# A Hybrid Wind Speed Forecasting System Based on a ‘Decomposition and Ensemble’ Strategy and Fuzzy Time Series

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review and Discussion for Previous Works

- A hybrid forecasting system is developed including two modules—data pre-processing and forecasting. Unlike previous time series models that dealt with continuous numbers, the fuzzy time series model is handled by fuzzy sets, which solve the weakness of traditional models requiring extensive historical data and assumptions. The effectiveness of this hybrid system is tested and is found to significantly enhance forecasting performance.
- The pre-processing of raw data for wind speed forecasting makes significant contribution to forecasting accuracy. However, in most extant studies, the forecasting was often based on original data, which was not pre-processed. The volatility of and noise in unprocessed data seriously influence the forecasting accuracy and stability. The proposed hybrid system employs the ‘decomposition and ensemble’ strategy to effectively reduce noise in the wind speed time series signal. The results prove that eliminating the noise and uncertainty components from the original chaotic time series by pre-processing the raw data can remarkably improve the forecasting performance.
- The forecasting performance of the fuzzy time series model is always influenced by the interval length, which in turn, depends on the discretization method. Therefore, to search for the most suitable discretization method for wind speed forecasting, four different interval partitioning methods of fuzzy time series have been discussed and compared. The results indicate that supervised discretization methods outperform unsupervised methods in most cases.
- To obtain the best settings of the system, sensitivity analysis of the parameters of the hybrid system is performed, which demonstrates that by appropriately selecting the ensemble number, the white noise amplitude is found to increase forecasting accuracy.
- The Diebold–Mariano (DM) test and forecasting effectiveness (FE) have been selected as testing methods, and the variance in the error is used to measure the stability of the forecasting results in addition to common evaluation metrics thereby enabling a more thorough evaluation of the proposed hybrid system.

## 3. Method

#### 3.1. Data Pre-Processing Method—Ensemble Empirical Mode Decomposition

- Step 1:
- Add the normal distribution white noise series to the signal that is to be decomposed.
- Step 2:
- Decompose the signal with the added normal distribution white noise series into several intrinsic mode functions.
- Step 3:
- Repeat Step 1 and Step 2, and add a new white noise series each time.
- Step 4:
- Regard the ensemble means of intrinsic mode functions that are obtained during decompositions as the final result.

#### 3.2. Forecasting Method—Weighted Fuzzy Time Series (FTS) Algorithm

**Definition**

**1.**

_{j}(t) are constructed based on it. Then

**F**(t), a set of f

_{1}(t), f

_{2}(t) …, is regarded as the fuzzy time series which is defined on Y(t).

**Definition**

**2.**

**F**(t) is assumed to be only caused by

**F**(t − 1). A forecasting model is described as

**F**(t) =

**F**(t − 1) *

**R**(t − 1, t), where

**F**(t − 1) and

**F**(t) are fuzzy sets and

**R**(t − 1, t) is the fuzzy logical relationship (FLR).

**Definition**

**3.**

**F**(t − 1) =

**A**and

_{i}**F**(t) =

**A**. The fuzzy logical relationship (FLR) between two fuzzy values can be expressed as ${\mathit{A}}_{\mathit{i}}\to {\mathit{A}}_{\mathit{j}}$ where

_{j}**A**and

_{i}**A**represent the left-hand side (LHS) and right-hand side (RHS) of the FLR, respectively.

_{j}**Definition**

**4.**

- Step 1:
- Determine the universe of discourse
**U**= [min − a, max + a], and then partition them into several intervals according to the interval partitioning methods mentioned above. From this, continuous data for further observations could be assigned linguistic values. - Step 2:
- Set a fuzzy membership function, and obtain the fuzzy set for actual continuous values. The fuzzy set
**A**is defined based on intervals, as in [63]._{i}$$\begin{array}{l}{\mathit{A}}_{1}=1/{u}_{1}+0.5/{u}_{2}+0/{u}_{3}+0/{u}_{4}+0/{u}_{5}+0/{u}_{6}+0/{u}_{7}+0/{u}_{8}+0/{u}_{9}+0/{u}_{10}\\ {\mathit{A}}_{2}=0.5/{u}_{1}+1/{u}_{2}+0.5/{u}_{3}+0/{u}_{4}+0/{u}_{5}+0/{u}_{6}+0/{u}_{7}+0/{u}_{8}+0/{u}_{9}+0/{u}_{10}\\ \begin{array}{cc}& \text{}\vdots \end{array}\\ {\mathit{A}}_{10}=0/{u}_{1}+0/{u}_{2}+0/{u}_{3}+0/{u}_{4}+0/{u}_{5}+0/{u}_{6}+0/{u}_{7}+0/{u}_{8}+0.5/{u}_{9}+1/{u}_{10}\end{array}$$ - Step 3:
- Fuzzify observations. For example, the fuzzified result of one data is
**A**when the maximum degree of membership of this data is in_{j}**A**._{j} - Step 4:
- Determine the fuzzy logical relationships and group them. For example, if ${\mathit{A}}_{\mathit{i}}\to {\mathit{A}}_{\mathit{j}},{\mathit{A}}_{\mathit{i}}\to {\mathit{A}}_{\mathit{k}},{\mathit{A}}_{\mathit{i}}\to {\mathit{A}}_{\mathit{l}}$ can be grouped as ${\mathit{A}}_{\mathit{i}}\to {\mathit{A}}_{\mathit{j}},{\mathit{A}}_{\mathit{k}},{\mathit{A}}_{\mathit{l}}$.
- Step 5:
- Establish weights. From step 4 above, the weight matrix can be obtained and further standardized. The defuzzified matrix can then be calculated by applying the centroid defuzzification method.
- Step 6:
- Calculate forecasting results. Forecasting results can be calculated by multiplication of the defuzzified and standardized weighting matrices defined as follows:$$\begin{array}{l}\mathit{W}\_\mathit{s}\left(t\right)=\left({\widehat{W}}_{1},{\widehat{W}}_{2},\cdots ,{\widehat{W}}_{k}\right)\\ {\widehat{W}}_{i}={W}_{i}/{\displaystyle \sum _{i=1}^{k}{W}_{i}}\end{array}$$$$\mathit{F}\left(t\right)=\mathit{D}\left(t-1\right)\ast \mathit{W}\_\mathit{s}\left(t-1\right)$$
**W_s**is the standardized weighting matrix,**D**is the defuzzified matrix. Wrepresents the unstandardized weighting matrix elements, while ${\widehat{W}}_{i}$ represents standardized ones, and_{i}**F**(t) is the forecasting result. - Step 7:
- Lastly, forecasted values obtained above are amended by employing Equation (3) to obtain the ultimate forecasting result.$$\mathit{F}\_\mathit{u}\left(t\right)=y(t-1)+\alpha \ast \left(\mathit{F}\left(t\right)-y\left(t-1\right)\right)$$
**F_s**is the ultimate forecasting value.

#### 3.3. Interval Partitioning Methods

#### 3.3.1. Equal Width Interval Algorithm

_{min}, X

_{max}) is divided into K equal sized intervals. Thus, the width of each interval is (X

_{max}− X

_{min})/K. However, when there exist points with considerable skewness, this method is not adaptive. The disadvantage of this method, caused by the uneven distribution of the time series, is that the data count in different intervals may vary significantly [72].

