A Design and Its Application of Multi-Granular Fuzzy Model with Hierarchical Tree Structures
Abstract
:1. Introduction
2. Generation of Information Granules and Design of the Granular Model
2.1. Context-Based Fuzzy C-Means Clustering
- [Step 1]
- Set the number of contexts to be generated and the cluster’s number to be estimated for each context. In addition, initialize the membership matrix having a value between 0 and 1.
- [Step 2]
- Generate the contexts via triangular fuzzy sets that are flexibly distributed in the output variable. The generation method of the context can be changed based on the user’s settings.
- [Step 3]
- Calculate the cluster centers for each context using Equation (4).
- [Step 4]
- Compute the objective function using the equation below. The above process stops if the value updated through the previous iteration is less than the threshold.
- [Step 5]
- Calculate a new membership matrix by the equation. Then, go to [Step 3].
Algorithm 1. The context based fuzzy C means clustering algorithm. |
Begin Choose the type of context to be created in the output space (e.g., type 1 = uniform, type 2 = flexible); Create p contexts; Randomly initialize v cluster centers; for to do Update the membership matrix U; Calculate the new cluster centers V Calculate the new objective function J; if (abs(Jt − Jt−1)<∈) then break; else Jt−1 = Jt; end if end for end |
2.2. CFCM-Based Granular Fuzzy Model
3. CFCM-Based Granular Model with Hierarchical Tree Structure
3.1. Hierarchical Tree Structures
3.2. CFCM-Based Multi-Granular Fuzzy Model (MGFM) with Hierarchical Tree Structure
- [Step 1]
- Calculate the positive and negative correlations of the input variables based on the correlations with the output variables in the database to be used.
- [Step 2]
- By using the positive and negative correlation ranks, designate input variables (input variables with high correlation ranks) to be input into the low-level granular model of the aggregated structure and input variables (input variables with low correlation ranks) to be input into the granular model of the incremental structure.
- [Step 3]
- Sets the context’s number to create in the low-level granular model and the cluster’s number to be produced per context. In addition, initialize the membership matrix .
- [Step 4]
- Create contexts and clusters using context-based FCM clustering to generate fuzzy rules automatically. The generated fuzzy rules are then used to calculate the output of the lower-level granular model.
- [Step 5]
- Process the output of the low-level granular model, the output variables of the database, and the input variables with low correlation ranks that will be used in the incremental structure so that they can be used as inputs to the high-level granular model.
- [Step 6]
- Use the processed database as input to the high-level granular model and generate fuzzy rules using CFCM clustering to calculate the final output.
3.3. Performance Evaluation of CFCM-Based MGFM
4. Experimental Results and Comments
4.1. Database
4.2. Experimental Methods and Results Analysis
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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m | Context | C | 2 | 3 | 4 | 5 | 6 | |
P | ||||||||
