Constructing a general weighted moving average from one exponent

Three inputs and one reusable smoother

A general weighted moving average can be built from three inputs: an ordered price series, a lookback length, and one exponent that shapes the weights. A weighted moving average is a lookback average in which each observation is multiplied by a rank-based weight before the weighted values are summed.

The general form uses one reusable rule. Raise each bar's recency rank to the exponent, then divide by a shared normalizing total.

How the weights are formed

The normalizing total is the sum of the integers from 1 through the lookback length, each raised to the chosen exponent. Each bar's weight is its recency rank raised to the exponent and then divided by that normalizing total, so nearer bars receive larger weights when the exponent is positive.

Once lookback length is fixed, the exponent is the only parameter that reshapes the weight curve.

Special cases of the exponent

An exponent of 0 reproduces a simple moving average. An exponent of 1 reproduces a conventional linearly weighted moving average.

An exponent of 2 is the square-weighted case, used in a twelve-period construction. An eight-period construction can use a fractional exponent of 0.6.

Sensitivity as the exponent rises

Raising the exponent increases sensitivity to the newest observations, shown on a nine-period average at exponents 0, 1, 2, 3, and 4. Sensitivity is how strongly the average responds to the newest observation as the exponent rises.

Longer lookbacks and price overlays

A longer lookback can be assembled by computing the denominator once and splitting the weighted sum into repeated terms that differ only in rank and lag. Overlaying the average on price requires the indicator scale to use the same high and low bounds as the price series.