# Performance Comparison of ANFIS Models by Input Space Partitioning Methods

^{*}

## Abstract

**:**

## 1. Introduction

_{2}performance and ZrO

_{2}and CaCO

_{3}nanoparticle solution transport and self-compression characteristics. Zamani [8] applied ANFIS to predict the ratio of diesel and gas in an oil reservoir. Selimiefendigi [9] applied ANFIS to predict the convection of internal CuO–water nanofluids through the thermal cycling of a circular cylinder. Ghademejad [10] applied ANFIS to predict farmyard, multipass, and water content for clay soil compaction. Mostafaei [11] applied ANFIS to predict the characteristics of biodiesel fuel owing to fatty acid composition.

## 2. Adaptive Neuro-Fuzzy Inference System (ANFIS)

#### 2.1. Structure of Adaptive Neuro-Fuzzy Inference System (ANFIS)

**Layer 1:**In the first layer, each node can output a value belonging to a linguistic level as an output, as shown in Equation (2).$${O}_{i}^{1}={u}_{{A}_{i}}\left(x\right),{O}_{i+1}^{1}={u}_{{b}_{i}}\left(y\right),i=1,2$$$${u}_{{A}_{i}}\left(x\right)=\mathrm{exp}\left\{-{\left(\frac{x-{c}_{i}}{{a}_{i}}\right)}^{2}\right\}$$**Layer 2:**In the second layer, each node receives the membership value shown in the conditional part of the fuzzy rule, and outputs it as the weight multiplied by the rule. The output of each node represents the fitness of the fuzzy rule.$${O}_{i}^{2}={w}_{i}={u}_{{A}_{i}}\left(x\right)\times {u}_{{B}_{i}}\left(y\right),i=1,2$$**Layer 3:**In the third layer, each node calculates the ratio of the ignition forces of the $i$th rule to the sum of all ignition forces through Equation (5). The value obtained is represented as a normalized value.$${O}_{i}^{3}=\overline{{w}_{i}}=\frac{{w}_{i}}{{w}_{1}+{w}_{2}},i=1,2$$**Layer 4:**In the fourth layer, each node performs an operation that multiplies the output function of the conclusion part of each rule by the standardized fitness.$${O}_{i}^{4}=\overline{{w}_{i}}{f}_{i}=\overline{{w}_{i}}\left({p}_{i}x+{q}_{i}y+{r}_{i}\right),i=1,2$$**Layer 5:**In the fifth and last layer, each node is composed of a single node. Based on all input values of the lower layer, the output value is calculated using Equation (7). The output value has a continuous-type value rather than a fuzzy set type.$${O}_{i}^{5}={y}_{i}^{*}={\displaystyle \sum}_{i=1}^{2}\overline{{w}_{i}}{f}_{i}=\frac{\sum {w}_{i}{f}_{i}}{\sum {w}_{i}}$$

#### 2.2. Learning Method of Adaptive Neuro-Fuzzy Inference System (ANFIS)

## 3. Fuzzy Rule Generation Method According to Input Space Partitioning Method

#### 3.1. Grid Partitioning Method

#### 3.2. Scatter Partitioning Method

#### 3.2.1. Subtractive Clustering (SC)

**Step 1:**The density of each data item in the input space is calculated using the density function of Equation (8).$${P}_{i}={\displaystyle \sum}_{j=1}^{m}\mathrm{exp}\left(-\frac{\Vert {x}_{i}-{x}_{j}\Vert}{{\left(\raisebox{1ex}{${r}_{a}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^{2}}\right)$$**Step 2:**Finds ${x}_{c1}$, which are the data having the highest density value from ${P}_{i}$, and this value becomes the center value of the first cluster.**Step 3:**The center of the cluster found in Step 2 is removed by Equation (9).$${P}_{i}={P}_{i}-{P}_{c1}\times \mathrm{exp}\left(-\frac{\Vert {x}_{i}-{x}_{j}\Vert}{{\left(\raisebox{1ex}{${r}_{a}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}^{2}}\right)$$**Step 4:**Repeat Step 2 to Step 3 until the highest density measurement value is smaller than the set value.

