# PSO with Dynamic Adaptation of Parameters for Optimization in Neural Networks with Interval Type-2 Fuzzy Numbers Weights

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basement

#### 2.1. Particle Swarm Optimization (PSO)

- $${x}_{i}\left(t+1\right)={x}_{i}\left(t\right)+{v}_{i}\left(t+1\right)$$
- ${x}_{i}$ is the particle $i$.
- $t$ is the current iteration.
- ${v}_{i}$ is the velocity of the particle $i$.

- $${v}_{i}\left(t+1\right)={v}_{ij}\left(t\right)+{c}_{1}{r}_{1}\left(t\right)\left[{y}_{ij}\left(t\right)-{x}_{ij}\left(t\right)\right]+{c}_{2}{r}_{2}\left(t\right)\left[{y}^{^}{}_{j}\left(t\right)-{x}_{ij}\left(t\right)\right]$$
- ${v}_{i}$ is the velocity of the particle $i$ in the dimension $j$.
- ${c}_{1}$ is the cognitive factor (importance of the best previous position of the particle).
- ${c}_{2}$ is the social factor (importance of the best global position of the swarm).
- ${r}_{1},{r}_{2}$ are random values in the range of [0, 1].
- ${y}_{ij}$ is the best position of the particle $i$ in the dimension $j$.
- ${x}_{ij}$ is the current position of the particle $i$ in the dimension $j$.
- ${y}^{^}{}_{j}$ is the best global position of the swarm in the dimension $j$.

#### 2.2. Type-2 Fuzzy Systems

- ${A}^{~}$ is a type-2 fuzzy set.
- ${\mu}_{{A}^{~}}\left(x,u\right)$ is a type-2 membership function.
- ${J}_{x}$ is called primary membership function of ${A}^{~}$.

#### 2.3. Fuzzy Neural Network

## 3. Antecedents Development

## 4. Proposed Method and Problem Description

- ${x}_{ij}\left(t\right)$: represents the particle in the iteration t.
- $\overline{{x}_{j}}\left(t\right)$: represents the best particle in the iteration t.

- minDiver: Minimum Euclidian distance for the particle.
- maxDiver: Maximum Euclidian distance for the particle.
- DiverNorm: Value obtained with Equation (13).$$DiverNorm=\frac{Diversity-minDiver}{maxDiver-minDiver}$$

- If (Iteration is Low) and (Diversity is Low) then (${c}_{1}$ is VeryHigh) (${c}_{2}$ is VeryLow).
- If (Iteration is Low) and (Diversity is Medium) then (${c}_{1}$ is High) (${c}_{2}$ is Medium).
- If (Iteration is Low) and (Diversity is High) then (${c}_{1}$ is High) (${c}_{2}$ is Low).
- If (Iteration is Medium) and (Diversity is Low) then (${c}_{1}$ is High) (${c}_{2}$ is Low).
- If (Iteration is Medium) and (Diversity is Medium) then (${c}_{1}$ is Medium) (${c}_{2}$ is Medium).
- If (Iteration is Medium) and (Diversity is High) then (${c}_{1}$ is Low) (${c}_{2}$ is High).
- If (Iteration is High) and (Diversity is Low) then (${c}_{1}$ is Medium) (${c}_{2}$ is VeryHigh).
- If (Iteration is High) and (Diversity is Medium) then (${c}_{1}$ is Low) (${c}_{2}$ is High).
- If (Iteration is High) and (Diversity is High) then (${c}_{1}$ is VeryLow) (${c}_{2}$ is VeryHigh).

## 5. Simulation Results

## 6. Discussion of Results

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 10.**Illustration of the real data against the prediction data of the Germany stock exchange time series for the fuzzy neural network using Hamacher optimized with dynamic PSO.

**Figure 11.**Illustration of the real data against the prediction data of the Mexican stock exchange time series for the fuzzy neural network using Frank optimized with dynamic PSO.