#### 3.3.2. Equal Frequency Interval Algorithm

#### 3.3.3. Entropy-Based Discretization Algorithm

- Step 1:
- Define the entropy of intervals. For an object set T, the entropy function is calculated as under:$$\mathit{E}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}\left(T\right)=-{\displaystyle \sum _{i=1}^{n}{p}_{i}\cdot \mathrm{log}\left({p}_{i}\right)}$$
_{i}is the probability of class i. - Step 2:
- Apply all possible cut points to divide the data into two parts, and from all possible cut methods, find the one with minimum entropy. For each cut point, the entropy of each split is defined as:$$\mathit{E}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}\left(T|split\right)={p}_{left}\cdot \mathit{E}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}\left({T}_{left}\right)+{p}_{right}\cdot \mathit{E}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}\left({T}_{right}\right)$$
_{left}and p_{right}represent probabilities of the left (T_{left}) and right (T_{right}) sets, respectively. - Step 3:
- Regard the two intervals obtained in step 2 as independent intervals and then repeat step 1.
- Step 4:
- Run iterations, but stop the process when the set criterion is achieved.

#### 3.3.4. Chi-Square-Based Discretization Algorithm

^{2}) is a discretization algorithm based on the value of Chi-square, which measures the relationship between a class and adjacent intervals. The Chi-square-based discretization algorithm splits the data set based on user-defined significance levels. This algorithm includes the top-down (Chi-split) and bottom-up (Chi-merge) methods, both of which are based on Chi-square. The top-down method regards the entire interval value as a discrete value and then split this interval into two adjacent sub-intervals. The process then runs into iterations and stops once a set criterion is achieved. When the Chi-square test is significant, the split must continue; otherwise, it should be stopped. contrary to the top-down approach, the bottom-up method regards each attribute value as a discrete value and then repeatedly merges adjacent attribute values, if the two are statistically similar, until the stopping condition is met. The stopping criterion is determined by a Chi-square threshold defined by user to stop the merge operation when two adjacent intervals cannot be proven to be sufficiently similar [66].

^{2}) is a statistic to test the independence between row and column variables in a contingency table, as presented in Table 1. In the Chi-Square-based discretization algorithm, the formula to calculate χ

^{2}statistic at a cut point for two adjacent intervals is described in Equation (6) [75].

- c is the classes number.
- O
_{ij}is the example number in the ith interval and jth class. - E
_{ij}is the expected frequency in the ith interval and jth class, computed by E_{ij}= (R_{i}C_{j})/N. - R
_{i}represents the example number in the ith interval. - C
_{j}represents the examples number in the jth class.

## 4. Data Description and Setup

**A**–

_{1}**A**. Taking the Chi-square-based discretization of Dataset III, the fuzzy relationship groups are summarized in Table 4. Each number in the matrix indicates the occurrence of a fuzzy logic relationship. Based on this matrix and Equation (1), the weight matrix can be calculated, as presented in Table 4 and Table 5. Ultimately, forecasting values can be calculated by Equations (2) and (3). After repeated tests, the weight in Equation (3) was set as 0.5.

_{10}## 5. Experimental Results for Datasets

- (1)
- The top half of Figure 3 presents forecasting results of the original data and that of data preprocessed via ensemble empirical mode decomposition employing fuzzy time series forecasting methods—entropy-based discretization, Chi-square-based discretization, equal frequency interval discretization, and equal width interval discretization. It is obvious that forecasting results obtained using fuzzy time series under supervised discretization methods tend to match actual values more closely compared to the unsupervised methods. The details of parts A and B in Figure 3 illustrate the local enlargement comparison of the different methods.
- (a)
- As shown in Figure 3, compared to equal width interval discretization, forecasting curves of the entropy- and Chi-square-based discretization more closely follow the shape of the actual testing curve. Equal frequency interval discretization demonstrates the worst performance. Thus, supervised discretization methods are, in general, found to be superior to unsupervised methods.
- (b)
- Better forecasting is achieved when the wind speed is steady without any sudden change. Evidently, the forecasting system perform better between sample numbers 130–170 and 300–350, and better follow the shape of the actual testing curve.
- (c)
- Comparing the curves of the original and pre-pre-processing data, the degree of overlap of the curves in the second picture is evidently superior to that in the first. Thus, it can be seen that data pre-processing plays a vital role in wind speed forecasting.
- (d)
- As shown in parts A and B in Figure 3, the degree of overlap of the curves near the local maximum forecasting value is better than that near the local minimum forecasting value. Near the local minimum forecasting value, the curve corresponding to equal frequency interval discretization, when compared to other curves, deviates considerably from the actual value curve.

- (2)
- The lower part of Figure 3 demonstrates the forecasting error (forecast value minus actual value) for the four different interval partitioning methods described in this paper.
- (a)
- In terms of individual forecasting values, the forecasting error is notably large, such as that calculated for sample numbers 100, 250, and 300, wherein there exist large fluctuations in wind speed. It is conclude that the performance of forecasting methods is poor when large fluctuations are present in data.
- (b)
- It is noteworthy that the forecasting error for pre-processed data is significantly less compared to original data. All points distribute around a zero-scale line. The points in the right image are also more concentrated than in the left one. It is to be noted that most points, which deviate from the zero-scale line, further belong to the equal frequency interval discretization method.

## 6. Analysis and Discussion

^{2}; and VAR measures the stability of the methods. Furthermore, MAE, RMSE, MAPE, and VAR are negative indicators; i.e., the lower the better, while IA is a positive indicator.

#### 6.1. Experiment I: The Data Pre-Processing for Fuzzy Time Series Forecasting

**Remark**

**1:**

#### 6.2. Experiment II: The Comparison of Fuzzy Time Series, Artificial Neural Network, Statistical Models and Support Vector Regression

**Remark**

**2:**

#### 6.3. Experiment III: Forecasting Performance of the Fuzzy Time Series with Different Interval Partitioning Methods

^{2}), entropy based discretization, equal frequency interval discretization, and equal width interval discretization. Most of the metrics indicate that the Chi-square-based discretization performs the best for Datasets I and III. For dataset II, the entropy-based discretization method demonstrates the best forecasting performance for original data, while the equal frequency interval discretization rules the roost in handling pre-processed data. Figure 4 shows the forecasting results graphic of the three datasets. From Table 10 and Figure 4a, it can be concluded that supervised discretization methods possess better stability and forecasting accuracy compared to unsupervised methods. In Figure 4b, scatter plot of the observations and values forecasted by the proposed hybrid forecasting system indicates that the proposed system demonstrates great performance.

**Remark**

**3:**

#### 6.4. Experiment IV: Testing Based on the DM Test and Forecasting Effectiveness

#### 6.4.1. DM Test

- ${\epsilon}_{t+h}^{}$ denotes the forecasting error
- S
^{2}denotes the estimation value for the variance of ${d}_{h}=L\left({\epsilon}_{t+h}^{(i)}\right)-L\left({\epsilon}_{t+h}^{(j)}\right)$ - $L$ denotes a loss function that is utilized to represent the forecasting accuracy of the model.

_{α}

_{/2}is the critical value of the standard normal distribution when the significance level is α.

