Fuzzy inference system (FIS) is a process of fuzzy logic reasoning to produce the output based on fuzzified inputs. The system starts with identifying input from data, applying the fuzziness to input using membership functions (MF), generating fuzzy rules for the fuzzy sets and obtaining the output. There are several types of input MFs which can be introduced in FIS, commonly chosen based on the type of real data, sensitivity of certain rule implied and computational limits. This paper focuses on the construction of interval type 2 (IT2) trapezoidal shape MF from fuzzy C Means (FCM) that is used for fuzzification process of mamdani FIS. In the process, upper MF (UMF) and lower MF (LMF) of the MF need to be identified to get the range of the footprint of uncertainty (FOU). This paper proposes Genetic tuning process, which is a part of genetic algorithm (GA), to adjust parameters in order to improve the behavior of existing system, especially to enhance the accuracy of the system model. This novel process is a hybrid approach which produces Genetic Fuzzy System (GFS) that helps to enhance fuzzy classification problems and performance. The approach provides a new method for the construction and tuning process of the IT2 MF, based on the FCM outcomes. The result is compared to Gaussian shape IT2 MF and trapezoid IT2 MF generated by the classic GA method. It is shown that the proposed approach is able to outperform the mentioned benchmarked approaches. The work implies a wider range of IT2 MF types, constructed based on FCM outcomes, and an optimum generation of the FOU so that it can be implemented in practical applications such as prediction, analytics and rule-based solutions.

The three main types of fuzzy membership function (MF) are Gaussian [

The purpose of Genetic Algorithm (GA) as part of producing the FIS is to generate IT2 trapezoidal MF (referred as trapMF in this paper) by using the theory of genetic tuning and lateral adjustment [

In GFS, the concept of weak ignorance function is introduced. It is used to evaluate or measure the uncertainty of membership values [

The composed MFs are usually obtained by a normalization process and they remain the same during the rule based construction process [

Tuning process in GFS is commonly applied to improve the performance of fuzzy model. It introduces an alterations of MF shapes to improve global interaction which induces better cooperation among rules [

The tuning steps will involve a simultaneous step of adjusting the amplitude with varying range, by moving only two parameters (left and right) of trapMF. At the same time, lateral adjustment based on 2-tuples fuzzy linguistic representation of the MF is also performed, where the trapMF parameters are adjusted to the left or right position based on varying range in order to achieve better adaptation of the fuzzy partitions. In order to prove the enhancement in term of performance of this methodology, the result of the fuzzy modeling through FIS will be compared with the generic tuning of trapMF whereby it follows the initial situation of genetic weak ignorance tuning [

This paper is structured as follows; next section contains literature review, the explanation of FIS which becomes the research methodology is presented in Section 3, while results and discussion are contained in Section 4. The paper ends with conclusion in Section 5 in which the research is summarized as well as some future works are outlined.

The modification of the LMF amplitude of trapMF is based on ignorance function [

The approximation of the genetic weak tuning in trapMF is initialized with the parameters for UMF. The equation of the straight line between parameter

Total certainty (

Maximum uncertainty (

Based on the above situation, the initial value set for

The initial FT1 trapMF is built first from the previous section, which is represented by the MF representation below:

In order to calculate the range of FOU for IT2 trapMF, the proposition in [

Based on

Obtain each cluster center and sigma:

Point

Point

Point (

From A, the equation for amplitude of straight line (

B =

In order to achieve the best possible range of FOU through this weak tuning, a range of parameter

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

0.05 | 0.10 | 0.15 | 0.18 | 0.20 | 0.25 | 0.5 (Maximum ignorance) |

In [

The adjustment is performed by considering two parameters, α and β, which represents the lateral displacement and the amplitude variation. Next, each label can be represented by a 3-tuple (

_{s2}, is computed in

where _{S} and _{S} are respectively the right and the left extreme of the support of _{S} is the size of the support of

To show the enhancement of the tuning method, the lateral adjustment is modified so that it has similar range with the applied amplitude, to ensure better stability and accuracy of the MF. Hence, the proposition of the alignment of amplitude and lateral adjustment of the IT2 trapMF is α =

Based on

Based on the Gaussian output, the parameter center,

FIS process in

Based on _{L}) and upper limit (_{R}). It is frequently referred as the IT2 centroid, which is based on

where

_{i} = ith output value

_{umf} = UMF

_{lmf} = LMF

To estimate the values for _{L} and _{R}, iterative type-reduction methods are used [