1.5 | Uniform | 2 | 0 | 0 | 0 | 0 | 0 | |
3 | 0.2687 | 0.2701 | 0.2855 | 0.3002 | 0.2826 | |||
4 | 0.3647 | 0.3690 | 0.3838 | 0.3790 | 0.3431 | |||
5 | 0.4600 | 0.4101 | 0.4177 | 0.4143 | 0.3932 | |||
6 | 0.4526 | 0.4651 | 0.4304 | 0.3800 | 0.4182 | |||
Context | C | 2 | 3 | 4 | 5 | 6 | ||
P | ||||||||
Flexible | 2 | 0 | 0 | 0 | 0 | 0 | ||
3 | 0.2871 | 0.2597 | 0.2246 | 0.2805 | 0.3187 | |||
4 | 0.3893 | 0.3701 | 0.3843 | 0.3690 | 0.3533 | |||
5 | 0.4105 | 0.3902 | 0.4050 | 0.4099 | 0.3793 | |||
6 | 0.4346 | 0.4419 | 0.3970 | 0.3890 | 0.3915 | |||
m | Context | C | 2 | 3 | 4 | 5 | 6 | |
P | ||||||||
2 | Uniform | 2 | 0 | 0 | 0 | 0 | 0 | |
3 | 0.3027 | 0.2907 | 0.2945 | 0.3069 | 0.3115 | |||
4 | 0.4294 | 0.4017 | 0.4014 | 0.3759 | 0.3828 | |||
5 | 0.4278 | 0.4028 | 0.4481 | 0.4036 | 0.4165 | |||
6 | 0.4091 | 0.4445 | 0.4447 | 0.4213 | 0.4619 | |||
Context | C | 2 | 3 | 4 | 5 | 6 | ||
P | ||||||||
Flexible | 2 | 0 | 0 | 0 | 0 | 0 | ||
3 | 0.3248 | 0.3184 | 0.2914 | 0.3055 | 0.3004 | |||
4 | 0.4494 | 0.4301 | 0.4605 | 0.4197 | 0.4520 | |||
5 | 0.4771 | 0.4160 | 0.4203 | 0.4689 | 0.4453 | |||
6 | 0.4219 | 0.4416 | 0.4311 | 0.4534 | 0.4258 |
m | Context | C | 2 | 3 | 4 | 5 | 6 | |
P | ||||||||
1.5 | Uniform | 2 | 0 | 0 | 0 | 0 | 0 | |
3 | 0.2493 | 0.2766 | 0.2808 | 0.2966 | 0.2747 | |||
4 | 0.3915 | 0.3568 | 0.3854 | 0.3833 | 0.3573 | |||
5 | 0.4505 | 0.3713 | 0.3921 | 0.4036 | 0.4398 | |||
6 | 0.4538 | 0.4602 | 0.4559 | 0.4060 | 0.4247 | |||
Context | C | 2 | 3 | 4 | 5 | 6 | ||
P | ||||||||
Flexible | 2 | 0 | 0 | 0 | 0 | 0 | ||
3 | 0.2696 | 0.2620 | 0.2132 | 0.2714 | 0.3021 | |||
4 | 0.3860 | 0.3593 | 0.3432 | 0.3509 | 0.3809 | |||
5 | 0.4098 | 0.4505 | 0.4110 | 0.4170 | 0.3844 | |||
6 | 0.4055 | 0.4408 | 0.4080 | 0.4090 | 0.3865 | |||
m | Context | C | 2 | 3 | 4 | 5 | 6 | |
P | ||||||||
2 | Uniform | 2 | 0 | 0 | 0 | 0 | 0 | |
3 | 0.3001 | 0.2874 | 0.2909 | 0.2805 | 0.2962 | |||
4 | 0.4068 | 0.4149 | 0.3938 | 0.4254 | 0.4290 | |||
5 | 0.4641 | 0.4599 | 0.4435 | 0.4600 | 0.4513 | |||
6 | 0.4553 | 0.4737 | 0.4530 | 0.4479 | 0.4710 | |||
Context | C | 2 | 3 | 4 | 5 | 6 | ||
P | ||||||||
Flexible | 2 | 0 | 0 | 0 | 0 | 0 | ||
3 | 0.2935 | 0.2837 | 0.3133 | 0.3320 | 0.2877 | |||
4 | 0.4395 | 0.4266 | 0.4178 | 0.4270 | 0.4334 | |||
5 | 0.4458 | 0.4610 | 0.4351 | 0.4850 | 0.4734 | |||
6 | 0.4597 | 0.4909 | 0.4798 | 0.4864 | 0.4724 |
m | Context | C | 2 | 3 | 4 | 5 | 6 | |
P | ||||||||
1.5 | Uniform | 2 | 0 | 0 | 0 | 0 | 0 | |
3 | 0.2679 | 0.2735 | 0.2797 | 0.2616 | 0.2577 | |||
4 | 0.3767 | 0.3430 | 0.3439 | 0.3382 | 0.3763 | |||
5 | 0.3823 | 0.3163 | 0.3502 | 0.3501 | 0.3392 | |||
6 | 0.4212 | 0.4330 | 0.4835 | 0.4001 | 0.3689 | |||
Context | C | 2 | 3 | 4 | 5 | 6 | ||
P | ||||||||
Flexible | 2 | 0 | 0 | 0 | 0 | 0 | ||
3 | 0.2920 | 0.2473 | 0.2734 | 0.2831 | 0.2607 | |||
4 | 0.3854 | 0.3441 | 0.3670 | 0.3759 | 0.3284 | |||
5 | 0.3646 | 0.3487 | 0.3548 | 0.3914 | 0.3714 | |||
6 | 0.3517 | 0.3646 | 0.3558 | 0.4101 | 0.3671 | |||
m | Context | C | 2 | 3 | 4 | 5 | 6 | |
P | ||||||||
2 | Uniform | 2 | 0 | 0 | 0 | 0 | 0 | |
3 | 0.2723 | 0.2745 | 0.2793 | 0.2834 | 0.2897 | |||
4 | 0.3823 | 0.3520 | 0.3711 | 0.3909 | 0.3778 | |||
5 | 0.3387 | 0.3305 | 0.3478 | 0.3670 | 0.3605 | |||
6 | 0.4158 | 0.3834 | 0.4181 | 0.4238 | 0.4370 | |||
