# A Hybrid Model Based on Principal Component Analysis, Wavelet Transform, and Extreme Learning Machine Optimized by Bat Algorithm for Daily Solar Radiation Forecasting

^{*}

## Abstract

**:**

## 1. Introduction

- The factors affecting solar radiation contain meteorological indicators and historical data on solar radiation in this paper;
- ELM is a new type of neural network that has been applied in solar radiation prediction, which avoids the shortcomings of slow learning, large training samples, and over-fitting in previous studies;
- The BA-optimized ELM application further improves the robustness and prediction accuracy of the model;
- Implementation of WT greatly reduces the difficulty of solar radiation prediction;
- This paper focuses on the correlation between influencing factors and uses PCA to reduce the dimensionality to improve computational efficiency and prediction accuracy;
- This may be the first paper to study solar radiation prediction methods that can be applied to different parts of the world at the same time.

## 2. Methodology

#### 2.1. Wavelet Transform

_{f}(a,b) is the result of the wavelet transform.

#### 2.2. Bat Algorithm

_{max}and F

_{min}are the max and min sonic wave frequency of the bat i. In the flight, each initial bat is allocated one frequency in conformity with [F

_{min}, F

_{max}] randomly.

_{0}is the solution which is randomly selected in current best disaggregation, ${A}^{t}$ represents the mean of current bat populations, and μ is the D-dimensional vector within [−1, 1].

#### 2.3. Extreme Learning Machine

**n**input variables

**x**…

_{1}**x**. The hidden layer neuron number is L; the output layer neuron number is m, corresponding to m input variables y1… y

_{n}_{n}.

#### 2.4. The Proposed Model

## 3. Empirical Analysis

#### 3.1. Data

#### 3.2. Input Selection

#### 3.2.1. Selection of Meteorological Indexes by Pearson Coefficient Test

#### 3.2.2. Decomposition of Solar Radiation Series by WT

#### 3.2.3. Determination of the Lags by PACF

_{i}as the output variable and apply x

_{i-k}as one of the input variables if the PACF at lag k exceeds the 95% confidence interval. Table 3 presents the chosen variables of solar radiation in Beijing, New York, Melbourne, and São Paulo after WT.

#### 3.2.4. Reduction of Dimensionality by PCA

_{t-i}represents the i-th lag of historical solar radiation data.

#### 3.3. Parameters Setting and Forecasting Evaluation Criteria

^{2}. The formulas are represented as follows.

_{i}and y

_{i}* are actual and predicted values, respectively.

#### 3.4. Solar Radiation Forecasting

#### 3.4.1. The Case of Beijing

^{2}are given respectively in Figure 10. Referring to Figure 10, Table 6, the following can be obtained:

- (a)
- The MAE, MAPE, and RMSE of PCA-WT-BA-ELM are the minimum and the R
^{2}is the maximum, which demonstrates its performance sufficiently; - (b)
- The predicted carbon price curve is closest to the actual carbon price curve, which is better than the Single-LSSVM and Single-BPNN. Single ELM’s MAE, MAPE, RMSE and R
^{2}surpasses Single-LSSVM and Single-BPNN, showing that Single-ELM has the best predictive performance. In addition, as can be discovered in Table 6, the learning speed of Single-ELM is the shortest, reflecting that in the part of prediction accuracy and learning speed, Single-ELM exceeds Single-LSSVM and Single-BPNN; - (c)
- When Comparing with single ELM, hybrid models (including PSO-ELM and BA-ELM) have smaller MAE, MAPE, RMSE, and larger R
^{2}, which shows that it makes sense to optimize the ELM parameters. BA-ELM′s MAE, MAPE, and RMSE are smaller, and BA-ELM’s R^{2}is larger than PSO-ELM’s R^{2}, reflecting that BA-ELM is more precious in the whole, and BA is superior to PSO in the part of optimizing the parameter of ELM; - (d)
- After the comparison with BA-ELM, the predicted solar radiation curve of WT-BA-ELM is closest to the actual one. For that the solar radiation series is highly uncertain, nonlinear, dynamic, and complex, it may not be appropriate to predict straight without decomposition. It can therefore be seen that the MAE, MAPE, RMSE, and R
^{2}of WT-BA-ELM are better than BA-ELM; - (e)
- The predicted solar radiation curve of PCA-WT-BA-ELM is closer to the actual solar radiation curve than that of WT-BA-ELM. The MAE, MAPE, RMSE, and R
^{2}of PCA-WT-BA-ELM are better than WT-BA-ELM. All of this can verify the need to use PCA to reduce the dimensions of the BA-ELM input.

