Unit-invariant trend filters and the c-test

A pretest before historical comparison

A lookback trend filter turns an ordered price series into a forecast of relative trend strength or direction over a defined sampling interval. It can be rejected on methodological grounds, before any statistical trial, if a dimensional-coherency check applied to its formulas shows that the forecast changes when only the units of measure change.

Dimensional-coherency is the requirement that the forecast not depend on an arbitrary choice of price unit or an independently restated time unit. Price-time-heterogeneity is the reason that check is not automatic: price units and time units need not be rescaled together, so a valid rule must survive changing either axis on its own.

Price and time are not interchangeable

Price and time are heterogeneous inputs. A valid rule must keep the same meaning when a price increment is restated in another unit over the same sampling interval.

Angle rules fail the invariance check

Angle-based trend rules fail that invariance. The same ordered prices produce different right-angle or inclination readings when only the vertical price scale is altered.

A rule that the second leg of a rise must incline at twice the first-leg angle is incoherent, because a price-axis rescale can push the first leg far enough that the required second-leg angle would reverse time.

Connecting observed prices on given dates is an incidence construction, not an angle filter. The crossing price and date can stay fixed while chart angles change with the price axis.

Mixed efficiency scores can reverse order

Mixing price increments with elapsed-time terms in one lookback efficiency score can invert the ranking of two three-observation series after every price is multiplied by the same positive constant used in the c-test. Readings that looked nearly tied can switch order once only the price unit changes.

How the c-test is applied

The c-test is a formula-level screen. The coherency procedure multiplies every price term by a positive constant other than one and leaves nonprice terms unchanged. The filter passes only if its indications remain the same.

n-velocity is net price change over a lookback of n observations, divided by n minus one. Under the c-test that raw level is itself rescaled with price, so it is not a coherent standalone forecast.

The ratio of two such velocities is invariant. Dividing the velocity by current price also leaves the indication unchanged. Those two rewrites are the velocity-ratio and the percent-of-price-velocity.

What still fails after a pass

A velocity-ratio is a dimensionally consistent comparison of relative trend strength, but it becomes erratic when the denominator is near zero. A percent-of-price-velocity passes the unit check yet still has different typical ranges across futures contracts.