# 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 |

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**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