<P> 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), \ t> 0 \ end (aligned))) </P> <P> where α (\ displaystyle \ alpha) is the smoothing factor, and 0 <α <1 (\ displaystyle 0 <\ alpha <1). </P> <P> Intuitively, the simplest way to smooth a time series is to calculate a simple, or unweighted, moving average . This is known as using a rectangular or "boxcar" window function . The smoothed statistic s is then just the mean of the last k observations: </P> <Dl> <Dd> s t = 1 k ∑ n = 0 k − 1 x t − n = x t + x t − 1 + x t − 2 + ⋯ + x t − k + 1 k = s t − 1 + x t − x t − k k, (\ displaystyle s_ (t) = (\ frac (1) (k)) \, \ sum _ (n = 0) ^ (k - 1) x_ (t-n) = (\ frac (x_ (t) + x_ (t - 1) + x_ (t - 2) + \ cdots + x_ (t - k + 1)) (k)) = s_ (t - 1) + (\ frac (x_ (t) - x_ (t-k)) (k)),) </Dd> </Dl>

Exponential smoothing is one of the most commonly used techniques