Unsupervised Attribute Reduction Algorithm for Mixed Data Based on Fuzzy Optimal Approximation Set
Abstract
:1. Introduction
- (1)
- We extend the existing theory of optimal approximation sets to create a theory of fuzzy optimal approximation sets. This enables their application to mixed data types.
- (2)
- We introduce improved algorithms for solving optimal approximation sets, known as IOAS and FIOAS. These enhanced methods enable higher similarity to the target set and produce smaller optimal approximation sets.
- (3)
- We propose an unsupervised attribute reduction algorithm based on fuzzy optimal approximation sets. This algorithm is capable of selecting fewer attributes while maintaining the accuracy of classification and clustering tasks.
2. Relevant Concepts
- Boundedness, , , and only if ;
- Symmetry, ;
- , , ,
3. Improvement Based on Optimal Approximation Set
3.1. Optimal Approximation Set
U | d | |||
---|---|---|---|---|
1 | 0 | 1 | 0 | |
1 | 0 | 1 | 0 | |
1 | 0 | 1 | 0 | |
1 | 0 | 1 | 0 | |
1 | 0 | 1 | 0 | |
1 | 0 | 1 | 1 | |
1 | 0 | 1 | 1 | |
0 | 1 | 0 | 1 | |
0 | 1 | 0 | 1 | |
0 | 1 | 0 | 1 | |
0 | 1 | 0 | 1 | |
0 | 1 | 0 | 0 | |
0 | 1 | 0 | 0 | |
0 | 1 | 0 | 0 | |
0 | 1 | 0 | 0 | |
0 | 1 | 0 | 0 | |
0 | 0 | 1 | 1 | |
0 | 0 | 1 | 1 |
3.2. Improvement of Optimal Approximation Set
- For the target set X, let ;
- Let denote , and denote ;
- Let denote .
Algorithm 1: IOAS |
4. Improved Algorithm Based on Fuzzy Optimal Approximation Set
4.1. Relative Definition
- Reflexivity ;
- Symmetry ;
- Transitivity ,
- ;
- ;
- .
4.2. Improved Fuzzy Optimal Approximation Set Model
Algorithm 2: FIOAS |
U | |||
---|---|---|---|
1 | 0.2 | 12.1 | |
1 | 0.3 | 11.1 | |
2 | 0.7 | 3.2 | |
2 | 0.7 | 3.3 | |
3 | 0.8 | 2.6 |
5. Unsupervised Algorithm Based on Fuzzy Optimal Approximation Set
5.1. Relative Definitions and Model
- ,
5.2. Complexity Analysis
Algorithm 3: FOUAR |
5.3. Specific Example
- ,
- and are also calculated to be 1. Following the same procedure, we can see that
- ,
- ,
- ⋯
- ,
- .
- Similarly, we can obtain , , for .
- ,
- , ,
- , .
- Thus, the relevance degree can be calculated as
- .
- Similarly, we can obtain:
- , ,
- , .
- Therefore, can be obtained as follows:
- .
- Hence, can be selected as the attribute corresponding to the maximum attribute relevance and added to R. At this point, .
- ,
- , ,
- , .
- Then, calculate the relevance .
- We can similarly obtain that ,
- , and .
- Since , we select the attribute corresponding to the maximum and obtain .
