# Interval Type-3 Fuzzy Inference System Design for Medical Classification Using Genetic Algorithms

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Type-3 Fuzzy Logic

## 3. Genetic Algorithms

## 4. Proposed Method

#### 4.1. Description of the Method

_{Low}, MF

_{Medium}, and MF

_{High}). In this work, the Trapezoidal membership function is used due to its ability to represent a triangular membership function by joining its central points [31]. The optimal parameters of the Type-3 Trapezoidal MF (a

_{1}, b

_{1}, c

_{1}, d

_{1}, LowerScale, and LowerLag) and the fuzzy if-then rules are designed by a real-coded GA. In Figure 5, a representation of the proposed method is shown, where a Sugeno Model is applied with n attributes to finally obtain a final result.

#### 4.2. Datasets Description

#### 4.3. Application to Medical Classification

#### 4.4. Description of the GA

_{a1}, p

_{b1}, p

_{c1}, p

_{d1}, ${p}_{\lambda}$, ${p}_{\mathcal{l}1}$, and ${p}_{\mathcal{l}2}$) and there are three Type-3 Trapezoidal MF in each fuzzy input. Three membership functions are used because previous works have shown that this number of membership functions allows good results for classification problems [30,31], as well as in other applications [47]. The three constants of the output are added to this multiplication, and finally, the multiplication of TR by 2 (consequents and activation status). For example, Haberman’s Survival dataset has three attributes, n = 3. Therefore, TR = 27, so the chromosome size is 119 genes. A representation of the chromosome applied for the Interval Type-3 FIS design is shown in Figure 8.

- The dataset is divided into design and testing sets.
- The range of the fuzzy input and the maximum number of fuzzy rules are established based on the design test, the search space of the GA is determined, and the individuals are established with random values.
- Each individual allows the design of each fuzzy inference system. The parameters of the membership functions are established.
- The same individual is allowed to know if a fuzzy rule will be added to the fuzzy inference system using the genes assigned to this task. The values of these genes are values between 0 and 1. If the value is equal to or less than 0.5, the fuzzy rules are omitted, and if the value is greater than 0.5, the fuzzy rules are added, and a consequent is assigned.
- When the fuzzy inference system is fully designed, the testing set is used to prove the FIS, where each instance is evaluated for the fuzzy inference system, and the resulting value determines its class.

## 5. Experimental Results

#### 5.1. Haberman’s Survival Dataset Results

#### 5.2. PIMA Indian Diabetes Dataset Results

## 6. Discussion

#### 6.1. Statistical Comparison

_{0}is rejected. There is enough evidence to affirm that the proposed method is better than T1FIS and IT2FIS applied to the PIMA Indian Diabetes dataset using only five attributes and 30% of the instances for the evaluation of the FIS designed.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Fuzzy inputs for Haberman’s Survival dataset: (

