Ambiente Núcleo
A) El relato en la historia política.
data
Let (x1, y1), ..., (xn, yn) be the interval data set to be used to create T-I fuzzy set. The method is as follows.
Step 1: Sort the (left) first elements of each interval in ascending order and store them as
(a1: an) = sort(ascend(x1...xn))
5.3. Construction of Type-I Fuzzy Sets from Features 122
them as
(b1: bn) = sort(descend(y1...yn))
Step 3: If any(a ≥= b) then delete that entry of (a, b), this eliminates any empty intervals that result in NU LL with the original method intervals
Step 4: Combine both results as interval data and save them as final result=(a1, b1), ..., (an, bn)
This algorithm is an approximation of the original algorithm in many cases giving
identical result. The major advantage of the proposed method is that it is much faster than
the original method making it practical in real world applications where high dimensional
data of large volume needs to be processed in computationally efficient manner. Now we
consider a few examples based on synthetic data to create T-I fuzzy sets with the proposed
method and compare the results with the original method.
5.3.2
Examples from Synthetic Data
To explain the method, we consider some examples based on synthetic data. As we are
dealing with interval data, it is possible that input data can be either completely overlap-
ping, partially overlapping or mainly non-overlapping. Completely overlapping means
that all intervals have some common values, partially overlapping means that some en-
tries will not have any common values and non-overlapping means that all entries will be
isolated and nothing will be common between them. We consider all of these scenarios
and provide the details of the method.
First we take an example of interval data that is completely overlapping. Table 5.2
shows a set of three overlapping interval data. Figure 5.6 shows that all three entries
of data have some area common in them. We apply both the original method and our
5.3. Construction of Type-I Fuzzy Sets from Features 123
Table 5.2: Example of completely overlapping data
Interval Number Interval data
1 (3,6)
2 (2,7)
3 (4,8)
Figure 5.6: Plot of completely overlapping data
First we apply the original method described in Chapter 2 using Equation 2.27 for T-I fuzzy set creation.
µ(A) = y1/([3, 6] ∪ [2, 7] ∪ [4, 8]) + y2/(([3, 6] ∩ [2, 7]) ∪ ([3, 6] ∩ [4, 8]) ∪ ([2, 7] ∩ [4, 8])) + y3/([3, 6] ∩ [2, 7] ∩ [4, 8])
=y1/[2, 8] + y2/[3, 7] + y3/[4, 6]
Now we apply our proposed method.
Step 1: Sort[3, 2, 4] in ascending order = [2, 3, 4] Step 2: Sort[6, 7, 8] in descending order = [8, 7, 6] Step 3: Check if (2≥ 8 or 3 ≥ 7 or 4 ≥ 6)= Null
Step 4: Combine both sorting results =[2, 8], [3, 7], [4, 6]
5.3. Construction of Type-I Fuzzy Sets from Features 124
that both methods have produced equal results in case of completely overlapping data. It
indicates that the proposed method is working well when all intervals overlap. Figure 5.7
shows the created T-I fuzzy set by applying both methods. The x-axis shows the domain
values for the fuzzy set and y-axis shows the membership grade values.
Table 5.3: Result of overlapping data
Interval Number Original Method Proposed Method
1 (2, 8) (2, 8)
2 (3, 7) (3, 7)
3 (4, 6) (4, 6)
Figure 5.7: T-I fuzzy set for overlapping data
Now, we take an example of data where some of the entries do not overlap. Table 5.4
shows the data where interval one is completely non-overlapping while intervals 2 and 3
overlap. It can also be seen in Figure 5.8.
Table 5.4: Example of partially overlapping data
Interval Number Interval data
1 (1, 3)
2 (5, 9)
5.3. Construction of Type-I Fuzzy Sets from Features 125
Figure 5.8: Plot of partially overlapping data
Firstly, we apply the original method.
µ(A) = y1/([1, 3] ∪ [5, 9] ∪ [4, 8]) + y2/(([1, 3] ∩ [5, 9]) ∪ ([1, 3] ∩ [4, 8]) ∪ ([5, 9] ∩ [4, 8])) + y3/([1, 3] ∩ [4, 8] ∩ [5, 9])
=y1/[1, 3], [4, 9] + y2/[5, 8] + y3/NULL Now we apply our proposed method.
Step 1: Sort[1, 5, 4] in ascending order = [1, 4, 5] Step 2: Sort[3, 9, 8] in descending order = [9, 8, 3] Step 3: Check if (1≥ 9 or 4 ≥ 8 or 5 ≥ 3)= delete [5, 3] Step 4: Combine both sorting results =[1, 9], [4, 8]
The result of the original and proposed method are shown in Table 5.5. It can be seen
that the result is a sub-normal fuzzy set. Both methods are behaving differently in this
example and this shows that if the interval data involves non-overlapping data then the
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Table 5.5: Result of data with some non-overlapped entries
Interval Number Original Method Proposed Method
1 (1, 3),(4, 9) (1, 9)
2 (5, 8) (4,8)
3 - -
To further investigate a complex situation, we take an example of interval data where
all of the data is non-overlapping. Table 5.6 shows the data. The plot of non-overlapping
data is shown in Figure 5.9. It is evident from the figure that all three entries have no
common area between them. Table 5.7 shows a comparison of result of the two methods.
Table 5.6: Example of completely non-overlapping data
Interval Number Interval data
1 (2,3)
2 (6,7)
3 (4,5)
Figure 5.9: Plot of completely non-overlapping data
Firstly, we apply the original method.
5.3. Construction of Type-I Fuzzy Sets from Features 127
y3/([2, 3] ∩ [6, 7] ∩ [4, 5])
=y1/[2, 3], [4, 5], [6, 7] + y2/NULL + y3/NULL Now we apply our proposed method.
Step 1: Sort[2, 6, 4] in ascending order = [2, 4, 6] Step 2: Sort[3, 7, 5] in descending order = [7, 5, 3] Step 3: Check if (2≥ 7 or 4 ≥ 5 or 6 ≥ 3)= delete [6, 3] Step 4: Combine both sorting results =[2, 7], [4, 5]
The results of proposed and original methods are shown in Table 5.7. It can be seen
that the proposed method results are completely different from the original method. It
indicates that the proposed method is not suitable for completely non-overlapping data.
Table 5.7: Result of completely non-overlapping data
Interval data Number Original Method Proposed Method
1 (2,3),(4,5),(6,7) (2,7)
2 - (4,5)
The examples for overlapping, partial overlapping and non-overlapping synthetic data
are shown to illustrate the working of the proposed method. Further experiments were
conducted with more higher order synthetic data for three scenarios and on the data set
used by Miller et al. [76] to have more confidence in the method. The results showed
that the proposed method produces an acceptable approximation when data is completely
overlapping. In case of non-overlapping data, different scenarios produce different results.
As we are dealing with spectral data within a certain region or wave numbers, the chances
are minimal that the data will be completely non-overlapped. We believe that for the
majority of the time our proposed method will provide near equivalent results.
5.3. Construction of Type-I Fuzzy Sets from Features 128