U | |||||
---|---|---|---|---|---|
A | 2 | 38.0 | 100.4 | 2 | |
A | 1 | 36.2 | 97.2 | 1 | |
A | 1 | 36.1 | 97.0 | 1 | |
A | 1 | 36.3 | 97.3 | 1 | |
A | 1 | 36.4 | 97.5 | 1 | |
B | 2 | 39.2 | 102.6 | 1 | |
B | 2 | 39.3 | 102.6 | 1 | |
B | 2 | 38.4 | 101.1 | 1 | |
B | 2 | 39.2 | 102.6 | 1 | |
B | 2 | 38.1 | 100.6 | 1 | |
B | 2 | 36.1 | 96.9 | 1 | |
B | 2 | 36.4 | 97.5 | 1 | |
B | 2 | 36.3 | 97.3 | 1 | |
B | 2 | 36.3 | 97.3 | 1 | |
B | 2 | 36.7 | 98.0 | 5 | |
B | 2 | 36.2 | 97.2 | 5 | |
B | 2 | 36.3 | 97.9 | 5 | |
C | 2 | 36.1 | 100.58 | 5 |
6. Experiments and Analyses
6.1. Experimental Preparation
6.2. Validation Experiment of IOAS
6.3. Classification Experiments
6.4. Clustering Experiment
6.5. Reduction Experiment
6.6. Parameter Experiment
6.7. Hypothesis Testing
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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ID | Dataset | Object | Conditional Attribute | Decision Class | Type |
---|---|---|---|---|---|
1 | Anneal | 798 | 38 | 6 | Mixed |
2 | Autos | 205 | 25 | 6 | Mixed |
3 | Bands | 531 | 39 | 2 | Mixed |
4 | Cmc | 1473 | 10 | 3 | Mixed |
5 | Cleveland | 303 | 13 | 5 | Mixed |
6 | Credit approval | 690 | 15 | 2 | Mixed |
7 | German | 1000 | 20 | 2 | Mixed |
8 | Heart disease | 270 | 13 | 2 | Mixed |
9 | Hepatitis | 155 | 19 | 2 | Mixed |
10 | SCADI | 70 | 205 | 7 | Mixed |
11 | Dermatology | 366 | 34 | 6 | Numeric |
12 | Ecoli | 336 | 7 | 8 | Numeric |
13 | Ionosphere | 351 | 33 | 2 | Numeric |
14 | Iris | 150 | 5 | 2 | Numeric |
15 | Movement libras | 360 | 90 | 15 | Numeric |
16 | Sonar | 208 | 60 | 2 | Numeric |
17 | Parkinsons | 768 | 22 | 2 | Numeric |
18 | WDBC | 569 | 31 | 2 | Numeric |
19 | Chess | 3196 | 36 | 2 | Nominal |
20 | Lymphography | 148 | 18 | 4 | Nominal |
21 | Monks | 432 | 6 | 2 | Nominal |
22 | Audiology | 226 | 69 | 6 | Nominal |
ID | Dataset | Object | Conditional Attribute | Decision Class | Type |
---|---|---|---|---|---|
1 | Bank | 4521 | 16 | 2 | Mixed |
2 | Car | 1728 | 6 | 4 | Nominal |
3 | Mushroom | 8124 | 22 | 2 | Nominal |
4 | Diagnosis | 120 | 6 | 2 | Nominal |