**a**) age of patient; (

**b**) year of operation; and (

**c**) number of positive axillary nodes detected.

**Figure 11.**Best Type-3 fuzzy inputs variables for Haberman’s Survival dataset: (

**a**) age of patient; (

**b**) year of operation; and (

**c**) number of positive axillary nodes detected.

**Figure 13.**Best Type-3 fuzzy inputs variables for PIMA Indian Diabetes dataset: (

**a**) glucose; (

**b**) blood pressure; (

**c**) body mass index; (

**d**) diabetes pedigree function; and (

**e**) age.

Dataset | Attributes | Instances |
---|---|---|

Haberman’s Survival | 3 | 306 |

PIMA Indian Diabetes | 5 and 7 | 336 |

Cryotherapy | 7 | 90 |

Immunotherapy | 8 | 90 |

PIMA Indian Diabetes | 8 | 768 |

Indian Liver | 9 | 583 |

Breast Cancer Coimbra | 10 | 116 |

Attributes | R_{min} | R_{max} |
---|---|---|

Age (attr1) | 27 | 73.70 |

Op_Year (attr2) | 52.20 | 75.9 |

Axil_Nodes (attr3) | 0 | 57.2 |

Parameters | Minimum | Maximum | |
---|---|---|---|

Trapezoidal MFs (a_{1}, b_{1}, c_{1}, d_{1}) | - | - | |

LowerScale ($\lambda $) | 0.1 | 0.9 | |

LowerLag ($\mathcal{l}$_{1}, $\mathcal{l}$_{2}). | 0.1 | 0.9 | |

Output | 0 | 1 | |

Fuzzy if-then rules | Consequents | 1 | 3 |

(Activation or deactivation) | 0 | 1 |

Rule | Antecedents | Consequent | ||
---|---|---|---|---|

Age | Output | Axil_Nodes | Output | |

1 | MF_{Low} | MF_{Low} | MF_{High} | MF_{High} |

2 | MF_{Low} | MF_{Medium} | MF_{Low} | MF_{Low} |

3 | MF_{Low} | MF_{Medium} | MF_{Medium} | MF_{Medium} |

4 | MF_{Low} | MF_{Medium} | MF_{High} | MF_{Medium} |

5 | MF_{Low} | MF_{High} | MF_{Low} | MF_{Low} |

6 | MF_{Medium} | MF_{Low} | MF_{Low} | MF_{Medium} |

7 | MF_{Medium} | MF_{Medium} | MF_{Low} | MF_{Medium} |

8 | MF_{Medium} | MF_{High} | MF_{Medium} | MF_{High} |

9 | MF_{High} | MF_{Low} | MF_{Low} | MF_{Low} |

10 | MF_{High} | MF_{Medium} | MF_{Low} | MF_{Medium} |

11 | MF_{High} | MF_{Medium} | MF_{Medium} | MF_{Medium} |

12 | MF_{High} | MF_{Medium} | MF_{High} | MF_{High} |

13 | MF_{High} | MF_{High} | MF_{Low} | MF_{Medium} |

Design | Testing | ||
---|---|---|---|

(Best) | (Average) | (Best) | (Average) |

79.89% | 77.14% | 77.05% | 75.30% |

Design | Testing | ||
---|---|---|---|

(Best) | (Average) | (Best) | (Average) |

86.38% | 83.65% | 83.17% | 81.52% |

Rule | Antecedents | Consequent | ||||
---|---|---|---|---|---|---|

Glucose | BP | BMI | DPF | AGE | Output | |

1 | MF_{Low} | MF_{Medium} | MF_{Low} | MF_{Medium} | MF_{High} | MF_{Low} |

2 | MF_{Low} | MF_{Medium} | MF_{Low} | MF_{High} | MF_{Medium} | MF_{Low} |

3 | MF_{Low} | MF_{Medium} | MF_{Medium} | MF_{Medium} | MF_{High} | MF_{High} |

4 | MF_{Low} | MF_{Medium} | MF_{Medium} | MF_{High} | MF_{Medium} | MF_{High} |

5 | MF_{Medium} | MF_{Low} | MF_{Medium} | MF_{Medium} | MF_{Medium} | MF_{High} |

6 | MF_{Medium} | MF_{Medium} | MF_{Low} | MF_{Medium} | MF_{Low} | MF_{Low} |

7 | MF_{Medium} | MF_{Medium} | MF_{Low} | MF_{High} | MF_{Low} | MF_{Medium} |

8 | MF_{Medium} | MF_{Medium} | MF_{Medium} | MF_{Medium} | MF_{Low} | MF_{High} |

9 | MF_{Medium} | MF_{Medium} | MF_{Medium} | MF_{Medium} | MF_{Medium} | MF_{Medium} |

10 | MF_{Medium} | MF_{High} | MF_{Medium} | MF_{Low} | MF_{Medium} | MF_{High} |

11 | MF_{High} | MF_{Low} | MF_{Low} | MF_{High} | MF_{High} | MF_{High} |

12 | MF_{High} | MF_{Medium} | MF_{Low} | MF_{Medium} | MF_{Medium} | MF_{High} |

13 | MF_{High} | MF_{Medium} | MF_{Medium} | MF_{Low} | MF_{Low} | MF_{High} |

14 | MF_{High} | MF_{Medium} | MF_{Medium} | MF_{Medium} | MF_{High} | MF_{High} |

15 | MF_{High} | MF_{Medium} | MF_{High} | MF_{Low} | MF_{Medium} | MF_{Medium} |

16 | MF_{High} | MF_{High} | MF_{Medium} | MF_{Low} | MF_{Low} | MF_{Medium} |

17 | MF_{High} | MF_{High} | MF_{Medium} | MF_{Low} | MF_{Low} | MF_{Medium} |

18 | MF_{High} | MF_{High} | MF_{Medium} | MF_{Low} | MF_{Low} | MF_{Medium} |

19 | MF_{High} | MF_{High} | MF_{Medium} | MF_{Low} | MF_{Low} | MF_{Medium} |

20 | MF_{High} | MF_{Medium} | MF_{Medium} | MF_{High} | MF_{Medium} | MF_{Medium} |

21 | MF_{High} | MF_{Medium} | MF_{Medium} | MF_{High} | MF_{High} | MF_{Medium} |