#### 3.2.2. Fuzzy C-Means (FCM) Clustering

**Step 1:**Initialize the parameter and the membership matrix to have a random value between 0 and 1 satisfying Equation (10).$${u}_{ij}={\left[{\displaystyle \sum}_{k=1}^{cx}\frac{{\Vert {x}_{j}-{v}_{i}\Vert}^{\frac{2}{m-1}}}{\Vert {x}_{j}-{v}_{i}\Vert}\right]}^{-1}$$$${d}_{ik}=d\left({x}_{k}-{v}_{i}\right)={\left[{\displaystyle \sum}_{j=1}^{n}{\left({x}_{ki}-{v}_{ij}\right)}^{2}\right]}^{\frac{1}{2}}$$**Step 2:**The center value of the new cluster is calculated by the value $E=\left\{{e}_{1},{e}_{2},\dots ,{e}_{k}\right\}$ of the input data and the previously obtained membership function ${u}_{ik}$.$${v}_{ik}=\frac{{{\displaystyle \sum}}_{k=1}^{n}{\left({u}_{ik}\right)}^{m}{x}_{kj}}{{{\displaystyle \sum}}_{k=1}^{n}{\left({u}_{ik}\right)}^{m}}$$**Step 3:**Using the center value ${v}_{ij}$ obtained in Step 2 and the input data $E$, the membership matrix ${u}_{ik}$ is continuously updated while increasing the number of iterations $r$.$${u}_{ik}^{\left(r+1\right)}=\frac{1}{{{\displaystyle \sum}}_{j=1}^{c}{\left[\frac{{d}_{ik}^{r}}{{d}_{jk}^{r}}\right]}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$m-1$}\right.}}$$**Step 4:**The above procedure is repeated until the error of the repeated membership matrix ${U}^{r}$ and ${U}^{r+1}$ is smaller than the arbitrary threshold value $\Delta $.$$\Delta =\Vert {U}^{r+1}-{U}^{r}\Vert =\underset{ik}{\mathrm{max}}\left|{u}_{ik}^{r+1}-{u}_{ik}^{r}\right|$$

#### 3.2.3. Context-Based Fuzzy C-Means (CFCM) Clustering

**Step 1:**Let $m(1m\infty )$ and the number of clusters $c\left(2\le c\le n\right)$ be set.**Step 2:**Set the initial partitioning matrix $U$ and the threshold $\epsilon $, and choose the number of iterations.$$U\left(\left[{u}_{ij}\right]i=1,\dots ,c,j=1,\dots ,n\right)$$**Step 3:**Compute the center of each cluster ${c}_{i}\left(i=1,2,\dots ,c\right)$ using the membership matrix $U$ and Equation (16).$${c}_{i}=\frac{{{\displaystyle \sum}}_{j=1}^{n}{u}_{ij}^{m}{x}_{j}}{{{\displaystyle \sum}}_{j=1}^{n}{u}_{ij}^{m}}$$**Step 4:**The partitioning matrix $U$ is updated using the center value of the cluster $c$ and Equation (17).$${u}_{ij}=\frac{{f}_{j}}{{{\displaystyle \sum}}_{k=1}^{c}{\left(\frac{{d}_{ij}}{{d}_{kj}}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\left(m-1\right)$}\right.}}$$**Step 5:**If $\Vert {J}^{r}-{J}^{r+1}\Vert \le \epsilon $ is satisfied, the above procedure is stopped. If not, proceed from Step 3 again.$$J={\displaystyle \sum}_{j=1}^{n}{\displaystyle \sum}_{i=1}^{c}{u}_{ij}^{m}{\Vert {x}_{j}-{c}_{i}\Vert}^{2}$$