5 | Vote | 435 | 16 | 2 | Nominal |
6 | Chess | 3196 | 37 | 2 | Nominal |
7 | Flag | 194 | 29 | 8 | Nominal |
8 | Flare | 323 | 12 | 2 | Nominal |
9 | Letter-recognition | 20,000 | 17 | 1 | Nominal |
10 | Lymphography | 148 | 18 | 4 | Nominal |
Dataset | Similarity before Optimization | Similarity after Optimization | Improvement in Similarity | Compression Rate |
---|---|---|---|---|
bank | 0.1179 | 0.1732 | 147% | 15% |
car | 0.1128 | 0.1244 | 110% | 19% |
mushroom | 0.0948 | 0.4516 | 476% | 50% |
diagnosis | 0.6120 | 0.6270 | 103% | 24% |
Vote | 0.392 | 0.9166 | 234% | 40% |
Chess | 0.5765 | 0.6738 | 117% | 24% |
Flag | 0.0909 | 0.2380 | 262% | 84% |
Flare | 0.0287 | 0.1200 | 413% | 67% |
Letter-recognition | 0.0724 | 0.0961 | 133% | 28% |
Lymphography | 0.0533 | 0.2222 | 417% | 50% |
Average | 0.2151 | 0.3643 | 241.2% | 40.1% |
Dataset | Raw | CART (Number) | |
---|---|---|---|
Anneal | 38 | 3,12,32,37,1,35,17,7,36,9,5,33,8,13,27,38,4(17) | 0.4 |
Autos | 25 | 2,17,6,5,15,4,7,22,18,19,10,11,1,23,13,16,9,12,25(19) | 0.5 |
Bands | 39 | 2,22,24,29(4) | 2.7 |
Cmc | 10 | 2,7,8,3,6,5,4,1(8) | 0.9 |
Cleveland | 13 | 3,12,11,13,7,2,9,6(8) | 0.2 |
Credit approval | 15 | 6,4,10,7,12,1,9,13,11,15(10) | 0.1 |
German | 20 | 4,7,3,12,2,1,6(7) | 2.0 |
Heart disease | 13 | 3,11,13,7,2,9,6,12(8) | 0.2 |
Hepatitis | 19 | 6,4,10,18,2,19,12,15,16,7,5,11,17,3(14) | 1.6 |
SCADI | 205 | 105,34,48,2,200,179,201,167,193,81,123,145,1,66(14) | 2.3 |
Dermatology | 34 | 27,8,11,30,13,10,15,26,9,5,24,18,23,14,4,21,3,34,32,19,28,1,17,7,31,2(26) | 0.5 |
Ecoli | 7 | 7,1,5,2,6,3(6) | 2.3 |
Ionosphere | 33 | 1,6,27,26,8,16,14,4,18,13,20,12,24,28,22,10,2,9(18) | 0.3 |
Iris | 5 | 3,2,4(3) | 0.8 |
Movement libras | 90 | 8,26,1,62,59,82,29,89,42,41,90,17,32,48,71,18,47,72,56,39,85,44,86,6,9,65,37,52,74,21,16,46,57,13(34) | 0.9 |
Sonar | 60 | 58,46,11,16,30,6,38,34,53,22,5,60,52,7,4(15) | 1.7 |
Parkinsons | 22 | 9,18,17,2,21,22,16,3,1,10,2,11(12) | 2.3 |
WDBC | 31 | 22,7,23,26,10,16,30,6,19,11,20,2,31,28(14) | 2.4 |
Lymphography | 18 | 14,18,12,15,1,2,16,8,11,3(10) | - |