22 | MF_{High} | MF_{High} | MF_{High} | MF_{Low} | MF_{High} | MF_{Medium} |

23 | MF_{High} | MF_{Medium} | MF_{High} | MF_{High} | MF_{Medium} | MF_{High} |

24 | MF_{High} | MF_{High} | MF_{High} | MF_{Low} | MF_{Medium} | MF_{High} |

25 | MF_{High} | MF_{High} | MF_{Medium} | MF_{Medium} | MF_{Low} | MF_{Medium} |

26 | MF_{High} | MF_{High} | MF_{Medium} | MF_{Medium} | MF_{Medium} | MF_{Medium} |

27 | MF_{High} | MF_{High} | MF_{High} | MF_{Medium} | MF_{Medium} | MF_{Low} |

Dataset | k-Fold | % Design Set | % Testing Set | Design | Testing | ||
---|---|---|---|---|---|---|---|

Mean | Std Dev | Mean | Std Dev | ||||

Haberman’s Survival | - | 60 | 40 | 77.14 | 1.3906 | 75.30 | 0.6361 |

5 | 80 | 20 | 76.78 | 1.3645 | 76.69 | 1.1509 | |

10 | 90 | 10 | 76.55 | 0.8739 | 77.73 | 2.0233 | |

Cryotherapy | - | 60 | 40 | 85.12 | 4.0829 | 87.13 | 1.5446 |

5 | 80 | 20 | 85.42 | 3.1570 | 86.67 | 2.0286 | |

10 | 90 | 10 | 89.17 | 2.1510 | 89.67 | 1.6605 | |

Immunotherapy | - | 60 | 40 | 84.26 | 2.4194 | 82.04 | 2.2759 |

5 | 80 | 20 | 84.86 | 1.8391 | 85.56 | 2.0951 | |

10 | 90 | 10 | 84.40 | 2.3520 | 83.78 | 1.8295 | |

PIMA Indian Diabetes | - | 60 | 40 | 74.26 | 1.4760 | 77.76 | 1.1085 |

5 | 80 | 20 | 74.11 | 1.7514 | 77.18 | 0.3064 | |

10 | 90 | 10 | 74.46 | 1.8471 | 77.66 | 1.7584 | |

Indian Liver | - | 60 | 40 | 72.45 | 0.7416 | 71.86 | 0.4014 |

5 | 80 | 20 | 71.85 | 0.4863 | 72.40 | 0.6870 | |

10 | 90 | 10 | 71.93 | 0.7086 | 72.26 | 0.9167 | |

Breast Cancer Coimbra | - | 60 | 40 | 63.19 | 4.5129 | 71.06 | 2.1350 |

5 | 80 | 20 | 66.19 | 3.1440 | 72.78 | 2.2471 | |

10 | 90 | 10 | 75.58 | 1.2419 | 74.91 | 1.4342 | |

PIMA Indian Diabetes (5 attributes) | - | 70 | 30 | 83.65 | 0.9194 | 81.52 | 1.1133 |

Method | Design | Testing | ||
---|---|---|---|---|

(Best) | (Average) | (Best) | (Average) | |

T1FIS | 83.44% | 81.46% | 80.20% | 76.34% |

IT2FIS | 86.68% | 82.45% | 83.17% | 78.68% |

IT3FIS | 86.38% | 83.65% | 83.17% | 81.52% |

Dataset | Hold-Out | 5-Fold | 10-Fold |
---|---|---|---|

Haberman’s Survival | 00:04:22 | 00:16:36 | 00:32:54 |

Cryotherapy | 00:02:54 | 00:11:08 | 00:24:11 |

Immunotherapy | 00:02:32 | 00:10:41 | 00:21:04 |

PIMA Indian Diabetes | 00:24:46 | 01:43:04 | 03:50:29 |

Indian Liver | 00:23:32 | 02:04:30 | 03:38:27 |

Breast Cancer Coimbra | 00:05:45 | 00:23:34 | 00:45:00 |

PIMA Indian Diabetes (5 attributes) | 05:24:34 | - | - |

Parameter | Value |
---|---|

Significance | 0.95 |

H_{0} | µ1 = µ2 |

H_{1} | µ1 > µ2 |

Critical Value (Z-test/T-test) | 1.645/1.812 |

Method | N | Mean | Std Dev | z-Value | p-Value |
---|---|---|---|---|---|

IT3FIS | 30 | 81.52 | 1.1133 | 9.7750 | 1.36 × 10^{−21} |

T1FIS | 30 | 76.34 | 2.6814 | ||

IT3FIS | 30 | 81.52 | 1.1133 | 6.6860 | 1.54 × 10^{−10} |

IT2FIS | 30 | 78.68 | 2.0429 |

Dataset | Method | N | Mean | Std Dev | z-Value | p-Value |
---|---|---|---|---|---|---|