#### 6.4.2. Forecasting Effectiveness

_{n}denotes the discrete probability distribution at time n. As any prior information of the discrete probability distribution is unknown, Q

_{n}is defined as 1/N. A

_{n}is the forecasting accuracy defined as:

**Remark**

**4:**

## 7. Sensitivity Analysis of Parameters in the Proposed Hybrid Forecasting System

#### 7.1. Setting the Ensemble Number for Ensemble Empirical Mode Decomposition

#### 7.2. Setting Amplitude of Added Noise

## 8. Further Experiments for Hourly Time Horizon

**Remark**

**5:**

## 9. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

AIC | A-Information Criterion |

ARIMA | Autoregressive Integrated Moving Average |

BPNN | Back Propagation Neural Network |

Chi^{2} | Chi-square |

CRO | Coral Reefs Optimization |

DES | Double Exponential Smoothing |

DM | Diebold–Mariano |

EF | equal frequency |

ELM | Extreme Learning Machine |

EW | equal width |

FE | forecasting effectiveness |

FLR | Fuzzy Logical Relationship |

FSP | Feature Selection Problem |

FTS | Fuzzy time series algorithm |

HS | Harmony Search |

IA | Index of agreement of forecasting results |

LHS | Left-hand side |

MAE | Mean absolute error |

MAPE | Mean Absolute Percentage Error |

R | Correlation coefficient |

RBFNN | Radial Basis Function Neural Network |

RHS | Right-hand side |

RMSE | Root Mean Square Error |

SAM | Seasonal Adjustment Method |

SVR | Support Vector Regression |

VAR | Variance of the error |

## References

- Baños, R.; Manzano-Agugliaro, F.; Montoya, F.G.; Gil, C.; Alcayde, A.; Gómez, J. Optimization methods applied to renewable and sustainable energy: A review. Renew. Sustain. Energy Rev.
**2011**, 15, 1753–1766. [Google Scholar] [CrossRef] - Harmsen, J.H.M.; Roes, A.L.; Patel, M.K. The impact of copper scarcity on the efficiency of 2050 global renewable energy scenarios. Energy
**2013**, 50, 62–73. [Google Scholar] [CrossRef] - Yesilbudak, M.; Sagiroglu, S.; Colak, I. A new approach to very short term wind speed prediction using k -nearest neighbor classification. Energy Convers. Manag.
**2013**, 69, 77–86. [Google Scholar] [CrossRef] - Wang, J.; Jiang, H.; Zhou, Q.; Wu, J.; Qin, S. China’s natural gas production and consumption analysis based on the multicycle Hubbert model and rolling Grey model. Renew. Sustain. Energy Rev.
**2016**, 53, 1149–1167. [Google Scholar] [CrossRef] - Hernández-Escobedo, Q.; Saldaña-Flores, R.; Rodríguez-García, E.R.; Manzano-Agugliaro, F. Wind energy resource in Northern Mexico. Renew. Sustain. Energy Rev.
**2014**, 32, 890–914. [Google Scholar] [CrossRef] - Oh, K.Y.; Kim, J.Y.; Lee, J.K.; Ryu, M.S.; Lee, J.S. An assessment of wind energy potential at the demonstration offshore wind farm in Korea. Energy
**2012**, 46, 555–563. [Google Scholar] [CrossRef] - Montoya, F.G.; Manzano-Agugliaro, F. Wind turbine selection for wind farm layout using multi-objective evolutionary algorithms. Expert Syst. Appl.
**2014**, 41, 6585–6595. [Google Scholar] [CrossRef] - Manzano-Agugliaro, F.; Alcayde, A.; Montoya, F.G.; Zapata-Sierra, A.; Gil, C. Scientific production of renewable energies worldwide: An overview. Renew. Sustain. Energy Rev.
**2013**, 18, 134–143. [Google Scholar] [CrossRef] - World Wind Energy Association. Available online: http://www.wwindea.org/11961-2/ (accessed on 24 July 2017).
- Ma, X.; Jin, Y.; Dong, Q. A generalized dynamic fuzzy neural network based on singular spectrum analysis optimized by brain storm optimization for short-term wind speed forecasting. Appl. Soft Comput. J.
**2017**, 54, 296–312. [Google Scholar] [CrossRef] - State Grid. Nb/T 31046 Function Specification of Wind Power Prediction; China Electric Power Press: Beijing, China, 2013. [Google Scholar]
- Chunyan, Y. Research on Wind Speed and Wind Power Forecasting Related Issue; Huazhong University of Science and Technology: Wuhan, China, 2013. [Google Scholar]
- Hernandez-escobedo, Q.; Manzano-agugliaro, F.; Gazquez-parra, J.A.; Zapata-sierra, A. Is the wind a periodical phenomenon? The case of Mexico. Renew. Sustain. Energy Rev.
**2011**, 15, 721–728. [Google Scholar] [CrossRef] - Ackermann, T.; Söder, L. Wind energy technology and current status: A review. Renew. Sustain. Energy Rev.
**2011**, 4, 315–374. [Google Scholar] [CrossRef] - Chang, P.C.; Yang, R.Y.; Lai, C.M. Potential of Offshore Wind Energy and Extreme Wind Speed Forecasting on the West Coast of Taiwan. Energies
**2015**, 8, 1685–1700. [Google Scholar] [CrossRef] - Safat, A. A Physical Approach to Wind Speed Prediction for Wind Energy Forecasting. Available online: http://www.iawe.org/Proceedings/CWE2006/MC4-01.pdf (accessed on 24 July 2017).
- Yamaguchi, A.; Enoki, K.; Ishihara, T.; Fukumoto, Y.; Okino, M.; Iba, S.; Ohya, Y.; Karasudani, T.; Watanabe, K.; Noda, M.; et al. Wind Power Forecasting with Physical Model and Multi Time Scale Model. J. Wind Eng.
**2010**, 2007, 251–264. [Google Scholar] [CrossRef] - Filik, T. Improved Spatio-Temporal Linear Models for Very Short-Term Wind Speed Forecasting. Energies
**2016**, 9, 168. [Google Scholar] [CrossRef] - Lei, M.; Luan, S.; Jiang, C.; Liu, H.; Yan, Z. A review on the forecasting of wind speed and generated power. Renew. Sustain. Energy Rev.
**2009**, 13, 915–920. [Google Scholar] [CrossRef] - Shukur, O.B.; Lee, M.H. Daily wind speed forecasting through hybrid AR-ANN and AR-KF models. J. Teknol.
**2015**, 72, 89–95. [Google Scholar] [CrossRef] - Zhang, C.L. The Wind Speed Prediction Based on AR Model and BP Neural Network. Adv. Mater. Res.
**2012**, 450–451, 1593–1596. [Google Scholar] - Torres, J.L.; García, A.; De Blas, M.; De Francisco, A. Forecast of hourly average wind speed with ARMA models in Navarre (Spain). Sol. Energy
**2005**, 79, 65–77. [Google Scholar] [CrossRef] - Cadenas, E.; Rivera, W.; Campos-Amezcua, R.; Heard, C. Wind Speed Prediction Using a Univariate ARIMA Model and a Multivariate NARX Model. Energies
**2016**, 9, 109. [Google Scholar] [CrossRef] - Wang, G.Q.; Wang, S.; Liu, H.Y.; Xue, Y.D.; Ping, Z.; Amp, E. Self-adaptive and dynamic cubic ES method for wind speed forecasting. Power Syst. Prot. Control
**2014**, 42, 117–122. [Google Scholar] - Booth, D.E. Time Series (3rd ed.). J. Technometrics
**1992**, 34, 118–119. [Google Scholar] - Li, G.; Shi, J. On comparing three artificial neural networks for wind speed forecasting. Appl. Energy
**2010**, 87, 2313–2320. [Google Scholar] [CrossRef] - Hyperbolic tangent basis function neural networks training by hybrid evolutionary programming for accurate short-term wind speed prediction. In Proceedings of the Ninth Intelligent Systems Design and Applications Conference (ISDA’09), Pisa, Italy, 30 November–2 December 2009.
- Salcedo-sanz, S.; Pastor-sánchez, A.; Prieto, L.; Blanco-aguilera, A.; García-herrera, R. Feature selection in wind speed prediction systems based on a hybrid coral reefs optimization—Extreme learning machine approach. Energy Convers. Manag.
**2014**, 87, 10–18. [Google Scholar] [CrossRef] - Salcedo-Sanz, S.; Pastor-Sánchez, A.; Ser J, D.e.l.; Prieto, L.; Geem, Z.W. A Coral Reefs Optimization algorithm with Harmony Search operators for accurate wind speed prediction. Renew. Energy
**2015**, 75, 93–101. [Google Scholar] [CrossRef] - Zhang, Q.; Lai, K.K.; Niu, D.; Wang, Q.; Zhang, X. A Fuzzy Group Forecasting Model Based on Least Squares Support Vector Machine (LS-SVM) for Short-Term Wind Power. Energies
**2012**, 5, 3329–3346. [Google Scholar] [CrossRef] - Salcedo-Sanz, S.; Ortiz-García, E.G.; Pérez-Bellido, A.M.; Portilla-Figueras, E.; Prieto, L.; Paredes, D.; Correoso, F. Performance comparison of Multilayer Perceptrons and Support vector Machines in a Short-term Wind speed Prediction Problem. Neural Netw. World
**2009**, 19, 37–51. [Google Scholar] - Ortiz-García, E.G.; Salcedo-Sanz, S.; Pérez-Bellido, Á.M.; Gascón-Moreno, J.; Portilla-Figueras, J.A.; Prieto, L. Short-term wind speed prediction in wind farms based on banks of support vector machines. Wind Energy
**2011**, 14, 193–207. [Google Scholar] [CrossRef] - Salcedo-Sanz, S.; Ortiz-Garcı, E.G.; Pérez-Bellido, Á.M.; Portilla-Figueras, A.; Prieto, L. Short term wind speed prediction based on evolutionary support vector regression algorithms. Expert Syst. Appl.
**2011**, 38, 4052–4057. [Google Scholar] [CrossRef] - Jiang, Y.; Song, Z.; Kusiak, A. Very short-term wind speed forecasting with Bayesian structural break model. Renew. Energy
**2013**, 50, 637–647. [Google Scholar] [CrossRef] - Troncoso, A.; Salcedo-sanz, S.; Casanova-mateo, C.; Riquelme, J.C.; Prieto, L. Local models-based regression trees for very short-term wind speed prediction. Renew. Energy
**2015**, 81, 589–598. [Google Scholar] [CrossRef] - Pourmousavi Kani, S.A.; Ardehali, M.M. Very short-term wind speed prediction: A new artificial neural network-Markov chain model. Energy Convers. Manag.
**2011**, 52, 738–745. [Google Scholar] [CrossRef] - Khashei, M.; Bijari, M.; Ardali, G.A.R. Improvement of Auto-Regressive Integrated Moving Average models using Fuzzy logic and Artificial Neural Networks (ANNs). Neurocomputing
**2009**, 72, 956–967. [Google Scholar] [CrossRef] - Salcedo-Sanz, S.; Prieto, L.; Prieto, L.; Correoso, F. Letters: Accurate short-term wind speed prediction by exploiting diversity in input data using banks of artificial neural networks. Neurocomputing
**2009**, 72, 1336–1341. [Google Scholar] [CrossRef] - Chang, W.Y. Short-Term Wind Power Forecasting Using the Enhanced Particle Swarm Optimization Based Hybrid Method. Energies
**2013**, 6, 4879–4896. [Google Scholar] [CrossRef] - Salcedo-Sanz, S.; Pérez-Bellido, Á.M.; Ortiz-García, E.G.; Portilla-Figueras, A.; Prieto, L.; Paredes, D. Hybridizing the fifth generation mesoscale model with artificial neural networks for short-term wind speed prediction. Renew. Energy
**2009**, 34, 1451–1457. [Google Scholar] [CrossRef] - Sanz, S.S.; Prieto, L.; Paredes, D.; Correoso, F. Short-term wind speed prediction by hybridizing global and mesoscale forecasting models with artificial neural networks. In Proceedings of the Eighth International Conference on Hybrid Intelligent Systems (HIS’08), Barcelona, Spain, 10–12 September 2008. [Google Scholar]
- Hervás-Martínez, C.; Salcedo-Sanz, S.; Gutiérrez, P.A.; Ortiz-García, E.G.; Prieto, L. Evolutionary product unit neural networks for short-term wind speed forecasting in wind farms. Neural Comput. Appl.
