Exponential Smoothing (One Update Step)

Exponential smoothing steadies a noisy stream of readings. Each time a new reading $x_t$ arrives, the new smoothed value $s_t$ is a blend of that reading and the previous smoothed value $s_{t-1}$: a fraction $\alpha$ of the new, and the rest of the old. It is a rule for a running average, a procedure you choose to apply, not a law of nature, so the thing to judge is whether it suits the job. It is used to steady a sensor feed or to forecast one step ahead.

$\alpha$ sits between 0 and 1, and it sets how much weight the newest reading gets. A small $\alpha$ smooths hard and responds slowly. At $\alpha = 1$ the formula returns $x_t$ unchanged, so the raw data pass straight through. The name comes from what you get when you unroll the recursion: a reading $k$ steps back carries the weight $\alpha(1 - \alpha)^k$, so old readings are forgotten geometrically, never all at once. Start the series by setting the first smoothed value equal to the first reading. The three values $s_t$, $x_t$ and $s_{t-1}$ share the unit of the readings, and $\alpha$ has no unit.

Take $\alpha = 0.3$, a previous smoothed value of 50 and a latest reading of 60: $0.3 \times 60 + 0.7 \times 50 = 18 + 35 = 53$. The smoothed value moves from 50 toward 60 but only reaches 53, because the new reading gets a weight of 0.3 and the old value 0.7. The relation also turns around. A smoothed value of 52 from 50, with a latest reading of 60, means $\alpha$ was 0.2, which is the fraction of the way the smoothed value travelled from the old value toward the new reading. A smoothed value of 53 from 50 at $\alpha = 0.3$ means the latest reading was 60.

If readings are spaced $\Delta t$ apart and you want a time constant $\tau$, set $\alpha = 1 - e^{-\Delta t/\tau}$. That ties the smoothing to a time scale instead of to a count of readings, so it can be kept the same when the spacing of the readings changes.