# A Novel Hybrid Machine Learning Model for Wind Speed Probabilistic Forecasting

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

- (1)
- A new machine learning method named LGB is used to predict wind-speed sequences, which can provide accurate wind-speed prediction results.
- (2)
- A novel hybrid model combining LGB and GPR is proposed for wind-speed probability prediction.
- (3)
- The proposed hybrid model is applied to a real case in the United States and compared with eight contrasting models.

## 2. Methodology

#### 2.1. Light Gradient Boosting Machine

#### 2.1.1. Model Formulation

_{1}, y

_{1}), (x

_{2}, y

_{2}), $\dots $, (x

_{N}, y

_{N})}, x

_{i}$\in $X, y

_{i}$\in $Y. x

_{i}is the n-dimensional feature vector, X is the input space, y

_{i}is the one-dimensional label, Y is the output space, and N is the number of samples. The model can be expressed as follows:

_{1}, R

_{2}, $\dots $, R

_{J}, there is a certain output value corresponding to each region c

_{j}, and the regression tree can be expressed as:

_{1}, c

_{1}), (R

_{2}, c

_{2}), $\dots $, (R

_{J}, c

_{J})} is the divided area of the tree and the output value on the corresponding area, J is the complexity of the tree, that is, the number of leaf nodes of the tree.

#### 2.1.2. Model Optimization Mechanism

- (1)
- Gradient-based One-Side Sampling (GOSS): Without changing the distribution of sample data, some samples with small gradients can be eliminated, and only the remaining samples with larger gradients can be retained to estimate information gain, thereby reducing the number of training samples. Since samples with smaller gradients also contribute little to information gain, GOSS technology can make the LGB model faster while ensuring accuracy.
- (2)
- Exclusive Feature Bundling: In practical applications, high-dimensional data is often sparse. LGB model adopts the histogram (Histogram) algorithm to merge those mutually exclusive features after discretizing continuous features to form new features, reduce feature dimension, reduce memory usage, and speed up model training.
- (3)
- Leaf-wise Tree Growth with Depth Limit: Change the level-wise tree growth adopted by most decision tree models to a leaf-wise growth strategy. Compared to the original, each leaf node was split, but now only the leaf node with the largest split gain is split, which reduces unnecessary overhead. In the case of the same number of classifications, the latter is more accurate than the former. LGB avoids model overfitting by setting the maximum tree depth parameter. The growth diagram of the decision tree is shown in Figure 1:

#### 2.1.3. Model Implementation Process

- (1)
- Initialize, find the constant value that minimizes the overall loss function.$${f}_{0}\left(x\right)=\mathrm{arg}\underset{c}{\mathrm{min}}{\displaystyle \sum _{i=1}^{N}L\left({y}_{i},c\right)}$$
- (2)
- For m = 1, 2, $\dots $, M.
- (a)
- For i = 1, 2, $\dots $, N, the residual is estimated by the negative gradient of the loss function.$${r}_{mi}=-{\left[\frac{\partial L\left({y}_{i},f\left({x}_{i}\right)\right)}{\partial f\left({x}_{i}\right)}\right]}_{f\left(x\right)={f}_{m-1}\left(x\right)}$$
- (b)
- Fit a regression tree to r
_{m}to obtain the leaf node area R_{mj}of the m-th tree, that is, j = 1, 2, $\dots $, J. - (c)
- For j = 1, 2, $\dots $, J, estimate the value of the leaf node region using a linear search fit to minimize the loss function.$${c}_{mj}=\mathrm{arg}\underset{c}{\mathrm{min}}{\displaystyle \sum _{{x}_{i}\in {R}_{mj}}L\left({y}_{i},{f}_{m-1}\left({x}_{i}\right)+c\right)}$$
- (d)
- Iteratively update with the following formula.$${f}_{m}\left(x\right)={f}_{m-1}\left(x\right)+{\displaystyle \sum _{j=1}^{J}{c}_{mj}I\left(x\in {R}_{mj}\right)}$$

- (3)
- Get the final model.$${f}_{M}\left(x\right)={\displaystyle \sum _{m=1}^{M}{\displaystyle \sum _{j=1}^{J}{c}_{mj}I\left(x\in {R}_{mj}\right)}}$$

#### 2.2. Gaussian Process Regression

_{1}, x

_{2}, $\dots $, x

_{n}] represents the n-dimensional input feature vector, and Y = [y

_{1}, y

_{2}, $\dots $, y

_{n}] represents the predictor variable. Suppose x and y form the following regression model:

^{train}is:

^{train}, X

^{train}) is an n × n symmetric positive definite covariance matrix, I

_{n}is an n-dimensional identity matrix. The detailed expression of $\sum $(X

^{train}, X

^{train}) is as follows:

_{i,j}represents the covariance between feature i and feature j. Gaussian process kernel function $\kappa $ is introduced to simulate the covariance between each feature dimension, $\sum $(X

^{train}, X

^{train}) = ($\kappa $

_{ij}). In the paper, the radial basis kernel function is used, and the formula is as follows:

^{train}and y

^{test}is as follows:

^{train}and the test set feature input X

^{test}, $\sum ({X}^{test},{X}^{test})$ is the internal covariance matrix of the test set feature input.

