Incorporating Grey Total Influence into Tolerance Rough Sets for Classification Problems
Abstract
:1. Introduction
2. Tolerance Rough Sets
2.1. Rough Set Theory
2.2. Traditional Similarity Measure
2.3. Computational Steps of a TRS-Based Classifier
- Step 1.
- Determine 〈TC(x), TC(x)〉With x, TC(x) is composed of patterns certainly similar to x, and TC(x) is composed of patterns possibly similar to x. For subset and concept approximations, TC(x) is identical to TC(x), but TC(x) is not.
- Step 2.
- Classification using lower approximationsIf TC(x) = {x}, the classification of x can be left to the next step. If the cardinality of TC(x) is at least two, TC(x) − {x} is used to determine the relative frequency of the class inclusion of the training patterns in TC(x) − {x}. Then, x can be assigned to the class with the highest relative frequency by majority vote. However, if the highest relative frequency is not unique, the classification of x can be left until the next step.
- Step 3.
- Classification using upper approximationsThe boundary region BNDA(TC(x)) (TC(x) − TC(x)) of x can be used to determine the class label of x. Assume that patterns belonging to class Ci constitute Xi. With y in BNDA(TC(x)) ≠ φ, the rough membership function denoted by defined as:
3. Grey-Total-Influence-Based Tolerance Rough Sets
3.1. Studies Related to Measuring Total Influence
3.2. Grey Relational Analysis
3.3. Determining Grey Total Influence
3.3.1. Generating a Direct Influence Matrix Using GRA
3.3.2. Generating a Grey Direct Influence Matrix for Pattern Classification
- (1)
- Z11: z(x1l, x1p) (1 ≤ l ≤ m1) is obtained using x11, x12, …, as comparative sequences and x1i as a reference sequence, so that z(x1l, x1p) = ϒ(x1l, x1p).
- (2)
- Z12: z(x1p, x2q) is obtained using x11, x12, …, as comparative sequences and x2j as a reference sequence, so that z(x1p, x2q) = ϒ(x1p, x2q).
- (3)
- Z21: As the testing patterns are unseen by the training patterns, they do not have any impact on the training patterns. Therefore, z(x2q, x1p) is set to zero, so that Z21 = 0.
- (4)
- Z22: As the testing patterns are unseen, they do not have any impact on themselves. Therefore, z(x2k, x2q) (1 ≤ k ≤ m2) is set to zero, so that Z22 = 0.
3.3.3. Generating a Grey Total Influence Matrix
3.4. Grey-Total-Influence-Based Tolerance Relation
3.5. Illustrative Example
3.5.1. Training Phase
3.5.2. Testing Phase
4. Genetic-Algorithm-Based Learning Algorithm
Algorithm 1 The pseudo-code of the learning algorithm |
Set 0 to k; //1 ≤ k ≤ nmax |
Initialize population (k, nsize); |
Evaluate chromosomes (k, nsize); |
While not satisfying the stopping rule do |
Set k + 1 to k; |
Select (k, nsize); //Select generation k from generation k − 1 |
Crossover (k, nsize); |
Mutation (k, nsize); |
Elitist (k, nsize); |
Evaluate chromosome (k, nsize); |
End while |
- (1)
- Initialize population: The most common population size is between 50 and 500. Generate an initial population of nsize chromosome. Each parameter in a chromosome is assigned a real random value ranging from zero to one.
- (2)
- Evaluate chromosomes: Each chromosome corresponds to a GTI-TRSC that can be generated by the process shown in Figure 1. For each pattern, determine the lower and upper approximations for a GTI-based tolerance class. Furthermore, correct classification serves as a fitness function. Classification accuracy is the number of correct predictions made divided by the total number of predictions made, multiplied by 100 to turn it into a percentage.
- (3)
- Select: To produce generation k, randomly select two chromosomes from generation k − 1 by a binary tournament and place the one with higher fitness in a mating pool.
