Smooth a series with an exponentially weighted moving average

They are two ways of writing the same decay. span is the "N-period EMA" convention used by charting and finance tools, and it converts as alpha = 2 / (span + 1) — so a 12-period EMA is alpha = 0.153846. Use span when you are reproducing an N-day EMA, and alpha when you want a decay rate that no integer span can express, such as 0.25. The JSON report prints both, plus the equivalent half-life and center of mass.

Leave it on when you are summarising data and want the early points to be honest weighted averages of everything seen so far. Turn it off when you are reproducing a textbook simple exponential smoothing table or a finance EMA, both of which use the plain recursion y_t = (1 - alpha)·y_(t-1) + alpha·x_t seeded at the first observation. The two agree more and more as the series gets longer; they differ most in the first few periods, where adjust=true divides by the partial weight sum instead of leaning on the seed value.

It scans alpha across (0, 1] and then refines the best region with a golden-section search, minimising the sum of squared one-step-ahead forecast errors — the standard criterion for picking a simple exponential smoothing constant. A trending series usually fits a high alpha because following the data closely beats averaging it; noise around a stable mean fits a low alpha. Auto-fit needs at least two numeric values, and it honours whatever adjust, ignore_na and min_periods you have set.

A gap keeps its slot so your rows stay aligned: it comes back as null in the echoed values, while smoothed carries the last level forward — the smoothed level is defined at every period, and a missing observation simply does not update it. Gaps are also skipped when scoring, so they never contribute a forecast error. What changes is the weighting of the points around the gap. With ignore_na=false (the default) a gap still consumes one step of decay, so an observation after a long gap counts for less. With ignore_na=true the gap is skipped entirely and weights are assigned by position among the actual observations, as if it were never there.

No. This is single (simple) exponential smoothing, which models level only, so its forecast is flat at the last smoothed value. That is the correct forecast for a series without trend or seasonality, and it is still useful as a baseline for one that has them. Double exponential smoothing (Holt's linear trend) and triple exponential smoothing (Holt-Winters, with seasonality) are separate models and are not implemented here.

They score the one-step-ahead forecasts: for each period, the forecast is the smoothed level after the previous period, and the error is the actual value minus that forecast. sse is the sum of squared errors, mse its mean, rmse the square root of mse (in the units of your data), mae the mean absolute error, and mape the mean absolute percentage error. Lower is better, and comparing rmse across two alpha values is the usual way to decide which smoothing is doing more good than harm on your series.