**2012**, 21, 993–1005. [Google Scholar] [CrossRef] - Zhang, W.; Wang, J.; Wang, J.; Zhao, Z.; Tian, M. Short-term wind speed forecasting based on a hybrid model. Appl. Soft Comput. J.
**2013**, 13, 3225–3233. [Google Scholar] [CrossRef] - Hong, Y.Y.; Yu, T.H.; Liu, C.Y. Hour-Ahead Wind Speed and Power Forecasting Using Empirical Mode Decomposition. Energies
**2013**, 6, 6137–6152. [Google Scholar] [CrossRef] - Liu, H.; Chen, C.; Tian, H.Q.; Li, Y.F. A hybrid model for wind speed prediction using empirical mode decomposition and artificial neural networks. Renew. Energy
**2012**, 48, 545–556. [Google Scholar] [CrossRef] - Liu, H.; Tian, H.Q.; Liang, X.F.; Li, Y.F. Wind speed forecasting approach using secondary decomposition algorithm and Elman neural networks. Appl. Energy
**2015**, 157, 183–194. [Google Scholar] [CrossRef] - Liu, H.; Tian, H.; Liang, X.; Li, Y. New wind speed forecasting approaches using fast ensemble empirical model decomposition, genetic algorithm, Mind Evolutionary Algorithm and Artificial Neural Networks. Renew. Energy
**2015**, 83, 1066–1075. [Google Scholar] [CrossRef] - Tascikaraoglu, A.; Uzunoglu, M. A review of combined approaches for prediction of short-term wind speed and power. Renew. Sustain. Energy Rev.
**2014**, 34, 243–254. [Google Scholar] [CrossRef] - Jilani, T.A.; Burney, S.M.A. M-Factor High Order Fuzzy Time Series Forecasting for Road Accident Data. Adv. Soft Comput.
**2007**, 41, 246–254. [Google Scholar] - Jiang, P.; Wang, Y.; Wang, J. Short-term wind speed forecasting using a hybrid model. Energy
**2016**, 119, 561–577. [Google Scholar] [CrossRef] - Masrur, H.; Nimol, M. Short Term Wind Speed Forecasting Using Artificial Neural Network: A Case Study. In Proceedings of the International Conference on Innovations in Science, Engineering and Technology (ICISET), Dhaka, Bangladesh, 28–29 October 2016. [Google Scholar]
- Niazy, R.K.; Beckmann, C.F.; Brady, J.M.; Smith, S.M. Performance Evaluation of Ensemble Empirical Mode Decomposition. Adv. Adapt. Data Anal.
**2009**, 1, 231–242. [Google Scholar] [CrossRef] - Zhu, B. A Novel Multiscale Ensemble Carbon Price Prediction Model Integrating Empirical Mode Decomposition, Genetic Algorithm and Artificial Neural Network. Energies
**2012**, 5, 163–170. [Google Scholar] [CrossRef] - Zhaohua, W.U.; Huang, N.E. Ensemble Empirical Mode Decomposition: A Noise-Assisted Data Analysis Method. Adv. Adapt. Data Anal.
**2011**, 1, 1–41. [Google Scholar] - Yu, L.; Wang, Z.; Tang, L. A decomposition—Ensemble model with data-characteristic-driven reconstruction for crude oil price forecasting. Appl. Energy
**2015**, 156, 251–267. [Google Scholar] [CrossRef] - Chen, Y.S.; Cheng, C.H.; Tsai, W.L. Modeling fitting-function-based fuzzy time series patterns for evolving stock index forecasting. Appl. Intell.
**2014**, 41, 327–347. [Google Scholar] [CrossRef] - Wang, J.; Xiong, S. A hybrid forecasting model based on outlier detection and fuzzy time series—A case study on Hainan wind farm of China. Energy
**2014**, 76, 526–541. [Google Scholar] [CrossRef] - Li, S.T.; Cheng, Y.C. Deterministic fuzzy time series model for forecasting enrollments. Comput. Math. Appl.
**2007**, 53, 1904–1920. [Google Scholar] [CrossRef] - Lee, Y.C.; Wu, C.H.; Tsai, S.B. Grey system theory and fuzzy time series forecasting for the growth of green electronic materials. Int. J. Prod. Res.
**2014**, 52, 2931–2945. [Google Scholar] [CrossRef] - Sadaei, H.J.; Guimarães, F.G.; Da Silva, C.J.; Lee, M.H.; Eslami, T. Short-term load forecasting method based on fuzzy time series, seasonality and long memory process. Int. J. Approx. Reason.
**2017**, 83, 196–217. [Google Scholar] [CrossRef] - Song, Q.; Chissom, B.S. Fuzzy Time Series and Its Models; Elsevier North-Holland, Inc.: Amsterdam, The Netherlands, 1993. [Google Scholar]
- Yu, H.K. Weighted fuzzy time series models for TAIEX forecasting. Phys. A Stat. Mech. Appl.
**2012**, 349, 609–624. [Google Scholar] [CrossRef] - Abdullah, L.; Taib, I. High order fuzzy time series for exchange rates forecasting. In Proceedings of the 2011 3rd Conference on Data Mining and Optimization (DMO), Putrajaya, Malaysia, 28–29 June 2011; pp. 1–5. [Google Scholar]
- Chen, M.; Chen, B. A hybrid fuzzy time series model based on granular computing for stock price forecasting. Inf. Sci.
**2015**, 294, 227–241. [Google Scholar] [CrossRef] - Lu, W.; Chen, X.; Pedrycz, W.; Liu, X.; Yang, J. Using interval information granules to improve forecasting in fuzzy time series. Int. J. Approx. Reason.
**2015**, 57, 1–18. [Google Scholar] [CrossRef] - Dash, R.; Paramguru, R.L.; Dash, R. Comparative Analysis of Supervised and Unsupervised Discretization Techniques. Int. J. Adv. Sci. Technol.
**2011**, 2, 29–37. [Google Scholar] - Duda, J. Supervised and Unsupervised Discretization of Continuous Features. In Proceedings of the Twelfth International Conference on Machine Learning, Tahoe, CA, USA, 9–12 July 1995; Volume 12, pp. 194–202. [Google Scholar]
- Peng, L.; Wang, Q.; Yujia, G. Study on Comparison of Discretization Methods. In Proceedings of the International Conference on Artificial Intelligence and Computational Intelligence, 2009 (AICI’09), Shanghai, China, 7–8 November 2009; pp. 380–384. [Google Scholar]
- Hua, H.; Zhao, H. A Discretization Algorithm of Continuous Attributes Based on Supervised Clustering; Photoelectric Information Technology Research Room: Liaoning, China, 2009; pp. 1–5. [Google Scholar]
- Joiţa, D. Unsupervised Static Discretization Methods in Data Mining; Titu Maiorescu University: Bucharest, Romania, 2010. [Google Scholar]
- Schmidberger, G.; Frank, E. Unsupervised Discretization Using Tree-Based Density Estimation; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Wu, C.H.; Kao, S.C.; Okuhara, K. Examination and comparison of conflicting data in granulated datasets: Equal width interval vs. equal frequency interval. Inf. Sci.
**2013**, 239, 154–164. [Google Scholar] [CrossRef] - Fayyad, U.; Irani, K. Multi-Interval Discretization of Continuous-Valued Attributes for Classification Learning. In Proceedings of the Thirteenth International Joint Conference on Artificial Intelligence, Chambéry, France, 28 August–3 September 1993; pp. 1022–1027. [Google Scholar]
- Soares, C.; Knobbe, A. Entropy-based discretization methods for ranking data. Inf. Sci.
**2016**, 329, 921–936. [Google Scholar] - Boulle, M. Khiops: A Statistical Discretization Method of Continuous Attributes. Mach. Learn.
**2004**, 55, 53–69. [Google Scholar] [CrossRef] - Renani, E.T.; Elias, M.F.M.; Rahim, N.A. Using data-driven approach for wind power prediction: A comparative study. Energy Convers. Manag.
**2016**, 118, 193–203. [Google Scholar] [CrossRef] - Du, P.; Wang, J.; Guo, Z.; Yang, W. Research and application of a novel hybrid forecasting system based on multi-objective optimization for wind speed forecasting. Energy Convers. Manag.
**2017**, 150, 90–107. [Google Scholar] [CrossRef] - Xu, Y.; Yang, W.; Wang, J. Air quality early-warning system for cities in China. Atmos. Environ.
**2017**, 148, 239–257. [Google Scholar] [CrossRef] - Xiao, L.; Shao, W.; Wang, C.; Zhang, K.; Lu, H. Research and application of a hybrid model based on multi-objective optimization for electrical load forecasting. Appl. Energy
**2016**, 180, 213–233. [Google Scholar] [CrossRef] - Wang, S.; Zhang, N.; Wu, L.; Wang, Y. Wind speed forecasting based on the hybrid ensemble empirical mode decomposition and GA-BP neural network method. Renew. Energy
**2016**, 94, 629–636. [Google Scholar] [CrossRef] - Wang, Y.H.; Yeh, C.H.; Young, H.W.V.; Hu, K.; Lo, M.T. On the computational complexity of the empirical mode decomposition algorithm. Phys. A Stat. Mech. Appl.
**2014**, 400, 159–167. [Google Scholar] [CrossRef] - Ma, X.; Liu, D. Comparative Study of Hybrid Models Based on a Series of Optimization Algorithms and Their Application in Energy System Forecasting. Energies
**2016**, 9, 640. [Google Scholar] [CrossRef]

