# Comparison of T-Norms and S-Norms for Interval Type-2 Fuzzy Numbers in Weight Adjustment for Neural Networks

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Proposed Methodology

#### 3.1. Architecture of the Traditional Neural Network

#### 3.2. Architecture of the Fuzzy Neural Network with Interval Type-2 Fuzzy Numbers Weights

#### 3.3. Proposed Adjustment for Interval Type-2 Fuzzy Numbers with Backpropagation Learning

**Stage****1:**- The Nguyen-Widrow algorithm is utilized to initialize the lower and upper values of the interval type-2 fuzzy numbers weights for the neural network.
**Stage****2:**- The input pattern and the wanted output for the neural network is established.
**Stage****3:**- The output of the neural network is calculated. In the first instance, the inputs for the network are introduced and the output of the network is obtained performing the calculations of the outputs from the input layer until the output layer.
**Stage****4:**- Determine the error terms for the neurons of the layers. In the output layer, the calculation of lower ($\underset{\_}{{\delta}_{pk}^{O}}$) and upper ($\overline{{\delta}_{pk}^{O}}$) delta for each neuron “k” is performed with the follow equations:$$\underset{\_}{{\delta}_{pk}^{O}}=\left({d}_{pk}-{y}_{pk}\right){f}_{k}^{O\prime}(\underset{\_}{y})$$$$\overline{{\delta}_{pk}^{O}}=\left({d}_{pk}-{y}_{pk}\right){f}_{k}^{O\prime}\left(\overline{y}\right)$$In the hidden layer, the calculation of lower ($\underset{\_}{{\delta}_{pj}^{h}}$) and upper ($\underset{\_}{\overline{{\delta}_{pj}^{h}}}$) delta for each neuron “j” is perform with the follow equations:$$\underset{\_}{{\delta}_{pj}^{\mathrm{h}}}={f}_{j}^{h\prime}\left(\underset{\_}{Net}\right){\displaystyle \sum}_{k}\underset{\_}{{\delta}_{pk}^{O}}\underset{\_}{{w}_{kj}}$$$$\overline{{\delta}_{pj}^{\mathrm{h}}}={f}_{j}^{h\prime}\left(\overline{Net}\right){\displaystyle \sum}_{k}\overline{{\delta}_{pk}^{O}}\overline{{w}_{kj}}$$
**Stage****5:**- The utilization of a recursive algorithm allows the actualization of the interval type-2 fuzzy number weights, beginning from the output neurons and updating backwards until the neurons in the input layer. The adjustment is described as follows:The calculation of the change of interval type-2 fuzzy number weights is achieved with the equations described as follows:Calculations of the output neurons:$$\underset{\_}{\u2206{w}_{kj}\left(t+1\right)}=\underset{\_}{{\delta}_{pk}^{O}}\underset{\_}{{y}_{pj}}$$$$\overline{\u2206{w}_{kj}\left(t+1\right)}=\overline{{\delta}_{pk}^{O}}\overline{{y}_{pj}}$$Calculations of the hidden neurons:$$\underset{\_}{\u2206{w}_{ji}\left(t+1\right)}=\underset{\_}{{\delta}_{pj}^{h}}{x}_{pi}$$$$\overline{\u2206{w}_{ji}\left(t+1\right)}=\overline{{\delta}_{pj}^{h}}{x}_{pi}$$
**Stage****6:**- The method is recurrent until for each of the learned patterns the error terms are small enough.$${E}_{p}=\frac{1}{2}{\displaystyle \sum}_{k=1}^{M}{\delta}_{pk}^{2}$$

## 4. Simulation Results

#### 4.1. Neural Network with Interval Type-2 Fuzzy Numbers Weights (NNIT2FNW) for T-Norm and S-Norm of Sum-Product

#### 4.2. NNIT2FNW for T-Norm and S-Norm of Dombi

#### 4.3. NNIT2FNW for T-Norm and S-Norm of Hamacher

#### 4.4. NNIT2FNW for T-Norm and S-Norm of Frank

#### 4.5. Comparison of Traditional Neural Network Against NNIT2FNW for T-Norm and S-Norm

#### 4.6. Comparison of the Proposed Methods for Mackey-Glass Data with Noise

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Scheme of the proposed structure and equations of the neuron with interval type-2 fuzzy number weights.

