## 3 Methodology
### Feature Extraction
#### 3.3.4 Exponential Smoothing Features (ESF)  
The exponential smoothing technique uses exponential functions to smooth time series data. It is used to make short-term forecasts based on prior assumptions. It also helps in producing slightly unreliable long-term forecasts. Exponential smoothing techniques are straightforward yet effective methods that are commonly applied in various business scenarios like inventory management. There are three distinct techniques for ESF: basic exponential smoothing (SES), double exponential smoothing (DES), and triple exponential smoothing (TES). Simple exponential smoothing is a budget-friendly approach utilized when historical data do not exhibit cyclical patterns or trends. This technique involves generating a smooth curve by incorporating one actual previous value and one forecasted previous value.  
\[T_{t}=\alpha\,z_{t-1}+(1-\beta)T_{t-1} \tag{17}\]  
In this context, \(T_{t}\) is used to signify the estimated value (or smoothed value) at \(T=t\), while \(T_{t-1}\) represents the smoothed value when \(T=t-1,\ z_{t-1}\) denotes the actual value of the series at T = t - 1. The smoothing parameter, denoted as \(\alpha\) and varying between 0 and 1.  
1, is responsible for determining the level of smoothness within the data series. Despite \(T_{t}\) taking into consideration the influence of all past data, only two numerical values \(z_{t-1}\) and \(U_{t-1}\), are required for the actual computation process. Hence, one of the main benefits of using exponential smoothing is its ability to efficiently handle considerable amounts of data.  
The method of double exponential smoothing, also referred to as [PERSON]'s exponential smoothing, is frequently used in the field of economics to forecast values and predict trends. The mathematical formula for [PERSON]'s exponential smoothing can be stated as follows:  
\[U_{t}=\alpha z_{t}+(1-\alpha)(U_{t-1}+T_{t-1}) \tag{18}\]  
\[T_{t}=\gamma(U_{t}+U_{t-1})+(1-\gamma)T_{t-1} \tag{19}\]  
\[\overset{\wedge}{z_{t}}(l)=U_{t}+lT_{t} \tag{20}\]  
In this context, \(U_{t}\) represents the smoothed level, and \(T_{t}\) represents the trend. The smoothing parameters are denoted by \(\alpha\) and \(\gamma\). The variable \(z_{t}\) denotes the actual value of the time series at a certain time period \(t\), while \(\overset{\wedge}{z_{t}}(l)\) shows that the forecast made \(l\) steps ahead of the origin of the forecast. The initial values for the trend and level are determined by the governing equations provided in the following section.  
\[T_{t}=0 \tag{21}\]  
\[U_{t}=z_{2}-z_{1} \tag{22}\]  
The Holt-Winters method, sometimes called triple exponential smoothing, is a valuable tool for analyzing seasonal time series since it can accurately represent both trend and seasonality trends in past data. This approach dissects the level, trend, and seasonal trends as part of its procedure. While widely used, it is limited in that it can only fit one seasonal pattern, which can be a drawback for load forecasting. The Holt-Winters technique offers two computational models: one for additive and one for multiplicative calculations. Additive calculations are appropriate for data with consistent seasonal patterns, while multiplicative models are suitable for data with changing seasonal patterns. Equations (23)-(26) are used for the additive model.  
\[M_{t}=\alpha(z_{t}-U_{t-1})+(1-\alpha)(M_{t-1}-T_{t-1}) \tag{23}\]  
\[T_{t}=\beta(M_{t}+M_{t-1})+(1-\beta)T_{t-1} \tag{24}\]  
\[U_{t}=\gamma(z_{t}+M_{t-1})+(1-\gamma)U_{t-1} \tag{25}\]  
\[\overset{\wedge}{z_{t+q}}=M_{t}+qT_{t}+U_{t+m+q} \tag{26}\]  
In this context, \(M\) indicates the estimated level affected by the quantity of \(\alpha\), \(T\) refers to the estimated trend influenced by the quantity of \(\beta\), and \(U\) represents the estimate of seasonality influenced by the quantity of \(\gamma\). \(q\) Signifies the seasonal period, \(m\) represents the length, \(z\) is the current data, and \(\overset{\wedge}{z}\) represents the forecasted value for the next period.