**Figure 3.**Forecasting results and error in fuzzy time series with different interval lengths using original and pre-processing data in Dataset I.

**Figure 4.**Comparison of forecasting results obtained using different models for Dataset I. (

**a**) Comparison of the forecasting results obtained from original and pre-processing data; (

**b**) Comparison of actual and forecasting values of hybrid forecasting system; (

**c**) Comparison of forecasting performance for different models

Class 1 | Class 2 | …… | Class c | Sum | |
---|---|---|---|---|---|

Interval 1 | O_{11} | O_{12} | …… | O_{1c} | R_{1} |

Interval 2 | O_{21} | O_{22} | …… | O_{2c} | R_{2} |

Sum | C_{1} | C_{2} | …… | C_{c} | N |

Datasets | Numbers | Statistical Indicators | |||||
---|---|---|---|---|---|---|---|

Maximum (m/s) | Minimum (m/s) | Mean (m/s) | Interquartile Range (m/s) | Std. (m/s) | |||

Equation | - | - | - | $\mathit{M}\mathit{e}\mathit{a}\mathit{n}=\raisebox{1ex}{$\sum _{\mathit{i}=1}^{\mathit{N}}{\mathit{x}}_{\mathit{i}}$}\!\left/ \!\raisebox{-1ex}{$\mathit{N}$}\right.$ | ${\mathit{Q}}_{\mathit{d}}={\mathit{Q}}_{\mathit{U}-}{\mathit{Q}}_{\mathit{L}}$ | $\mathit{S}=\sqrt{\frac{1}{\mathit{N}}{\displaystyle \sum _{\mathit{i}=1}^{\mathit{N}}{\left({\mathit{x}}_{\mathit{i}}-\overline{\mathit{x}}\right)}^{2}}}$ | |

