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Wilder's Smoothed Moving Average (RMA)

Summary

Wilder's smoothed moving average: an exponential average whose smoothing factor is the reciprocal of the period rather than the 2/(n+1) of a classic EMA, seeded with a simple average of the first window. J. Welles Wilder Jr. introduced it in 1978 as the smoothing inside RSI, ATR and the directional-movement family; this exposes it as a moving average in its own right.

Because the smoothing factor is smaller than an EMA's at the same period, RMA reacts more slowly and gives noticeably more weight to old data: it takes about twice the period to shed the influence of a bar. Read it as a slow trend line — direction and slope matter, individual crossings much less than on a faster average.

RMA is also selectable as a moving-average type (TA_MAType_RMA) wherever an optInMAType parameter is accepted (MA, BBANDS, STOCH, MACDEXT, ...).

It travels under five names for one object: RMA (TradingView, pandas-ta), SMMA (MetaTrader), Wilder's Smoothing or Wilder's Average (thinkorswim), wilders (Tulip), WilderMA (Wealth-Lab).

Formula

alpha = 1 / N, beta = 1 - alpha, N = optInTimePeriod

seed at bar N-1: RMA = ( x[0] + x[1] + ... + x[N-1] ) / N

for i >= N: RMA[i] = alpha * x[i] + beta * RMA[i-1]

Notes

  • Wilder's own writing uses a period of 14, and pandas-ta defaults to 10. The default here is the one the rest of the moving-average family carries, so a call that swaps one MA for another keeps its period.
  • The smoothing factor being 1/N is sometimes quoted as "an RMA of N is an EMA of 2N-1". The factors really are identical, since 2/((2N-1)+1) is 1/N, but the two seed over different windows: the series differ through the warm-up and only converge as the seed's influence decays.
  • TA_RMA over TA_TRANGE is TA_ATR, bit for bit. The recurrence is spelled here exactly as ATR spells it.
  • The recurrence is the alpha * x + (1 - alpha) * prev form, which is TradingView Pine's ta.rma and pandas' ewm(adjust=False) kernel. Implementations that spell it prev + (x - prev) * alpha or (prev * (N-1) + x) / N are algebraically the same average and differ from this one only at the last bits.
  • Being recursive, an output depends on how much history precedes it: the same bar computed from an earlier start differs while the seed still carries weight, and that difference decays by a factor of 1 - 1/N per bar. The unstable period is how much of the warm-up to discard.

Inputs

  • inReal — Data on which to compute the average

Outputs

  • outReal — Wilder's smoothed moving average of the input

Parameters

ParameterTypeDefaultAccepted valuesDescription
optInTimePeriodinteger301–100000Number of bars in the seed window, and the reciprocal of the smoothing factor

Properties

Numerical Stability: Initial Unstable Period

✅ Overlap Input i
☐ Independent Y-Axis
☐ Candlestick
☐ Can Output NaN or ±Inf
✅ Identity at Period 1 i

Implementation

TA-Lib Definition: rma.c · rma.yaml

NativeFile
Cta_RMA.c
Rustrma.rs
JavaCore_RMA.java

TA-Lib is also available for Python, R and more using a wrapper.

Aliases

Wilder's Smoothed Moving Average, Smoothed Moving Average, SMMA, Wilder's Smoothing, WilderMA

See Also

EMA · SMA · ATR · RSI · DEMA

References

  • J. Welles Wilder Jr., New Concepts in Technical Trading Systems, Trend Research, 1978. The original definition, given as the smoothing inside RSI, ATR and the directional-movement family.
  • Steven B. Achelis, Technical Analysis from A to Z, page 366 — a worked Wilder-smoothing series.
  • thinkorswim, WildersSmoothing — "smoothing factor of 1/length"; "The value for the first period is an SMA".
  • Incredible Charts, Wilder Moving Average — the 1/n versus 2/(n+1) comparison.
  • TradingView Pine Script, ta.rma — the same recurrence, SMA-seeded.
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