#### 3.4.2. The Case of New York, Melbourne, and São Paulo

^{2}. It can forecast the solar radiation of different parts of the world.

## 4. Conclusions

- (a)
- ELM is superior to BPNN and LSSVM in predicting accuracy and learning speed. Because the ELM parameter, which the users have to make appropriate adjustments of, is the just number of hidden nodes. After stochastically installing the input weight and the hidden layer deviation, the output weight of the ELM can be analytically determined by solving the linear system according to the Moore-Penrose (MP) generalized inverse idea.
- (b)
- In terms of prediction precision, both BA-ELM and PSO-ELM are superior to ELM, and BA-ELM is better than PSO-ELM. Therefore, it makes sense to optimize the parameters of the ELM through optimization method, and BA is more competitive than PSO;
- (c)
- The model using decomposition method WT-BA-ELM performs better than that without it, which means that the decomposition method is able to ameliorate the forecasting performance, and it is essential to denoise the solar radiation sequence through WT as its uncertain, nonlinear, dynamic and complex features;
- (d)
- Compared with the model not using dimensionality reduction method WT-BA-ELM, the model with PCA-WT-BA-ELM is superior. It shows that the dimension reduction method is able to enhance the forecasting performance, and it is a necessity to decrease the dimension of many input indicators of BA-ELM through PCA;
- (e)
- The PCA-WT-BA-ELM model is superior to other methods in solar radiation prediction in Beijing, New York, Melbourne, and São Paulo. It can therefore be inferred that the proposed hybrid model can be utilized to predict solar radiation in different parts of the world at the same time, and it greatly expands the application of the model.

_{3}, and Air Quality Index (AQI). For the fact of the serious air pollution, these indicators are very significant for Beijing solar radiation prediction. Hence, there are still several directions to study in the future.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**(

**a**) The wavelet transform (WT) decomposed results of Beijing solar radiation series; (

**b**) The WT decomposed results of New York solar radiation series; (

**c**) The WT decomposed results of Melbourne solar radiation series; (

**d**) The WT decomposed results of São Paulo solar radiation series.

**Figure 6.**(

**a**) Partial autocorrelation function (PACF) results of Beijing solar radiation series after WT; (

**b**) PACF results of New York solar radiation series after WT; (

**c**) PACF results of Melbourne solar radiation series after WT; (

**d**) The PACF results of São Paulo solar radiation series after WT.

**Figure 7.**Scree plot of Beijing, New York, Melbourne, and São Paulo in principal component analysis (PCA) analysis.

Indicator | Abbreviation | Unit | Beijing | New York | Melbourne | São Paulo |
---|---|---|---|---|---|---|

Precipitation | PRECTOT | Mm/day | 0.118 ** | 0.089 ** | −0.020 | 0.101 ** |

Specific Humidity at 2 M | SH2M | kg/kg | 0.911 ** | 0.883 ** | 0.834 ** | 0.895 ** |

Relative Humidity at 2 M | RH2M | % | 0.446 ** | 0.562 ** | −0.446 ** | 0.502 ** |

Surface Pressure | SP | kPa | −0.779 ** | −0.290 ** | −0.568 ** | −0.658 ** |

Temperature Range at 2 M | T2M_RANGE | C | −0.162 ** | -0.449 ** | 0.237 ** | −0.572 ** |