**Figure 12.**Illustration of the real data against the prediction data of the Dow-Jones stock exchange time series for the fuzzy neural network using Hamacher optimized with dynamic PSO.

**Figure 13.**Illustration of the real data against the prediction data of the London stock exchange time series for the fuzzy neural network using Hamacher without optimization.

**Figure 14.**Illustration of the real data against the prediction data of theNasdaq stock exchange time series for the fuzzy neural network using Hamacher optimized with dynamic PSO.

**Figure 15.**Illustration of the real data against the prediction data of the Shanghai stock exchange time series for the fuzzy neural network using Hamacher and optimized with the dynamic PSO.

**Figure 16.**Illustration of the real data against the prediction data of Taiwan stock exchange time series for the fuzzy neural network using Hamacher optimized with dynamic PSO.

Parameters | Values |
---|---|

Particles | 50 |

Dimensions | 2 |

Iterations | 50 |

Inertia Weight | 0.1 |

Constriction Coefficient (C) | 1 |

R1, R2 | Random in [0, 1] |

${c}_{1}$, ${c}_{2}$ | Dynamic adjust IT2FIS |

Error | Results | Germany | Mexican | Dow-Jones | London | Nasdaq | Shanghai | Taiwan |
---|---|---|---|---|---|---|---|---|

MAE | Best | 1089.19 | 733.50 | 920.89 | 167.05 | 468.81 | 361.95 | 383.00 |

Average | 1138.48 | 755.34 | 956.03 | 168.76 | 470.43 | 367.90 | 412.52 | |

RMSE | Best | 1264.68 | 917.93 | 1075.70 | 200.13 | 550.95 | 561.29 | 458.28 |

Average | 1332.28 | 935.58 | 1109.73 | 203.29 | 558.95 | 580.00 | 480.11 |

**Table 3.**Results of the IT2FNWNN with the S-norm and T-norm of the sum of the product for the financial time series.

Error | Results | Germany | Mexican | Dow-Jones | London | Nasdaq | Shanghai | Taiwan |
---|---|---|---|---|---|---|---|---|

MAE | Best | 1038.84 | 739.26 | 736.26 | 155.51 | 585.70 | 391.21 | 321.86 |

Average | 1148.27 | 760.49 | 937.09 | 159.20 | 725.07 | 403.97 | 339.18 | |

RMSE | Best | 1338.21 | 915.45 | 840.21 | 192.14 | 786.84 | 458.75 | 499.62 |

Average | 1414.93 | 939.70 | 1098.61 | 196.65 | 884.07 | 474.99 | 615.84 |

**Table 4.**Results of the IT2FNWNN with the S-norm and T-norm of Hamacher for the financial time series.

Error | Results | Germany | Mexican | Dow-Jones | London | Nasdaq | Shanghai | Taiwan |
---|---|---|---|---|---|---|---|---|

MAE | Best | 1055.06 | 730.13 | 686.05 | 151.76 | 463.77 | 314.71 | 349.78 |

Average | 1093.54 | 756.95 | 787.61 | 158.96 | 711.63 | 334.81 | 392.70 | |

RMSE | Best | 1216.24 | 912.65 | 800.68 | 188.46 | 570.36 | 569.32 | 449.41 |

Average | 1334.42 | 945.19 | 925.43 | 198.98 | 854.65 | 626.49 | 472.24 |

Error | Results | Germany | Mexican | Dow-Jones | London | Nasdaq | Shanghai | Taiwan |
---|---|---|---|---|---|---|---|---|

MAE | Best | 1058.87 | 730.96 | 853.69 | 153.35 | 663.02 | 354.07 | 362.56 |

Average | 4439.15 | 756.19 | 944.12 | 159.58 | 741.04 | 336.64 | 399.85 | |

RMSE | Best | 1256.37 | 915.84 | 972.34 | 186.68 | 730.97 | 607.68 | 448.20 |

Average | 1402.82 | 936.95 | 1101.39 | 197.71 | 902.34 | 640.26 | 470.09 |

**Table 6.**Results of the optimization with Dynamic PSO for Hamacher IT2FNWNN for the financial time series.

Financial Time Series | Prediction Error | No. Neurons | Parameter $\mathit{\gamma}$ |
---|---|---|---|

