# Performance Evaluation of Automobile Fuel Consumption Using a Fuzzy-Based Granular Model with Coverage and Specificity

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## Abstract

**:**

## 1. Introduction

## 2. GM

#### 2.1. CFCM Clustering

- [Step 1]
- The number of linguistic contexts (2 to 20) and the number of clusters to be created in each context (2 to 20) was selected. The belonging matrix $U$ was initialized to an arbitrary value between 0 and 1.
- [Step 2]
- A linguistic context was created using a triangular membership function that was evenly distributed in the output space.
- [Step 3]
- For each context, the cluster center $c$ and the belonging value $u$ were calculated.$${c}_{i}=\frac{{{\displaystyle \sum}}_{k=1}^{N}{u}_{ik}^{m}{x}_{k}}{{{\displaystyle \sum}}_{k=1}^{N}{u}_{ik}^{m}}$$
- [Step 4]
- The objective function was calculated, as given by Equation (6), and if the degree of improvement obtained through the previous iteration wasless than the threshold value, the process was stopped.$$J={\displaystyle \sum}_{i=1}^{c}{\displaystyle \sum}_{k=1}^{N}{u}_{ik}^{m}{d}_{ik}^{2}$$$$\left|{J}^{p}-{J}^{p-1}\right|\le \u03f5$$
- [Step 5]
- The new membership matrix U was calculated from Equation (3), and control was returned to [Step 3].

#### 2.2. Structure of the GM

#### 2.3. Structure of the GM

## 3. Performance Evaluation Method

#### 3.1. Performance Evaluation Method Suitable for the GM

#### 3.1.1. Coverage

#### 3.1.2. Specificity

## 4. Experimental Results

#### 4.1. Auto MPG Database

#### 4.2. Experiment Method and Analysis of Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Conceptual description of the context-based fuzzy C-means (CFCM) clustering method: (

**a**) Linguistic context generated in the output space; (

**b**) clusters estimated for each context.

**Figure 9.**Predictive performance of different GMs: (

**a**) The GM that evenly divides the linguistic context; (

**b**) the GM that flexibly divides the linguistic context.

**Figure 10.**RMSE performance results on the training dataset for the GM that flexibly splits the linguistic context.

**Figure 11.**Root mean square error (RMSE) performance results on the test dataset for the GM that flexibly splits the linguistic context.

**Figure 12.**Predictive performance for the GM that flexibly divides the linguistic context using the method proposed by Hu [30] (using training data).

**Figure 13.**Performance index of the GM by the variation of the number of contexts and clusters (flexible contexts).

**Figure 14.**Performance index of the GM by the variation of the number of contexts (flexible contexts).

PI (Performance Index) Methods | Equations | |
---|---|---|

Hu [30] | Coverage | $Cov=\frac{1}{N}{\displaystyle {\displaystyle \sum}_{k=1}^{N}}incl\left({y}_{k},{Y}_{k}\right)$ |

Specificity | $Spec=\left(\frac{1}{N}{\displaystyle {\displaystyle \sum}_{k=1}^{N}}exp\left(-\left|{y}_{k}^{+}-{y}_{k}^{-}\right|\right)\right){10}^{4}$ | |

Performance index | $PI=Cov\xb7Spec$ | |

Zhu [31] | Coverage | $Cov=\frac{1}{N}{\displaystyle {\displaystyle \sum}_{k=1}^{N}}cov\left(tarke{t}_{k},{Y}_{k}\right)$ |

Specificity | $Spec=\frac{1}{N}{\displaystyle {\displaystyle \sum}_{k=1}^{N}}spec\left({Y}_{k}\right),$ $spec\left({Y}_{k}\right)=\mathrm{max}\left(0,1-\frac{\left|{y}_{K}^{+}-{y}_{k}^{-}\right|}{range}\right)$ | |

Performance index | $PI=\mathrm{arg}max\left(Cov\xb7Spec\right)$ | |

Galaviz [32] | Coverage | ${f}_{1}\left(cov\right)=\frac{1}{N}{\displaystyle {\displaystyle \sum}_{k=1}^{N}}\left({t}_{k}\in {Y}_{k}\right)$ |