CFCM (context-based fuzzy c-means) Clustering Algorithm |

Begin |

Fixed $c,2c10$; |

Fixed $p,p=6$; |

Fixed $m,m=1.5$; |

Fixed max iterations; |

Randomly initialize ${v}_{ij},{v}_{ij}=clustercenters$; |

for $t=1$ to max iterations do |

Update the membership matrix ${u}_{ij}$ using Equation (12); |

Calculate the new cluster lefts ${v}_{ij}$ using Equation (10); |

Calculate the new objective function ${J}_{m}$ using Equation (13); |

if $\left(abs\left({J}^{r}-{J}^{r+1}\right)<\epsilon \right)$ then |

break; |

else |

${J}_{m}^{t-1}={J}_{m}^{t}$; |

end if |

end for |

end |

## 4. Experimental Method and Result Analysis

#### 4.1. Database

#### 4.2. Experiment Method

#### 4.3. Combined Cycle Power Plant Database

#### 4.4. Auto-MPG Database

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Input space partitioning methods: (

**a**) grid partitioning method and (

**b**) scatter partitioning method.

**Figure 5.**Fuzzy C-means clustering method: (

**a**) number of clusters = 2 and (

**b**) number of clusters = 3.

**Figure 6.**Difference between FCM (fuzzy c-means) clustering method and CFCM (context-based fuzzy c-means) clustering method.

**Figure 7.**Context-based fuzzy C-means clustering method: (

**a**) contexts generated from output space, and (

**b**) center of cluster corresponding to each context.

**Table 1.**Experimental results of ANFIS1 model. RMSE (root-mean square error), AT (time-averaged ambient temperature), AP (ambient pressure), RH (relative humidity), and V (exhaust vacuum), ANFIS (adaptive neuro-fuzzy inference system).

Algorithm | Input Combination | Membership Functions | Number of Rules | Training RMSE | Testing RMSE |
---|---|---|---|---|---|

ANFIS1 | 2 Inputs (AT, RH) | 2 | 4 | 4.49 | 4.58 |

3 | 9 | 4.43 | 4.57 | ||

4 | 16 | 4.45 | 4.52 | ||

5 | 25 | 4.40 | 4.60 | ||

3 Inputs (AT, V, AP) | 2 | 8 | 4.27 | 4.35 | |

3 | 27 | 4.04 | 4.24 | ||

4 | 64 | 3.90 | 4.21 | ||

5 | 125 | 3.92 | 4.13 | ||

All Inputs | 2 | 16 | 4.02 | 4.17 | |

3 | 81 | 3.84 | 4.21 | ||

4 | 256 | 3.49 | 4.45 | ||

5 | 625 | 3.22 | 5.19 |

Algorithm | Radii | Number of Rules | Training RMSE | Testing RMSE |
---|---|---|---|---|

ANFIS2 | 0.2 | 42 | 3.80 | 4.18 |

0.3 | 17 | 3.99 | 4.13 | |

0.4 | 7 | 4.07 | 4.22 | |

0.5 | 5 | 4.09 | 4.26 | |

0.6 | 2 | 4.24 | 4.26 | |

0.7 | 2 | 4.13 | 4.35 | |

0.8 | 3 | 4.14 | 4.22 | |

0.9 | 2 | 4.20 | 4.26 |

Algorithm | Number of Cluster (c) | Number of Rules | Training RMSE | Testing RMSE |
---|---|---|---|---|

ANFIS3 | 5 | 5 | 4.03 | 4.17 |

10 | 10 | 3.93 | 4.16 | |

15 | 15 | 3.91 | 4.16 | |

20 | 20 | 3.80 | 4.15 | |

25 | 25 | 3.77 | 4.13 | |

30 | 30 | 3.86 | 4.10 | |

35 | 35 | 3.76 | 4.11 | |

40 | 40 | 3.78 | 4.10 | |

45 | 45 | 3.76 | 4.11 | |

50 | 50 | 3.84 | 4.08 |

Algorithm | Context (p) & Cluster (c) | Number of Rules | Training RMSE | Testing RMSE | |
---|---|---|---|---|---|

