Constructing a decycler from high-pass cancellation

A trend as a longer-period cycle

A price series can be treated as a continuum of cycle lengths with differing amplitudes. An apparent trend is a segment of a longer-period cycle rather than a separate process. That longer-period oscillation is the dominant cycle inside the continuous spectrum of cycle lengths.

Lag from low-pass smoothers

Conventional trend identification uses low-pass smoothers that pass long-period components and suppress shorter fluctuations. Those smoothers add substantial display lag: about half the window length for a simple moving average, or roughly 25 bars on a 50-bar average.

Cancellation construction of the decycler

A high-pass filter passes short-period fluctuations and rejects long-period components, which keeps computational lag between input and output small.

A decycler is built by subtracting the high-pass output from the original series. Shared high-frequency content cancels, while long-period content that never entered the high-pass output remains as a low-lag trend overlay. Cancellation construction builds that trend filter by subtracting two series that share high-frequency content rather than by smoothing the input directly.

The simple decycler has a single cutoff period that should still pass as low-frequency content. Larger settings, on the order of a half-year lookback with a supplied default of 125 bars, are the intended range for trend work.

Hysteresis bands

A hysteresis band is a pair of percentage envelopes around the decycler. Bands at plus and minus 0.5 percent classify the series as an uptrend above the upper band, a downtrend below the lower band, and trend-neutral inside the band.

The decycler oscillator

A further high-pass applied to the decycler removes residual very-long-period content that still contributes lag and yields a scaled oscillator. That second high-pass is the decycler oscillator.

A two-line oscillator setup uses one instance at the chosen high-pass period with amplitude scale 1 and a second instance at 80 percent of that period with amplitude scale 1.2, illustrated as 125 versus 100. The shorter line crossing above the longer line is read as an upside reversal, and crossing below is read as a downside reversal.

The high-pass coefficient

The high-pass coefficient is computed from cosine and sine of 0.707 times 360 divided by the cutoff period. It is then applied as a second-order recursion on successive closes, or on the decycler itself at the oscillator stage.