Average directional index construction: recursive smoothing and lookback offset

A construction experiment

The finished average directional index is the completed output of a directional-movement construction. This article holds that finished series still, freezes the published 14-bar recursion so a replica can be checked, and then changes only the averaging choices that sit underneath it.

A true drop-off window and a complete first-bar offset are the two swaps under review. The question at this stage is which of those choices move the same inputs, not which choice later wins an out-of-sample comparison.

Default lookback

A 14-bar lookback is presented as the default half-cycle length corresponding to an observed 28-bar cycle. That default is treated as changeable, so the construction can be rebuilt on another named window without changing what the finished output is meant to be.

How the finished output is assembled

Plus and minus directional indicators are the corresponding 14-bar directional-movement totals divided by the 14-bar true-range total and scaled by 100. Those two series are then drawn as crossing lines.

Daily directional index is the absolute gap between those two indicators divided by their sum and scaled by 100. The first average directional index is the 14-bar mean of that daily index. Later values blend thirteen parts of the prior average with the newest daily index.

Recursive smoothing versus a drop-off window

The published 14-bar sum updates by subtracting one-fourteenth of the prior sum and adding the newest daily true range. It does not drop the oldest observation from a fixed window. Directional movement is accumulated on that same 14-bar recursion.

Comparing that recursive sum with a conventional moving sum produced differences as large as 15 percent when the observation that would have left the window was far from the series average.

Lookback offset in the first window

Lookback offset is the choice of which first row enters a named 14-bar window. Starting the first 14-bar true-range total from the first price row, instead of the first true-range row, leaves only 13 true-range observations in that initial window.

Applying the one-fourteenth subtraction from the first update onward keeps later 14-bar averages on a 13-observation base. A complete-window alternative differed by as much as 13 percent.

Staggered starting rows

Completing the module requires different components to start on different rows, including the first, second, fourteenth, and twenty-eighth observations. True range, directional movement, and the average directional index do not share one offset, so a replica has to keep those starts separate if the finished series is to be checked.

What the algebraic gaps do not prove

Algebraic discrepancies alone are not treated as proof that the published shortcuts help or hurt a later trading rule. That question is deferred to tests of finished variants on multiple contracts.