# Smooth a series with an exponentially weighted moving average Smooth a numeric series with an EWMA. Set alpha, span, half-life or center of mass, auto-fit alpha, forecast ahead, and export JSON, CSV or SVG. ## Run it - **CLI:** `gizza tool exponential-smoother "12 14 13 17 16 20 19 24"` - **Web:** https://gizza.ai/tools/exponential-smoother/ - **Agents:** machine-readable descriptor (parameters JSON Schema) at https://gizza.ai/tools/exponential-smoother/tool.json ## Inputs - `series` — Series _(field)_ - `mode` — Decay given as _(field)_ - `alpha` — Alpha _(field)_ - `span` — Span _(field)_ - `halflife` — Half-life _(field)_ - `com` — Center of mass _(field)_ - `adjust` — Bias-corrected weighting (adjust) _(field)_ - `ignore_na` — Ignore gaps when weighting _(field)_ - `min_periods` — Minimum periods _(field)_ - `forecast` — Forecast periods _(field)_ - `output` — Output format _(field)_ ## Output - Smoothed series (text) ## Query parameters Open the tool pre-filled and auto-run via URL: - `series` — Series - `mode` — Decay given as - `alpha` — Alpha - `span` — Span - `halflife` — Half-life - `com` — Center of mass - `adjust` — Bias-corrected weighting (adjust) - `ignore_na` — Ignore gaps when weighting - `min_periods` — Minimum periods - `forecast` — Forecast periods - `output` — Output format Example: `https://gizza.ai/tools/exponential-smoother/?series=12%0A14%0A13%0A17%0A16%0A20%0A19%0A24&mode=alpha&alpha=0.3&span=5&halflife=3&com=2&adjust=true&ignore_na=true&min_periods=0&forecast=0&output=json` --- ## About this tool Exponential Smoother applies an exponentially weighted moving average (EWMA) to a numeric series. Every past observation still counts, but its weight decays geometrically, so the result tracks the level of noisy data without the lag and the hard cut-off of a fixed-window average. The decay is one number, `alpha`, and you can supply it however your reference does: - `mode=alpha` — the smoothing factor itself, `0 < alpha <= 1` - `mode=span` — the familiar N-period EMA setting, `alpha = 2 / (span + 1)` - `mode=halflife` — periods until an observation's weight halves, `alpha = 1 - exp(-ln2 / halflife)` - `mode=com` — center of mass, `alpha = 1 / (1 + com)` - `mode=auto` — fit alpha by minimising the sum of squared one-step-ahead forecast errors Whichever you pick, the report echoes all four equivalents so you can move between conventions. Set `adjust=false` for the plain simple-exponential-smoothing recursion `y_t = (1 - alpha)·y_(t-1) + alpha·x_t` used by finance EMAs, or leave `adjust=true` for the bias-corrected form that divides by the decaying weight sum. Everything runs locally in your browser — the numbers never leave the device. ### Worked example Input: ```text 12 14 13 17 16 20 19 24 ``` With `mode=alpha`, `alpha=0.3` and `adjust=true`, the smoothed series comes back as `12`, `13.1765`, `13.0959`, `14.6372`, `15.1286`, `16.7849`, `17.5091`, `19.5755`. The first value is just the first observation. The second is `(14 + 0.7·12) / (1 + 0.7) = 13.1765` and the third `(13 + 0.7·14 + 0.49·12) / 2.19 = 13.0959`, so the dip at period 3 barely moves the level. The final `level` is `19.5755`. The JSON report also carries the equivalent `span` of `5.66667`, a `halflife` of `1.94336` and a `com` of `2.33333`, plus one-step-ahead errors (`rmse` `3.62333`, `mae` `3.00297`, `mape` `15.741`) so you can compare two settings on the same data. Set `forecast=6` to project six periods ahead. Simple exponential smoothing has a flat forecast function, so all six sit at that final level. ### Limits and edge cases - Accepts up to 20,000 data points and 2 MB of input text; `forecast` is capped at 1,000 periods. - `alpha` must be greater than 0 and at most 1. `alpha=1` reproduces the input exactly; there is no valid `alpha=0`. - `span` must be at least 1, `halflife` greater than 0, and `com` 0 or greater. - A leading text header row (for example `value`) is skipped; `na`, `n/a`, `nan`, `null`, `none`, `-`, `.` and `?` mark a missing observation. A gap is echoed as `null` in `values`, but `smoothed` carries the previous level forward. - Warm-up: `min_periods=0` and `min_periods=1` both emit a value from the first observation. Higher values return `null` until that many observations have been seen. - Error metrics are one-step-ahead: the forecast for a period is the smoothed level after the previous period. `mape` skips periods whose actual value is 0. - The series is treated as evenly spaced. Irregular timestamps and date-based half-lives are not supported — resample to a fixed interval first. - All reported numbers are rounded to 6 significant digits. ## FAQ
What is the difference between alpha and span? 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.
Should I turn "adjust" on or off? 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.
How does auto-fit choose alpha? 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.
How are missing values handled? 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.
Can it forecast a trend or a seasonal pattern? 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.
What do the error metrics mean? 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.
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