doi: 10.3934/jimo.2020129

A polyhedral conic functions based classification method for noisy data

Department of Industrial Engineering, Faculty of Engineering, Eskisehir Technical University Eskisehir, 26555, Turkey

* Corresponding author; The author published under the name Rafail N. Gasimov until 2007

Received  November 2019 Revised  July 2020 Published  August 2020

This paper presents a robust binary classification method, which is an extended version of the Modified Polyhedral Conic Functions (M-PCF) algorithm, earlier developed by Gasimov and Ozturk. The new version presented in this paper, has new features in comparison to the original algorithm. The mathematical model used in the new version, is relaxed by allowing some inaccuracies in an optimal way. By this way, it is aimed to reduce the overfitting and improve the generalization property. In the original version, the sublevel set of a separating function generated at every iteration, does not contain any element of the other set. This is changed in the new version, where the sublevel sets of separating functions generated by the new algorithm, are allowed to contain some elements from other set. On the other hand, the new algorithm uses a tolerance parameter which prevents generating "less productive separating functions". In the original version, the algorithm continues till all points of the "first" set are separated from the second one, where a separating function is generated if there still exist unseparated elements regardless the number of such elements. In the new version, the tolerance parameter is used to terminate iterations if there are only a few unseparated elements. By this way, it is aimed to improve the generalization property of the algorithm, and therefore the new version is called Parameterized Polyhedral Conic Functions (P-PCF) method. The performance and efficiency of the proposed algorithm is demonstrated on well-known datasets from the literature and on noisy data.

Citation: Müge Acar, Refail Kasimbeyli. A polyhedral conic functions based classification method for noisy data. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020129
References:
[1]

A. Astorino and M. Gaudioso, Polyhedral separability through successive LP, Journal of Optimization Theory and Applications, 112 (2002), 265-293.  doi: 10.1023/A:1013649822153.  Google Scholar

[2]

K. Bache and M. Lichman, UCI Machine Learning Repository. University of California, School of Information and Computer Science, (2013)., http://archive.ics.uci.edu/ml Google Scholar

[3]

A. M. Bagirov, Max–min separability, Optimization Methods and Software, 20 (2005), 277-296.  doi: 10.1080/10556780512331318263.  Google Scholar

[4]

A. M. BagirovG. Ozturk and R. Kasimbeyli, A sharp augmented Lagrangian-based method in constrained non-convex optimization, Optimization Methods and Software, 34 (2019), 462-488.  doi: 10.1080/10556788.2018.1496431.  Google Scholar

[5]

A. M. BagirovJ. UgonD. WebbG. Ozturk and and R. Kasimbeyli, A novel piecewise linear classifier based on polyhedral conic and max–min separabilities, TOP, 21 (2013), 3-24.  doi: 10.1007/s11750-011-0241-5.  Google Scholar

[6]

K. P. Bennett and O. L. Mangasarian, Robust linear programming discrimination of two linearly inseparable sets, Optimization Methods and Software, 1 (1992), 23-34.   Google Scholar

[7]

C. E. Brodley and M. A. Friedl, Identifying mislabeled training data, Journal of Artificial Intelligence Research, 11 (1999), 131-167.   Google Scholar

[8]

E. Cimen and G. Ozturk, O-PCF algorithm for one-class classification, Optimization Methods and Software, (2019), 1–15. Google Scholar

[9]

W. W. Cohen, Fast effective rule induction, Proceedings of the Twelfth International Conference on Machine Learning, ML95, San Francisco, CA, 115–123. Google Scholar

[10]

C. Cortes and V. Vapnik, Support-vector networks, Machine Learning, 20 (1995), 273-297.  doi: 10.1007/BF00994018.  Google Scholar

[11]

R. N. Gasimov and G. Ozturk, Separation via polyhedral conic functions, Optimization Methods and Software, 21 (2006), 527-540.  doi: 10.1080/10556780600723252.  Google Scholar

[12]

R. N. Gasimov and O. Ustun, Solving the quadratic assignment problem using F-MSG algorithm, Journal of Industrial and Management Optimization, 3 (2007), 173-191.  doi: 10.3934/jimo.2007.3.173.  Google Scholar