The approximation of the genetic weak tuning in trapMF is initialized with the parameters for UMF. The equation of the straight line between parameter

For the purpose of testing the theory, the same dataset to generate FT1 trapMF from Gaussian output is used to ensure the continuation of research objective, which is to generate IT2 trapMF from FCM. The total of six datasets, (QoS, Iris, Abalone, HCV, BirchSet, Europe), which are popular among benchmark datasets are prepared to test for result accuracy [

First amplitude parameter W 0.05 is performed in all datasets, and according to result, IT2 trapMF is failed to be generated as the number of uncertainties becomes close 0, with values of UMF and LMF are the same except for very slight difference.

Since this method comes from GA components, further evaluation of the output is performed using MSE to test for accuracy of FIS using the IT2 trapMF. The MSE is measured to identify the best

From the error measurements, it is proven that genetic weak tuning can be used to construct IT2 trapMF, with the cooperation of lateral adjustment.

Total error | MSE | % Error | |
---|---|---|---|

LA1 | 1778.53 | 18.935 | 35.46 |

LA2 | 1824.16 | 19.427 | 36.36 |

Accuracy | Precision | Recall | F-Measure | |
---|---|---|---|---|

IT2 WTLA1-TRAP | 70.331 | 69.342 | 79.633 | 74.132 |

IT2 WTLA2-TRAP | 69.324 | 72.345 | 63.823 | 67.817 |

IT2 FCMTrap-GA | 70.121 | 71.923 | 62.934 | 67.129 |

IT2FCM | 64.233 | 69.012 | 60.123 | 64.262 |

% IT2 WTLA1-TRAP/IT2 WTLA2-TRAP | 1.43 | −4.33 | 19.85 | 8.52 |

% IT2 WTLA1-TRAP/IT2 FCMTrap-GA | 0.30 | −3.72 | 20.97 | 9.45 |

% IT2 WTLA1-TRAP/IT2FCM | 8.67 | 0.48 | 24.50 | 13.31 |

% IT2 WTLA2-TRAP/IT2 FCMTrap-GA | −1.15 | 0.58 | 1.39 | 1.02 |

% IT2 WTLA2-TRAP/IT2FCM | 7.34 | 4.61 | 5.80 | 5.24 |

IT2 WTLA1-TRAP | 87.332 | 78.233 | 75.231 | 76.703 |

IT2 WTLA2-TRAP | 82.103 | 79.023 | 71.213 | 74.915 |

IT2 FCMTrap-GA | 77.234 | 64.234 | 62.923 | 63.572 |

IT2FCM | 66.123 | 60.623 | 59.444 | 60.028 |

% IT2 WTLA1-TRAP/IT2 WTLA2-TRAP | 5.99 | −1.01 | 5.34 | 2.33 |

% IT2 WTLA1-TRAP/IT2 FCMTrap-GA | 1.29 | 1.56 | 1.59 | 1.57 |

% IT2 WTLA1-TRAP/IT2FCM | 24.29 | 22.51 | 20.98 | 21.74 |

% IT2 WTLA2-TRAP/IT2 FCMTrap-GA | 5.93 | 18.71 | 11.64 | 15.14 |

% IT2 WTLA2-TRAP/IT2FCM | 19.46 | 23.28 | 16.53 | 19.87 |

IT2 WTLA1-TRAP | 67.623 | 71.274 | 78.233 | 74.592 |

IT2 WTLA2-TRAP | 60.823 | 63.673 | 60.213 | 61.895 |

IT2 FCMTrap-GA | 60.342 | 59.234 | 58.232 | 58.729 |

IT2FCM | 60.344 | 60.456 | 59.001 | 59.720 |

% IT2 WTLA1-TRAP/IT2 WTLA2-TRAP | 10.06 | 10.66 | 23.03 | 17.02 |

% IT2 WTLA1-TRAP/IT2 FCMTrap-GA | 1.66 | 1.69 | 1.72 | 1.70 |

% IT2 WTLA1-TRAP/IT2FCM | 10.76 | 15.18 | 24.58 | 19.94 |

% IT2 WTLA2-TRAP/IT2 FCMTrap-GA | 0.79 | 6.97 | 3.29 | 5.12 |

% IT2 WTLA2-TRAP/IT2FCM | 0.79 | 5.05 | 2.01 | 3.51 |

In this section, the proposed method that construct IT2 Trap for FCM using both tuning methods is implemented in FIS. The output of FIS is then evaluated against the other FIS with input MFs from classic IT2FCM and trapezoidal input MFs optimized using GA (IT2 FCMTrap-GA). The proposed algorithm with the discussed two tuning methods are named IT2WTLA1-TRAP and IT2WTLA2-TRAP respectively.

