# Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Scale and Move Transformation-Based Interpolation

#### 2.1.1. N Closest Rules Selection

#### 2.1.2. Intermediate Rule Construction

#### 2.1.3. Scale Transformation

#### 2.1.4. Move Transformation

#### 2.2. Type-2 Fuzzy Sets

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Proposed Interval Type-2 Transformation-Based Interpolation

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

- Calculate representative values:The lower and upper representative values $\mathrm{Rep}{({\tilde{A}}^{K})}_{x}$ and $\mathrm{Rep}{({\tilde{A}}^{K})}_{y}$ of a given interval type-2 fuzzy set $\tilde{A}$ are calculated first using Equation (19). The shape diversity factors ${f}_{\tilde{A}}^{K}$ and weight factors ${w}_{\tilde{A}}^{K}$ are computed according to Equations (20) and (21), respectively. The overall Rep $\mathrm{Rep}(\tilde{A})$ is then obtained by Equation (22), $K=L,U$. The calculations for all of the antecedent variables of all rules and their counterparts in the observation follow the same procedure.
- Choose closest N rules:
- Construct intermediate ruleThe normalised weight ${w}_{{\tilde{A}}_{ij}}^{\prime}$ of the j-th antecedent of the i-th chosen rule, which is calculated by Equation (4), together with the parameter ${\delta}_{{\tilde{A}}_{j}}$, which is calculated by Equation (6), are used in Equation (5) to obtain the value of each antecedent variable ${x}_{j}$ within the intermediate rule ${\tilde{A}}_{j}^{\prime}$, $i\in \{1,\dots ,N\}$, $j\in \{1,\dots ,M\}$. From this, two parameters ${w}_{{\tilde{B}}_{i}}^{\prime}$ and ${\delta}_{\tilde{B}}$ are computed using Equation (8) and are then used to construct ${\tilde{B}}^{\prime}$ from Equation (7), resulting in the intermediate rule ${\tilde{A}}_{1}^{\prime}\wedge \cdots {\tilde{A}}_{j}^{\prime}\wedge \cdots {\tilde{A}}_{M}^{\prime}\Rightarrow {\tilde{B}}^{\prime}$.
- Perform scale, move and height transformations:In conjunction with the given ${\tilde{A}}_{j}^{*}$ for each antecedent variable ${x}_{j}$, the rates ${s}_{j}^{K}$, ${m}_{j}^{K}$ and ${h}_{j}$, $K\in \{L,U\}$, can then be calculated using Equations (9), (11) and (23). Due to the extra uncertainty encountered in the membership functions, a further transformation on the height of the LMF is needed (because the LMFs of different interval type-2 fuzzy sets may have different heights), while the height of the UMF remains the same owing to its normality. This additional transformation is introduced to transform the heights of ${\tilde{A}}_{j}^{{}^{\prime}L}$ to those of ${\tilde{A}}_{j}^{*L}$, with the height rate h being calculated by:$${h}_{j}=\frac{{H}_{{\tilde{A}}_{j}}^{*L}}{{H}_{{\tilde{A}}_{j}}^{{}^{\prime}L}}$$
- Derive interpolated conclusion:The second intermediate term ${\tilde{B}}^{\u2033}$ and the interpolated result ${\tilde{B}}^{*}$ can then be estimated by the combined ${s}_{\tilde{B}}^{K}$, ${m}_{\tilde{B}}^{K}$ and ${h}_{\tilde{B}}$, $K\in \{L,U\}$. Here, ${s}_{\tilde{B}}^{K}$ and ${m}_{\tilde{B}}^{K}$ are computed following Equations (9)–(13), respectively, and ${h}_{\tilde{B}}$ is computed according to Equation (23) such that:$${h}_{\tilde{B}}=\frac{1}{M}\sum _{j=1}^{M}{h}_{j}$$
- Implement modified procedure:To obtain intuitive interpolated conclusions for interval type-2 fuzzy sets, the relative location between the LA and UA of an interval type-2 fuzzy set should be considered [39]. For this purpose, ${B}^{\u2033}$ is modified into ${B}_{c}^{\u2033}$ to maintain the relative location both before and after the scale transformation. Here, a relative location factor $\theta $ is defined by:$$\theta =\frac{{B}^{{}^{\prime}L}}{{B}^{{}^{\prime}U}}=\frac{{B}^{{}^{\u2033}L}}{{B}_{n}^{{}^{\u2033}U}}=\frac{{B}_{n}^{{}^{\u2033}L}}{{B}^{{}^{\u2033}U}}$$$${B}_{c}^{{}^{\u2033}K}=\frac{{B}^{{}^{\u2033}K}+{B}_{n}^{{}^{\u2033}K}}{2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}K=L,U$$Similarly, the final interpolated conclusion can also be modified from ${B}^{*}$ to ${B}_{c}^{*}$ using the same $\theta $ to maintain the relative location both before and after the move transformation.

## 4. Experimentation and Discussion

#### 4.1. Case 1

#### 4.2. Case 2

#### 4.3. Case 3

#### 4.4. Case 4

## 5. Type-2 Fuzzy Sets vs. Rough-Fuzzy Sets for T-FRI

#### 5.1. Conceptual Comparison

#### 5.2. Practical Application and Comparison

#### 5.2.1. Application Problem

#### 5.2.2. Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FOU | Footprint of uncertainty |

FRI | Fuzzy rule interpolation |

LA | Lower approximation |

LMF | Lower membership function |

T-FRI | Scale and move transformation-based fuzzy rule interpolation approach |

UA | Upper approximation |

UMF | Upper membership function |

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**Figure 1.**Different membership functions for some eye contact perceived by different people [23].