Dataset I | All | 2000 | 12.8 | 2.1 | 6.9815 | 2.7 | 1.8202 |

Training | 1500 | 12.8 | 2.1 | 7.2781 | 2.7 | 1.8852 | |

Testing | 500 | 10.2 | 3.1 | 6.0918 | 1.5 | 1.2401 | |

Dataset II | All | 2000 | 15.3 | 2.6 | 8.7764 | 3.7 | 2.3675 |

Training | 1500 | 15.3 | 2.6 | 9.1389 | 3.9 | 2.4516 | |

Testing | 500 | 12.2 | 3.9 | 7.6890 | 2.8 | 1.6787 | |

Dataset III | All | 2000 | 16.2 | 2.9 | 8.7374 | 4.9 | 2.8693 |

Training | 1500 | 16.2 | 2.9 | 9.1703 | 4.9 | 2.9067 | |

Testing | 500 | 15.9 | 3.7 | 7.4384 | 3.5 | 2.312 |

Methods | Equal Width | Equal Frequency | Entropy Based | Chi-Square Based |
---|---|---|---|---|

u_{1} | (2.00, 3.45) | (2.00, 5.00) | (2.00, 3.90) | (2.00, 3.90) |

u_{2} | (3.45, 4.90) | (5.00, 5.80) | (3.90, 5.00) | (3.90, 5.10) |

u_{3} | (4.90, 6.35) | (5.80, 6.50) | (5.00, 6.40) | (5.10, 6.20) |

u_{4} | (6.35, 7.80) | (6.50, 7.10) | (6.40, 7.40) | (6.20, 7.30) |

u_{5} | (7.80, 9.25) | (7.10, 7.80) | (7.40, 8.90) | (7.30, 8.80) |

u_{6} | (9.25, 10.70) | (7.80, 8.60) | (8.90, 9.80) | (8.80, 9.70) |

u_{7} | (10.70, 12.15) | (8.60, 9.50) | (9.80, 10.90) | (9.70, 10.60) |

u_{8} | (12.15, 13,60) | (9.50, 10.40) | (10.90, 12.80) | (10.60, 11.90) |

u_{9} | (13.60, 15.05) | (10.40, 11.70) | (12.80, 15.30) | (11.90, 13.60) |

u_{10} | (15.05, 16.50) | (11.70, 16.20) | (15.30, 16.20) | (13.60, 16.20) |

P_{t}_{-1} | P_{t} | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | A_{6} | A_{7} | A_{8} | A_{9} | A_{10} | |

A_{1} | 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

A_{2} | 6 | 114 | 18 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |

A_{3} | 1 | 20 | 119 | 29 | 3 | 0 | 0 | 0 | 0 | 0 |

A_{4} | 0 | 0 | 32 | 97 | 27 | 1 | 0 | 1 | 0 | 0 |

A_{5} | 0 | 0 | 3 | 29 | 106 | 26 | 7 | 0 | 0 | 0 |

A_{6} | 0 | 0 | 0 | 0 | 31 | 60 | 35 | 8 | 0 | 0 |

A_{7} | 0 | 0 | 0 | 0 | 4 | 34 | 50 | 52 | 5 | 0 |

A_{8} | 0 | 0 | 0 | 0 | 0 | 13 | 50 | 137 | 56 | 2 |

A_{9} | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 56 | 145 | 27 |

A_{10} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 25 | 42 |

A_{1} | A_{2} | A_{3} | A_{4} | A_{5} | A_{6} | A_{7} | A_{8} | A_{9} | A_{10} | |
---|---|---|---|---|---|---|---|---|---|---|

A_{1} | 0.5882 | 0.4118 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

A_{2} | 0.0426 | 0.8085 | 0.1277 | 0.0213 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

A_{3} | 0.0058 | 0.1163 | 0.6919 | 0.1686 | 0.0174 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

A_{4} | 0.0000 | 0.0000 | 0.2025 | 0.6139 | 0.1709 | 0.0063 | 0.0000 | 0.0063 | 0.0000 | 0.0000 |

A_{5} | 0.0000 | 0.0000 | 0.0175 | 0.1696 | 0.6199 | 0.1520 | 0.0409 | 0.0000 | 0.0000 | 0.0000 |

A_{6} | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.2313 | 0.4478 | 0.2612 | 0.0597 | 0.0000 | 0.0000 |

A_{7} | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0276 | 0.2345 | 0.3448 | 0.3586 | 0.0345 | 0.0000 |

A_{8} | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0504 | 0.1938 | 0.5310 | 0.2171 | 0.0078 |

A_{9} | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0172 | 0.2414 | 0.6250 | 0.1164 |

A_{10} | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0563 | 0.3521 |

Metric | Definition | Equation |
---|---|---|

MAE | The mean absolute error of forecasting results | $\mathbf{MAE}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}\left|{y}_{i}-{\widehat{y}}_{i}\right|}$ |

RMSE | The root mean square value of the errors | $\mathbf{RMSE}=\sqrt{\frac{1}{N}\times {\displaystyle \sum _{i=1}^{N}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}}}$ |

MAPE | The average of absolute percentage error | $\mathbf{MAPE}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}\left|\frac{{y}_{i}-{\widehat{y}}_{i}}{{y}_{i}}\right|}\times 100\%$ |

IA | The index of agreement of forecasting results | $\mathbf{IA}=1-{\displaystyle \sum _{i=1}^{N}{\left({\widehat{y}}_{i}-{y}_{i}\right)}^{2}}/{\displaystyle \sum _{i=1}^{N}{\left(\left|{\widehat{y}}_{i}-\overline{y}\right|+\left|{y}_{i}+\overline{y}\right|\right)}^{2}}$ |

VAR | The variance of the forecasting error | $\mathbf{Var}=E{\left(\widehat{y}-E\left(\widehat{y}\right)\right)}^{2}$ |

MAE | RMSE | MAPE | IA | VAR | |
---|---|---|---|---|---|

Dataset I | |||||

FTS-Chi^{2} | 36.82% | 37.89% | 37.14% | 4.69% | 61.33% |

FTS-Entropy | 36.68% | 35.16% | 35.63% | 4.54% | 58.92% |

FTS-EF | 35.46% | 37.29% | 35.66% | 4.73% | 60.59% |

FTS-EW | 37.65% | 38.86% | 37.90% | 5.30% | 62.43% |

Dataset II | |||||

FTS-Chi^{2} | 31.81% | 33.64% | 31.17% | 1.79% | 55.94% |

FTS-Entropy | 33.13% | 34.61% | 31.89% | 1.79% | 57.23% |

FTS-EF | 29.62% | 31.38% | 29.09% | 1.66% | 52.89% |

FTS-EW | 28.27% | 29.99% | 26.99% | 1.76% | 50.86% |

Dataset III | |||||

FTS-Chi^{2} | 32.09% | 33.65% | 29.93% | 1.12% | 55.97% |

FTS-Entropy | 34.54% | 35.82% | 31.22% | 1.28% | 59.58% |

FTS-EF | 32.25% | 33.15% | 30.72% | 1.15% | 55.31% |

FTS-EW | 32.08% | 33.45% | 31.17% | 1.16% | 55.56% |

Model | Experimental Parameter | Value |
---|---|---|

BPNN | Maximum number of iteration times | 1000 |

Learning rate | 0.01 | |

Training accuracy goal | 0.00001 | |

Node-point number of input layer | 5 | |

Node-point number of hidden layer | 2 | |

Node-point number of output layer | 1 | |

ELM | Node-point number of input layer | 5 |

Node-point number of hidden layer | 20 | |

Node-point number of output layer | 1 | |

Elman | Node-point number of input layer | 5 |

Node-point number of hidden layer | 14 | |

Node-point number of output layer | 1 | |

Iteration number of display once in an image | 20 | |

Maximum number of iteration times | 1000 | |

SVR | Node point number of input layer | 5 |

Node point number of output layer | 1 | |

Type of SVR model | epsilon-SVR | |

Type of kernel function | RBF | |

Parameter of epsilon-SVR | 4 | |

ARIMA (p, d, q) | Autoregressive term (p) | 4 |

Moving average number (q) | 5 | |

Difference times (d) | 1 | |

DES | Smoothing coefficient | 0.9 |

**Table 9.**Comparison of the hybrid forecasting system against artificial intelligence, statistical, and persistence model.