^{test}of the test set can be obtained by Bayesian inference.

^{test}is the predicted mean of the test set; ${\sigma}_{{y}^{test}}^{2}$ is the variance of the Gaussian distribution.

#### 2.3. LGB-GPR

**Step1**: Train LGB model with $\left[{X}_{1}^{ta},{X}_{2}^{ta},\dots ,{X}_{Ta}^{ta}\right]$ and $\left[{y}_{1}^{ta},{y}_{2}^{ta},\dots ,{y}_{Ta}^{ta}\right]$ as features and labels, respectively.

**Step2**: Taking $\left[{X}_{1}^{ta},{X}_{2}^{ta},\dots ,{X}_{Ta}^{ta}\right]$ and $\left[{X}_{1}^{te},{X}_{2}^{te},\dots ,{X}_{Te}^{te}\right]$ as input, respectively, use the trained LGB model to obtain $\left[{y}_{1,1}^{ta},{y}_{1,2}^{ta},\dots ,{y}_{1,Ta}^{ta}\right]$ and $\left[{y}_{1,1}^{te},{y}_{1,2}^{te},\dots ,{y}_{1,Te}^{te}\right]$.

**Step3**: Train the GPR model with $\left[{y}_{1,1}^{ta},{y}_{1,2}^{ta},\dots ,{y}_{1,Ta}^{ta}\right]$ and $\left[{y}_{1}^{ta},{y}_{2}^{ta},\dots ,{y}_{Ta}^{ta}\right]$.

**Step4**: Using the trained GPR model to get $\left[{y}_{2,1}^{te},{y}_{2,2}^{te},\dots ,{y}_{2,Te}^{te}\right]$, evaluate model prediction accuracy based on $\left[{y}_{1}^{te},{y}_{2}^{te},\dots ,{y}_{Te}^{te}\right]$ and $\left[{y}_{2,1}^{te},{y}_{2,2}^{te},\dots ,{y}_{2,Te}^{te}\right]$.

## 3. Scoring Metrics

#### 3.1. Deterministic Forecasting Evaluation Metrics

^{2}), root mean square error (RMSE), and mean absolute percent error (MAPE) are employed to evaluate the deterministic forecasting results:

^{2}is to 1, the better the prediction results.

#### 3.2. Probabilistic Forecasting Evaluation Metric

## 4. Case Study

#### 4.1. Case Introduction

#### 4.2. Data Processing

#### 4.2.1. Data Normalized

#### 4.2.2. Feature Selection

#### 4.3. Model Selection and Hyperparameter Optimization

## 5. Result and Discussion

#### 5.1. Deterministic Prediction Result Evaluation

^{2}, RMSE, and MAPE, were used to evaluate the accuracy of the model’s deterministic forecast results. The scoring results of the three metrics for different models on different datasets are shown in Table 2, Table 3 and Table 4. The best scores in the table are highlighted in bold.

^{2}value of LGB reaches 0.958, which is significantly higher than ANN’s 0.94, RF’s 0.941, and LSTM’s 0.953. In the second dataset, LGB achieved the highest R

^{2}value of 0.963, higher than SVR’s 0.878, LGB’s 0.961, and GPR’s 0.936. The above results show that the LGB model has higher R

^{2}values than the other models on all three datasets. As can be seen from Table 3, compared with the LGB model, LGB-GPR improves the accuracy of the three datasets by 1.6%, 1.7%, and 1.8%, respectively. Compared with the GPR model, LGB-GPR improves the accuracy by 24.9%, 24.1%, and 6.9% on the three datasets, respectively. The above results prove that, after combining the LGB and GPR models, the model prediction accuracy can be effectively improved. In Table 4, the performance of LGB is second only to LGB-GPR, achieving MAPE scores of 0.156, 0.130, and 0.119 on three datasets. This phenomenon shows that the LGB model itself has excellent wind-speed prediction performance, and the combination with the GPR further improves its prediction accuracy.

