# Kernel Ridge Regression Model Based on Beta-Noise and Its Application in Short-Term Wind Speed Forecasting

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## Abstract

**:**

## 1. Introduction

## 2. Bayesian Principle to Beta-Noise Empirical Risk Loss

## 3. $\mathit{KRR}$ Model Based on Beta-Noise

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Note:**The $KRR$ of the Gaussian-noise characteristic ($GN-KRR$) was discussed in [9,10,11]. The Gaussian empirical risk loss in the sample point $({x}_{i},{y}_{i})\in {D}_{N}$ is $c({\xi}_{i})=\frac{1}{2}{\xi}_{i}^{2}$, thus the dual Problem of model $KRR$ based on Gaussian-noise characteristic ($GN-KRR$) is

## 4. Solution Based on Genetic Algorithm

- (1)
- Let training samples ${D}_{N}=\{({x}_{1},{y}_{1}),({x}_{2},{y}_{2}),\dots ,({x}_{N},{y}_{N})\}$, where ${x}_{i}\in X={R}^{n}$, ${y}_{i}\in R$ ($i=1,\dots ,N$).
- (2)
- Select the appropriate positive $C,u,v$ and the suitable kernel $K(\u2022,\u2022)$.
- (3)
- Solve optimization Problem (18), gain optimal Solution $\alpha =({\alpha}_{1},\cdots ,{\alpha}_{N})$.
- (4)
- Construct the decision-making function$$f(x)={\varpi}^{T}\xb7\mathsf{\Phi}(x)+b=\sum _{i=1}^{N}({\alpha}_{i}\xb7K({x}_{i},x))+b,$$

## 5. Short-Term Wind Speed and Wind Power Forecasting with Real Data-Set

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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Model | MAE | RMSE | MAPE(%) | SEP(%) |
---|---|---|---|---|

$\nu $-SVR | 0.4280 | 0.5833 | 7.02 | 7.02 |

GN-KRR | 0.4219 | 0.5768 | 7.94 | 7.06 |

BN-KRR | 0.3668 | 0.4233 | 6.84 | 5.23 |

Model | MAE | RMSE | MAPE(%) | SEP(%) |
---|---|---|---|---|

$\nu $-SVR | 0.7979 | 1.0116 | 23.36 | 12.53 |

GN-KRR | 0.7109 | 0.9226 | 17.17 | 11.43 |

BN-KRR | 0.6640 | 0.8417 | 18.82 | 10.43 |

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**MDPI and ACS Style**

Zhang, S.; Zhou, T.; Sun, L.; Liu, C.
Kernel Ridge Regression Model Based on Beta-Noise and Its Application in Short-Term Wind Speed Forecasting. *Symmetry* **2019**, *11*, 282.
https://doi.org/10.3390/sym11020282

**AMA Style**

Zhang S, Zhou T, Sun L, Liu C.
Kernel Ridge Regression Model Based on Beta-Noise and Its Application in Short-Term Wind Speed Forecasting. *Symmetry*. 2019; 11(2):282.
https://doi.org/10.3390/sym11020282

**Chicago/Turabian Style**

Zhang, Shiguang, Ting Zhou, Lin Sun, and Chao Liu.
2019. "Kernel Ridge Regression Model Based on Beta-Noise and Its Application in Short-Term Wind Speed Forecasting" *Symmetry* 11, no. 2: 282.
https://doi.org/10.3390/sym11020282