Exponential smoothing

Exponential smoothing or exponential moving average (EMA) is a <a href="/facts/Rule_of_thumb/RopfGsJ0">rule of thumb</a> technique for smoothing <a href="/facts/Time_series/fSXPR817">time series</a> data using the exponential <a href="/facts/Window_function/5Xj7h2ah">window function</a>. Whereas in the <a href="/facts/Simple_moving_average/tWY74500">simple moving average</a> the past observations are weighted equally, exponential functions are used to assign exponentially decreasing weights over time. It is an easily learned and easily applied procedure for making some determination based on prior assumptions by the user, such as seasonality. Exponential smoothing is often used for analysis of time-series data.
Exponential smoothing is one of many <a href="/facts/Window_functions/5Xj7h2ah">window functions</a> commonly applied to smooth data in <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, acting as <a href="/facts/Low-pass_filter/QFsUjw7h">low-pass filters</a> to remove high-frequency <a href="/facts/Noise/6ueEsAG0">noise</a>. This method is preceded by <a href="/facts/Sim%25C3%25A9on_Denis_Poisson/zmsDITrY">Poisson</a>'s use of recursive exponential window functions in convolutions from the 19th century, as well as <a href="/facts/Kolmogorov%25E2%2580%2593Zurbenko_filter/wJeh655p">Kolmogorov and Zurbenko's use of recursive moving averages</a> from their studies of turbulence in the 1940s.
The raw data sequence is often represented by 
 
 
 
 {
 
 x
 
 t
 
 
 }
 
 
 {\textstyle \{x_{t}\}}
 
 beginning at time 
 
 
 
 t
 =
 0
 
 
 {\textstyle t=0}
 
, and the output of the exponential smoothing algorithm is commonly written as 
 
 
 
 {
 
 s
 
 t
 
 
 }
 
 
 {\textstyle \{s_{t}\}}
 
, which may be regarded as a best estimate of what the next value of 
 
 
 
 x
 
 
 {\textstyle x}
 
 will be. When the sequence of observations begins at time 
 
 
 
 t
 =
 0
 
 
 {\textstyle t=0}
 
, the simplest form of exponential smoothing is given by the following formulas:

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

where 
 
 
 
 α
 
 
 {\textstyle \alpha }
 
 is the smoothing factor, and 
 
 
 
 0
 <
 α
 <
 1
 
 
 {\textstyle 0<\alpha <1}
 
. If 
 
 
 
 
 s
 
 t
 −
 1
 
 
 
 
 {\textstyle s_{t-1}}
 
 is substituted into 
 
 
 
 
 s
 
 t
 
 
 
 
 {\textstyle s_{t}}
 
 continuously so that the formula of 
 
 
 
 
 s
 
 t
 
 
 
 
 {\textstyle s_{t}}
 
 is fully expressed in terms of 
 
 
 
 {
 
 x
 
 t
 
 
 }
 
 
 {\textstyle \{x_{t}\}}
 
, then exponentially decaying weighting factors on each raw data 
 
 
 
 
 x
 
 t
 
 
 
 
 {\textstyle x_{t}}
 
 is revealed, showing how exponential smoothing is named.
The simple exponential smoothing is not able to predict what would be observed at 
 
 
 
 t
 +
 m
 
 
 {\textstyle t+m}
 
 based on the raw data up to 
 
 
 
 t
 
 
 {\textstyle t}
 
, while the double exponential smoothing and triple exponential smoothing can be used for the prediction due to the presence of 
 
 
 
 
 b
 
 t
 
 
 
 
 {\displaystyle b_{t}}
 
 as the sequence of best estimates of the linear trend.

Exponential smoothing open-in-new

Exponential smoothing