p | c | ||||

ANFIS4 | 6 | 2 | 12 | 4.11 | 4.15 |

3 | 18 | 4.11 | 4.14 | ||

4 | 24 | 4.04 | 4.20 | ||

5 | 30 | 4.11 | 4.13 | ||

6 | 36 | 4.14 | 4.09 | ||

7 | 42 | 4.11 | 4.08 | ||

8 | 48 | 4.11 | 4.08 | ||

9 | 54 | 4.02 | 4.18 | ||

10 | 60 | 4.11 | 4.08 | ||

11 | 66 | 4.11 | 4.08 | ||

12 | 72 | 4.10 | 4.10 | ||

13 | 78 | 4.07 | 4.12 | ||

14 | 84 | 4.07 | 4.12 | ||

15 | 90 | 4.06 | 4.15 | ||

16 | 96 | 4.20 | 4.00 | ||

17 | 112 | 4.20 | 4.00 | ||

18 | 118 | 4.10 | 4.11 | ||

19 | 124 | 4.09 | 4.10 | ||

20 | 130 | 4.10 | 4.10 | ||

10 | 2 | 20 | 4.11 | 4.14 | |

3 | 30 | 4.10 | 4.14 | ||

4 | 40 | 4.10 | 4.14 | ||

5 | 50 | 4.09 | 4.11 | ||

6 | 60 | 4.13 | 4.08 | ||

7 | 70 | 4.12 | 4.08 | ||

8 | 80 | 4.11 | 4.08 | ||

9 | 90 | 4.01 | 4.19 | ||

10 | 100 | 4.12 | 4.08 | ||

11 | 110 | 4.12 | 4.08 | ||

12 | 120 | 4.10 | 4.10 | ||

13 | 130 | 4.07 | 4.12 | ||

14 | 140 | 4.07 | 4.12 | ||

15 | 150 | 4.06 | 4.15 | ||

16 | 160 | 4.20 | 4.00 | ||

17 | 170 | 4.20 | 3.99 | ||

18 | 180 | 4.09 | 4.11 | ||

19 | 190 | 4.10 | 4.10 | ||

20 | 200 | 4.10 | 4.10 |

Algorithms | Input Combination | MF, Radii, c, p | Number of Rules | Training RMSE | Testing RMSE |
---|---|---|---|---|---|

ANFIS1 | 3 Inputs | MF = 5 | 125 | 3.92 | 4.13 |

ANFIS2 | All Inputs | Radii = 0.3 | 17 | 3.99 | 4.13 |

ANFIS3 | All Inputs | c = 50 | 50 | 3.85 | 4.08 |

ANFIS4 | All Inputs | p = 10, c = 17 | 170 | 4.20 | 3.99 |

Algorithm | Input Combination | Membership Functions | Number of Rules | Training RMSE | Testing RMSE |
---|---|---|---|---|---|

ANFIS1 | 2 Inputs (displacement, acceleration) | 2 | 4 | 3.35 | 3.33 |

3 | 9 | 3.24 | 3.47 | ||

4 | 16 | 3.16 | 3.93 | ||

5 | 25 | 2.78 | 5.07 | ||

3 Inputs (displacement, weight, acceleration) | 2 | 8 | 2.69 | 3.28 | |

3 | 27 | 2.29 | 7.01 | ||

4 | 64 | 1.84 | 11.14 | ||

5 | 125 | 0.98 | 19.90 | ||

4 Inputs (displacement, horsepower, weight, acceleration) | 2 | 16 | 2.04 | 3.37 | |

3 | 81 | 1.25 | 10.34 | ||

4 | 256 | 0.11 | 19.69 | ||

5 | 625 | 0.01 | 12.44 |

Algorithm | Radii | Number of Rules | Training RMSE | Testing RMSE |
---|---|---|---|---|

ANFIS2 | 0.2 | 104 | 0.00 | 3.78 |

0.3 | 43 | 0.48 | 10.87 | |

0.4 | 19 | 1.15 | 14.25 | |

0.5 | 11 | 1.99 | 6.84 | |

0.6 | 6 | 2.32 | 3.20 | |

0.7 | 4 | 2.52 | 2.85 | |

0.8 | 4 | 2.53 | 2.82 | |

0.9 | 3 | 2.72 | 2.92 |

Algorithm | Number of Clusters (c) | Number of Rules | Training RMSE | Testing RMSE |
---|---|---|---|---|