**Figure 6.**Illustration of the real data against the prediction data of the Mackey-Glass time series for the fuzzy neural network.

**Figure 8.**Illustration of the prediction data of the FNNIT2FND against the real data for the Mackey-Glass time series.

**Figure 10.**Illustration of the prediction data of the FNNIT2FNH against the real data for the Mackey-Glass time series.

**Figure 12.**Illustration of the prediction data of the FNNIT2FNF against the real data for the Mackey-Glass time series.

**Figure 14.**Illustration of the prediction data against for the traditional neural network the real data of the Mackey-Glass time series.

**Figure 15.**Illustration of the convergence curves in the training process for traditional neural network.

**Figure 16.**Illustration of the results of prediction error of the TNN against the results of FNNIT2FNSp, FNNIT2FND, FNNIT2FNH, and FNNIT2FNF for data with Gaussian noise of the Mackey-Glass time series for MAE.

**Table 1.**Results for the fuzzy neural network with interval type-2 fuzzy numbers with T-norm of sum-product in time series prediction using Mackey-Glass time series.

No. Neurons | Best Prediction Error MAE | Average MAE |
---|---|---|

5 | 0.0187 | 0.0240 |

6 | 0.0197 | 0.0245 |

7 | 0.0188 | 0.0250 |

8 | 0.0172 | 0.0231 |

9 | 0.0198 | 0.0259 |

10 | 0.0170 | 0.0246 |

11 | 0.0190 | 0.0252 |

12 | 0.0192 | 0.0248 |

13 | 0.0198 | 0.0255 |

14 | 0.0191 | 0.0251 |

15 | 0.0185 | 0.0227 |

16 | 0.0149 | 0.0180 |

17 | 0.0180 | 0.0238 |

18 | 0.0202 | 0.0242 |

19 | 0.0205 | 0.0239 |

20 | 0.0164 | 0.0247 |

21 | 0.0201 | 0.0243 |

22 | 0.0189 | 0.0241 |

23 | 0.0178 | 0.0249 |

24 | 0.0195 | 0.0250 |

25 | 0.0195 | 0.0259 |

26 | 0.0189 | 0.0233 |

27 | 0.0195 | 0.0246 |

28 | 0.0191 | 0.0248 |

29 | 0.0175 | 0.0245 |

30 | 0.0149 | 0.0233 |

31 | 0.0193 | 0.0245 |

32 | 0.0182 | 0.0259 |

33 | 0.0195 | 0.0252 |

34 | 0.0170 | 0.0243 |

35 | 0.0195 | 0.0241 |

36 | 0.0188 | 0.0251 |

37 | 0.0209 | 0.0248 |

38 | 0.0187 | 0.0243 |

39 | 0.0195 | 0.0254 |

40 | 0.0190 | 0.0246 |

41 | 0.0188 | 0.0263 |

42 | 0.0172 | 0.0233 |

43 | 0.0188 | 0.0249 |

44 | 0.0192 | 0.0237 |

45 | 0.0192 | 0.0247 |

46 | 0.0157 | 0.0247 |

47 | 0.0188 | 0.0252 |

48 | 0.0189 | 0.0246 |

49 | 0.0204 | 0.0247 |

50 | 0.0151 | 0.0246 |

51 | 0.0190 | 0.0250 |

52 | 0.0179 | 0.0239 |

53 | 0.0191 | 0.0242 |

54 | 0.0177 | 0.0240 |

55 | 0.0168 | 0.0240 |

56 | 0.0202 | 0.0251 |

