# Hybrid Approach of Fractional Generalized Pareto Motion and Cosine Similarity Hidden Markov Model for Solar Radiation Forecasting

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## Abstract

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## 1. Introduction

#### 1.1. Research Background

#### 1.2. Literature Review

#### 1.3. Research Highlights

#### 1.4. Organization of the Article

## 2. Difference Iterative Prediction Model, Based on fGPm

#### 2.1. Definition and Simulation of the fGPm

#### 2.2. Long-Range Dependence and Self-Similarity of the fGPm

#### 2.3. The Difference and Iteration in the fGPm Predictive Model

#### 2.4. Parameter Estimation

## 3. Combination of the Hidden Markov Model and Cosine-Similarity

## 4. Hybrid Algorithm, Based on fGPm and CS-HMM

#### 4.1. Calculation for the Number of Prediction Steps

#### 4.2. Hybrid Predictive Algorithm

- (1)
- The MPS is obtained by calculating the maximum Lyapunov exponent.
- (2)
- The long-range dependence condition for the fGPm difference iterative prediction model is tested.
- (3)
- The initial prediction of the solar radiation is carried out, based on the CS-HMM algorithm.
- (4)
- The initial prediction results are fed to the SDE, driven by fGPm to predict future increments.
- (5)
- The Maruyama notation is employed in the discretized SDE.
- (6)
- The prediction is finalized with the difference and iteration.

## 5. Case Study

#### 5.1. Dataset of Solar Irradiation

#### 5.2. The Number of Prediction Steps for Solar Irradiation Prediction

#### 5.3. Self-Similarity and the Long-Range Dependence of Solar Radiation Prediction

#### 5.4. Evaluation of the Hybrid Predictive Model

#### 5.5. Discussion and Future Work

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CS-HMM | cosine similarity hidden Markov model |

fGPm | fractional generalized Pareto motion |

GPm | generalized Pareto motion |

SDE | stochastic differentiate equation |

MPS | maximum prediction steps |

RMSE | root mean square error |

MAE | mean absolute error |

R/S | rescaled range method |

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**Figure 5.**Training dataset of solar irradiance. The x-axis corresponds to time (day) and the y-axis corresponds to the effective daily surface radiation (MJ/(m

^{2}∙d)).

**Figure 8.**(

**a**) Forecasting the results for the hybrid approach; (

**b**) forecasting results for the fGPm approach; (

**c**) forecasting results for the CS-HMM algorithm.

Predictive Method | MAE | RMSE | Relative Error | |
---|---|---|---|---|

Max (%) | Mean (%) | |||

Hybrid | 0.8984 | 1.9864 | 2.97 | 1.13 |

fGPm | 0.9265 | 2.0752 | 3.06 | 1.22 |

CS-HMM | 0.9574 | 2.1224 | 3.14 | 1.31 |

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**MDPI and ACS Style**

Song, W.; Deng, W.; Chen, D.; Jin, R.; Kudreyko, A.
Hybrid Approach of Fractional Generalized Pareto Motion and Cosine Similarity Hidden Markov Model for Solar Radiation Forecasting. *Fractal Fract.* **2023**, *7*, 93.
https://doi.org/10.3390/fractalfract7010093

**AMA Style**

Song W, Deng W, Chen D, Jin R, Kudreyko A.
Hybrid Approach of Fractional Generalized Pareto Motion and Cosine Similarity Hidden Markov Model for Solar Radiation Forecasting. *Fractal and Fractional*. 2023; 7(1):93.
https://doi.org/10.3390/fractalfract7010093

**Chicago/Turabian Style**

Song, Wanqing, Wujin Deng, Dongdong Chen, Rong Jin, and Aleksey Kudreyko.
2023. "Hybrid Approach of Fractional Generalized Pareto Motion and Cosine Similarity Hidden Markov Model for Solar Radiation Forecasting" *Fractal and Fractional* 7, no. 1: 93.
https://doi.org/10.3390/fractalfract7010093