Chess | 36 | 9,7,11,35,10,34,13,33,24,6,5,15,36,26,18,12,21,20,4,22,23,16,30,31,1,27,17,2,8,3,19,25,28(33) | - |
Monks | 6 | 5,1,2,4,3(5) | - |
Audiology | 69 | 64,66,4,6,2,60,15,1,10,5,57,40,7,38,11,25,17,53,24,41,44,59,52,47(24) | - |
Average | 36.73 | 14.05 | - |
Dataset | Raw Data | FSFS | USQR | UEBR | UFRFS | FRUAR | FOUAR |
---|---|---|---|---|---|---|---|
Anneal | 92.11 ± 0.60 | 92.08 ± 0.82 | 92.18 ± 0.78 | 90.28 ± 0.51 | 92.29 ± 0.72 | 92.07 ± 0.65 | 92.97 ± 0.57 |
Autos | 72.88 ± 2.76 | 72.09 ± 3.08 | 71.38 ± 2.27 | 74.16 ± 2.55 | 67.63 ± 2.57 | 74.81 ± 2.19 | 75.61 ± 2.24 |
Bands | 74.24 ± 2.04 | 73.28 ± 1.76 | 68.34 ± 1.54 | 69.09 ± 1.83 | 71.65 ± 1.17 | 72.29 ± 2.16 | 72.77 ± 1.88 |
Cmc | 50.36 ± 0.83 | 50.46 ± 0.98 | 50.92 ± 1.01 | 51.15 ± 1.09 | 51.08 ± 1.01 | 50.55 ± 0.77 | 51.36 ± 0.50 |
Cleveland | 51.42 ± 1.78 | 51.68 ± 2.06 | 50.12 ± 2.08 | 50.89 ± 2.85 | 51.00 ± 1.18 | 52.14 ± 1.90 | 53.36 ± 1.33 |
Credit | 81.39 ± 0.53 | 72.16 ± 1.27 | 81.72 ± 1.35 | 81.30 ± 1.33 | 81.33 ± 1.20 | 85.07 ± 0.66 | 85.59 ± 0.75 |
German | 70.04 ± 1.40 | 69.51 ± 0.99 | 68.98 ± 0.56 | 70.30 ± 1.35 | 67.59 ± 1.16 | 70.33 ± 1.64 | 71.41 ± 1.26 |
Heart | 76.56 ± 1.08 | 73.41 ± 2.14 | 76.19 ± 1.86 | 72.00 ± 1.80 | 76.41 ± 1.50 | 80.67 ± 1.91 | 80.78 ± 1.30 |
Hepatitis | 56.07 ± 2.17 | 60.12 ± 3.37 | 56.25 ± 3.53 | 56.96 ± 3.28 | 59.07 ± 2.56 | 59.09 ± 3.63 | 60.85 ± 4.00 |
SCADI | 77.14 ± 1.78 | 70.86 ± 1.93 | 75.43 ± 1.76 | 77.71 ± 1.20 | 70.57 ± 1.81 | 80.71 ± 1.81 | 81.29 ± 1.42 |
Dermatology | 93.96 ± 0.55 | 91.36 ± 0.61 | 94.15 ± 0.47 | 90.57 ± 0.44 | 92.02 ± 0.52 | 95.87 ± 0.64 | 96.21 ± 0.33 |
Ecoli | 81.39 ± 0.88 | 75.11 ± 1.01 | 81.52 ± 1.34 | 81.14 ± 1.04 | 80.97 ± 1.21 | 81.91 ± 0.84 | 82.61 ± 1.17 |
Ionosphere | 87.86 ± 0.97 | 90.37 ± 0.88 | 87.77 ± 0.74 | 88.69 ± 1.04 | 86.24 ± 1.51 | 88.04 ± 1.12 | 88.78 ± 0.57 |
Iris | 95.00 ± 1.05 | 93.93 ± 0.73 | 93.67 ± 0.65 | 95.00 ± 0.72 | 94.53 ± 1.29 | 95.40 ± 0.49 | 95.40 ± 0.49 |
Movement L | 66.25 ± 1.27 | 58.03 ± 1.74 | 63.28 ± 1.67 | 57.36 ± 1.17 | 40.47 ± 2.04 | 68.33 ± 1.42 | 68.97 ± 1.68 |
