--- title: "Autocorrelation Periodogram: Formula" description: "The autocorrelation periodogram, popularized by John Ehlers, computes the autocorrelation of a filtered price series at many lags and then takes a discrete..." canonical: https://www.getswoopr.com/indicators/cycle-and-signal-processing/autocorrelation-periodogram/ source: "Swoopr Investment: https://www.getswoopr.com" --- ## Direct Answer > The autocorrelation periodogram, popularized by John Ehlers, computes the autocorrelation of a filtered price series at many lags and then takes a discrete Fourier transform of those correlations. The resulting power at each period shows which cycle length dominates. The period with the highest power is the dominant cycle estimate. # Autocorrelation Periodogram: Formula, Meaning, Signals, Examples and How to Use It ## How Is Autocorrelation Periodogram Calculated? The autocorrelation periodogram, popularized by John Ehlers, computes the autocorrelation of a filtered price series at many lags and then takes a discrete Fourier transform of those correlations. The resulting power at each period shows which cycle length dominates. **Formula:** `R(lag) = Pearson correlation of the series with itself shifted by lag; Power(P) = (sum over lag of R(lag)*cos(2*pi*lag/P))^2 + (sum over lag of R(lag)*sin(2*pi*lag/P))^2` **Inputs:** - Close prices, high-pass and smoothed - Maximum lag - Range of candidate periods **Worked example.** For an ideal sine wave with a 20-bar period, the autocorrelation at a lag of 10 bars is about -1.0. The inputs are illustrative numbers, not market data. **Reading it.** The period with the highest power is the dominant cycle estimate. A sharp peak means a clear cycle, a flat or broad spectrum means none. Heat-map displays show how the dominant period shifts over time. **Limits.** Resolution at long periods is limited by the number of lags used. The autocorrelation estimate is noisy for short windows, so dominant peaks can shift between bars. ## What is Autocorrelation Periodogram? The autocorrelation periodogram, popularized by John Ehlers, computes the autocorrelation of a filtered price series at many lags and then takes a discrete Fourier transform of those correlations. The resulting power at each period shows which cycle length dominates. ## How to use Autocorrelation Periodogram The period with the highest power is the dominant cycle estimate. A sharp peak means a clear cycle, a flat or broad spectrum means none. Heat-map displays show how the dominant period shifts over time. A reading alone is never a buy or sell signal; it needs context from price structure, volume, volatility and the broader market regime. ## Strengths - Correlating the series with lagged copies of itself makes the input normalized, so amplitude does not drive the spectrum. - Evaluating power at each candidate period yields a full profile of cycle lengths, not one winner, so competing cycles are visible. - Reproducible when the formula and data convention are fixed. ## Weaknesses and limitations - Each lag's correlation is estimated from fewer overlapping samples as the lag grows, so long-lag values are the least reliable. - The power surface is computed anew each bar, so reading it over time needs extra smoothing or selection to avoid a jumpy dominant period. - Correlated indicators can create false confidence by repeating the same underlying information. ## Editorial and risk note This page is educational content, not individualized investment advice. It should not imply guaranteed prediction, accuracy or outperformance. ## Frequently Asked Questions Is Autocorrelation Periodogram a buy or sell signal? No. The period with the highest power is the dominant cycle estimate. Resolution at long periods is limited by the number of lags used. A trade built on it still needs a trigger, an invalidation rule, a position size and realistic execution assumptions. What is the best setting for Autocorrelation Periodogram? Autocorrelation Periodogram takes these inputs: close prices, high-pass and smoothed; maximum lag; range of candidate periods. There is no universal best setting for the tunable ones, so test them across instruments, regimes and out-of-sample periods and favor stable parameter regions over one historical winner. Can Autocorrelation Periodogram be used by itself? Resolution at long periods is limited by the number of lags used. The autocorrelation estimate is noisy for short windows, so dominant peaks can shift between bars. For that reason it is read next to measures of direction, participation, valuation, liquidity and risk, not alone. Does Autocorrelation Periodogram work on every timeframe? The calculation (R(lag) = Pearson correlation of the series with itself shifted by lag; Power(P) = (sum over lag of R(lag)*cos(2*pi*lag/P))^2 + (sum over lag of R(lag)*sin(2*pi*lag/P))^2) can be run on any bar size, but behavior is not portable: a reading on a five-minute chart describes a very different market window from the same reading on a daily chart. Validate the exact timeframe before relying on it. Why does Autocorrelation Periodogram give false signals? Each lag's correlation is estimated from fewer overlapping samples as the lag grows, so long-lag values are the least reliable. The power surface is computed anew each bar, so reading it over time needs extra smoothing or selection to avoid a jumpy dominant period. Lag, noise and regime changes add to this, because market participants react to new information after the reading is calculated. How should Autocorrelation Periodogram be backtested? Rebuild Autocorrelation Periodogram from close prices, high-pass and smoothed using point-in-time data, write the rules before testing, include delisted securities where relevant, model realistic fills and costs, reserve a validation sample and compare with a simpler baseline. S Swoopr Editorial Team The Swoopr Editorial Team produces independent investment education, research, and tools for understanding markets, evaluating opportunities, and managing risk. All content is educational only and does not constitute personalized investment advice. See our [editorial policy](https://www.getswoopr.com/legal/editorial-policy/) and [corrections policy](https://www.getswoopr.com/legal/corrections-policy/).