Germany | 417.33 | 39 | 1.6471 |

Mexican | 744.34 | 21 | 0.9997 |

Dow-Jones | 517.58 | 15 | 1.9270 |

London | 155.72 | 22 | 1.0016 |

Nasdaq | 227.53 | 56 | 0.6822 |

Shanghai | 211.40 | 99 | 1.7397 |

Taiwan | 236.15 | 28 | 1.1997 |

**Table 7.**Results of the IT2FNWNN with the S-norm and T-norm of Hamacher optimized with dynamic PSO for the financial time series.

Error | Results | Germany | Mexican | Dow-Jones | London | Nasdaq | Shanghai | Taiwan |
---|---|---|---|---|---|---|---|---|

MAE | Best | 417.33 | 728.61 | 517.58 | 154.87 | 227.53 | 211.40 | 236.15 |

Average | 813.21 | 751.48 | 881.97 | 160.47 | 342.14 | 313.86 | 401.49 | |

RMSE | Best | 507.43 | 909.39 | 769.15 | 192.81 | 283.92 | 443.85 | 331.32 |

Average | 1172.56 | 930.47 | 1290.34 | 197.99 | 530.34 | 543.39 | 472.44 |

**Table 8.**Results of the optimization with dynamic PSO for Frank IT2FNWNN for the financial time series.

Financial Time Series | Prediction Error | No. Neurons | Parameter $\mathit{\gamma}$ |
---|---|---|---|

Germany | 935.77 | 89 | 1.7796 |

Mexican | 726.84 | 22 | 1.4607 |

Dow-Jones | 768.81 | 119 | 3.1526 |

London | 153.60 | 32 | 1.2669 |

Nasdaq | 302.19 | 73 | 1.2910 |

Shanghai | 277.72 | 79 | 1.9962 |

Taiwan | 359.21 | 124 | 1.6347 |

**Table 9.**Results of the IT2FNWNN with the S-norm and T-norm of Frank optimized with dynamic PSO for the financial time series.

Error | Results | Germany | Mexican | Dow-Jones | London | Nasdaq | Shanghai | Taiwan |
---|---|---|---|---|---|---|---|---|

MAE | Best | 935.77 | 726.84 | 768.81 | 153.60 | 288.70 | 277.72 | 359.21 |

Average | 1050.87 | 773.60 | 863.79 | 163.12 | 391.39 | 373.38 | 399.28 | |

RMSE | Best | 1226.07 | 911.63 | 979.26 | 188.63 | 349.02 | 491.98 | 439.93 |

Average | 1302.99 | 963.74 | 1082.81 | 201.38 | 469.52 | 597.62 | 481.62 |

**Table 10.**Comparison of the best results of financial time series prediction with all neural networks.

Financial Time Series | Error | TNN | IT2FNWNNSp | IT2FNWNNH | IT2FNWNNF | IT2FNWNNH-PSO | IT2FNWNNF-PSO |
---|---|---|---|---|---|---|---|