[13]

M. HallE. FrankG. HolmesB. PfahringerP. Reutemann and I. H. Witten, The WEKA data mining software: An update, SIGKDD Explorations, 11 (2003), 10-18.   Google Scholar

[14]

N. Kasimbeyli and R. Kasimbeyli, A representation theorem for Bishop-Phelps cones, Pacific Journal of Optimization, 13 (2017), 55-74.   Google Scholar

[15]

R. Kasimbeyli, A nonlinear cone separation theorem and scalarization in nonconvex vector optimization, SIAM Journal on Optimization, 20 (2010), 1591-1619.  doi: 10.1137/070694089.  Google Scholar

[16]

R. Kasimbeyli, Radial epiderivatives and set-valued optimization, Optimization, 58 (2009), 521-534.  doi: 10.1080/02331930902928310.  Google Scholar

[17]

R. Kasimbeyli and M. Karimi, Separation theorems for nonconvex sets and application in optimization, Operations Research Letters, 47 (2019), 569-573.  doi: 10.1016/j.orl.2019.09.011.  Google Scholar

[18]

R. Kasimbeyli and M. Mammadov, Optimality conditions in nonconvex optimization via weak subdifferentials, Nonlinear Analysis: Theory, Methods and Applications, 74 (2011), 2534-2547.  doi: 10.1016/j.na.2010.12.008.  Google Scholar

[19]

R. KasimbeyliO. Ustun and A. Rubinov, The modified subgradient algorithm based on feasible values, Optimization, 58 (2009), 535-560.  doi: 10.1080/02331930902928419.  Google Scholar

[20]

D. T. Larose and C. D. Larose, Discovering knowledge in data: An introduction to data mining, John Wiley & Sons, Hoboken, NJ, 2005.  Google Scholar

[21]

C. J. Mantas and J. Abell'an, Credal-C4.5 decision tree based on imprecise probabilities to classify noisy data. Expert Systems with Applications, 41(10) (2014), 4625-4637. Google Scholar

[22]

G. J. McLachlan, Discriminant Analysis and Statistical Pattern Recognition, John Wiley & Sons, Inc., New York, 1992. doi: 10.1002/0471725293.  Google Scholar

[23]

G. OzturkA. M. Bagirov and R. Kasimbeyli, An incremental piecewise linear classifier based on polyhedral conic separation, Machine Learning, 101 (2015), 397-413.  doi: 10.1007/s10994-014-5449-9.  Google Scholar

[24]

G. Ozturk and M. T. Ciftci, Clustering based polyhedral conic functions algorithm in classification, Journal of Industrial and Management Optimization, 11 (3) (2015), 921-932.  doi: 10.3934/jimo.2015.11.921.  Google Scholar

[25]

J. R. Quinlan, The effect of noise on concept learning, Machine Learning, (1986), 149–166. Google Scholar

[26]

A. M. Rubinov and R. N. Gasimov, Strictly increasing positively homogeneous functions with applications to exact penalization, Optimization, 52 (2003), 1-28.  doi: 10.1080/0233193021000058931.  Google Scholar

[27]

J. A. SáezM. GalarJ. Luengo and F. Herrera, Tackling the problem of classification with noisy data using multiple classifier systems: Analysis of the performance and robustness, Information Sciences, 247 (2013), 1-20.   Google Scholar

[28]

X. Zhu and X. Wu, Class noise vs. attribute noise: A quantitative study, Artificial Intelligence Review, 22 (2004), 177-210.   Google Scholar

show all references

References:
[1]

A. Astorino and M. Gaudioso, Polyhedral separability through successive LP, Journal of Optimization Theory and Applications, 112 (2002), 265-293.  doi: 10.1023/A:1013649822153.  Google Scholar

[2]

K. Bache and M. Lichman, UCI Machine Learning Repository. University of California, School of Information and Computer Science, (2013)., http://archive.ics.uci.edu/ml Google Scholar

[3]

A. M. Bagirov, Max–min separability, Optimization Methods and Software, 20 (2005), 277-296.  doi: 10.1080/10556780512331318263.  Google Scholar

[4]

