Constructing filters by nesting offsets and variable weights

The same closes, three assemblies

The archive builds filters by assembling a few reusable parts on ordered closes. Exponential smoothing is a recursive blend of the latest observation with the prior average, using a decay constant taken from a chosen smoothing length. A moving average is a lookback mean of ordered price observations, used both as a plotted baseline and as a displaced reference window inside a scaled ratio. An adaptive moving average is a recursive average whose blending weight is scaled by a contemporaneous up-versus-down change oscillator, so the effective length shortens when net directional change is large and lengthens when it is small.

Editorial reading: those parts are interchangeable components. The constructions below keep the same ordered closes and change only how a window is offset, how a smooth is nested, or how a weight is allowed to vary.

A displaced reference window

Anchored momentum is a scaled ratio-minus-one that compares the close, or an exponential average of the close, with a simple moving average used as a fixed reference window.

One construction is 100 times a ratio-minus-one that places either the close or an exponential average of the close over a simple moving average of the close read at offset ((simple-average length - 1) / 2) - momentum length.

A second construction keeps the same 100 times ratio-minus-one form but uses a simple moving average whose length is twice the momentum period plus one as the denominator.

A nested exponential step

A generalized double-exponential step is a weighted mix of a single exponential average and a double exponential average that sits between those two smooths, controlled by a factor on the unit interval. It is constructed as (1 + weighting factor) times a single exponential average of price minus the same factor times a double exponential average of that price and period.

T3 is that generalized double-exponential step applied three times in succession, each pass using the prior pass as its price input with unchanged period and factor.

Supplied T3 constructions use a weighting factor of 0.7 with a period of 6 in one recipe and a period of 7 in another, and the factor is entered on the unit interval.

A variable blending weight

A variable-length exponential average is a recursive exponential update whose weight is the product of a fixed exponential constant and the absolute value of a summed up-close versus down-close oscillator. After a warm-up of period minus 1 observations that returns the close, the next value equals the prior value plus the product of an absolute up/down-change oscillator, the constant 2 / (smoothing length + 1), and the close-minus-prior gap.

That oscillator is the absolute value of (sum of up-close changes - sum of down-close changes) / (sum of up-close changes + sum of down-close changes) over a stated lookback. Two recipes pair an average length of 21 with a smoothing length of 5.

Closes are classified against bands at 1.01 and 0.99 times the variable-length average to mark observations above, below, or inside the channel.

A companion line on the price scale

IE/2 is the arithmetic mean of a linear-regression slope plus a simple average of price and the endpoint linear-regression value of price, computed over one shared lookback. The supplied recipe uses a shared lookback of 15.

T3 and IE/2 constructions attach an alert when the close crosses the plotted filter, and both plots use the same vertical scale as price.