Linearity, commutation, and ratio smoothing in moving averages

A simple moving average is an equal-weight average of the most recent observations of a series. An exponential moving average is a recursive average that down-weights older observations with a fixed decay.

Linearity of the two averages

Linearity means that scaling and adding series before averaging equals averaging each series and then scaling and adding. For a constant scale factor, both a simple moving average and an exponential moving average treat a linear combination of two series as the same linear combination of the two separately averaged series.

Typical price is the bar midpoint formed from high, low, and close. Averaging a typical-price series built from high, low, and close is interchangeable with averaging those three series first and then taking their mean.

Order of stacked layers

Successive smoothing means applying more than one moving average, of any mix of types and lengths, to the same series. When simple and exponential moving averages are stacked, the order of the layers does not change the final series.

A five-period exponential average of a ten-period simple average of a fifteen-period simple average of closes equals the same three averages applied in reverse order.

Age-weighting after successive smoothing

Age-weighting is the share of the result attributed to each past observation, ordered by how old that observation is. Repeated averaging produces a smoother series whose percentage weights by age form a spreading, more symmetric bell rather than matching a single longer average.

Each extra average shifts weight from newer observations toward older ones, so the bell of significant weights moves toward older ages and covers a wider span. A mixed stack of simple and exponential averages still yields the same bell-shaped age-weighting pattern as stacks of one type alone.

Ratio smoothing

Ratio smoothing means smoothing a quotient either by averaging the quotient itself or by averaging the numerator and denominator separately and then dividing. For a positive denominator series, averaging the numerator and denominator separately and then dividing is not the same operator as averaging the raw ratio.

The separately averaged quotient is itself a weighted average of the raw ratio values, with larger weight on bars where the denominator is larger.