## 2 Theoretical Framework  
### Ses  
This is a suitable forecasting method based on data without a trend or a seasonal component [18] ([PERSON], 1959). In SES, \(\alpha\) ranging from 0 to 1 is used as a weight to indicate how similar the current and previous observations are: a weight value close to 1 emphasizes the current value whereas one close to 0 emphasizes the previous one [28]. The formula for SES is defined as the following:  
\[S_{t+1}=\alpha y_{t}+(1-\alpha)S_{t}, \tag{1}\]  
where \(S_{t}\) is the smoothed value of the time series at time \(t\), \(\alpha\) is the smoothing constant for \(\alpha\in[0,1]\), and \(y_{t}\) is the actual value of the time series at time \(t\).  
In the case of SES, the smoothed statistic is the forecasted value, which is derived as  
\[F_{t+1}=\alpha y_{t}+(1-\alpha)F_{t}, \tag{2}\]  
where \(F_{t+1}\) and \(F_{t}\) are the forecasted values of time series at time \(t+1\) and \(t\), respectively.  
Expanding Equation (2) means that  
\[F_{t}=\alpha y_{t-1}+(1-\alpha)F_{t-1},\]  
\[F_{t-1}=\alpha y_{t-2}+(1-\alpha)F_{t-2},\]  
\[F_{t-2}=\alpha y_{t-3}+(1-\alpha)F_{t-3},\]\[F_{t-3}=\alpha y_{t-4}+(1-\alpha)F_{t-4},\]  
etc.  
By substituting each of these into the following equation, we obtain  
\[\begin{array}{ll}F_{t+1}&=\alpha y_{t}+(1-\alpha)F_{t}=\alpha y_{t}+(1-\alpha )[\alpha y_{t-1}+(1-\alpha)F_{t-1}]\\ &=\alpha y_{t}+\alpha(1-\alpha)y_{t-1}+(1-\alpha)^{2}F_{t-1}\\ &=\alpha y_{t}+\alpha(1-\alpha)y_{t-1}+(1-\alpha)^{2}[\alpha y_{t-2}+(1- \alpha)F_{t-2}]\\ &=\alpha y_{t}+\alpha(1-\alpha)y_{t-1}+\alpha(1-\alpha)^{2}y_{t-2}+(1-\alpha) ^{3}F_{t-2}\\ &=\alpha y_{t}+\alpha(1-\alpha)y_{t-1}+\alpha(1-\alpha)^{2}y_{t-2}+(1-\alpha) ^{3}[\alpha y_{t-3}+(1-\alpha)F_{t-3}]\\ &=\alpha y_{t}+\alpha(1-\alpha)y_{t-1}+\alpha(1-\alpha)^{2}y_{t-2}+\alpha(1- \alpha)^{3}y_{t-3}+(1-\alpha)^{4}F_{t-3}.\end{array}\]  
Generalizing the above equation provides  
\[F_{t+1}=\sum_{j=0}^{t-1}\alpha(1-\alpha)^{j}y_{t-j}+(1-\alpha)^{t}F_{1}. \tag{3}\]  
Equation (3) represents the weighted moving average of all past observations with the weights decreasing exponentially; i.e., exponential smoothing [29]. It can be seen that large recent observations are assigned a larger weight, which can be interpreted as the weighted average for the most recent forecast from the most recent observations [1].  
The assumption of these weights are as follows:  
\[w_{t}\in[0,1]\text{ where }t=1,2,\ldots,n\]  
\[\sum_{t=1}^{n}w_{t}=1\text{ where }w_{1}\leq w_{2}\leq\ldots\leq w_{n}\]  
For forecasting from one step ahead at time \(t+1\), all of the observations in the series \(y_{1},y_{2},\ldots,y_{t}\) are used with an exponential weighting scheme that assigns the maximum weight to the most recent observation. Moreover, the weights decline systematically as the observations that are still included become older. Weighting in an exponential smoothing technique requires a given value of \(\alpha\), and the forecasted values vary depending on the value of this constant, as does the forecasting error. Hence, the critical process is determining the value of \(\alpha\) constrained by \(\alpha\in[0,1]\). Since mistakes in selecting the optimal value adversely affect the estimated results, obtaining the value of \(\alpha\) is conducted via trial and error while minimizing the sum of the squared error. As the value of \(\alpha\) controls the smoothing level adaptation speed, choosing the initial value must be conducted carefully, because a poor choice will require more time for the recursive smoothing formula to adapt and for its effect to dissipate. The impact is very tangible among small time series and those with a small value [30]. Considering Equation (3), the initial value \(F_{1}\) needs to be found via a prediction algorithm, which plays a critical role in computing all of the subsequent SES. In practice, different methods are applied to solve these problems.