Specificity | $f2\left(spec\right)={e}^{-\alpha \left(l/L\right)}$ | |

Performance index | $Q\left(PI\right)={f}_{1}\xb7{f}_{2}$ |

Algorithm | Performance Evaluation Method | ||
---|---|---|---|

Granular Model | RMSE | ||

Number of Contexts | Number of Clusters | Training RMSE | Testing RMSE |

10 | 2 | 3.96 | 4.15 |

3 | 3.98 | 4.18 | |

4 | 3.69 | 3.91 | |

5 | 3.72 | 3.90 | |

6 | 3.90 | 4.10 | |

7 | 3.89 | 4.07 | |

8 | 3.98 | 4.09 | |

9 | 3.95 | 4.15 | |

10 | 3.54 | 4.17 |

**Table 3.**RMSE prediction performance results for the GM that flexibly divides the linguistic context.

Algorithm | Performance Evaluation Method | ||
---|---|---|---|

Granular Model | RMSE | ||

Number of Contexts | Number of Clusters | Training RMSE | Testing RMSE |

10 | 2 | 3.75 | 3.79 |

3 | 3.65 | 3.80 | |

4 | 3.71 | 3.73 | |

5 | 3.95 | 3.93 | |

6 | 3.79 | 4.13 | |

7 | 3.87 | 4.12 | |

8 | 3.75 | 3.95 | |

9 | 3.89 | 4.31 | |

10 | 3.78 | 4.41 |

**Table 4.**Predictive performance for the GM that evenly divides the linguistic context using the method proposed by Hu [30].

Granular Model That Evenly Divides Linguistic Context (No. Context = 10) | |||
---|---|---|---|

Number of Clusters | Coverage | Specificity | Performance Index |

2 | 0.72 | 2.35 | 1.70 |

3 | 0.69 | 2.35 | 1.63 |

4 | 0.72 | 2.35 | 1.69 |

5 | 0.71 | 2.35 | 1.68 |

6 | 0.69 | 2.35 | 1.61 |

7 | 0.68 | 2.35 | 1.60 |

8 | 0.70 | 2.35 | 1.64 |

9 | 0.72 | 2.35 | 1.70 |

10 | 0.68 | 2.35 | 1.61 |

**Table 5.**Predictive performance for the GM that flexibly divides the linguistic context using the method proposed by Hu [30].

Granular Model That Flexibly Divides Linguistic Context (No. Context = 10) | |||
---|---|---|---|

Number of Clusters | Coverage | Specificity | Performance Index |

2 | 0.74 | 12.39 | 9.23 |

3 | 0.76 | 15.36 | 11.68 |

4 | 0.69 | 13.69 | 9.50 |

5 | 0.71 | 16.8 | 11.91 |

6 | 0.75 | 16.5 | 12.38 |

7 | 0.70 | 17.53 | 12.26 |

8 | 0.74 | 18.18 | 13.45 |

9 | 0.66 | 17.77 | 11.78 |

10 | 0.64 | 19.64 | 12.63 |

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**MDPI and ACS Style**

Yeom, C.-U.; Kwak, K.-C.
Performance Evaluation of Automobile Fuel Consumption Using a Fuzzy-Based Granular Model with Coverage and Specificity. *Symmetry* **2019**, *11*, 1480.
https://doi.org/10.3390/sym11121480

**AMA Style**

Yeom C-U, Kwak K-C.
Performance Evaluation of Automobile Fuel Consumption Using a Fuzzy-Based Granular Model with Coverage and Specificity. *Symmetry*. 2019; 11(12):1480.
https://doi.org/10.3390/sym11121480

**Chicago/Turabian Style**

Yeom, Chan-Uk, and Keun-Chang Kwak.
2019. "Performance Evaluation of Automobile Fuel Consumption Using a Fuzzy-Based Granular Model with Coverage and Specificity" *Symmetry* 11, no. 12: 1480.
https://doi.org/10.3390/sym11121480