#### 5.2. Probability Prediction Result Evaluation

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Comparison of deterministic prediction results in typical time periods: (

**a**) Deterministic prediction results on dataset 1; (

**b**) deterministic prediction results on dataset 2; (

**c**) deterministic prediction results on dataset 3.

**Figure 7.**Comparison of probabilistic prediction results in typical time periods: (

**a**) Probabilistic prediction results on dataset 1; (

**b**) probabilistic prediction results on dataset 2; (

**c**) probabilistic prediction results on dataset 3.

Model | Hyperparameter | Optimization Range |
---|---|---|

LGB-GPR | Tree Maximum depth | (3–15) |

Max number of leaves | (10–150) | |

Learning rate | (0.005–0.2) | |

Minimal number of data | (20–300) | |

Kernel | (‘RBF’, ‘W’, ‘RQ’) | |

LGB | Tree Maximum depth | (3–15) |

Max number of leaves | (10–150) | |

Learning rate | (0.005–0.2) | |

Minimal number of data | (20–300) | |

RF | Tree Maximum depth | (3–15) |

Learning rate | (0.005–0.2) | |

LSTM | Hidden layer nodes | (8–32) |

Dropout | (0.01–0.5) | |

Batch-size | (8–32) | |

Epochs | 100 | |

Optimizer | Adam | |

ANN | Hidden layer nodes | (8–32) |

Dropout | (0.01–0.5) | |

Batch-size | (8–32) | |

Epochs | 100 | |

Optimizer | Adam | |

SVR | Kernel | (‘rbf’, ‘poly’, ‘sigmoid’) |

GPR | Kernel | (‘C’, ‘RBF’, ‘RQ’) |

Model (R^{2}) | Dataset 1 | Dataset 2 | Dataset 3 | Average |
---|---|---|---|---|

SVR | 0.851 | 0.878 | 0.818 | 0.849 |

LR | 0.947 | 0.956 | 0.949 | 0.951 |

RF | 0.932 | 0.951 | 0.940 | 0.941 |

ANN | 0.940 | 0.956 | 0.924 | 0.940 |

LSTM | 0.946 | 0.960 | 0.953 | 0.953 |

LGB | 0.950 | 0.961 | 0.952 | 0.954 |

GPR | 0.917 | 0.936 | 0.949 | 0.934 |

LGB-GPR | 0.954 | 0.963 | 0.956 | 0.958 |

Model (RMSE) | Dataset 1 | Dataset 2 | Dataset 3 | Average |
---|---|---|---|---|

SVR | 0.539 | 0.521 | 0.569 | 0.543 |

LR | 0.321 | 0.311 | 0.301 | 0.311 |

RF | 0.364 | 0.331 | 0.328 | 0.341 |

ANN | 0.342 | 0.311 | 0.368 | 0.340 |

LSTM | 0.326 | 0.299 | 0.290 | 0.305 |

LGB | 0.305 | 0.291 | 0.285 | 0.294 |

GPR | 0.401 | 0.377 | 0.301 | 0.360 |

LGB-GPR | 0.300 | 0.286 | 0.280 | 0.288 |

Model (MAPE) | Dataset 1 | Dataset 2 | Dataset 3 | Average |
---|---|---|---|---|

SVR | 0.409 | 0.350 | 0.367 | 0.375 |

LR | 0.156 | 0.138 | 0.136 | 0.143 |

RF | 0.182 | 0.143 | 0.142 | 0.156 |

ANN | 0.170 | 0.138 | 0.170 | 0.159 |

LSTM | 0.167 | 0.133 | 0.141 | 0.147 |

LGB | 0.156 | 0.130 | 0.119 | 0.135 |

GPR | 0.209 | 0.170 | 0.130 | 0.170 |

LGB-GPR | 0.152 | 0.128 | 0.118 | 0.133 |

Model (ICPC) | Dataset 1 | Dataset 2 | Dataset 3 | Average |
---|---|---|---|---|

GPR | 0.868 | 0.961 | 0.798 | 0.876 |

LGB-GPR | 0.945 | 0.980 | 0.989 | 0.971 |

Model (CRPS) | Dataset 1 | Dataset 2 | Dataset 3 | Average |
---|---|---|---|---|

GPR | 0.225 | 0.209 | 0.165 | 0.200 |

LGB-GPR | 0.165 | 0.157 | 0.148 | 0.157 |

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**MDPI and ACS Style**

Liu, G.; Wang, C.; Qin, H.; Fu, J.; Shen, Q. A Novel Hybrid Machine Learning Model for Wind Speed Probabilistic Forecasting. *Energies* **2022**, *15*, 6942.
https://doi.org/10.3390/en15196942

**AMA Style**

Liu G, Wang C, Qin H, Fu J, Shen Q. A Novel Hybrid Machine Learning Model for Wind Speed Probabilistic Forecasting. *Energies*. 2022; 15(19):6942.
https://doi.org/10.3390/en15196942

**Chicago/Turabian Style**

Liu, Guanjun, Chao Wang, Hui Qin, Jialong Fu, and Qin Shen. 2022. "A Novel Hybrid Machine Learning Model for Wind Speed Probabilistic Forecasting" *Energies* 15, no. 19: 6942.
https://doi.org/10.3390/en15196942