57 | 0.0196 | 0.0255 |

58 | 0.0181 | 0.0250 |

59 | 0.0192 | 0.0248 |

60 | 0.0173 | 0.0239 |

61 | 0.0168 | 0.0236 |

62 | 0.0188 | 0.0239 |

63 | 0.0168 | 0.0240 |

64 | 0.0183 | 0.0238 |

65 | 0.0169 | 0.0252 |

66 | 0.0185 | 0.0250 |

67 | 0.0174 | 0.0253 |

68 | 0.0171 | 0.0230 |

69 | 0.0185 | 0.0244 |

70 | 0.0186 | 0.0248 |

71 | 0.0210 | 0.0251 |

72 | 0.0182 | 0.0249 |

73 | 0.0206 | 0.0247 |

74 | 0.0169 | 0.0249 |

75 | 0.0170 | 0.0240 |

76 | 0.0174 | 0.0233 |

77 | 0.0206 | 0.0245 |

78 | 0.0185 | 0.0244 |

79 | 0.0190 | 0.0247 |

80 | 0.0178 | 0.0246 |

81 | 0.0179 | 0.0247 |

82 | 0.0185 | 0.0243 |

83 | 0.0192 | 0.0254 |

84 | 0.0170 | 0.0237 |

85 | 0.0178 | 0.0242 |

86 | 0.0186 | 0.0260 |

87 | 0.0197 | 0.0233 |

88 | 0.0197 | 0.0256 |

89 | 0.0178 | 0.0252 |

90 | 0.0191 | 0.0257 |

91 | 0.0183 | 0.0265 |

92 | 0.0193 | 0.0240 |

93 | 0.0199 | 0.0240 |

94 | 0.0166 | 0.0242 |

95 | 0.0206 | 0.0248 |

96 | 0.0181 | 0.0236 |

97 | 0.0191 | 0.0252 |

98 | 0.0199 | 0.0248 |

99 | 0.0173 | 0.0249 |

100 | 0.0181 | 0.0248 |

101 | 0.0168 | 0.0237 |

102 | 0.0173 | 0.0250 |

103 | 0.0198 | 0.0245 |

104 | 0.0191 | 0.0237 |

105 | 0.0205 | 0.0245 |

106 | 0.0197 | 0.0246 |

107 | 0.0179 | 0.0256 |

108 | 0.0185 | 0.0244 |

109 | 0.0189 | 0.0241 |

110 | 0.0164 | 0.0242 |

111 | 0.0190 | 0.0254 |

112 | 0.0198 | 0.0250 |

113 | 0.0173 | 0.0245 |

114 | 0.0203 | 0.0244 |

115 | 0.0168 | 0.0248 |

116 | 0.0170 | 0.0233 |

117 | 0.0199 | 0.0254 |

118 | 0.0188 | 0.0252 |

119 | 0.0196 | 0.0247 |

120 | 0.0189 | 0.0250 |

**Table 2.**Results for the fuzzy neural network with interval type-2 fuzzy numbers with T-norm of Dombi in time series prediction using Mackey-Glass time series.

Experiment | Prediction Error |
---|---|

1 | 0.0457 |

2 | 0.0466 |

3 | 0.0549 |

4 | 0.0581 |

5 | 0.0599 |

6 | 0.0636 |

7 | 0.0656 |

8 | 0.0671 |

9 | 0.0675 |

10 | 0.0694 |

Average | 0.0622 |

**Table 3.**Results for the fuzzy neural network with interval type-2 fuzzy numbers with T-norm of Hamacher in time series prediction using Mackey-Glass time series.

Experiment | Prediction Error |
---|---|

1 | 0.0130 |

2 | 0.0138 |

3 | 0.0149 |

4 | 0.0154 |

5 | 0.0163 |

6 | 0.0165 |

7 | 0.0170 |

8 | 0.0175 |

9 | 0.0177 |

10 | 0.0183 |

Average | 0.0164 |

**Table 4.**Results for the fuzzy neural network with interval type-2 fuzzy numbers with T-norm of Frank in time series prediction using Mackey-Glass time series.

