Engineering spectral cycle detection and moving-average harmony

Start from a no-cycle null

TradersWeek editorial framing: treat market-cycle claims as a falsifiable engineering problem. Start from a null that prices contain no cycles, extract candidate periods with a filtered spectrum, and only then ask whether those same periods reappear.

Common technical methods for locating cycles include descriptive statistics, visual chart reading, trigonometric constructions such as centered simple moving averages, and traditional oscillators. Published cycle-duration tables from different analysts do not agree on a single set of market periods.

Extract candidate periods with a filtered spectrum

A spectrum analyzer with high-pass detrending, low-pass smoothing, and bandpass period selection can be used to test a null that market prices contain no cycles. Spectral analysis, in this usage, is a filtered frequency scan of ordered market observations that maps how much energy sits at each candidate period over time.

The dominant cycle is the period that currently carries the strongest spectral energy, which can weaken, drift, or be replaced by another period. On S&P 500 weekly data from October 2009 to September 2010, the scan showed a fairly stable 16-to-17-week period, which corresponds to an 80-to-85-day cycle on daily bars.

Retest the same periods on other intervals and indexes

The same weekly scan on Toronto Stock Exchange data also showed a 16-week period, while a larger period near 22 weeks became stronger than the 16-week period at the beginning of August 2010.

Detected cycles are not stationary: a dominant period can weaken, be replaced, and fluctuate, as when the Toronto weekly period moved between 18 and 15 weeks.

Translate cycle fractions into moving-average windows

A weaker 106-week period can appear below a 160-week lookback, and half and one-eighth of a longer cycle were plotted as 53-week and 13-week simple moving averages on S&P 500 weekly prices from 2009 to September 2010.

A moving average here is a centered or simple average whose lookback is taken from a detected cycle half-period or related fraction, used as a baseline overlay rather than a standalone rule. Those 53-week and 13-week windows translate into roughly the familiar 200-day and 50-day simple moving averages, offering a cycle-based reason those two averages are commonly paired.

Read the result as a rejected null, not a proof

The exercise does not prove that cycles exist, but it does reject a strict claim that market prices contain no cycles, while treating sinusoidal fits as expandable, contractible, and time-varying in amplitude.