# A Naive Bayesian Wind Power Interval Prediction Approach Based on Rough Set Attribute Reduction and Weight Optimization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed Approach for Forecasting Wind Power Intervals and General theory

#### 2.1. Basic Theory of Rough Sets

#### 2.2. Naive Bayesian Classifier

#### 2.3. The PSO Algorithm

## 3. Establishing the RS-PSO-NBC Wind Power Intervals Model

#### 3.1. Rough Set Selects Criteria Attribute

- (1)
- Establish a decision table. For a sample $t$, the wind speed ${V}_{t+1}$; ${V}_{t}$; ${V}_{t-1}$, the power of ${P}_{t}$; ${P}_{t-1}$; ${P}_{t-2}$; and ${P}_{t-3}$ are taken as original condition attributes, $C=\{{C}_{1},{C}_{2},\cdots ,{C}_{n}\}$, ${C}_{i}\left(i=1,2,\cdots ,7\right)$. Wind power output ${P}_{t+1}$ is selected as the decision attribute $D$. One element of universe $U$ can be defined as ${x}_{t}$, ${x}_{t}=\{{C}_{1},{C}_{2},\cdots ,{C}_{5};{D}_{t}\}$
- (2)
- Discretize the decision tables. Rough sets can only deal with discrete information, so the decision table needs to be discretized. In this paper, an equidistant interval algorithm is used. According to the maximum and minimum values for wind power, the value interval of ${C}_{i}\left(i=4,5,\cdots ,7\right)$ is divided into 20 discrete intervals, and the values falling in each interval are equal to 1, 2, 3, ..., 20 respectively. ${C}_{i}\left(i=1,2,3\right)$ is also discretized according to the maximum and minimum values of the wind speed.
- (3)
- Calculate the attribute significance of each condition attribute and determine the input for NBC.

#### 3.2. The Naive Bayesian Classifer Infers the Power Class

#### 3.3. PSO Optimizes Output Weight $\beta $

#### 3.3.1. Optimizing the Objective Function

#### 3.3.2. Weight optimization by PSO

- (1)
- Initialize the particle swarm, via random initialization of the particles.
- (2)
- Calculate the fitness of each particle according to the objective function F, as shown in (13).
- (3)
- For each particle, its fitness is compared with its historical optimum fitness, and if the current fitness is better, that fitness is denoted as the historical optimum value.
- (4)
- For each particle, compare its fitness and the fitness of the best position experienced by the swarm; if better, it is optimal as a swarm.
- (5)
- The velocity and position of the particle are evolved according to the velocity and position update Equations (9)–(10).
- (6)
- If the end condition is reached (an optimal solution or the maximum number of iterations), then the swarm optimal position is the optimal output weight $\beta $, otherwise go to step (2).

#### 3.4. The Prediction Process

- (1)
- Firstly, the rough set is used to reduce the input variables, and the selected condition attribute is taken as the input for the NBC interval prediction model. The data is pre-processed. The data is divided into a training data set and test data set. Training data output (wind power) fluctuates up and down slightly, as the upper and lower bounds for the initial prediction model to determine the initial output weight ${\beta}_{\mathrm{int}}$.
- (2)
- The conditional attribute selected by the rough set is taken as the input of the Naive Bayesian, and the Naive Bayesian model is established by using training data.
- (3)
- Initialization parameters of PSO are established, including set population and iteration, initial particles position around ${\beta}_{\mathrm{int}}$, random initial velocity, and individual and global optimum position.
- (4)
- The wind power is divided into power partitions with equal intervals, and the different power segments are optimized by particle swarm to find the respective optimum values of output weight. The fitness, the speed, the position, and the global optimal value of each particle are calculated according to relative equations in each iteration. After the iteration, the optimal output weight ${\beta}_{best}$ obtained.
- (5)
- Applying the trained Naive Bayesian prediction intervals to the test data, the output result of the wind prediction intervals is calculated, and the PIs are evaluated by the evaluation index.