- (4)
- Crossover: Let be randomly selected chromosomes (1 ≤ i, j ≤ nsize) from generation k. Prc determines whether crossover can be performed on any two real-valued parameters. Two new chromosomes, are generated and are added into Pk+1. The related crossover operations are performed as:
- (5)
- Mutation: Prm determines whether a mutation can be performed on each real-valued parameter of a newly generated chromosome. With a mutation, the affected gene is altered by adding a random number selected from a prespecified interval, such as (−0.01, 0.01). A smaller Prm is required to avoid excessive perturbation.
- (6)
- Elitist strategy: Randomly remove ndel chromosomes from generation k. Insert ndel chromosomes with the maximum fitness from generation k − 1. A smaller ndel is required to generate a smaller perturbation in generation k.
- (7)
- Stopping rule: When nmax generations have been created, the algorithm reaches the stopping condition.
5. Computer Simulations
5.1. Evaluating Classification Performance
- (1)
- HLM: The lattice machine generates hypertuples as a model of the data. Some more general hypertuples can be used in the hierarchy that covers objects covered by the hypertuples. The covering hypertuples locate various levels of the hierarchy.
- (2)
- RSES-O: RSES-O is implemented in RSES. An optimal threshold for the positive region is used to shorten all decision rules with a minimal number of descriptors.
- (3)
- RSES-H: RSES-H can be obtained by constructing a hierarchy of rule-based classifiers. The levels of the hierarchy are defined by different levels of minimal rule shortening. A new pattern can be classified by a single hierarchy of the classifier.
- (4)
- RIONA: RIONA is also implemented in RSES. It uses the nearest neighbor method to induce distance-based rules. For a new pattern, the patterns most similar to it can vote for its decision, but patterns that do not match any rule are excluded from voting.
- (1)
- GTRSC: Instead of a simple distance measure used to evaluate the proximity of any two patterns, the GRG (grey relational grade) is used here to implement a relationship-based similarity measure that generates a tolerance class for each pattern. As mentioned above, only direct relationships were considered in the GTRSC.
- (2)
5.2. Statistical Analysis
- (1)
- GTI-TRSC-SU significantly outperformed TRSC-CO (9.15 − 2 = 7.15), TRSC-SU (9.55 − 2 = 7.55), RSES-H (7.90 − 2 = 5.90), RSES-O (8.80 − 2 = 6.00), RIONA (8.80 − 2 = 6.80), and HLM (9.20 − 2 = 7.2).
- (2)
- GTI-TRSC-CO significantly outperformed TRSC-CO (9.15 − 1.80 = 7.35), TRSC-SU (9.55 − 1.80 = 7.75), RSES-H (7.90 − 1.80 = 6.10), RSES-O (8.80 − 1.80 = 6.20), RIONA (8.80 − 1.80 = 7.00), and HLM (9.20 − 1.80 = 7.40).
- (3)
- There was no significant difference between GTI-TRSC and GTRSC for both set and concept approximations. Even so, GTI-TRSC outperformed GTRSC on seven out of ten datasets.
- (4)
- Although GTI-TRSC did not significantly outperform the FTRSC, the difference between GTI-TRSC-CO and FTRSC-SU was slightly less than CD (6.35 − 1.80 = 4.50). Therefore, it is reasonable to conclude that GTI-TRSC-CO was superior to FTRSC-SU. Even so, it is interesting to investigate the applications that can render GTI-TRSC and FTRSC significantly different.