Sonar | 71.11 ± 2.83 | 69.83 ± 2.77 | 69.82 ± 2.47 | 67.64 ± 3.08 | 55.09 ± 1.87 | 76.42 ± 2.04 | 76.68 ± 2.60 |
Parkinsons | 85.59 ± 1.70 | 84.26 ± 2.05 | 85.38 ± 1.92 | 87.44 ± 0.78 | 85.33 ± 1.21 | 86.93 ± 2.43 | 87.89 ± 2.07 |
WDBC | 92.64 ± 0.55 | 93.04 ± 0.52 | 92.51 ± 0.58 | 92.79 ± 0.70 | 93.00 ± 0.44 | 93.62 ± 0.84 | 93.94 ± 0.27 |
Lymphography | 74.93 ± 1.84 | 77.13 ± 1.94 | 76.50 ± 2.55 | 78.05 ± 1.78 | 74.96 ± 2.52 | 77.27 ± 1.78 | 79.48 ± 2.54 |
Chess | 99.34 ± 0.11 | 80.97 ± 0.32 | 99.07 ± 0.12 | 98.31 ± 0.12 | 99.11 ± 0.09 | 99.14 ± 0.09 | 99.16 ± 0.09 |
Monks | 100.00 ± 0.00 | 77.77 ± 0.02 | 100.00 ± 0.00 | 100.00 ± 0.00 | 100.00 ± 0.00 | 100.00 ± 0.00 | 100.00 ± 0.00 |
Audiology | 77.10 ± 1.23 | 57.77 ± 1.74 | 65.68 ± 1.63 | 64.17 ± 1.63 | 73.76 ± 1.19 | 65.26 ± 1.80 | 74.52 ± 1.01 |
Average | 78.52 | 74.33 | 77.31 | 77.05 | 75.64 | 79.36 | 80.44 |
Dataset | Raw Data | FSFS | USQR | UEBR | UFRFS | FRUAR | FOUAR |
---|---|---|---|---|---|---|---|
Anneal | 91.38 ± 0.59 | 91.53 ± 0.29 | 91.21 ± 0.54 | 90.85 ± 0.60 | 91.02 ± 0.22 | 91.70 ± 0.41 | 93.00 ± 0.33 |
Autos | 72.85 ± 0.81 | 72.68 ± 1.60 | 73.99 ± 1.06 | 72.75 ± 1.35 | 71.47 ± 1.16 | 76.06 ± 2.27 | 77.92 ± 1.33 |
Bands | 78.64 ± 0.93 | 77.78 ± 0.70 | 77.57 ± 0.85 | 77.69 ± 0.82 | 77.65 ± 0.95 | 78.19 ± 0.61 | 78.30 ± 1.09 |
Cmc | 43.65 ± 0.26 | 47.54 ± 0.67 | 43.56 ± 0.54 | 43.52 ± 0.48 | 43.83 ± 0.25 | 42.75 ± 0.83 | 44.00 ± 0.54 |
Cleveland | 54.35 ± 0.93 | 51.68 ± 2.06 | 50.12 ± 2.08 | 50.89 ± 2.85 | 54.48 ± 0.76 | 52.14 ± 1.90 | 53.36 ± 1.33 |
CreditA | 80.83 ± 0.54 | 65.22 ± 0.50 | 80.84 ± 0.78 | 81.20 ± 0.56 | 80.32 ± 0.69 | 82.19 ± 0.67 | 81.19 ± 0.59 |
German | 68.66 ± 0.58 | 66.76 ± 0.49 | 69.81 ± 0.38 | 68.47 ± 0.74 | 67.95 ± 0.74 | 70.00 ± 0.35 | 70.15 ± 0.57 |
Heart disease | 75.04 ± 0.93 | 71.37 ± 0.87 | 74.37 ± 0.57 | 73.04 ± 1.29 | 74.70 ± 1.28 | 78.48 ± 0.99 | 78.52 ± 0.83 |
Hepatitis | 60.35 ± 1.97 | 62.52 ± 1.36 | 53.22 ± 1.30 | 58.58 ± 1.62 | 58.52 ± 2.07 | 60.82 ± 2.13 | 63.51 ± 1.45 |
SCADI | 79.86 ± 0.45 | 63.57 ± 1.21 | 71.86 ± 1.36 | 79.00 ± 0.69 | 64.43 ± 1.84 | 73.14 ± 0.90 | 79.14 ± 1.00 |