A. M. BagirovG. Ozturk and R. Kasimbeyli, A sharp augmented Lagrangian-based method in constrained non-convex optimization, Optimization Methods and Software, 34 (2019), 462-488.  doi: 10.1080/10556788.2018.1496431.  Google Scholar

[5]

A. M. BagirovJ. UgonD. WebbG. Ozturk and and R. Kasimbeyli, A novel piecewise linear classifier based on polyhedral conic and max–min separabilities, TOP, 21 (2013), 3-24.  doi: 10.1007/s11750-011-0241-5.  Google Scholar

[6]

K. P. Bennett and O. L. Mangasarian, Robust linear programming discrimination of two linearly inseparable sets, Optimization Methods and Software, 1 (1992), 23-34.   Google Scholar

[7]

C. E. Brodley and M. A. Friedl, Identifying mislabeled training data, Journal of Artificial Intelligence Research, 11 (1999), 131-167.   Google Scholar

[8]

E. Cimen and G. Ozturk, O-PCF algorithm for one-class classification, Optimization Methods and Software, (2019), 1–15. Google Scholar

[9]

W. W. Cohen, Fast effective rule induction, Proceedings of the Twelfth International Conference on Machine Learning, ML95, San Francisco, CA, 115–123. Google Scholar

[10]

C. Cortes and V. Vapnik, Support-vector networks, Machine Learning, 20 (1995), 273-297.  doi: 10.1007/BF00994018.  Google Scholar

[11]

R. N. Gasimov and G. Ozturk, Separation via polyhedral conic functions, Optimization Methods and Software, 21 (2006), 527-540.  doi: 10.1080/10556780600723252.  Google Scholar

[12]

R. N. Gasimov and O. Ustun, Solving the quadratic assignment problem using F-MSG algorithm, Journal of Industrial and Management Optimization, 3 (2007), 173-191.  doi: 10.3934/jimo.2007.3.173.  Google Scholar

[13]

M. HallE. FrankG. HolmesB. PfahringerP. Reutemann and I. H. Witten, The WEKA data mining software: An update, SIGKDD Explorations, 11 (2003), 10-18.   Google Scholar

[14]

N. Kasimbeyli and R. Kasimbeyli, A representation theorem for Bishop-Phelps cones, Pacific Journal of Optimization, 13 (2017), 55-74.   Google Scholar

[15]

R. Kasimbeyli, A nonlinear cone separation theorem and scalarization in nonconvex vector optimization, SIAM Journal on Optimization, 20 (2010), 1591-1619.  doi: 10.1137/070694089.  Google Scholar

[16]

R. Kasimbeyli, Radial epiderivatives and set-valued optimization, Optimization, 58 (2009), 521-534.  doi: 10.1080/02331930902928310.  Google Scholar

[17]

R. Kasimbeyli and M. Karimi, Separation theorems for nonconvex sets and application in optimization, Operations Research Letters, 47 (2019), 569-573.  doi: 10.1016/j.orl.2019.09.011.  Google Scholar

[18]

R. Kasimbeyli and M. Mammadov, Optimality conditions in nonconvex optimization via weak subdifferentials, Nonlinear Analysis: Theory, Methods and Applications, 74 (2011), 2534-2547.  doi: 10.1016/j.na.2010.12.008.  Google Scholar

[19]

R. KasimbeyliO. Ustun and A. Rubinov, The modified subgradient algorithm based on feasible values, Optimization, 58 (2009), 535-560.  doi: 10.1080/02331930902928419.  Google Scholar

[20]

D. T. Larose and C. D. Larose, Discovering knowledge in data: An introduction to data mining, John Wiley & Sons, Hoboken, NJ, 2005.  Google Scholar

[21]

C. J. Mantas and J. Abell'an, Credal-C4.5 decision tree based on imprecise probabilities to classify noisy data. Expert Systems with Applications, 41(10) (2014), 4625-4637. Google Scholar

[22]

G. J. McLachlan, Discriminant Analysis and Statistical Pattern Recognition, John Wiley & Sons, Inc., New York, 1992. doi: 10.1002/0471725293.  Google Scholar

[23]