6. Discussion and Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Pattern | Conditional Attribute | Decision Attribute | |||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||
x1 | 51.1 | 35.2 | 14.0 | 2.0 | 1 |
x2 | 53.0 | 37.0 | 15.0 | 2.0 | 1 |
x3 | 50.0 | 32.1 | 12.0 | 2.0 | 2 |
x4 | 52.0 | 27.0 | 39.0 | 14.6 | 1 |
x5 | 59.0 | 30.0 | 42.3 | 15.0 | 2 |
x6 | 56.7 | 25.4 | 39.0 | 11.0 | 2 |
Data | # Patterns | # Attributes | # Classes |
---|---|---|---|
Australian approval | 690 | 14 | 2 |
Glass | 214 | 9 | 6 |
Hepatitis | 155 | 19 | 2 |
Iris | 150 | 4 | 3 |
Pima Indian diabetes | 768 | 8 | 2 |
Sonar | 208 | 60 | 2 |
Statlog Heart | 270 | 13 | 2 |
Tic-Tac-Toe | 958 | 9 | 2 |
Voting | 435 | 16 | 2 |
Wine | 178 | 13 | 3 |
Dataset | Classification Methods | |||||
---|---|---|---|---|---|---|
HLM | RSES-H | RSES-O | RIONA | TRSC-SU | TRSC-CO | |
Australian approval | 92.0 | 87.0 | 86.4 | 85.7 | 85.9 | 87.1 |
Glass | 71.3 | 63.4 | 61.2 | 66.1 | 65.7 | 68.1 |
Hepatitis | 78.7 | 81.9 | 82.6 | 82.0 | 83.9 | 83.5 |
Iris | 94.1 | 95.5 | 94.9 | 94.4 | 95.7 | 95.2 |
Diabetes | 72.6 | 73.8 | 73.8 | 75.4 | 74.1 | 73.6 |
Sonar | 73.7 | 75.3 | 74.3 | 86.1 | 74.3 | 75.0 |
Statlog Heart | 79.0 | 84.0 | 83.8 | 82.3 | 82.9 | 83.3 |
TTT | 95.0 | 99.1 | 99.0 | 93.6 | 82.3 | 82.3 |
Voting | 95.4 | 96.5 | 96.4 | 95.3 | 93.4 | 94.0 |
Wine | 92.6 | 91.2 | 90.7 | 95.4 | 93.0 | 95.3 |
Average rank | 9.20 | 7.90 | 8.80 | 8.80 | 9.55 | 9.15 |
Dataset | Classification Methods | |||||
---|---|---|---|---|---|---|
FTRSC-SU | FTRSC-CO | GTRSC-SU | GTRSC-CO | GTI-TRSC-SU | GTI-TRSC-CO | |
Australian approval | 88.0 | 87.7 | 89.3 | 89.1 | 91.0 | 90.9 |
Glass | 69.1 | 69.4 | 70.1 | 69.9 | 79.8 | 79.7 |
Hepatitis | 85.6 | 84.3 | 86.0 | 87.0 | 88.7 | 89.8 |
Iris | 95.7 | 96.2 | 96.1 | 96.3 | 96.3 | 96.4 |
Diabetes | 75.7 | 75.9 | 76.5 | 76.0 | 77.9 | 81.6 |
Sonar | 78.8 | 79.5 | 83.0 | 82.8 | 86.7 | 87.8 |
Statlog Heart | 83.9 | 84.1 | 84.4 | 84.0 | 86.9 | 86.1 |
TTT | 97.3 | 97.8 | 98.5 | 98.5 | 98.9 | 98.5 |
Voting | 96.6 | 96.3 | 96.0 | 96.2 | 97.4 | 97.7 |
Wine | 93.2 | 95.1 | 97.4 | 97.9 | 97.9 | 98.1 |
Average rank | 6.35 | 5.60 | 4.40 | 4.55 | 2.00 | 1.80 |
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Hu, Y.-C.; Chiu, Y.-J. Incorporating Grey Total Influence into Tolerance Rough Sets for Classification Problems. Appl. Sci. 2018, 8, 1173. https://doi.org/10.3390/app8071173
Hu Y-C, Chiu Y-J. Incorporating Grey Total Influence into Tolerance Rough Sets for Classification Problems. Applied Sciences. 2018; 8(7):1173. https://doi.org/10.3390/app8071173
Chicago/Turabian StyleHu, Yi-Chung, and Yu-Jing Chiu. 2018. "Incorporating Grey Total Influence into Tolerance Rough Sets for Classification Problems" Applied Sciences 8, no. 7: 1173. https://doi.org/10.3390/app8071173