Answer:

The RAW data sequence is often represented by {\displaystyle \{x_{t}\}}\{x_{t}\} beginning at time {\displaystyle t=0}t=0, and the output of the exponential smoothing algorithm is commonly WRITTEN as {\displaystyle \{s_{t}\}}\{s_{t}\}, which may be regarded as a best estimate of what the next value of {\displaystyle x}x will be. When the sequence of observations begins at time {\displaystyle t=0}t=0, the simplest form of exponential smoothing is given by the formulas:[1]

{\displaystyle {\begin{aligned}s_{0}&=x_{0}\\s_{t}&=\alpha x_{t}+(1-\alpha )s_{t-1},\ t>0\end{aligned}}}{\begin{aligned}s_{0}&=x_{0}\\s_{t}&=\alpha x_{t}+(1-\alpha )s_{t-1},\ t>0\end{aligned}}

where {\displaystyle \alpha }\alpha  is the smoothing factor, and {\displaystyle 0<\alpha <1}0<\alpha <1.\

he simplest form of exponential smoothing is given by the formula:

{\displaystyle s_{t}=\alpha x_{t}+(1-\alpha )s_{t-1}=s_{t-1}+\alpha (x_{t}-s_{t-1})}{\displaystyle s_{t}=\alpha x_{t}+(1-\alpha )s_{t-1}=s_{t-1}+\alpha (x_{t}-s_{t-1})}.

where α is the smoothing factor, and 0 < α < 1. In other words, the smoothed statistic st' is a simple weighted average of the current observation xt and the previous smoothed statistic st−1. The term smoothing factor applied to α here is something of a misnomer, as larger values of α actually reduce the level of smoothing, and in the limiting case with α = 1 the output series is just the current observation. Simple exponential smoothing is easily applied, and it produces a smoothed statistic as soon as two observations are available.

Values of α close to ONE have less of a smoothing effect and give greater WEIGHT to recent changes in the data, while values of α closer to zero have a greater smoothing effect and are less responsive to recent changes. There is no formally correct procedure for choosing α. Sometimes the statistician's judgment is used to CHOOSE an appropriate factor. Alternatively, a statistical technique may be used to optimize the value of α. For example, the method of least squares might be used to determine the value of α for which the sum of the quantities {\displaystyle (s_{t}-x_{t+1})^{2}}{\displaystyle (s_{t}-x_{t+1})^{2}} is minimized.[6]