Dermatology | 95.56 ± 0.12 | 89.05 ± 0.53 | 88.03 ± 0.37 | 86.04 ± 0.66 | 88.49 ± 0.99 | 95.50 ± 0.47 | 95.93 ± 0.36 |
Ionosphere | 86.47 ± 0.68 | 87.43 ± 0.43 | 87.95 ± 0.45 | 89.06 ± 0.64 | 87.69 ± 0.36 | 87.66 ± 0.48 | 89.34 ± 0.43 |
Ecoli | 81.08 ± 0.54 | 75.32 ± 0.76 | 80.85 ± 0.75 | 80.90 ± 0.93 | 81.01 ± 1.07 | 81.01 ± 0.62 | 81.17 ± 0.85 |
Movement L | 85.97 ± 0.76 | 81.67 ± 0.49 | 85.06 ± 0.89 | 85.11 ± 0.51 | 86.11 ± 0.47 | 87.69 ± 1.01 | 87.17 ± 0.79 |
Iris | 95.20 ± 0.53 | 91.13 ± 1.18 | 90.87 ± 0.55 | 95.40 ± 0.38 | 95.53 ± 0.32 | 95.27 ± 0.49 | 95.27 ± 0.80 |
Sonar | 87.02 ± 0.87 | 87.15 ± 1.07 | 79.01 ± 1.47 | 77.18 ± 1.74 | 54.12 ± 1.08 | 88.46 ± 0.69 | 86.50 ± 1.06 |
Parkinsons | 96.26 ± 0.61 | 94.36 ± 0.49 | 93.33 ± 0.72 | 95.98 ± 0.73 | 92.65 ± 0.90 | 96.46 ± 0.71 | 96.72 ± 0.87 |
WDBC | 95.40 ± 0.25 | 95.19 ± 0.28 | 94.31 ± 0.29 | 94.76 ± 0.33 | 94.15 ± 0.27 | 95.80 ± 0.21 | 96.03 ± 0.34 |
Lymphography | 78.19 ± 1.39 | 67.24 ± 0.88 | 78.15 ± 1.05 | 78.31 ± 1.40 | 73.23 ± 1.66 | 77.80 ± 1.03 | 78.60 ± 1.26 |
Chess | 84.23 ± 0.31 | 67.58 ± 0.15 | 84.34 ± 0.16 | 84.56 ± 0.24 | 83.95 ± 0.37 | 84.83 ± 0.24 | 84.48 ± 0.14 |
Monks | 91.95 ± 0.65 | 77.78 ± 0.02 | 99.44 ± 0.43 | 91.06 ± 0.55 | 90.90 ± 0.51 | 99.45 ± 0.64 | 99.54 ± 0.38 |
Audiology | 66.07 ± 1.09 | 44.44 ± 0.76 | 58.06 ± 1.10 | 60.19 ± 0.56 | 61.25 ± 1.02 | 57.80 ± 0.64 | 62.18 ± 1.12 |
Average | 79.50 | 74.04 | 77.54 | 77.93 | 76.07 | 79.69 | 80.55 |
Dataset | Raw Attribute | K-Means (Number) | |
---|---|---|---|
Anneal | 38 | 3,32,34,33,12,8,35,5,4,36,37,1,9,17(14) | 3.0 |
Autos | 25 | 2,17,6,5,15,4,7,14(8) | 0.1 |
Bands | 39 | 2,3,14,25,24(5) | 1.4 |
Cmc | 10 | 2,7,8,3,1,6,4,5(8) | 2.1 |
Cleveland | 13 | 3,12,11,13,7,2,9,6(8) | 1.1 |
Credit approval | 15 | 6,4,10,7,12,9,1,2,3,14,8,13,11,15(14) | 1.1 |
German | 20 | 4,7,3,12,8,1,11,13,6,17,14,9(12) | 1.8 |
Heart disease | 13 | 3,11,13,7,2,9,6,12,8,4(10) | 0.3 |
Hepatitis | 19 | 6,4,10,12,7,1,11,9,5,8,13,3,14,19,17(15) | 0.2 |
SCADI | 70 | 105,34,48,5,123,172,1,2,201,68,81,187,181,145,66,167,178(17) | 1.0 |
