# Direct Multistep Wind Speed Forecasting Using LSTM Neural Network Combining EEMD and Fuzzy Entropy
## 2 Methodology
### EMD and EEMD  
EMD was proposed by [PERSON] in 1998 [PERSON] (1998). It is a self-adaptive and efficient method for analyzing nonlinear and nonstationary signals. Since the wind speed series is nonlinear and nonstationary, EMD is efficient to analyze the wind speed signal. The basic idea of EMD is to smooth the fluctuating signals adaptively and to decompose them into fluctuations or trends of different scales. After decomposition, a finite and small number of intrinsic mode functions (IMFs) and a residual component are obtained. An IMF is a function that satisfies the following two conditions: (a) in the whole data set, the number of extrema and the number of zero crossings must either be equal or differ at most by one; and (b) at any point, the average of the envelopes defined by the local maxima and the local minima must be zero [PERSON] (1998). The specific steps for decomposing time series by using the EMD algorithm are detailed in Ref. [PERSON] (1998).  
Mode mixing is the most significant drawback of EMD [PERSON] (1998), which means that a single IMF consists of signals with dramatically disparate scales or a signal of the same scale appears in different IMF components. The mode mixing compromises the stationarity of IMFs and therefore limits the effectiveness of the EMD algorithm. To solve the mode mixing problem in EMD, a new noise-assisted data analysis method EEMD is proposed. In EEMD, the white Gaussian noise is added into the original time series, and the EMD is then applied on noise added time series to obtain IMFs that are free from mode mixing ([PERSON] et al., 2016; [PERSON] et al., 2016). However, the resulting IMFs include white noise, which is then removed by obtaining the mean of multiple trials. Therefore, the true IMF components are defined as the mean of an ensemble of trails and each trail consists of the decomposition results of the signal plus a white Gaussian noise of finite amplitude ([PERSON] et al., 2016).  
For the wind speed series {\(v(t)\), \(t\) = 1, 2,..., \(N\)}, the main steps of the EEMD algorithm are described as follows:  
* Add the white Gaussian noise series \(\varepsilon_{i}(t)\) to the original wind speed series \(v(t)\) and obtain a new series \(V_{j}(t)\).
* Decompose the new series \(V_{j}(t)\) into several IMFs and a residue by using the EMD algorithm.
* For \(j\) = 1, 2,..., \(N_{E}\), repeat Step 1 and Step 2, and add different white Gaussian noise series each time. \(N_{E}\) is the number of repeated procedures.
* Take the mean of all IMF components and the mean of residual components as the final results.  
After the decomposition by using EEMD algorithm, the wind speed series {\(v(t)\)} can be expressed as:  
\[v(t)=\sum_{i=1}^{n}c_{i}(t)+r_{n}(t) \tag{1}\]  
where {\(c_{i}(t)\), \(I\) = 1, 2,..., \(n\)} represents the different IMF component, and \(r_{n}(t)\) is the residue after \(n\) IMFs are derived. These IMFs contain components of different time characteristic scales of the wind speed series. Their scales range from small to large, and the frequencies range from high to low.