Bench-testing smoothers with cycle phase and power

Cycle length, frequency, and phase

Cycle length is the number of ordered observations that complete one full oscillation. Frequency is how fast the cycle advances. It is the reciprocal of cycle length, and it can also be written as degrees per sampling interval. A 64-observation cycle has frequency 1/64, and a 16-observation cycle has frequency 1/16.

Angular advance equals 360 degrees divided by cycle length. A 64-period cycle therefore moves 5.625 degrees per sampling interval. Phase is measured in degrees, with 360 degrees completing one cycle, and it describes whether one series leads or lags another.

Exponential smoothing on two known cycles

Exponential smoothing is a recursive average of ordered observations that can delay turning points and reduce amplitude in a frequency-dependent way. The dominant cycle is the oscillation length a model is meant to retain when a filter is applied to price, volume, or breadth.

A five-period exponential moving average of a 64-period sine wave lags that wave by about 11 to 12 degrees and reduces its amplitude only slightly. The same five-period exponential moving average applied to a 16-period cycle at 22.5 degrees per interval lags by about 33 degrees and reduces amplitude more.

First difference advances phase

The first difference is the change between consecutive observations, which advances phase instead of delaying it. The first difference of a 64-period sine wave shifts the plot leftward, introducing phase lead rather than lag.

Residual power at each frequency

Power equals amplitude squared and is used to quantify how much of a cycle a filter still passes. Spectral analysis measures retained power and phase at each frequency so a filter's effect on a known cycle can be checked.

On a 5.625-degree-per-interval cycle a five-period exponential smoother retains about 95% of power, leaving amplitude near 0.97. On a 22.5-degree-per-interval cycle the same five-period exponential smoother retains about 0.52 of power, matching a peak amplitude near 0.72. A moving-average filter attenuates some frequencies more than others, so a construction check is whether the intended dominant cycle keeps most of its amplitude while other components are reduced.