# Predicting Stock Market Volatility Using MODWT with HyFIS and FS.HGD Models

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## Abstract

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## 1. Introduction

## 2. Literature Review

^{2}compared to 46.34 W/m

^{2}. This study used data from the solar radiation measurement network in Portugal between 2007 and 2013. In a related study, Bhagat et al. (2022) discussed the establishment of a dynamic evolving neural fuzzy inference system model for natural air temperature prediction. This model leverages the HyFIS algorithm and the adaptive neuro-fuzzy inference system to achieve superior accuracy and adaptability compared to other models. The authors provide a comprehensive explanation of the model’s architecture and training process, including the use of the heuristic gradient descent (HGD) algorithm. These results demonstrate the effectiveness of the HyFIS algorithm in improving the performance of neural fuzzy inference systems for prediction tasks. Overall, the HyFIS model offers an attractive solution for accurate and reliable predictions in a variety of fields.

## 3. Methods and Mathematical Models

#### 3.1. Wavelet Transform

#### 3.2. HyFIS Model

#### 3.3. FS.HGD Model

#### 3.3.1. Fuzzy System

#### 3.3.2. Heuristic Method

#### 3.3.3. Learning Method

- Step 1: Specify the initial value ${\omega}_{j}^{init}$ of ${\omega}_{j}$, the value of $\beta $ and the maximum iteration number ${t}_{max}.$ Let $t:=0$.
- Step 2: For $p=\mathrm{1,2},\dots ,m$, adjust each ${\omega}_{j}$ by (8). Let $t:=t$+ 1.
- Step 3: If $t\ge {t}_{max}$, then stop this procedure, else go to Step 2.

- 4.
- ${\omega}_{j}^{init}=0,j=\mathrm{1,2},\dots ,N$,
- 5.
- ${\omega}_{j}^{init}$ = ${\omega}_{j}^{HM},J=\mathrm{1,2},\dots ,N$.

#### 3.4. Accuracy Criteria

#### 3.4.1. Algorithm of Self-Tuning

#### 3.4.2. Error Criteria Test

## 4. Data Description

## 5. Empirical Results and Discussion

#### 5.1. Selecting Variables

#### 5.2. Results of FS.HGD and HyFIS

## 6. Limitations and Future Work

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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LSCS | Repo | Loil | |
---|---|---|---|

Sample size | 2026 | 2026 | 2026 |

Arithmetic mean | 6.749 | 0.696 | 4.299 |

Standard deviation | 0.692 | 0.280 | 0.354 |

Skewness | −2.099 | 2.006 | −0.175 |

Kurtosis | 4.263 | 22.797 | −1.107 |

Variables | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | Collinearity Statistics | ||
---|---|---|---|---|---|---|---|

B | Std. Error | Beta | Tolerance | VIF | |||

(Constant) | 12.495 | 0.139 | 89.634 | 0.000 | |||

Repo | 0.198 | 0.043 | 0.080 | 4.621 | 0.000 | 0.893 | 1.120 |

Loil | −1.369 | 0.034 | −0.699 | −40.355 | 0.000 | 0.893 | 1.120 |

WT Function | ME | MAE | MAPE | |
---|---|---|---|---|

Haar | ARIMA(0,1,1) with drift | 0.000396455 | 0.004130676 | 0.08892953 |

Db4 | ARIMA(0,1,0) | 0.000575673 | 0.004708187 | 0.09547983 |

LA-8 | ARIMA(1,1,0) | 0.00000532 | 0.003214182 | 0.06449683 |

BL-14 | ARIMA(1,1,0) with drift | 0.000009564 | 0.003294034 | 0.06557786 |

C6 | ARIMA (1,1,0) with drift | 0.000032995 | 0.003314655 | 0.06604977 |

Models | RMSE | MAE | MAPE |
---|---|---|---|

FS.HGD | 0.105636185 | 0.076441481 | 1.092438331 |

MODWT-Haar-FS.HGD | 0.129052214 | 0.093689045 | 1.347518627 |

MODWT-d4-FS.HGD | 0.058661444 | 0.046339464 | 0.648079708 |

MODWT-LA8-FS.HGD | 0.048260312 | 0.038441829 | 0.538406565 |

MODWT-bl14-FS.HGD | 0.050900152 | 0.042638941 | 0.597125838 |

MODWT-C6-FS.HGD | 0.0829242 | 0.075660166 | 1.055475213 |

FS.HGD+ARIMA direct | 0.085056996 | 0.075359705 | 1.050702605 |

HyFIS | 0.086024702 | 0.081597959 | 1.150437522 |

MODWT-Haar-HyFIS | 0.092197458 | 0.085709326 | 1.198456099 |

MODWT-d4-HyFIS | 0.090689813 | 0.084132196 | 1.177323058 |

MODWT-LA8-HyFIS | 0.604894468 | 0.423032849 | 6.794333688 |

MODWT-bl14-HyFIS | 0.082887834 | 0.071352358 | 0.995263056 |

MODWT-C6-HyFIS | 0.091249745 | 0.083464919 | 1.165024215 |

HyFIS+ARIMA direct | 0.086024702 | 0.081597959 | 1.150437522 |

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**MDPI and ACS Style**

Alenezy, A.H.; Ismail, M.T.; Wadi, S.A.; Jaber, J.J.
Predicting Stock Market Volatility Using MODWT with HyFIS and FS.HGD Models. *Risks* **2023**, *11*, 121.
https://doi.org/10.3390/risks11070121

**AMA Style**

Alenezy AH, Ismail MT, Wadi SA, Jaber JJ.
Predicting Stock Market Volatility Using MODWT with HyFIS and FS.HGD Models. *Risks*. 2023; 11(7):121.
https://doi.org/10.3390/risks11070121

**Chicago/Turabian Style**

Alenezy, Abdullah H., Mohd Tahir Ismail, Sadam Al Wadi, and Jamil J. Jaber.
2023. "Predicting Stock Market Volatility Using MODWT with HyFIS and FS.HGD Models" *Risks* 11, no. 7: 121.
https://doi.org/10.3390/risks11070121