## 4. Simulation Results and Analysis

#### 4.1. Significant Condition Attributes Reduced by Rough Set

#### 4.2. Results of Predictive Intervals

#### 4.3. Results of Optimizing Weights for Each Power Segment by PSO

#### 4.4. Comparison with Other Methods

## 5. Conclusions

- (1)
- The Naive Bayesian method is used to obtain the output power probability intervals, making use of the prior knowledge and distribution hypothesis of known data, and to reason from the observed data according to these probabilities and distributions to make the optimal judgment.
- (2)
- Rough set theory is used to reduce the inputs of the Naive Bayesian prediction model and to improve input selection accuracy, which improves the accuracy of the wind power prediction intervals.
- (3)
- Different power segments have different characteristics, and the output weights of the Naive Bayesian Classifier prediction model for these power segments are also different. Using the particle swarm optimization algorithm to find the optimal power output weights, respectively, higher coverage and narrower average bandwidth for the wind power forecasting intervals can be obtained.
- (4)
- In this paper, we use two evaluation indices: the predicted interval coverage probability and the average bandwidth of the intervals. The interval coverage probability indicates reliability, and the average bandwidth can be used to evaluate the interval coverage probability on the basis of their accuracy. Finally, a comparison between NBC and RS-NBC shows the superior interval prediction of the proposed approach.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 8.**The prediction intervals of segmented optimization and non-segmented optimization at an 85% confidence level: (

**a**) segmented optimization and (

**b**) non-segmented optimization.

Object | Condition Attribute | Decision Attribute | ||||||
---|---|---|---|---|---|---|---|---|

$U$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{3}$ | ${C}_{4}$ | ${C}_{5}$ | ${C}_{6}$ | ${C}_{7}$ | $D$ |

1 | 5 | 3 | 5 | 7 | 3 | 10 | 5 | 3 |

2 | 6 | 5 | 6 | 3 | 8 | 18 | 9 | 6 |

3 | 4 | 5 | 4 | 9 | 6 | 7 | 5 | 4 |

$4$ | 11 | 10 | 14 | 19 | 6 | 7 | 10 | 14 |

5 | 8 | 12 | 9 | 10 | 18 | 14 | 15 | 9 |

6 | 10 | 8 | 8 | 16 | 13 | 12 | 15 | 11 |

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ |

Wind Speed ${\mathit{V}}_{\mathit{t}+1}$ | Wind Speed ${\mathit{V}}_{\mathit{t}}$ | Wind Speed ${\mathit{V}}_{\mathit{t}-1}$ | Power ${\mathit{P}}_{\mathit{t}}$ | Power ${\mathit{P}}_{\mathit{t}-1}$ | Power ${\mathit{P}}_{\mathit{t}-2}$ | Power ${\mathit{P}}_{\mathit{t}-3}$ |
---|---|---|---|---|---|---|

0.2778 | 0.1333 | 0.1222 | 0.1889 | 0.1667 | 0.0667 | 0.0444 |

$\mathit{N}$ | ${\mathit{\beta}}_{\mathit{u}\mathit{p}}$ | ${\mathit{\beta}}_{\mathit{l}\mathit{o}\mathit{w}}$ |
---|---|---|

1 | 1.5922 | 0.4444 |

2 | 1.1994 | 0.7783 |

3 | 1.2241 | 0.8524 |

4 | 1.1370 | 0.8137 |

5 | 1.2439 | 0.9198 |

6 | 1.0971 | 0.8911 |

7 | 1.1475 | 0.9324 |

8 | 1.1355 | 0.9447 |

9 | 1.0187 | 0.8761 |

10 | 1.1016 | 0.8751 |

Confidence Level/% | Method | PICP/% | PINAW |
---|---|---|---|

80 | NBC | 79.56 | 237.3430 |

RS-NBC | 80.22 | 217.4590 | |

RS-PSO-NBC | 80.87 | 190.2462 | |

85 | NBC | 84.84 | 269.3378 |

RS-NBC | 85.27 | 272.8840 | |

RS-PSO-NBC | 85.51 | 228.8553 | |

90 | NBC | 89.67 | 307.1498 |

RS-NBC | 90.33 | 296.1640 | |

RS-PSO-NBC | 90.45 | 271.6239 |

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**MDPI and ACS Style**

Yang, X.; Fu, G.; Zhang, Y.; Kang, N.; Gao, F.
A Naive Bayesian Wind Power Interval Prediction Approach Based on Rough Set Attribute Reduction and Weight Optimization. *Energies* **2017**, *10*, 1903.
https://doi.org/10.3390/en10111903

**AMA Style**

Yang X, Fu G, Zhang Y, Kang N, Gao F.
A Naive Bayesian Wind Power Interval Prediction Approach Based on Rough Set Attribute Reduction and Weight Optimization. *Energies*. 2017; 10(11):1903.
https://doi.org/10.3390/en10111903

**Chicago/Turabian Style**

Yang, Xiyun, Guo Fu, Yanfeng Zhang, Ning Kang, and Feng Gao.
2017. "A Naive Bayesian Wind Power Interval Prediction Approach Based on Rough Set Attribute Reduction and Weight Optimization" *Energies* 10, no. 11: 1903.
https://doi.org/10.3390/en10111903