**Figure 2.**A type-2 fuzzy set corresponding to the situation depicted by Figure 1.

**Figure 3.**Lower membership function ${\tilde{A}}^{L}$ and upper membership function ${\tilde{A}}^{U}$ of a triangular interval type-2 fuzzy set $\tilde{A}$.

**Figure 8.**Causal diagram of the simplified application problem [23].

**Figure 9.**Interpolated results from conventional FRI [23].

**Figure 10.**Interpolated results from interval type-2 interpolation [23].

**Figure 11.**Interpolated results from rough-fuzzy interpolation (taken from [23]).

Rule 1 | ${\tilde{A}}_{1}=<(1,3.5,4;1),(0,3.5,5;1)>$ |

${\tilde{B}}_{1}=<(1.5,2,3;1),(0,2,5;1)>$ | |

Rule 2 | ${\tilde{A}}_{2}=<(12,13,13.5;1),(11,13,14;1)>$ |

${\tilde{B}}_{2}=<(11,11.5,12;1),(10,11.5,13;1)>$ | |

Observation | ${\tilde{A}}^{*}=<(6.5,8,9.5;1),(6,8,10;1)>$ |

Rule 1 | ${\tilde{A}}_{11}=<(3,3,3;1),(3,3,3;1)>$ |

${\tilde{A}}_{12}=<(2.5,2.5,2.5;1),(2.5,2.5,2.5;1)>$ | |

${\tilde{B}}_{1}=<(4,4,4;1),(4,4,4;1)>$ | |

Rule 2 | ${\tilde{A}}_{21}=<(12,13,13.5;0.6),(11,13,14.5;1)>$ |

${\tilde{A}}_{22}=<(11.5,12.5,13.5;0.5),(10,12,14.5;1)>$ | |

${\tilde{B}}_{2}=<(10.5,11.5,12;0.5),(9,11.5,13;1)>$ | |

Observation | ${\tilde{A}}_{1}^{*}=<(6,7,8;0.6),(5,7.5,9;1)>$ |

${\tilde{A}}_{2}^{*}=<(5.5,6,7;0.5),(4,6,8;1)>$ |

Rule 1 | ${\tilde{A}}_{11}=<(1,2,3;0.7),(0,1.5,4;1)>$ |

${\tilde{A}}_{12}=<(2,3,4;0.5),(1,2.5,5;1)>$ | |

${\tilde{A}}_{13}=<(7,8,9;0.6),(6,7.5,10;1)>$ | |

${\tilde{B}}_{1}=<(1,1.5,2.5;0.6),(0,1.5,3.5;1)>$ | |

Rule 2 | ${\tilde{A}}_{21}=<(16,17.5,19;0.7),(15,17.5,20;1)>$ |

${\tilde{A}}_{22}=<(11.5,12.5,13;0.5),(10,13,15;1)>$ | |

${\tilde{A}}_{23}=<(21,22,23;0.6),(20,22,23.5;1)>$ | |

${\tilde{B}}_{2}=<(21.5,23,23.5;0.6),(20.5,23,24;1)>$ | |

Rule 3 | ${\tilde{A}}_{31}=<(21.5,22.5,24;0.7),(20,22,25;1)>$ |

${\tilde{A}}_{32}=<(6.5,7,8.5;0.5),(5.5,7.5,9;1)>$ | |

${\tilde{A}}_{33}=<(12,12.5,14;0.6),(11,13,15;1)>$ | |

${\tilde{B}}_{3}=<(16.5,17,18;0.6),(15,17,20;1)>$ | |

Rule 4 | ${\tilde{A}}_{41}=<(11.5,12,13;0.7),(10.5,12,14;1)>$ |

${\tilde{A}}_{42}=<(22,23,23.5;0.5),(21,22.5,24;1)>$ | |

${\tilde{A}}_{43}=<(17,18.5,19;0.6),(16,18.5,19.5;1)>$ | |

${\tilde{B}}_{4}=<(12,13,14;0.6),(11.5,13.5,14.5;1)>$ | |

Observation | ${\tilde{A}}_{1}^{*}=<(6,6.5,7.5,;0.7),(5,6.5,9;1)>$ |

${\tilde{A}}_{2}^{*}=<(16.5,18,19.5;0.5),(15,18,20;1)>$ | |

${\tilde{A}}_{3}^{*}=<(1.5,2.5,4;0.6),(0.5,3,5;1)>$ |

Rule 1 | ${\tilde{A}}_{1}=<(0,5,6;1),(0,5,6;1)>$ |

${\tilde{B}}_{1}=<(0,2,4;1),(0,2,4;1)>$ | |

Rule 2 | ${\tilde{A}}_{2}=<(11,13,14;1),(11,13,14;1)>$ |

${\tilde{B}}_{2}=<(10,11,13;1),(10,11,13;1)>$ | |

Observation | ${\tilde{A}}^{*}=<(7,8,9;1),(7,8,9;1)>$ |

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**MDPI and ACS Style**

Chen, C.; Shen, Q.
Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets. *Algorithms* **2017**, *10*, 91.
https://doi.org/10.3390/a10030091

**AMA Style**

Chen C, Shen Q.
Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets. *Algorithms*. 2017; 10(3):91.
https://doi.org/10.3390/a10030091

**Chicago/Turabian Style**

Chen, Chengyuan, and Qiang Shen.
2017. "Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets" *Algorithms* 10, no. 3: 91.
https://doi.org/10.3390/a10030091