Dataset I | Hybrid Forecasting System | Artificial Neural Network | SVR | Statistical | Persistence Model | ||||||

Chi^{2} | Entropy | EF | EW | BPNN | ELM | Elman | ARIMA (4,1,5) | DES | |||

MAE | 0.304745 | 0.314015 | 0.310712 | 0.311066 | 0.480500 | 0.464689 | 0.488986 | 0.474854 | 0.468817 | 0.618175 | 0.458800 |

RMSE | 0.393810 | 0.398935 | 0.422710 | 0.396898 | 0.624371 | 0.612871 | 0.626326 | 0.633335 | 0.608993 | 0.860441 | 0.617997 |

MAPE (%) | 5.199317 | 5.384246 | 5.300246 | 5.310732 | 8.313917 | 7.834408 | 8.664982 | 8.114452 | 7.966683 | 10.328762 | 7.749842 |

IA | 0.973831 | 0.973006 | 0.971658 | 0.973076 | 0.925417 | 0.930872 | 0.921294 | 0.927558 | 0.931646 | 0.890443 | 0.934266 |

VAR | 0.155138 | 0.159302 | 0.173445 | 0.156431 | 0.382100 | 0.374668 | 0.370987 | 0.399316 | 0.371603 | 0.741841 | 0.382683 |

Dataset II | Hybrid Forecasting System | Artificial Neural Network | SVR | Statistical | Persistence Model | ||||||

Chi^{2} | Entropy | EF | EW | BPNN | ELM | Elman | ARIMA (4,1,5) | DES | |||

MAE | 0.303159 | 0.305047 | 0.292344 | 0.327601 | 0.429400 | 0.419508 | 0.485842 | 0.437043 | 0.422248 | 0.531832 | 0.411800 |

RMSE | 0.385926 | 0.394417 | 0.382119 | 0.420453 | 0.572332 | 0.559468 | 0.623497 | 0.598708 | 0.563047 | 0.722019 | 0.554923 |

MAPE (%) | 4.040435 | 4.063754 | 3.933468 | 4.372152 | 5.727318 | 5.513481 | 8.556761 | 5.842899 | 5.607781 | 6.913070 | 5.417698 |

IA | 0.986645 | 0.986254 | 0.987135 | 0.984222 | 0.969177 | 0.970732 | 0.923628 | 0.966562 | 0.969659 | 0.955681 | 0.971915 |

VAR | 0.149199 | 0.155871 | 0.146307 | 0.177116 | 0.325634 | 0.312639 | 0.374678 | 0.356416 | 0.317654 | 0.522351 | 0.308557 |

Dataset III | Hybrid forecasting System | Artificial Neural Network | SVR | Statistical | Persistence Model | ||||||

Chi^{2} | Entropy | EF | EW | BPNN | ELM | Elman | ARIMA (4,1,5) | DES | |||

MAE | 0.319252 | 0.326514 | 0.336218 | 0.334258 | 0.465963 | 0.452667 | 0.521913 | 0.489951 | 0.465122 | 0.601174 | 0.456400 |

RMSE | 0.419349 | 0.431265 | 0.435344 | 0.425761 | 0.628820 | 0.613350 | 0.695743 | 0.650611 | 0.623680 | 0.829816 | 0.618935 |

MAPE (%) | 4.655949 | 4.692781 | 5.037214 | 4.771814 | 6.583105 | 6.338355 | 7.535347 | 6.924305 | 6.555370 | 8.308627 | 6.386741 |

IA | 0.991649 | 0.991099 | 0.991225 | 0.991315 | 0.980192 | 0.981376 | 0.974440 | 0.979072 | 0.971571 | 0.968741 | 0.981649 |

VAR | 0.176199 | 0.186327 | 0.184881 | 0.181607 | 0.396132 | 0.376780 | 0.481291 | 0.423972 | 0.389664 | 0.689968 | 0.383807 |

Dataset I | Original Data | Hybrid Forecasting System | ||||||

FTS-Chi^{2} | FTS-Entropy | FTS-EF | FTS-EW | EEMD-FTS-Chi^{2} | EEMD-FTS-Entropy | EEMD-FTS-EF | EEMD-FTS-EW | |

MAE | 0.482308 | 0.486552 | 0.490665 | 0.498894 | 0.304745 | 0.314015 | 0.310712 | 0.311066 |

RMSE | 0.634074 | 0.651953 | 0.636203 | 0.649179 | 0.39381 | 0.42271 | 0.398935 | 0.396898 |

MAPE (%) | 8.270632 | 8.36813 | 8.234151 | 8.551709 | 5.199317 | 5.384246 | 5.300246 | 5.310732 |

IA | 0.930179 | 0.929037 | 0.929454 | 0.924069 | 0.973831 | 0.973006 | 0.971658 | 0.973076 |

VAR | 0.401151 | 0.404214 | 0.422243 | 0.416326 | 0.155138 | 0.159302 | 0.173445 | 0.156431 |

Dataset II | Original Data | Hybrid Forecasting System | ||||||

FTS-Chi^{2} | FTS-Entropy | FTS-EF | FTS-EW | EEMD-FTS-Chi^{2} | EEMD-FTS-Entropy | EEMD-FTS-EF | EEMD-FTS-EW | |

MAE | 0.444548 | 0.433445 | 0.437201 | 0.456687 | 0.303159 | 0.305047 | 0.292344 | 0.327601 |

RMSE | 0.581522 | 0.584344 | 0.57475 | 0.600577 | 0.385926 | 0.382119 | 0.394417 | 0.420453 |

MAPE (%) | 5.870509 | 5.730971 | 5.774887 | 5.988826 | 4.040435 | 4.063754 | 3.933468 | 4.372152 |

IA | 0.969249 | 0.970143 | 0.96978 | 0.967156 | 0.986645 | 0.986254 | 0.987135 | 0.984222 |

VAR | 0.338608 | 0.330869 | 0.34208 | 0.360426 | 0.149199 | 0.155871 | 0.146307 | 0.177116 |

Dataset III | Original Data | Hybrid Forecasting System | ||||||

FTS-Chi^{2} | FTS-Entropy | FTS-EF | FTS-EW | EEMD-FTS-Chi^{2} | EEMD-FTS-Entropy | EEMD-FTS-EF | EEMD-FTS-EW | |

MAE | 0.470124 | 0.481921 | 0.513594 | 0.492126 | 0.319252 | 0.326514 | 0.336218 | 0.334258 |

RMSE | 0.632072 | 0.678365 | 0.645095 | 0.639762 | 0.419349 | 0.435344 | 0.431265 | 0.425761 |

MAPE (%) | 6.644708 | 6.774126 | 7.323562 | 6.932627 | 4.655949 | 4.692781 | 5.037214 | 4.771814 |

IA | 0.980658 | 0.979836 | 0.97873 | 0.979918 | 0.991649 | 0.991099 | 0.991225 | 0.991315 |

VAR | 0.400191 | 0.416966 | 0.457436 | 0.408696 | 0.176199 | 0.186327 | 0.184881 | 0.181607 |

Datasets | Models | BPNN | ELM | Elman | SVR | ARIMA | DES |
---|---|---|---|---|---|---|---|

