# Optimization by Context Refinement for Development of Incremental Granular Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. IGM

#### 2.1. Linear Regression

_{1}, a

_{2}]

^{T}. The improvement of the model occurs in which the local part is based on the input-error data $\{{x}_{k},{e}_{k}\}$, where the error is ${e}_{k}={y}_{k}-{z}_{k}$. Subsequently, the if–then fuzzy rules and contexts of triangular membership functions are obtained by the context-based fuzzy clustering approach.

#### 2.2. Context-Based Fuzzy C-Means Clustering

#### 2.3. Local GFM

#### 2.4. IGM

- [Step 1]
- Use linear regression on the numerical data points. Subsequently, the errors ${e}_{k}={y}_{k}-{z}_{k}$ are obtained by the difference between the desired and linear regression outputs.
- [Step 2]
- Obtain the input and error pairs $\{{x}_{k},{e}_{k}\}$. These error values are employed as the output data in the use of the local GFM. Subsequently, the contexts in the error space are generated. The linguistic contexts are produced as shown in Figure 1.
- [Step 3]
- Estimate the clusters using context-based fuzzy clustering approach.
- [Step 4]
- Calculate the aggregation values by the linear summation of the activation levels and the context weight. Consequently, the model output results in fuzzy number with a triangular type.
- [Step 5]
- Integrate the linear regression output and granular results of the GFM. Hence, the prediction result is expressed as $\stackrel{\u02c6}{Y}=z\oplus \stackrel{\u02c6}{E}$

## 3. Refinement of Contexts in IGM Design

## 4. Experimental Results

#### 4.1. Coagulant Dosing in Water Purification Plant

#### 4.2. Automobile MPG Prediction

#### 4.3. Boston Housing Data

#### 4.4. Discussion

- -
- The incremental granular model has high prediction performance by combining linear regression and local granular fuzzy model.
- -
- The local granular fuzzy model generates the automatic if-then rules using context-based fuzzy clustering method from numerical data set.
- -
- The incremental granular model can enhance the prediction performance by combining the derivative-based optimization and context-based fuzzy clustering.
- -
- In contrast to the conventional back-propagation method, after adjusting the contexts by steepest descent method, the cluster centers in the premise part are estimated by using context-based fuzzy clustering method.

- -
- The number of contexts is obtained by trial and error method.
- -
- The number of cluster center per context are obtained by trial and error method.
- -
- As the number of data points increase, the number of rules also increase
- -
- The specific context can include the small data points, when the distribution of context is uniform.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Generation of contexts by statistical distribution from error obtained by linear regression. (

**a**) Error histogram (

**b**) probability density function (PDF) (

**c**) conditional density function (CDF) (

**d**) six linguistic contexts.

**Figure 8.**Performance of PAC prediction for training and testing datasets. (

**a**) prediction for training data (

**b**) prediction for testing data.

**Figure 10.**Cluster centers corresponding to each context (square marks denote cluster center). (

**a**) context no.1 (

**b**) context no.2 (

**c**) context no.3 (

**d**) context no.4 (

**e**) context no.5 (

**f**) context no.6 (

**g**) context no.7 (

**h**) context no.8.

**Figure 11.**Performance of MPG prediction for training and testing datasets. (

**a**) prediction for training data (

**b**) prediction for testing data.

**Figure 13.**Cluster centers corresponding to each context (square marks = cluster center). (

**a**) context no.1 (

**b**) context no.2 (

**c**) context no.3 (

**d**) context no.4 (

**e**) context no.5 (

**f**) context no.6 (

**g**) context no.7 (

**h**) context no.8.

No. of Rule (*: No. of Node) | Trn_RMSE | Tst_RMSE | ||
---|---|---|---|---|

Linear regression | - | 3.508 | 3.578 | |

Multilayer perceptron | 45 * | 3.191 | 3.251 | |

RBFN based on context-based fuzzy c-means clustering [20] | 45 * | 3.048 | 3.219 | |

Linguistic model [4] (p = 8) | c = 8 | 64 | 2.427 | 2.800 |

IGM [13] | p = c = 8 | 64 | 1.790 | 2.009 |

The proposed model | p = c = 6 | 36 | 2.124 | 2.271 |

p = c = 7 | 49 | 1.862 | 2.093 | |

p = c = 8 | 64 | 1.640 | 1.976 | |

p = c = 9 | 81 | 1.631 | 2.105 |

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Lee, M.-W.; Kwak, K.-C.
Optimization by Context Refinement for Development of Incremental Granular Models. *Symmetry* **2020**, *12*, 1916.
https://doi.org/10.3390/sym12111916

**AMA Style**

Lee M-W, Kwak K-C.
Optimization by Context Refinement for Development of Incremental Granular Models. *Symmetry*. 2020; 12(11):1916.
https://doi.org/10.3390/sym12111916

**Chicago/Turabian Style**

Lee, Myung-Won, and Keun-Chang Kwak.
2020. "Optimization by Context Refinement for Development of Incremental Granular Models" *Symmetry* 12, no. 11: 1916.
https://doi.org/10.3390/sym12111916