G. OzturkA. M. Bagirov and R. Kasimbeyli, An incremental piecewise linear classifier based on polyhedral conic separation, Machine Learning, 101 (2015), 397-413.  doi: 10.1007/s10994-014-5449-9.  Google Scholar

[24]

G. Ozturk and M. T. Ciftci, Clustering based polyhedral conic functions algorithm in classification, Journal of Industrial and Management Optimization, 11 (3) (2015), 921-932.  doi: 10.3934/jimo.2015.11.921.  Google Scholar

[25]

J. R. Quinlan, The effect of noise on concept learning, Machine Learning, (1986), 149–166. Google Scholar

[26]

A. M. Rubinov and R. N. Gasimov, Strictly increasing positively homogeneous functions with applications to exact penalization, Optimization, 52 (2003), 1-28.  doi: 10.1080/0233193021000058931.  Google Scholar

[27]

J. A. SáezM. GalarJ. Luengo and F. Herrera, Tackling the problem of classification with noisy data using multiple classifier systems: Analysis of the performance and robustness, Information Sciences, 247 (2013), 1-20.   Google Scholar

[28]

X. Zhu and X. Wu, Class noise vs. attribute noise: A quantitative study, Artificial Intelligence Review, 22 (2004), 177-210.   Google Scholar

Figure 1.  The illustration of M-PCF algorithm on the training set for Example 1
Figure 2.  The illustration of M-PCF algorithm on the test set for Example 1
Figure 3.  The illustration of P-PCF algorithm on the training set for Example 1
Figure 4.  The illustration of P-PCF algorithm on the test set for Example 1
Table 1.  Data illustrating different noise types
No. Attribute 1 Attribute 2 Class Status
1 0.26 small A
2 0.25 small A
3 0.29 small B class noise
4 1.02 large B
5 1.05 large B
6 0.30 large B attribute noise
No. Attribute 1 Attribute 2 Class Status
1 0.26 small A
2 0.25 small A
3 0.29 small B class noise
4 1.02 large B
5 1.05 large B
6 0.30 large B attribute noise
Table 2.  Data for Example 1
Training set $ A $ Test set $ A $ Training set $ B $ Test set $ B $
(x, y) (x, y) (x, y) (x, y)
(2, 4) (3, 5) (4, 19) (5, 20)
(2, 6) (3, 7) (6, 19) (7, 20)
(2, 8) (3, 9) (8, 19) (11, 20)
(2, 10) (3, 11) (10, 19) (13, 20)
(2, 12) (3, 13) (12, 19) (15, 20)
(2, 14) (5, 5) (14, 19) (17, 20)
(4, 4) (5, 7) (16, 19) (17, 18)
(4, 8) (5, 9) (18, 19) (17, 16)
(4, 10) (5, 11) (16, 17) (17, 14)
(4, 14) (5, 13) (18, 17) (17, 12)
(6, 4) (7, 5) (16, 15) (17, 10)
(6, 6) (7, 7) (18, 15) -
(6, 8) (7, 9) (16, 13) -
(6, 10) (7, 11) (18, 13) -
(6, 12) (7, 13) (16, 11) -
(6, 14) (9, 5) (16, 9) -
(8, 4) (9, 7) (18, 9) -
(8, 6) (9, 9) (17, 8) -
(8, 8) (9, 11) (17, 6) -
(8, 10) (9, 13) (17, 4) -
