A polyhedral conic functions based classification method for noisy data

Müge Acar; Refail Kasimbeyli

doi:10.3934/jimo.2020129

Article Contents

2021, Volume 17, Issue 6: 3493-3508. Doi: 10.3934/jimo.2020129

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A polyhedral conic functions based classification method for noisy data

Müge Acar and
Refail Kasimbeyli^,

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

Received: November 2019

Revised: July 2020

Early access: August 2020

Published: November 2021

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 62H30, 65K0590C25, 90C90; Secondary: 68Q32.

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Figure 1. The illustration of M-PCF algorithm on the training set for Example 1

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Figure 2. The illustration of M-PCF algorithm on the test set for Example 1

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Figure 3. The illustration of P-PCF algorithm on the training set for Example 1

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Figure 4. The illustration of P-PCF algorithm on the test set for Example 1

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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

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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)	-	-	-

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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)	-	-	-

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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)	-

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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

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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

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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

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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

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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

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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

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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

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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

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