Experiment | Prediction Error |
---|---|

1 | 0.0117 |

2 | 0.0140 |

3 | 0.0153 |

4 | 0.0156 |

5 | 0.0158 |

6 | 0.0163 |

7 | 0.0170 |

8 | 0.0175 |

9 | 0.0177 |

10 | 0.0179 |

Average | 0.0167 |

**Table 5.**Results for the traditional neural network (TNN) in the Mackey-Glass time series and the comparison against the FNNIT2FNSp, FNNIT2FND, FNNIT2FNH, and FNNIT2FNF.

Best Prediction Error | Average | |
---|---|---|

TNN | 0.0169 | 0.0203 |

FNNIT2FNSp | 0.0149 | 0.0180 |

FNNIT2FND | 0.0457 | 0.0622 |

FNNIT2FNH | 0.0130 | 0.0164 |

FNNIT2FNF | 0.0117 | 0.0167 |

**Table 6.**Results for the traditional neural network and fuzzy neural networks with all T-norms in the Mackey-Glass time series under different noise levels (n).

Noise Level | TNN | FNNIT2FNSp | FNNIT2FND | FNNIT2FNH | FNNIT2FNF |
---|---|---|---|---|---|

n = 0 | 0.0169 | 0.0149 | 0.0457 | 0.0130 | 0.0117 |

n = 0.1 | 0.0564 | 0.0617 | 0.0704 | 0.0556 | 0.0594 |

n = 0.2 | 0.1115 | 0.1135 | 0.0981 | 0.0960 | 0.0954 |

n = 0.3 | 0.1749 | 0.1275 | 0.1168 | 0.1171 | 0.1175 |

n = 0.4 | 0.2311 | 0.1554 | 0.1362 | 0.1360 | 0.1419 |

n = 0.5 | 0.3124 | 0.1661 | 0.1502 | 0.1536 | 0.1571 |

n = 0.6 | 0.3676 | 0.1897 | 0.1485 | 0.1576 | 0.1589 |

n = 0.7 | 0.4250 | 0.1866 | 0.1684 | 0.1770 | 0.1736 |

n = 0.8 | 0.4941 | 0.2018 | 0.1744 | 0.1811 | 0.1808 |

n = 0.9 | 0.5411 | 0.2077 | 0.1775 | 0.1887 | 0.1858 |

n = 1 | 0.5684 | 0.2075 | 0.1858 | 0.1920 | 0.1935 |

**Table 7.**Parameters used in the t-student statistical test for the TNN against FNNIT2FNH and FNNIT2FNF.

TNN | FNNIT2FNH | FNNIT2FNF | |
---|---|---|---|

No. Experiments | 30 | 30 | 30 |

Mean Data | 0.02028 | 0.01638 | 0.01665 |

Standard Deviation | 0.00158 | 0.00133 | 0.00123 |

Standard error of the mean | 0.00029 | 0.00024 | 0.00023 |

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**MDPI and ACS Style**

Gaxiola, F.; Melin, P.; Valdez, F.; Castillo, O.; Castro, J.R.
Comparison of T-Norms and S-Norms for Interval Type-2 Fuzzy Numbers in Weight Adjustment for Neural Networks. *Information* **2017**, *8*, 114.
https://doi.org/10.3390/info8030114

**AMA Style**

Gaxiola F, Melin P, Valdez F, Castillo O, Castro JR.
Comparison of T-Norms and S-Norms for Interval Type-2 Fuzzy Numbers in Weight Adjustment for Neural Networks. *Information*. 2017; 8(3):114.
https://doi.org/10.3390/info8030114

**Chicago/Turabian Style**

Gaxiola, Fernando, Patricia Melin, Fevrier Valdez, Oscar Castillo, and Juan R. Castro.
2017. "Comparison of T-Norms and S-Norms for Interval Type-2 Fuzzy Numbers in Weight Adjustment for Neural Networks" *Information* 8, no. 3: 114.
https://doi.org/10.3390/info8030114