(8, 12) (11, 5) (14, 20.5) -
(8, 14) (16, 7) (4, 6) -
(10, 4) (18, 7) (10, 6) -
(10, 8) (16, 5) (4, 12) -
(10, 10) (16, 3) (10, 12) -
(10, 14) (11, 7) (6, 21) -
(12, 4) (11, 9) (10, 21) -
(12, 6) (11, 11) (12, 21) -
(12, 8) (11, 13) (16, 21) -
(12, 10) - (18, 21) -
(12, 12) - - -
(12, 14) - - -
(8, 21) - - -
(14, 21) - - -
(18, 11) - - -
(18, 5) - - -
Training set $ A $ Test set $ A $ Training set $ B $ Test set $ B $
(x, y) (x, y) (x, y) (x, y)
(2, 4) (3, 5) (4, 19) (5, 20)
(2, 6) (3, 7) (6, 19) (7, 20)
(2, 8) (3, 9) (8, 19) (11, 20)
(2, 10) (3, 11) (10, 19) (13, 20)
(2, 12) (3, 13) (12, 19) (15, 20)
(2, 14) (5, 5) (14, 19) (17, 20)
(4, 4) (5, 7) (16, 19) (17, 18)
(4, 8) (5, 9) (18, 19) (17, 16)
(4, 10) (5, 11) (16, 17) (17, 14)
(4, 14) (5, 13) (18, 17) (17, 12)
(6, 4) (7, 5) (16, 15) (17, 10)
(6, 6) (7, 7) (18, 15) -
(6, 8) (7, 9) (16, 13) -
(6, 10) (7, 11) (18, 13) -
(6, 12) (7, 13) (16, 11) -
(6, 14) (9, 5) (16, 9) -
(8, 4) (9, 7) (18, 9) -
(8, 6) (9, 9) (17, 8) -
(8, 8) (9, 11) (17, 6) -
(8, 10) (9, 13) (17, 4) -
(8, 12) (11, 5) (14, 20.5) -
(8, 14) (16, 7) (4, 6) -
(10, 4) (18, 7) (10, 6) -
(10, 8) (16, 5) (4, 12) -
(10, 10) (16, 3) (10, 12) -
(10, 14) (11, 7) (6, 21) -
(12, 4) (11, 9) (10, 21) -
(12, 6) (11, 11) (12, 21) -
(12, 8) (11, 13) (16, 21) -
(12, 10) - (18, 21) -
(12, 12) - - -
(12, 14) - - -
(8, 21) - - -
(14, 21) - - -
(18, 11) - - -
(18, 5) - - -
Table 3.  Original data for Example 2 (from [11])
Training set $ A $ Test set $ A $ Training set $ B $ Test set $ B $
(x, y) (x, y) (x, y) (x, y)
(14.5, -2) (-0.5, 2) (1, 6) (20, -6)
(0.5, 2) (14.5, 2) (-6, -6) (-6, -1)
(2, -0.5) (13.5, -2) (8, -1) (-1, -6)
(-2, 2) - (15, -6) (8, 6)
(16, -0.5) - (-6, 6) (8, 1)
(16, 0.5) - (8, -6) (15, 6)
(12, -2) - (6, 6) (-6, 1)
(0.5, -2) - (6, 1) (1, -6)
(12, 0.5) - (20, 1) (6, 1)
(16, -2) - (20, -1) -
(2, 2) - (20, 6) -
(2, 0.5) - (13, 6) -
(-2, -2) - (-1, 6) -
(12, 2) - (6, -6) -
(13.5, 2) - (13, -6) -
(-0.5, -2) - - -
(2, -0.5) - - -
(16, 2) - - -
(-2, 0.5) - - -
(-2, -0.5) - - -
(12, -0.5) - - -
Training set $ A $ Test set $ A $ Training set $ B $ Test set $ B $
(x, y) (x, y) (x, y) (x, y)
(14.5, -2) (-0.5, 2) (1, 6) (20, -6)
(0.5, 2) (14.5, 2) (-6, -6) (-6, -1)
(2, -0.5) (13.5, -2) (8, -1) (-1, -6)
(-2, 2) - (15, -6) (8, 6)
(16, -0.5) - (-6, 6) (8, 1)
(16, 0.5) - (8, -6) (15, 6)
(12, -2) - (6, 6) (-6, 1)
(0.5, -2) - (6, 1) (1, -6)
(12, 0.5) - (20, 1) (6, 1)
(16, -2) - (20, -1) -
(2, 2) - (20, 6) -
(2, 0.5) - (13, 6) -
(-2, -2) - (-1, 6) -
(12, 2) - (6, -6) -
(13.5, 2) - (13, -6) -