Dataset I | Hybrid system1 | 9.5759 | 6.9282 | 9.6694 | 8.6703 | 9.6704 | 9.4034 |

Dataset II | 8.1057 | 5.2140 | 14.5774 | 6.4442 | 8.0502 | 8.9632 | |

Dataset III | 8.5758 | 6.8089 | 12.069 | 9.5842 | 8.5689 | 9.5542 | |

Dataset I | Hybrid system2 | 8.2046 | 9.1922 | 8.2829 | 7.4465 | 8.3739 | 9.2545 |

Dataset II | 8.5994 | 7.8156 | 14.7021 | 6.6676 | 8.5149 | 8.9278 | |

Dataset III | 7.4355 | 8.0469 | 11.9691 | 8.4399 | 7.3988 | 8.9997 | |

Dataset I | Hybrid system3 | 9.2870 | 7.9969 | 9.3517 | 8.4294 | 9.3695 | 9.3582 |

Dataset II | 7.9956 | 8.2683 | 14.2859 | 6.2895 | 7.8031 | 8.8359 | |

Dataset III | 8.3252 | 6.9085 | 12.1188 | 9.0679 | 8.2286 | 9.3185 | |

Dataset I | Hybrid system4 | 9.5094 | 8.9113 | 9.5849 | 8.5284 | 9.6266 | 9.3763 |

Dataset II | 7.4392 | 7.5030 | 13.8997 | 5.7378 | 7.1529 | 8.2366 | |

Dataset III | 7.7251 | 7.6462 | 11.9581 | 8.8024 | 7.6623 | 9.1974 |

Models | Dataset I | Dataset II | Data III | ||||
---|---|---|---|---|---|---|---|

First-Order | Second-Order | First-Order | Second-Order | First-Order | Second-Order | ||

Compared Models | BPNN | 0.9209 | 0.8558 | 0.9429 | 0.8943 | 0.9338 | 0.8789 |

ELM | 0.922 | 0.8563 | 0.9442 | 0.8986 | 0.9362 | 0.8825 | |

Elman | 0.9205 | 0.8557 | 0.8908 | 0.8028 | 0.8736 | 0.7760 | |

SVR | 0.9189 | 0.8487 | 0.9416 | 0.8868 | 0.9308 | 0.8740 | |

ARIMA | 0.9203 | 0.8565 | 0.9439 | 0.8969 | 0.9344 | 0.8803 | |

DES | 0.8967 | 0.8086 | 0.9309 | 0.8741 | 0.9169 | 0.8463 | |

Hybrid Forecasting System | Chi | 0.9480 | 0.9069 | 0.9596 | 0.9284 | 0.9534 | 0.9151 |

Entropy | 0.9462 | 0.9049 | 0.9594 | 0.9267 | 0.9531 | 0.9143 | |

EF | 0.9470 | 0.8994 | 0.9607 | 0.9272 | 0.9496 | 0.9058 | |

EW | 0.9469 | 0.9063 | 0.9563 | 0.9219 | 0.9523 | 0.9151 |

The Value of the Ensemble Number Is 200 | MAE | RMSE | MAPE (%) | IA | VAR | |
---|---|---|---|---|---|---|

The amplitude of added noise | 0.1 | 0.356060 | 0.469598 | 6.013668 | 0.961840 | 0.220853 |

0.2 | 0.304745 | 0.393810 | 5.199317 | 0.973831 | 0.155138 | |

0.5 | 0.335544 | 0.432928 | 5.720263 | 0.967039 | 0.187473 | |

White noise is 0.5 | MAE | RMSE | MAPE (%) | IA | VAR | |

The value of ensemble number | 50 | 0.340148 | 0.438051 | 5.774439 | 0.966492 | 0.192129 |

100 | 0.304745 | 0.393810 | 5.199317 | 0.973831 | 0.155138 | |

200 | 0.342039 | 0.441753 | 5.781073 | 0.966076 | 0.195446 |

MODELS | MAE | RMSE | MAPE | IA | VAR | |
---|---|---|---|---|---|---|

Hybrid forecasting system | Chi^{2} | 0.390194 | 0.055946 | 6.411913 | 0.953606 | 0.260964 |

Entropy | 0.416678 | 0.058084 | 6.928242 | 0.950354 | 0.264313 | |

EF | 0.437427 | 0.061288 | 7.264912 | 0.949147 | 0.312299 | |

EW | 0.432544 | 0.061552 | 7.003766 | 0.943157 | 0.315089 | |

Artificial Neural Network | BPNN | 0.825827 | 1.068416 | 14.24583 | 0.718107 | 1.154761 |

ELM | 0.859881 | 1.100304 | 14.36884 | 0.714523 | 1.221836 | |

Elman | 0.843463 | 1.083612 | 14.61775 | 0.72943 | 1.177776 | |

Statistical | ARIMA | 0.791711 | 1.024048 | 13.37103 | 0.727751 | 1.061284 |

DES | 1.205944 | 1.563405 | 20.08901 | 0.68189 | 2.472686 | |

SVR | 0.948013 | 1.225591 | 16.6055 | 0.646858 | 1.499271 |

DM Test | BP | ELM | Elman | SVR | ARIMA | DES | |
---|---|---|---|---|---|---|---|

Hybrid system 1 | Chi^{2} | 4.877626 | 4.950937 | 4.90825 | 4.686209 | 4.909181 | 5.366497 |

Hybrid system 2 | Entropy | 4.737063 | 4.798341 | 4.770708 | 4.573673 | 4.735968 | 5.327807 |

Hybrid system 3 | EF | 4.475986 | 4.527564 | 4.495222 | 4.397966 | 4.445831 | 5.258996 |

Hybrid system 4 | EW | 4.571886 | 4.634924 | 4.605242 | 4.467959 | 4.522839 | 5.257129 |

**Table 16.**Forecasting effectiveness of different forecasting models for hourly time horizon wind speed forecasting.

Forecasting Effectiveness | Chi^{2} | Entropy | EF | EW | BPNN |
---|---|---|---|---|---|

first-order | 0.93588 | 0.930718 | 0.927351 | 0.929962 | 0.855717 |

second-order | 0.88826 | 0.88145 | 0.870173 | 0.878382 | 0.747753 |

Forecasting effectiveness | ELM | Elman | SVR | ARIMA | DES |

first-order | 0.857697 | 0.853565 | 0.833945 | 0.862438 | 0.79911 |

second-order | 0.749865 | 0.745601 | 0.713875 | 0.758843 | 0.661372 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, H.; Jiang, Z.; Lu, H. A Hybrid Wind Speed Forecasting System Based on a ‘Decomposition and Ensemble’ Strategy and Fuzzy Time Series. *Energies* **2017**, *10*, 1422.
https://doi.org/10.3390/en10091422

**AMA Style**

Yang H, Jiang Z, Lu H. A Hybrid Wind Speed Forecasting System Based on a ‘Decomposition and Ensemble’ Strategy and Fuzzy Time Series. *Energies*. 2017; 10(9):1422.
https://doi.org/10.3390/en10091422

**Chicago/Turabian Style**

Yang, Hufang, Zaiping Jiang, and Haiyan Lu. 2017. "A Hybrid Wind Speed Forecasting System Based on a ‘Decomposition and Ensemble’ Strategy and Fuzzy Time Series" *Energies* 10, no. 9: 1422.
https://doi.org/10.3390/en10091422