Dermatology | 34 | 33,28,31,20,15,7,13,11,5,3,4,26,9,21,16,23,10,14,18,34,32,24,1,19(24) | 0.6 |
Ecoli | 7 | 3,7,6,5,2,1(6) | 0.7 |
Ionosphere | 33 | 1,6,27,26,8,16,14,4,18,13,20,12,24,28,22,10,2,9(18) | 0.3 |
Iris | 5 | 3,2,4(3) | 0.6 |
Movement libras | 90 | 8,15,55,72,31,44,79,18,85,45,9,38,40,28,80,58,39,48,89,67,1,46,14(24) | 3.0 |
Sonar | 60 | 3,60,2,4,31,21,30,20(8) | 0.1 |
Parkinsons | 22 | 9,18,17,20,21,22,16,3,1,10,2,11(12) | 2.3 |
WDBC | 31 | 1,15,22,18,29,19,30,7,26,9,23,11,10,27,3,6,12,16,4,5,25,24,2(23) | 0.1 |
Lymphography | 18 | 14,18,12,15,1,2,16,8,11,3(10) | - |
Chess | 36 | 9,7,11,35,10,34,13,33,24,6,5,15,36,26,18,12,21,20,4,22,23,16,30,31,1,27,17,2,8,3,19,25,28(33) | - |
Monks | 6 | 5,1,2,4,3 (5) | - |
Audiology | 69 | 64,66,4,6,2,60,15,1,10,5,57,40,7,38,11,25,17,53,24,41,44,59,52,47(24) | - |
Average | 36.72 | 13.68 | - |
Dataset | Raw Data | FSFS | USQR | UEBR | UFRFS | FRUAR | FOUAR |
---|---|---|---|---|---|---|---|
Anneal | 41.42 ± 6.13 | 35.85 ± 2.91 | 37.02 ± 4.64 | 36.70 ± 6.38 | 37.23 ± 3.45 | 36.50 ± 2.89 | 44.20 ± 0.66 |
Autos | 34.59 ± 1.55 | 28.05 ± 1.72 | 29.12 ± 1.83 | 28.68 ± 1.28 | 29.02 ± 2.31 | 28.20 ± 1.30 | 31.27 ± 2.64 |
Bands | 58.76 ± 0.02 | 58.76 ± 0.20 | 58.72 ± 0.19 | 59.19 ± 0.09 | 58.72 ± 0.19 | 58.64 ± 0.15 | 58.87 ± 0.15 |
Cmc | 41.50 ± 1.96 | 40.35 ± 2.19 | 41.12 ± 1.96 | 41.03 ± 2.26 | 41.89 ± 1.82 | 43.63 ± 1.86 | 43.96 ± 0.99 |
Cleveland | 35.21 ± 3.82 | 51.68 ± 2.06 | 50.12 ± 2.08 | 50.89 ± 2.85 | 36.80 ± 2.64 | 52.14 ± 1.90 | 53.36 ± 1.33 |
CreditA | 57.67 ± 3.09 | 56.59 ± 2.89 | 56.42 ± 2.72 | 55.62 ± 2.96 | 56.56 ± 2.89 | 56.72 ± 3.35 | 59.57 ± 3.06 |
German | 52.70 ± 0.00 | 53.35 ± 2.06 | 52.70 ± 0.00 | 52.70 ± 0.00 | 52.70 ± 0.00 | 52.70 ± 0.00 | 53.59 ± 2.81 |
Heart disease | 72.89 ± 7.18 | 76.30 ± 0.00 | 74.59 ± 5.39 | 59.78 ± 3.29 | 72.89 ± 7.18 | 76.30 ± 0.00 | 76.30 ± 0.00 |
Hepatitis | 61.61 ± 0.34 | 60.65 ± 0.00 | 59.68 ± 4.12 | 59.68 ± 0.55 | 61.03 ± 0.33 | 61.55 ± 1.02 | 61.68 ± 1.10 |
SCADI | 61.29 ± 7.48 | 39.29 ± 3.52 | 47.71 ± 10.57 | 52.71 ± 3.40 | 38.86 ± 2.92 | 47.57 ± 6.74 | 54.57 ± 8.85 |
