Constructing the average directional index from directional movement and true range

What the construction starts from

The average directional index is constructed from the idea that a trend is a sequence of price ranges that keep extending in one consistent direction. The constructed output is a recursively smoothed integer series built from successive directional-index readings.

How directional movement is assigned

Directional movement is the part of a bar that extends beyond the prior bar, classified only as positive, negative, or absent. When a bar extends both above and below the prior bar, the larger extension sets the sign.

If the current bar’s entire range lies inside the prior bar’s range, directional movement is recorded as zero.

Positive directional movement is the current high minus the prior high only when that upward extension exceeds the downward extension of the lows, and is otherwise zero. Negative directional movement uses the symmetric low-side test: the low-to-low extension when it exceeds the high-to-high extension, and otherwise zero.

How true range scales the extensions

Directional movement is expressed relative to true range. True range is the greatest of the current high minus the current low, the current high versus the prior close, and the current low versus the prior close. That is the largest of the current high-low span and the two distances from the prior close to the current high and the current low.

How 14-period totals are started and updated

Single-bar true range and directional-movement values are treated as too volatile to use raw, so the construction first totals 14 periods of each series.

After those initial totals, each 14-period series is updated by subtracting one fourteenth of the previous total and adding the newest observation. The updated total is then truncated so the arithmetic stays close to the original hand-worked form.

How the directional indicators become the average directional index

A directional indicator is an integer percentage that scales a 14-period directional-movement total by the matching 14-period true-range total. The plus and minus directional indicators are 100 times the corresponding 14-period directional-movement total divided by the 14-period true-range total, with the fractional part discarded.

The directional index is 100 times the absolute gap between the two directional indicators divided by their sum, stored as an integer. It is an integer percentage of how uneven the two directional indicators are relative to their combined size.

The average directional index begins as a 14-period mean of that directional-index series. Each later value is thirteen times the prior average plus the newest directional-index value, divided by 14 and converted to an integer.