Accuracy of the FIS are measured through several evaluation metrics, which are Classification accuracy, Precision, Recall, F-Measure. Confusion matrix is used to obtain the evaluation metrics. Other evaluation measurements to measure the performance are Mean Absolute Error (MAE) and Root Mean Square Error (RMSE).

Different tuning in the proposed method also being evaluated against each other to identify which tuning method has better performance. As for example, the calculation to get the percentage of difference is described in

Referring to

MAE | RMSE | |||
---|---|---|---|---|

Without noise | 5% Noise | Without noise | 5% Noise | |

IT2 WTLA1 TRAP | 0.4002 | 0.4122 | 0.4212 | 0.5092 |

IT2 WTLA2 TRAP | 0.4023 | 0.4132 | 0.5024 | 0.5087 |

IT2 FCM Trap-GA | 0.3923 | 0.4192 | 0.4942 | 0.5102 |

IT2 FCM Gaussian | 0.4566 | 0.4551 | 0.5261 | 0.5257 |

IT2 WTLA1 TRAP | 0.0234 | 0.0124 | 0.1832 | 0.1867 |

IT2 WTLA2 TRAP | 0.7234 | 0.7192 | 0.8232 | 0.8512 |

IT2 FCM Trap-GA | 1.9168 | 1.9221 | 2.7440 | 2.7102 |

IT2 FCM Gaussian | 2.2697 | 2.3023 | 2.8880 | 2.9023 |

IT2 WTLA1 TRAP | 22.8342 | 20.0421 | 23.6723 | 23.2823 |

IT2 WTLA2 TRAP | 23.2368 | 23.1928 | 28.9316 | 28.9011 |

IT2 FCM Trap GA | 22.7154 | 29.7639 | 55.8020 | 55.9350 |

IT2 FCM Gaussian | 40.2624 | 40.2104 | 73.5748 | 73.5348 |

Statistical test is carried out after obtaining all results to further analyze the significance of the experiment results. One-way analysis of variance or ANOVA (

IRIS | ||||||

Between groups | 207.1816 | 3 | 69.06053 | 191.7236 | 5.14E−87 | 2.61985358 |

Within groups | 214.6844 | 596 | 0.360209 | |||

Total | 421.866 | 599 | ||||

HCV | ||||||

Between groups | 24014458 | 3 | 8004819 | 446.1219 | 2.02E−229 | 2.60868638 |

Within groups | 42202218 | 2352 | 17943.12 | |||

Total | 66216676 | 2355 | ||||

ABALONE | ||||||

Between groups | 79632.73 | 3 | 26544.24 | 2583.307 | 0.002 | 2.60544073 |

Within groups | 171638.5 | 16704 | 10.2753 | |||

Total | 251271.3 | 16707 |

In this paper, we presented a method to generate IT2 trapMF by using approximation of genetic weak tuning and lateral adjustment. The UMF is initially prepared, and the LMF is approximated by the combination of the two elements of genetic algorithm. The result shows that it successfully generates symmetrical trapMF with minimum

There are several areas of improvement that can be incorporated in the present work and can be further extended. The possible future directions are discussed as follows:

Different genetic tuning methods other than amplitude and lateral adjustments can be explored to assist on IT2 MF constructions for different shapes of MFs.

Investigations on single amplitude tuning or lateral adjustments to construct IT2 trapMF with larger range without limited to weak ignorance function.

Introducing a method that can heuristically produce the correct trapezoidal shapes given any types of data. The improvement in term of accuracy or precision, for example, can be further explored.

Optimizing IT2 FIS results through the application of other than GA in MF generation processes.

Investigating of the IT2 MFs generated by FCM being applied in the general fuzzy type 2 FIS. The performance effects in various general fuzzy type 2 applications can be further explored.

The research is carried out under the funding of the Fundamental Research Grant Scheme, granted by the Ministry of Higher Education, Malaysia, and Yayasan Universiti Teknologi PETRONAS, granted by Universiti Teknologi PETRONAS.