Germany | MAE | 1089.19 | 1038.84 | 1055.06 | 1058.87 | 417.33 | 935.77 |

RMSE | 1264.68 | 1338.21 | 1216.24 | 1256.37 | 507.43 | 1226.07 | |

Mexican | MAE | 733.50 | 739.26 | 730.13 | 730.96 | 728.61 | 726.84 |

RMSE | 917.93 | 915.45 | 912.65 | 915.84 | 909.39 | 911.63 | |

Dow-Jones | MAE | 920.89 | 736.26 | 686.05 | 853.69 | 517.58 | 768.81 |

RMSE | 1075.70 | 840.21 | 800.68 | 972.34 | 769.15 | 979.26 | |

London | MAE | 167.05 | 155.51 | 151.76 | 153.35 | 154.87 | 153.60 |

RMSE | 200.13 | 192.14 | 188.46 | 186.68 | 192.81 | 188.63 | |

Nasdaq | MAE | 468.81 | 585.70 | 463.77 | 663.02 | 227.53 | 288.70 |

RMSE | 550.95 | 786.84 | 570.36 | 730.97 | 283.92 | 349.02 | |

Shanghai | MAE | 361.95 | 391.21 | 314.71 | 354.07 | 211.40 | 277.72 |

RMSE | 561.29 | 458.75 | 569.32 | 607.68 | 443.85 | 491.98 | |

Taiwan | MAE | 383.00 | 321.86 | 349.78 | 362.56 | 236.15 | 359.21 |

RMSE | 458.28 | 499.62 | 449.41 | 448.20 | 331.32 | 439.93 |

**Table 11.**Comparison of the best results of financial time series prediction with all neural networks.

Financial Time Series | Comparison Results | N | AVG | SD | SEM | ED | LLD 95% | T Value | P Value | GL |
---|---|---|---|---|---|---|---|---|---|---|

Germany | H1:TNN | 30 | 1332.3 | 22.8 | 4.2 | 159.7 | 66 | 2.9 | 0.004 | 29 |

H2: IT2FNWNNH-PSO | 30 | 1173 | 301 | 55 | ||||||

Mexican | H1:TNN | 30 | 935.6 | 12.5 | 2.3 | −28.15 | −35.27 | −5.83 | 1 | 44 |

H2: IT2FNWNNF-PSO | 30 | 9637 | 23.3 | 4.3 | ||||||

Dow-Jones | H1:TNN | 30 | 1082.8 | 47.1 | 8.6 | −208 | −491 | −1.24 | 0.888 | 29 |

H2: IT2FNWNNH-PSO | 30 | 1290 | 912 | 167 | ||||||

London | H1:TNN | 30 | 203.29 | 2.48 | 0.45 | 4.304 | 2.927 | 5.24 | 0.0004 | 50 |

H2: IT2FNWNNH | 30 | 198.98 | 3.76 | 0.69 | ||||||

Nasdaq | H1:TNN | 30 | 558.95 | 4.19 | 0.77 | 28.6 | −26 | 0.89 | 0.190 | 29 |

H2: IT2FNWNNH-PSO | 30 | 530 | 176 | 32 | ||||||

Shanghai | H1:TNN | 30 | 580 | 7.48 | 1.4 | 36.61 | 22.93 | 4.54 | 0.0004 | 30 |

H2: IT2FNWNNH-PSO | 30 | 543.4 | 43.5 | 7.9 | ||||||

Taiwan | H1:TNN | 30 | 480.11 | 7.31 | 1.3 | −1.51 | −5.88 | −0.58 | 0.717 | 47 |

H2: IT2FNWNNH-PSO | 30 | 481.6 | 12.2 | 2.2 |

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**MDPI and ACS Style**

Gaxiola, F.; Melin, P.; Valdez, F.; Castro, J.R.; Manzo-Martínez, A.
PSO with Dynamic Adaptation of Parameters for Optimization in Neural Networks with Interval Type-2 Fuzzy Numbers Weights. *Axioms* **2019**, *8*, 14.
https://doi.org/10.3390/axioms8010014

**AMA Style**

Gaxiola F, Melin P, Valdez F, Castro JR, Manzo-Martínez A.
PSO with Dynamic Adaptation of Parameters for Optimization in Neural Networks with Interval Type-2 Fuzzy Numbers Weights. *Axioms*. 2019; 8(1):14.
https://doi.org/10.3390/axioms8010014

**Chicago/Turabian Style**

Gaxiola, Fernando, Patricia Melin, Fevrier Valdez, Juan R. Castro, and Alain Manzo-Martínez.
2019. "PSO with Dynamic Adaptation of Parameters for Optimization in Neural Networks with Interval Type-2 Fuzzy Numbers Weights" *Axioms* 8, no. 1: 14.
https://doi.org/10.3390/axioms8010014