Earth Skin Temperature | EST | C | 0.922 ** | 0.768 ** | 0.800 ** | 0.691 ** |

Dew/Frost Point at 2 M | T2M_DEW | C | 0.967 ** | 0.906 ** | 0.828 ** | 0.901 ** |

Maximum Temperature at 2 M | T2M_MAX | C | 0.858 ** | 0.840 ** | 0.713 ** | 0.285 ** |

Temperature at 2 M | T2M | C | 0.926 ** | 0.844 ** | 0.893 ** | 0.868 ** |

Minimum Temperature at 2 M | T2M_MIN | C | 0.963 ** | 0.849 ** | 0.815 ** | 0.667 ** |

Wind Speed Range at 50 M | WS50M_RANGE | m/s | −0.139 ** | −0.167 ** | 0.361 ** | 0.042 |

Wind Speed Range at 10 M | WS10M_RANGE | m/s | −0.147 ** | −0.175 ** | 0.397 ** | 0.176 ** |

Minimum Wind Speed at 50 M | WS50M_MIN | m/s | −0.408 ** | −0.119 ** | −0.076 ** | −0.085 ** |

Minimum Wind Speed at 10 M | WS10M_MIN | m/s | −0.417 ** | −0.195 ** | −0.081 ** | −0.154 ** |

Maximum Wind Speed at 50 M | WS50M_MAX | m/s | −0.422 ** | −0.234 ** | 0.224 ** | −0.035 |

Maximum Wind Speed at 10 M | WS10M_MAX | m/s | −0.345 ** | −0.302 ** | 0.261 ** | 0.063 ** |

Wind Direction at 50 M | WD50M | m/s | −0.471 ** | −0.368 ** | −0.041 | 0.194 ** |

Wind Direction at 10 M | WD10M | m/s | −0.481 ** | −0.364 ** | −0.032 | 0.190 ** |

Wind Speed at 50 M | WS50M | m/s | −0.457 ** | −0.197 ** | 0.109 ** | −0.053 * |

Wind Speed at 10 M | WS10M | m/s | −0.426 ** | −0.278 ** | 0.154 ** | −0.073 ** |

Insolation Clearness Index | ICI | 0.048 * | 0.001 | 0.000 | −0.013 |

Region | Selected Indexes | Indexes Number |
---|---|---|

Beijing | SH2M, EST, T2M_DEW, T2M_MAX, T2M, T2M_MIN | 6 |

New York | SH2M, EST, T2M_DEW, T2M_MAX, T2M, T2M_MIN | 6 |

Melbourne | SH2M, EST, T2M_DEW, T2M_MAX, T2M, T2M_MIN | 6 |

São paulo | SH2M, SP, EST,T2M_DEW, T2M, T2M_MIN | 6 |

City | Lag |
---|---|

Beijing | (x_{t-1,} x_{t-2,} x_{t-3,} x_{t-4,} x_{t-5}) |

New York | (x_{t-1,} x_{t-2,} x_{t-3,} x_{t-4,} x_{t-5,} x_{t-6,} x_{t-7,} x_{t-8,} x_{t-9}) |

Melbourne | (x_{t-1,} x_{t-2,} x_{t-3,} x_{t-4,} x_{t-5,} x_{t-6,} x_{t-7}) |

São Paulo | (x_{t-1,} x_{t-2,} x_{t-3},x_{t-4,} x_{t-5,} x_{t-6,} x_{t-7}) |

Beijing | New York | Melbourne | São Paulo | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Component | PC1 | Component | PC1 | Component | PC1 | PC2 | PC3 | Component | PC1 | PC2 |