(-0.5, -2) - - -
(2, -0.5) - - -
(16, 2) - - -
(-2, 0.5) - - -
(-2, -0.5) - - -
(12, -0.5) - - -
Table 4.  Modified data with noise ratio of %60, for Example 2
Training set $ A $ Test set $ A $ Training set $ B $ Test set $ B $
(x, y) (x, y) (x, y) (x, y)
(16, 0.5) (12, -2) (-6, -6) (6, 6)
(-0.5, -2) (0.5, -2) (8, -1) (6, 1)
(16, 2) (12, 0.5) (15, -6) (20, 1)
(-2, 0.5) (16, -2) (-6, 6) (20, -1)
(12, -0.5) (2, 2) (6, -6) (20, 6)
(1, 6) (2, 0.5) (-6, -1) (13, 6)
(8, -6) - (8, 6) -
(-1, 6) - (8, 1) -
(13, -6) - (15, 6) -
(20, -6) - (6, -1) -
(-1, -6) - (14.5, -2) -
(-6, 1) - (0.5, 2) -
(1, -6) - (2, -2) -
- - (-2, 2) -
- - (16, -0, 5) -
- - (-2, -2) -
- - (12, 2) -
- - (13.5, 2) -
- - (2, -0.5) -
- - (-2, -0.5) -
- - (-0.5, 2) -
- - (14.5, 2) -
- - (13.5, -2) -
Training set $ A $ Test set $ A $ Training set $ B $ Test set $ B $
(x, y) (x, y) (x, y) (x, y)
(16, 0.5) (12, -2) (-6, -6) (6, 6)
(-0.5, -2) (0.5, -2) (8, -1) (6, 1)
(16, 2) (12, 0.5) (15, -6) (20, 1)
(-2, 0.5) (16, -2) (-6, 6) (20, -1)
(12, -0.5) (2, 2) (6, -6) (20, 6)
(1, 6) (2, 0.5) (-6, -1) (13, 6)
(8, -6) - (8, 6) -
(-1, 6) - (8, 1) -
(13, -6) - (15, 6) -
(20, -6) - (6, -1) -
(-1, -6) - (14.5, -2) -
(-6, 1) - (0.5, 2) -
(1, -6) - (2, -2) -
- - (-2, 2) -
- - (16, -0, 5) -
- - (-2, -2) -
- - (12, 2) -
- - (13.5, 2) -
- - (2, -0.5) -
- - (-2, -0.5) -
- - (-0.5, 2) -
- - (14.5, 2) -
- - (13.5, -2) -
Table 5.  Classification accuracies obtained for Example 2
P-PCF Algorithm M-PCF Algorithm
Training Test Training Test
Original Data 88.89 85.41 100 83.33
Noisy Data 61.80 56.25 100 52.08
P-PCF Algorithm M-PCF Algorithm
Training Test Training Test
Original Data 88.89 85.41 100 83.33
Noisy Data 61.80 56.25 100 52.08
Table 6.  Properties of datasets. Dataset description: $ N $ is the number of instances in the dataset, $ m $ is the number of instances in the first class, $ p $ is the number of instances in the second class, and $ n $ is the number of attributes
Dataset Short Name $ N $ $ m $ $ p $ $ n $
Wisconsin Breast Cancer Wis 683 444 239 10
German-Credit Ger 1000 700 300 21
Haberman Hab 306 225 81 4
Hearth-statlog Hea 270 137 160 14
Ionosphere Ion 351 126 225 35
Liver-disorders Liv 345 145 200 7
Sonar Son 208 111 107 61
Australian credit Aus 690 383 307 14
Monk Monk 432 228 204 6
Dataset Short Name $ N $ $ m $ $ p $ $ n $
Wisconsin Breast Cancer Wis 683 444 239 10
German-Credit Ger 1000 700 300 21
Haberman Hab 306 225 81 4
Hearth-statlog Hea 270 137 160 14
Ionosphere Ion 351 126 225 35
Liver-disorders Liv 345 145 200 7
Sonar Son 208 111 107 61
