A recursive median as shared state for trend rules

When averages absorb an isolated print

A mechanical-trading-system is a fully specified calculation path from raw prints through a short median, a recursive smooth, and an optional highpass. Every rule reads one output. Isolated extreme price or volume prints can distort the averages that path would otherwise treat as market state. A median window can ignore those extremes instead of folding them into the output.

How the recursive median is specified

A recursive-median-filter takes the median of a short lookback and then exponentially blends that median with the previous filter output. For trading, that can be specified as an exponential moving average of a five-bar median, so each new output reuses the prior output. The exponential blending constant lies between zero and one. It can be set from a critical-period with a cosine-sine identity, or approximated as five divided by that period. The critical-period is the design length that separates wavelengths the filter is meant to pass from those it is meant to reject.

Why the median is not an ordinary smoother

A median operator is nonlinear and is not a convolution, so its output has no Fourier representation, is not differentiable, and has no Taylor expansion. Those properties block ordinary polynomial curve-fitting of the filtered series. Repeating the median operation rapidly converges to a reduced series that matches the input except for computational lag. Impulsive-noise is the one-off extreme price or volume print that would dominate an average if it were blended into the filter instead of discarded.

An oscillator the rules can share

An oscillator-type rule input can be formed by sending the recursive median through a second-order highpass that removes the constant level and very long wavelengths, including long components associated with a fractal, pink-noise spectrum of market data. The result is a recursive-median-oscillator that can swing around zero as a rule input. The second-order highpass shortens its critical-period by about 70 percent relative to an exponential moving average of the same design period.

On a 2017 calendar-year equity-index example, a recursive-median-oscillator with a 40-bar highpass and 16-bar smoothing was compared with a 14-bar relative-strength calculation also smoothed at 16 bars and rescaled to swing from minus one to plus one. The median oscillator showed less lag and a faster response to large price moves.

Entry, stay-in, and abstention as one procedure

Trend-following is a signal procedure that uses shared rule inputs, market state, and execution constraints over the system holding period so the larger directional move, not an isolated extreme, governs whether the system is in, out, or flat. Rule-based-entry is a predefined, testable condition on that shared filtered state. The same condition family is used to allow a new position, keep it, or refuse it. When prints contain impulsive fluctuations, discarding outliers on this recursive-median path is specified as a way to smooth with limited lag so a trend procedure can read the larger move as shared state for entry, exit, and abstention.