Variable lookback and average true range as a trend-filter construction

A single lookback as a standing assumption

Many smoothing tools and 0-to-100 oscillators lock one lookback for every market on the assumption that a persistent cycle, such as a half-cycle of a 28-day rhythm, is always present.

First-difference autocorrelations did not supply a cycle

Daily first-difference autocorrelations were computed for eight futures and two equities, at lags from one through 40, on samples of about one thousand closing days. First-difference autocorrelation is the correlation of one-day price changes with the same series lagged by one through many sessions.

With a standard error near 0.03, only isolated lags exceeded a three-error significance cutoff, and those readings remained too small to treat as a usable predictive link. The long-sample autocorrelation maps stayed near zero, so a persistent short-term cycle was not available as a reason to keep one lookback across markets and eras. A twelve-month lag was noted as a separate seasonal case for some commodities.

Channel height against a random-walk yardstick

Average n-day channel height divided by one-day range, measured over 1,121 days, ran above the square root of n. The construction treats that extra displacement as movement beyond a pure random walk, and therefore as a trend yardstick. The channel-height ratio is that average n-day high-low span divided by the one-day average span, compared with the square root of n.

A random-walk index scaled by average true range

The rebuilt trend filter is a constructed reading that asks whether the latest high or low excursion is larger than a random-walk benchmark over a chosen lookback. The high-side and low-side indexes divide the current extreme versus the opposite extreme n days ago by average true range times the square root of n. That ratio is the random-walk index. A reading above 1 is classified as larger than the random-walk expectation.

Average true range is the volatility scale in the denominator, so a move must clear a range-based bound before it is treated as non-random.

Lookback taken from the current swing

Instead of locking one interval in advance, the construction evaluates many lookback lengths and keeps the largest index. Each session's working reading is the largest index across that set, so the sampling interval is taken from the most significant current swing rather than from a preset cycle length.

The histogram of lookbacks that produce that maximum bends between seven and eight days, which the construction uses as a split between short-horizon and longer-horizon trend readings.

True range keeps a gapped bar from shrinking the scale

True range is the greater of the bar's high-low span and the gap from the prior close to today's extreme. Average true range, built from that true-range input, is the volatility figure that keeps a gapped, narrow bar from shrinking the denominator used to scale the filter.