SH2M | 0.899 | SH2M | 0.87 | SH2M | 0.847 | −0.298 | 0.198 | SH2M | 0.902 | −0.162 |

EST | 0.973 | EST | 0.861 | EST | 0.81 | −0.056 | 0.458 | SP | 0.906 | −0.005 |

T2M_DEW | 0.954 | T2M_DEW | 0.854 | T2M_DEW | 0.782 | 0.22 | 0.513 | EST | 0.881 | 0.21 |

T2M_MAX | 0.932 | T2M_MAX | 0.856 | T2M_MAX | 0.774 | 0.415 | 0.321 | T2M_DEW | 0.857 | 0.38 |

T2M | 0.975 | T2M | 0.859 | T2M | 0.765 | 0.518 | 0.004 | T2M | 0.84 | 0.459 |

T2M_MIN | 0.985 | T2M_MIN | 0.861 | T2M_MIN | 0.741 | 0.517 | −0.255 | T2M_MIN | 0.816 | 0.455 |

Lag 1 x_{t-1} | 0.966 | Lag 1 x_{t-1} | 0.862 | Lag 1 x_{t-1} | 0.704 | 0.4 | −0.324 | Lag 1 x_{t-1} | 0.78 | 0.383 |

Lag 2 x_{t-2} | 0.96 | Lag 2 x_{t-2} | 0.854 | Lag 2 x_{t-2} | 0.779 | −0.451 | 0.034 | Lag 2 x_{t-2} | 0.928 | −0.152 |

Lag 3 x_{t-2} | 0.954 | Lag 3 x_{t-2} | 0.834 | Lag 3 x_{t-2} | 0.927 | −0.115 | −0.22 | Lag 3 x_{t-2} | −0.713 | 0.214 |

Lag 4 x_{t-4} | 0.95 | Lag 4 x_{t-4} | 0.93 | Lag 4 x_{t-4} | 0.776 | −0.459 | 0.021 | Lag 4 x_{t-4} | 0.813 | −0.417 |

Lag 5 x_{t-5} | 0.941 | Lag 5 x_{t-5} | 0.959 | Lag 5 x_{t-5} | 0.869 | −0.077 | −0.291 | Lag 5 x_{t-5} | 0.923 | −0.153 |

Lag 6 x_{t-6} | 0.938 | Lag 6 x_{t-6} | 0.924 | −0.259 | −0.15 | Lag 6 x_{t-6} | 0.922 | −0.311 | ||

Lag 7 x_{t-7} | 0.961 | Lag 7 x_{t-7} | 0.92 | −0.148 | −0.241 | Lag 7 x_{t-7} | 0.795 | −0.448 | ||

Lag8 x_{t-8} | 0.962 | |||||||||

Lag 9x_{t-9} | 0.965 |

Model | Parameters |
---|---|

BPNN | L = 10; learning rate = 0.0004 |

LSSVM | L = 10; γ = 50; σ^{2} = 2 |

ELM | L = 10; g(x) = ‘sig’; |

PSO-ELM | N = 10; N_iter = 500; c1 = c2 = 2; w = 1.5; rand = 0.8 |

BA-ELM | N = 10; N_iter = 500; A = 1.5; γ = θ = 0.9; R = 0.0001; F = [0, 2] |

Training Time (s) | Test Time (s) | |
---|---|---|

BPNN | 2621.112 | 0.265 |

LSSVM | 1784.593 | 0.153 |

ELM | 19.341 | 0.001 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Wei, Z.
A Hybrid Model Based on Principal Component Analysis, Wavelet Transform, and Extreme Learning Machine Optimized by Bat Algorithm for Daily Solar Radiation Forecasting. *Sustainability* **2019**, *11*, 4138.
https://doi.org/10.3390/su11154138

**AMA Style**

Zhang X, Wei Z.
A Hybrid Model Based on Principal Component Analysis, Wavelet Transform, and Extreme Learning Machine Optimized by Bat Algorithm for Daily Solar Radiation Forecasting. *Sustainability*. 2019; 11(15):4138.
https://doi.org/10.3390/su11154138

**Chicago/Turabian Style**

Zhang, Xing, and Zhuoqun Wei.
2019. "A Hybrid Model Based on Principal Component Analysis, Wavelet Transform, and Extreme Learning Machine Optimized by Bat Algorithm for Daily Solar Radiation Forecasting" *Sustainability* 11, no. 15: 4138.
https://doi.org/10.3390/su11154138