Australian credit Aus 690 383 307 14
Monk Monk 432 228 204 6
Table 7.  Training and test accuracies obtained by applying M-PCF and P-PCF methods for the original data
M-PCF Algorithm P-PCF Algorithm
Training Test Training Test
Wis 100 98.50 98.59 96.13
Ger 100 72.41 82.56 73.80
Hab 100 74.27 86.97 74.25
Hea 100 84.41 93.67 84.76
Ion 100 88.42 94.87 88.96
Liv 100 68.87 78.43 69.40
Son 100 70.24 80.47 71.09
Aus 100 85.42 87.2 86.23
Monk 100 99.82 100 99.02
M-PCF Algorithm P-PCF Algorithm
Training Test Training Test
Wis 100 98.50 98.59 96.13
Ger 100 72.41 82.56 73.80
Hab 100 74.27 86.97 74.25
Hea 100 84.41 93.67 84.76
Ion 100 88.42 94.87 88.96
Liv 100 68.87 78.43 69.40
Son 100 70.24 80.47 71.09
Aus 100 85.42 87.2 86.23
Monk 100 99.82 100 99.02
Table 8.  Test accuracies obtained for datasets with %0 noise
Datasets M-PCF P-PCF SVM 1-NN 3-NN C 4.5
Wis 98.50 96.13 95.91 91.21 95.61 92.39
Ger 72.41 73.80 70.35 68.50 67.70 74.5
Hab 74.27 74.25 73.82 68.48 68.28 69.42
Hea 84.41 84.76 78.88 69.99 68.47 70.73
Ion 88.42 88.96 90.48 90.22 89.98 89.87
Liv 68.87 69.40 61.12 59.17 58.87 58.96
Son 70.24 71.09 78.21 89.75 82.52 71.18
Aus 85.42 86.23 85.51 80.73 85.8 84.35
Monk2 99.82 92.02 80.56 75.69 97.92 99.5
Datasets M-PCF P-PCF SVM 1-NN 3-NN C 4.5
Wis 98.50 96.13 95.91 91.21 95.61 92.39
Ger 72.41 73.80 70.35 68.50 67.70 74.5
Hab 74.27 74.25 73.82 68.48 68.28 69.42
Hea 84.41 84.76 78.88 69.99 68.47 70.73
Ion 88.42 88.96 90.48 90.22 89.98 89.87
Liv 68.87 69.40 61.12 59.17 58.87 58.96
Son 70.24 71.09 78.21 89.75 82.52 71.18
Aus 85.42 86.23 85.51 80.73 85.8 84.35
Monk2 99.82 92.02 80.56 75.69 97.92 99.5
Table 9.  Test accuracies obtained for datasets with %5 noise
Datasets M-PCF P-PCF SVM 1NN 3NN C 4.5
Wis 86.84 96.14 96.34 89.16 94.29 92.80
Ger 70.8 72.40 73.37 68.44 65.98 63.01
Hab 63.9 74.44 72.17 67.29 66.25 68.47
Hea 68.89 78.44 77.84 62.58 67.03 69.99
Ion 67.85 85.84 89.18 88.02 89.10 88.28
Liv 61.46 67.76 55.29 59.52 53.85 59.45
Son 67.14 70.52 74.37 86.53 83.31 68.64
Aus 78.21 85.37 81.74 72.75 80.15 81.16
Monk 88.42 98.21 77.55 73.84 90.05 95.14
Datasets M-PCF P-PCF SVM 1NN 3NN C 4.5
Wis 86.84 96.14 96.34 89.16 94.29 92.80
Ger 70.8 72.40 73.37 68.44 65.98 63.01
Hab 63.9 74.44 72.17 67.29 66.25 68.47
Hea 68.89 78.44 77.84 62.58 67.03 69.99
Ion 67.85 85.84 89.18 88.02 89.10 88.28
Liv 61.46 67.76 55.29 59.52 53.85 59.45
Son 67.14 70.52 74.37 86.53 83.31 68.64
Aus 78.21 85.37 81.74 72.75 80.15 81.16
Monk 88.42 98.21 77.55 73.84 90.05 95.14
Table 10.  Test accuracies obtained for datasets with %10 noise
Datasets M-PCF P-PCF SVM 1NN 3NN C 4.5
