Constructing a repeated-median slope as a resistant trend filter

What a velocity system waits for

A velocity system issues trades only after the slope of a line fitted through N past prices crosses an explicit threshold. That threshold is the velocity barrier. The trend filter uses the barrier as a gate, so that signals are withheld while prices meander without a notable trend.

Why one bad print can change least-squares velocity

Ordinary least-squares slope is obtained by minimizing the sum of squared residuals between N prices and the line a + b*t. That estimator is the least-squares velocity. Its breakdown point is 1/N. The breakdown point is the smallest fraction of corrupted observations that can pull a regression coefficient far from the value implied by the uncontaminated data. Because the fraction is 1/N, a single bad observation can change the computed velocity.

Breakdown points for the median and least-absolute-deviation

The sample median of a set of numbers has a breakdown point of 50%, which is the highest breakdown point available. Once half the observations are corrupted, the good and bad values cannot be distinguished. Least-absolute-deviation regression minimizes the sum of absolute residuals rather than squared residuals. That fit has a breakdown point of 29.8%, so roughly one quarter of the price points can be bad before the fitted intercept and slope become erroneous.

How the repeated-median slope is assembled

A 1982 robust-regression construction for slope attains a 50% breakdown point. For each observation it takes the pairwise slopes through that point, reduces the point to the median of those pairwise slopes, and then takes the median of those pointwise medians. The resulting coefficient is the repeated-median slope.

What a 15-point demonstration shows

In a 15-point demonstration with two contaminated Y values at X positions 10 and 14, the first-stage pairwise slopes from the first point have a median of 1. That reading matches the uncontaminated unit slope. The same demonstration shows the contaminated points pulling the least-squares line toward the outliers, while the repeated-median line remains aligned with the uncontaminated trend.

Further nested-median passes on market prices

Even when the first pairwise-median calculation already recovers the correct slope, further nested-median passes are still required on market prices. Those series are not as regularly spaced as the textbook example.

Named methods kept in view

The moving average is a smoothing of ordered price, volume, or breadth observations over a defined sampling interval, and it belongs to the same construction family as the slope filters discussed here. The relative-strength index is a bounded oscillator built from ordered price observations over a defined lookback. It is retained here as a named method even though this installment concentrates on slope construction rather than oscillator formulas.