Context | C | 2 | 3 | 4 | 5 | 6 | ||
P | ||||||||
Flexible | 2 | 0 | 0 | 0 | 0 | 0 | ||
3 | 0.3031 | 0.3014 | 0.2885 | 0.2898 | 0.2949 | |||
4 | 0.3839 | 0.3653 | 0.3601 | 0.3878 | 0.3894 | |||
5 | 0.3518 | 0.3717 | 0.3843 | 0.3775 | 0.3998 | |||
6 | 0.3828 | 0.3627 | 0.3798 | 0.4072 | 0.4045 |
m | Context | C | 2 | 3 | 4 | 5 | 6 | |
P | ||||||||
1.5 | Uniform | 2 | 0 | 0 | 0 | 0 | 0 | |
3 | 0.2481 | 0.2718 | 0.2646 | 0.2651 | 0.2549 | |||
4 | 0.3917 | 0.3881 | 0.3783 | 0.4073 | 0.4010 | |||
5 | 0.3862 | 0.3426 | 0.3659 | 0.3699 | 0.3676 | |||
6 | 0.4478 | 0.4412 | 0.4603 | 0.4233 | 0.3852 | |||
Context | C | 2 | 3 | 4 | 5 | 6 | ||
P | ||||||||
Flexible | 2 | 0 | 0 | 0 | 0 | 0 | ||
3 | 0.2995 | 0.2966 | 0.2865 | 0.2663 | 0.2860 | |||
4 | 0.3765 | 0.3275 | 0.3391 | 0.3477 | 0.3425 | |||
5 | 0.3860 | 0.3396 | 0.3908 | 0.3997 | 0.3744 | |||
6 | 0.3698 | 0.4054 | 0.3892 | 0.3762 | 0.3348 | |||
m | Context | C | 2 | 3 | 4 | 5 | 6 | |
P | ||||||||
2 | Uniform | 2 | 0 | 0 | 0 | 0 | 0 | |
3 | 0.2836 | 0.2802 | 0.2890 | 0.2834 | 0.3195 | |||
4 | 0.3665 | 0.3729 | 0.3901 | 0.3969 | 0.3818 | |||
5 | 0.3668 | 0.3462 | 0.3414 | 0.3759 | 0.3304 | |||
6 | 0.3208 | 0.3850 | 0.4100 | 0.4161 | 0.4354 | |||
Context | C | 2 | 3 | 4 | 5 | 6 | ||
P | ||||||||
Flexible | 2 | 0 | 0 | 0 | 0 | 0 | ||
3 | 0.3127 | 0.3237 | 0.3116 | 0.3096 | 0.3310 | |||
4 | 0.3921 | 0.3912 | 0.3942 | 0.3945 | 0.3951 | |||
5 | 0.3701 | 0.3498 | 0.3630 | 0.3647 | 0.4252 | |||
6 | 0.3663 | 0.3749 | 0.3468 | 0.3475 | 0.3679 |
Databases | Models | m | Context | P | C | Rules | PI | Time (s) |
---|---|---|---|---|---|---|---|---|
Auto MPG | MGFM | 1.5 | Uniform | 6 | 3 | 18 | 0.4602 | 0.0133 |
Flexible | 5 | 3 | 15 | 0.4505 | 0.0132 | |||
2 | Uniform | 6 | 3 | 18 | 0.4737 | 0.0127 | ||
Flexible | 6 | 3 | 18 | 0.4909 | 0.0104 | |||
GFM | 2 | Uniform | 6 | 3 | 18 | 0.3986 | 0.0400 | |
Boston housing | MGFM | 1.5 | Uniform | 6 | 4 | 24 | 0.4603 | 1.2834 |
Flexible | 6 | 3 | 18 | 0.4053 | 1.2634 | |||
2 | Uniform | 6 | 6 | 36 | 0.4353 | 1.3044 | ||
Flexible | 5 | 6 | 30 | 0.4253 | 1.2937 | |||
GFM | 2 | Uniform | 5 | 6 | 30 | 0.3043 | 1.3427 | |
Energy efficiency | MGFM | 1.5 | Uniform | 5 | 6 | 30 | 0.4585 | 0.5562 |
Flexible | 5 | 5 | 25 | 0.4774 | 0.5472 | |||
2 | Uniform | 5 | 4 | 20 | 0.4824 | 0.2561 | ||
Flexible | 6 | 4 | 24 | 0.4952 | 0.2834 | |||
GFM | 2 | Uniform | 4 | 3 | 12 | 0.4561 | 1.9887 |
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Yeom, C.-U.; Kwak, K.-C. A Design and Its Application of Multi-Granular Fuzzy Model with Hierarchical Tree Structures. Appl. Sci. 2023, 13, 11175. https://doi.org/10.3390/app132011175
Yeom C-U, Kwak K-C. A Design and Its Application of Multi-Granular Fuzzy Model with Hierarchical Tree Structures. Applied Sciences. 2023; 13(20):11175. https://doi.org/10.3390/app132011175
Chicago/Turabian StyleYeom, Chan-Uk, and Keun-Chang Kwak. 2023. "A Design and Its Application of Multi-Granular Fuzzy Model with Hierarchical Tree Structures" Applied Sciences 13, no. 20: 11175. https://doi.org/10.3390/app132011175
APA StyleYeom, C.-U., & Kwak, K.-C. (2023). A Design and Its Application of Multi-Granular Fuzzy Model with Hierarchical Tree Structures. Applied Sciences, 13(20), 11175. https://doi.org/10.3390/app132011175