Haberman’s Survival | IT3FIS | 30 | 75.30 | 0.6361 | 2.8115 | 0.0025 |

GT2FIS | 30 | 74.01 | 2.4285 | |||

Cryotherapy | IT3FIS | 30 | 87.13 | 1.5446 | 2.0482 | 0.0203 |

GT2FIS | 30 | 85.52 | 4.0122 | |||

Immunotherapy | IT3FIS | 30 | 82.04 | 2.2759 | 2.2462 | 0.0123 |

GT2FIS | 30 | 78.79 | 7.5888 | |||

PIMA Indian Diabetes | IT3FIS | 30 | 77.76 | 1.1085 | 2.5702 | 0.0051 |

GT2FIS | 30 | 76.60 | 2.2156 | |||

Indian Liver | IT3FIS | 30 | 71.86 | 0.4014 | 0.7714 | 0.2202 |

GT2FIS | 30 | 71.48 | 2.6631 | |||

Breast Cancer Coimbra | IT3FIS | 30 | 71.06 | 2.1350 | 0.7483 | 0.2271 |

GT2FIS | 30 | 69.87 | 8.4725 |

**Table 14.**Values of t-tests using 20% of the instances for validation of the FIS with 5 cross-validations.

Dataset | Method | N | Mean | Std Dev | t-Value | p-Value |
---|---|---|---|---|---|---|

Haberman’s Survival | IT3FIS | 10 | 76.69 | 1.1509 | 6.1340 | 3.69 × 10^{−5} |

GT2FIS | 10 | 74.30 | 0.4450 | |||

Cryotherapy | IT3FIS | 10 | 86.67 | 2.0286 | 2.6674 | 0.0081 |

GT2FIS | 10 | 83.89 | 2.594 | |||

Immunotherapy | IT3FIS | 10 | 85.56 | 2.0951 | 18.5907 | 4.54 × 10^{−12} |

GT2FIS | 10 | 71.05 | 1.3027 | |||

PIMA Indian Diabetes | IT3FIS | 10 | 77.18 | 0.3064 | 6.3802 | 4.56 × 10^{−6} |

GT2FIS | 10 | 76.17 | 0.3970 | |||

Indian Liver | IT3FIS | 10 | 72.40 | 0.6870 | 2.7279 | 0.0074 |

GT2FIS | 10 | 71.36 | 0.9830 | |||

Breast Cancer Coimbra | IT3FIS | 10 | 72.78 | 2.2471 | 2.5457 | 0.0117 |

GT2FIS | 10 | 70.70 | 1.2927 |

**Table 15.**Values of t-tests using 10% of the instances for validation of the FIS with 10 cross-validations.

Dataset | Method | N | Mean | Std Dev | t-Value | p Value |
---|---|---|---|---|---|---|

Haberman’s Survival | IT3FIS | 10 | 77.73 | 2.0233 | 4.6205 | 4.75 × 10^{−4} |

GT2FIS | 10 | 74.67 | 0.5578 | |||

Cryotherapy | IT3FIS | 10 | 89.67 | 1.6605 | 5.4189 | 3.56 × 10^{−5} |

GT2FIS | 10 | 86.22 | 1.133 | |||

Immunotherapy | IT3FIS | 10 | 83.78 | 1.8295 | 7.0236 | 1.02 × 10^{−6} |

GT2FIS | 10 | 77.78 | 1.9876 | |||

PIMA Indian Diabetes | IT3FIS | 10 | 77.66 | 1.7584 | 2.8421 | 0.0097 |

GT2FIS | 10 | 76.05 | 0.338 | |||

Indian Liver | IT3FIS | 10 | 72.26 | 0.9167 | 2.1748 | 0.0252 |

GT2FIS | 10 | 71.57 | 0.4110 | |||

Breast Cancer Coimbra | IT3FIS | 10 | 74.91 | 1.4342 | 0.5277 | 0.3027 |

GT2FIS | 10 | 74.45 | 2.316 |

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**MDPI and ACS Style**

Melin, P.; Sánchez, D.; Castillo, O.
Interval Type-3 Fuzzy Inference System Design for Medical Classification Using Genetic Algorithms. *Axioms* **2024**, *13*, 5.
https://doi.org/10.3390/axioms13010005

**AMA Style**

Melin P, Sánchez D, Castillo O.
Interval Type-3 Fuzzy Inference System Design for Medical Classification Using Genetic Algorithms. *Axioms*. 2024; 13(1):5.
https://doi.org/10.3390/axioms13010005

**Chicago/Turabian Style**

Melin, Patricia, Daniela Sánchez, and Oscar Castillo.
2024. "Interval Type-3 Fuzzy Inference System Design for Medical Classification Using Genetic Algorithms" *Axioms* 13, no. 1: 5.
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