number of centroids, for example, by normalizing the summation of the distance by
the number of centroids (see step 2 in what follows). Examples for the classification
process as used in this paper include:
• Step1: Calculate the RBF for a given point.
• Step 2: Divide the result of step 1 by the number of centroids used for this
calculation.
• Step 3: If the result of step 2 is above a threshold (henceforth high RBF level
(HRL)), classify as true and classify others as false.
The numerical example given in the data set analysis section of this chapter will
illustrate this process.
Equation (2) gives an implicit form of this function. The term DET refers to
detection process as described above.
Equation 2: RBF with classification
h x
ð Þ ¼ Det
X N
m¼1
w m e
Àγ x m Àμ m
½
2
ð
Þ
(
)
ð2Þ
Once again, using a matrix the general notation yields Eq. 3.
Equation 3: General form of a RBF where the squared matrix is denoted by Φ
e
Àγ x 1 Àμ 1
½
2
e
Àγ x 1 Àμ 2
½
2
. . . e
Àγ x 1 Àμ N
½
2
e
Àγ x 2 Àμ 1
½
2
e
Àγ x 2 Àμ 2
½
2
. . . e
Àγ x 2 Àμ N
½
2
⋮
⋮
. . .
⋮
e
Àγ x K Àμ 1
½
2
e
Àγ x K Àμ 2
½
2
. . . e
Àγ x K Àμ N
½
2
2
6
6
6
6
4
3
7
7
7
7
5
w 1
w 2
⋮
w N
2
6
6
6
4
3
7
7
7
5
ð3Þ
The equation set above can be written in short as Φw ¼ y where Φ is the term in
the first squared parenthesis and w is the correspondent weights vector.
3 RBF Parameters Selection
As it was set out earlier, the values of W(s) and HRL should be defined or initialized
in some manner. We turn now into the last stage of the methodology. This is the
stage that involves setting the values of these parameters. Let us assume that the set
of known centroids (which will be called the training set) is based on known and
Using Radial Basis Function for Water Quality Events Detection
147