Wis 84.27 95.85 96.05 86.52 91.95 93.26
Ger 68.4 73.8 70.64 65.28 64.97 60.24
Hab 60.54 75.65 70.28 64.23 65.42 67.93
Hea 68.52 79.26 76.03 58.14 62.95 65.18
Ion 65.45 83.17 81.57 86.25 88.47 85.23
Liv 60.29 67.49 55.50 54.07 59.20 51.36
Son 62.64 68.56 73.97 81.18 81.11 52.36
Aus 72.01 83.48 77.54 69.42 74.93 74.89
Monk 78.56 96.98 74.77 69.91 81.48 89.35
Datasets M-PCF P-PCF SVM 1NN 3NN C 4.5
Wis 84.27 95.85 96.05 86.52 91.95 93.26
Ger 68.4 73.8 70.64 65.28 64.97 60.24
Hab 60.54 75.65 70.28 64.23 65.42 67.93
Hea 68.52 79.26 76.03 58.14 62.95 65.18
Ion 65.45 83.17 81.57 86.25 88.47 85.23
Liv 60.29 67.49 55.50 54.07 59.20 51.36
Son 62.64 68.56 73.97 81.18 81.11 52.36
Aus 72.01 83.48 77.54 69.42 74.93 74.89
Monk 78.56 96.98 74.77 69.91 81.48 89.35
Table 11.  Test accuracies obtained for datasets with %20 noise
Datasets M-PCF P-PCF SVM 1NN 3NN C 4.5
Wis 79.56 95.43 95.32 81.24 84.91 90.19
Ger 65.8 71.56 65.14 60.48 62.11 61.40
Hab 54.28 74.54 67.25 63.28 62.47 65.37
Hea 67.04 74.82 69.46 56.91 58.14 60.60
Ion 62.03 80.34 81.43 82.27 86.07 79.45
Liv 59.72 66.70 54.28 55.01 58.76 51.69
Son 59.64 65.98 70.84 73.91 78.21 63.9
Aus 64.34 82.03 70.73 63.04 65.36 68.41
Monk 71.78 92.34 66.9 62.73 70.83 77.55
Datasets M-PCF P-PCF SVM 1NN 3NN C 4.5
Wis 79.56 95.43 95.32 81.24 84.91 90.19
Ger 65.8 71.56 65.14 60.48 62.11 61.40
Hab 54.28 74.54 67.25 63.28 62.47 65.37
Hea 67.04 74.82 69.46 56.91 58.14 60.60
Ion 62.03 80.34 81.43 82.27 86.07 79.45
Liv 59.72 66.70 54.28 55.01 58.76 51.69
Son 59.64 65.98 70.84 73.91 78.21 63.9
Aus 64.34 82.03 70.73 63.04 65.36 68.41
Monk 71.78 92.34 66.9 62.73 70.83 77.55
Table 12.  Test accuracies obtained for datasets with %30 noise
Datasets M-PCF P-PCF SVM 1NN 3NN C 4.5
Wis 69.95 92.85 92.74 75.41 77.31 87.56
Ger 62.4 69.21 63.78 57.62 56.70 53.17
Hab 51.83 68.66 62.87 61.82 60.44 63.53
Hea 60.37 68.15 65.28 51.47 53.69 46.60
Ion 61.48 80.71 76.98 78.51 84.27 77.81
Liv 56.58 63.37 54.12 48.66 55.40 43.45
Son 52.64 60.01 68.64 65.23 72.80 65.75
Aus 52.78 73.77 62.46 55.65 55.8 58.84
Monk 51.47 90.61 60.19 53.7 60.42 66.44
Datasets M-PCF P-PCF SVM 1NN 3NN C 4.5
Wis 69.95 92.85 92.74 75.41 77.31 87.56
Ger 62.4 69.21 63.78 57.62 56.70 53.17
Hab 51.83 68.66 62.87 61.82 60.44 63.53
Hea 60.37 68.15 65.28 51.47 53.69 46.60
Ion 61.48 80.71 76.98 78.51 84.27 77.81
Liv 56.58 63.37 54.12 48.66 55.40 43.45
Son 52.64 60.01 68.64 65.23 72.80 65.75
Aus 52.78 73.77 62.46 55.65 55.8 58.84
Monk 51.47 90.61 60.19 53.7 60.42 66.44
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