Exponential smoothing

Exponential smoothing is a rule of thumb technique for smoothing time series data using the exponential window function. Whereas in the simple moving average 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 a rule of thumb technique for smoothing time series data using the exponential window function. Whereas in the simple moving average 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 window functions commonly applied to smooth data in signal processing, acting as low-pass filters to remove high frequency noise. This method is preceded by Poisson's use of recursive exponential window functions in convolutions from the 19th century, as well as Kolmogorov and Zurbenko's use of recursive moving averages from their studies of turbulence in the 1940s. The raw data sequence is often represented by { x t } {displaystyle {x_{t}}} beginning at time t = 0 {displaystyle t=0} , and the output of the exponential smoothing algorithm is commonly written as { s t } {displaystyle {s_{t}}} , which may be regarded as a best estimate of what the next value of x {displaystyle x} will be. When the sequence of observations begins at time t = 0 {displaystyle t=0} , the simplest form of exponential smoothing is given by the formulas: s 0 = x 0 s t = α x t + ( 1 − α ) s t − 1 ,   t > 0 {displaystyle {egin{aligned}s_{0}&=x_{0}\s_{t}&=alpha x_{t}+(1-alpha )s_{t-1}, t>0end{aligned}}} where α {displaystyle alpha } is the smoothing factor, and 0 < α < 1 {displaystyle 0<alpha <1} . The use of the exponential window function is first attributed to Poisson as an extension of a numerical analysis technique from the 17th century, and later adopted by the signal processing community in the 1940s. Here, exponential smoothing is the application of the exponential, or Poisson, window function. Exponential smoothing was first suggested in the statistical literature without citation to previous work by Robert Goodell Brown in 1956, and then expanded by Charles C. Holt in 1957. The formulation below, which is the one commonly used, is attributed to Brown and is known as “Brown’s simple exponential smoothing”. All the methods of Holt, Winters and Brown may be seen as a simple application of recursive filtering, first found in the 1940s to convert FIR filters to IIR filters.