ANFIS3 | 5 | 5 | 2.14 | 3.76 |

10 | 10 | 1.64 | 7.43 | |

15 | 15 | 1.39 | 8.68 | |

20 | 20 | 1.21 | 8.03 | |

25 | 25 | 1.00 | 8.35 | |

30 | 30 | 0.41 | 9.88 |

Algorithm | Context (p) & Cluster (c) | Number of Rules | Training RMSE | Testing RMSE | |
---|---|---|---|---|---|

p | c | ||||

ANFIS4 | 6 | 2 | 12 | 2.51 | 2.62 |

3 | 18 | 2.54 | 2.60 | ||

4 | 24 | 2.54 | 2.60 | ||

5 | 30 | 2.55 | 2.61 | ||

6 | 36 | 2.57 | 2.60 | ||

7 | 42 | 2.58 | 2.61 | ||

8 | 48 | 2.58 | 2.60 | ||

9 | 54 | 2.59 | 2.61 | ||

10 | 60 | 2.60 | 2.61 | ||

11 | 66 | 2.61 | 2.61 | ||

12 | 72 | 2.61 | 2.62 | ||

13 | 78 | 2.61 | 2.62 | ||

14 | 84 | 2.62 | 2.62 | ||

15 | 90 | 2.63 | 2.62 | ||

16 | 96 | 2.63 | 2.62 | ||

17 | 112 | 2.63 | 2.63 | ||

18 | 118 | 2.63 | 2.63 | ||

19 | 124 | 2.64 | 2.63 | ||

20 | 130 | 2.64 | 2.64 | ||

10 | 2 | 20 | 2.53 | 2.62 | |

3 | 30 | 5.55 | 2.61 | ||

4 | 40 | 2.57 | 2.61 | ||

5 | 50 | 2.57 | 2.61 | ||

6 | 60 | 2.58 | 2.61 | ||

7 | 70 | 2.58 | 2.61 | ||

8 | 80 | 2.55 | 2.61 | ||

9 | 90 | 2.60 | 2.61 | ||

10 | 100 | 2.60 | 2.62 | ||

11 | 110 | 2.61 | 2.62 | ||

12 | 120 | 2.61 | 2.62 | ||

13 | 130 | 2.62 | 2.62 | ||

14 | 140 | 2.63 | 2.62 | ||

15 | 150 | 2.63 | 2.63 | ||

16 | 160 | 2.63 | 2.63 | ||

17 | 170 | 2.63 | 2.63 | ||

18 | 180 | 2.64 | 2.63 | ||

19 | 190 | 2.64 | 2.63 | ||

20 | 200 | 2.65 | 2.63 |

Algorithms | Input Combination | MF, Radii, c, p | Number of Rules | Training RMSE | Testing RMSE |
---|---|---|---|---|---|

ANFIS1 | 3 Inputs | MF = 2 | 8 | 2.69 | 3.28 |

ANFIS2 | All Inputs | Radii = 0.8 | 4 | 2.53 | 2.82 |

ANFIS3 | All Inputs | c = 5 | 5 | 2.14 | 3.76 |

ANFIS4 | All Inputs | p = 6, c = 4 | 24 | 2.54 | 2.60 |

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**MDPI and ACS Style**

Yeom, C.-U.; Kwak, K.-C.
Performance Comparison of ANFIS Models by Input Space Partitioning Methods. *Symmetry* **2018**, *10*, 700.
https://doi.org/10.3390/sym10120700

**AMA Style**

Yeom C-U, Kwak K-C.
Performance Comparison of ANFIS Models by Input Space Partitioning Methods. *Symmetry*. 2018; 10(12):700.
https://doi.org/10.3390/sym10120700

**Chicago/Turabian Style**

Yeom, Chan-Uk, and Keun-Chang Kwak.
2018. "Performance Comparison of ANFIS Models by Input Space Partitioning Methods" *Symmetry* 10, no. 12: 700.
https://doi.org/10.3390/sym10120700