Dermatology | 68.03 ± 12.84 | 68.74 ± 10.26 | 68.99 ± 6.72 | 65.46 ± 6.83 | 70.77 ± 12.85 | 75.44 ± 15.70 | 78.17 ± 11.49 |
Ionosphere | 70.34 ± 2.10 | 68.83 ± 9.12 | 71.23 ± 0.00 | 69.23 ± 0.00 | 70.74 ± 3.64 | 70.37 ± 0.00 | 70.91 ± 0.09 |
Ecoli | 62.02 ± 9.18 | 54.55 ± 5.34 | 58.13 ± 3.61 | 58.90 ± 6.44 | 61.82 ± 6.87 | 61.10 ± 4.33 | 63.84 ± 11.62 |
Movement L | 44.42 ± 2.55 | 42.08 ± 2.25 | 42.53 ± 1.89 | 36.44 ± 2.32 | 31.44 ± 1.38 | 42.00 ± 1.71 | 44.31 ± 2.67 |
Iris | 88.33 ± 0.35 | 83.33 ± 0.00 | 80.73 ± 8.22 | 88.40 ± 0.34 | 79.00 ± 14.95 | 94.67 ± 0.00 | 94.67 ± 0.00 |
Sonar | 54.37 ± 1.46 | 55.96 ± 0.41 | 67.50 ± 1.71 | 52.02 ± 0.63 | 56.20 ± 0.27 | 65.19 ± 0.99 | 64.76 ± 0.23 |
Parkinsons | 62.77 ± 0.26 | 69.90 ± 5.89 | 68.51 ± 0.26 | 64.92 ± 4.63 | 74.97 ± 1.02 | 72.56 ± 3.23 | 72.72 ± 4.01 |
WDBC | 92.79 ± 0.00 | 87.08 ± 0.86 | 90.16 ± 9.70 | 92.86 ± 0.36 | 88.52 ± 9.12 | 92.92 ± 0.08 | 94.20 ± 0.19 |
Lymphography | 47.70 ± 6.54 | 42.91 ± 5.49 | 48.31 ± 6.72 | 43.38 ± 5.31 | 47.64 ± 6.34 | 50.27 ± 6.61 | 50.34 ± 5.81 |
Chess | 52.10 ± 1.75 | 58.09 ± 2.67 | 52.40 ± 3.07 | 52.07 ± 2.33 | 50.96 ± 0.04 | 51.34 ± 1.17 | 53.90 ± 3.93 |
Monks | 62.69 ± 3.81 | 58.52 ± 6.94 | 65.19 ± 4.10 | 63.15 ± 1.79 | 63.19 ± 1.56 | 63.06 ± 2.97 | 65.00 ± 3.51 |
Audiology | 28.85 ± 2.44 | 25.97 ± 1.46 | 27.35 ± 2.22 | 25.84 ± 1.86 | 27.70 ± 2.08 | 25.93 ± 2.70 | 28.14 ± 2.92 |
Average | 56.91 | 55.31 | 56.74 | 54.98 | 54.94 | 58.12 | 59.92 |
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Wen, H.; Zhao, S.; Liang, M. Unsupervised Attribute Reduction Algorithm for Mixed Data Based on Fuzzy Optimal Approximation Set. Mathematics 2023, 11, 3452. https://doi.org/10.3390/math11163452
Wen H, Zhao S, Liang M. Unsupervised Attribute Reduction Algorithm for Mixed Data Based on Fuzzy Optimal Approximation Set. Mathematics. 2023; 11(16):3452. https://doi.org/10.3390/math11163452
Chicago/Turabian StyleWen, Haotong, Shixin Zhao, and Meishe Liang. 2023. "Unsupervised Attribute Reduction Algorithm for Mixed Data Based on Fuzzy Optimal Approximation Set" Mathematics 11, no. 16: 3452. https://doi.org/10.3390/math11163452