QuanTAlib/lib/cycles/eacp/eacp.md

EACP: Ehlers Autocorrelation Periodogram

EACP estimates the dominant cycle period of a financial time series by computing autocorrelation across multiple lags and transforming the result into a power spectrum via the Wiener-Khinchin theorem. The output is a continuously updating cycle period measurement (in bars) that can adaptively tune other indicators to the market's current rhythm, making fixed-period assumptions unnecessary.

Historical Context

John Ehlers introduced the Autocorrelation Periodogram to solve the fundamental problem of cycle measurement in noisy financial data. Traditional spectral methods (FFT) assume stationarity and require long data windows, making them impractical for real-time trading. Ehlers leveraged the Wiener-Khinchin theorem, which establishes that a signal's autocorrelation function and its power spectral density form a Fourier transform pair. By computing autocorrelation in the time domain and transforming to frequency via a discrete cosine transform, the algorithm identifies spectral peaks corresponding to dominant periodicities. The center-of-gravity weighting of spectral peaks provides a robust, noise-tolerant period estimate. This enables truly adaptive trading systems where RSI, Stochastic, or moving average periods track the market's actual cycle length rather than relying on fixed parameters.

Architecture & Physics

1. Signal Pre-processing

A high-pass filter removes the DC (trend) component, and a Super-Smoother filter attenuates aliasing noise above the Nyquist frequency:

HP_t = (1 - \alpha_{HP}/2)^2 (P_t - 2P_{t-1} + P_{t-2}) + 2(1 - \alpha_{HP}) HP_{t-1} - (1 - \alpha_{HP})^2 HP_{t-2}

The Super-Smoother then applies a 2-pole Butterworth low-pass to HP_t.

2. Autocorrelation

For each lag k from 0 to MaxPeriod, the normalized Pearson autocorrelation is computed over an averaging window of M samples:

R_k = \frac{\sum_{i=0}^{M-1} (x_i - \bar{x})(x_{i-k} - \bar{x})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (x_{i-k} - \bar{x})^2}}

A high R_k at lag 20 implies a 20-bar cycle is present.

3. Power Spectrum (DFT of Autocorrelation)

For each candidate period p in [MinPeriod, MaxPeriod]:

P_p = \left(\sum_{k=0}^{M-1} R_k \cos\!\left(\frac{2\pi k}{p}\right)\right)^2

Smoothed with exponential decay: S_p = 0.2 \cdot P_p + 0.8 \cdot S_{p,prev}

4. Dominant Cycle Extraction

Center-of-gravity weighting across spectral peaks:

DC = \frac{\sum S_p \cdot p}{\sum S_p}

5. Optional Enhancement

When enhance=true, spectral values are cubed before CG weighting, sharpening peaks but increasing sensitivity to noise.

6. Complexity

O(N \times M) per bar where N is the period range and M is the averaging length. This is one of the most computationally expensive indicators due to nested correlation and DFT loops. Memory is O(N) for correlation and power arrays.

Mathematical Foundation

Parameters

Parameter	Description	Default	Constraint
minPeriod	Minimum period to evaluate	8	\geq 3
maxPeriod	Maximum period to evaluate	48	> minPeriod
enhance	Apply cubic emphasis to spectral peaks	true

Pseudo-code

function EACP(source, minPeriod, maxPeriod, enhance):
    N ← maxPeriod - minPeriod + 1
    M ← maxPeriod        // averaging window
    hpBuf ← HighPassFilter(source)
    ssfBuf ← SuperSmoother(hpBuf)

    power[N] ← {0}
    smoothPower[N] ← {0}

    for each bar:
        // Autocorrelation for each lag
        corr[0..maxPeriod] ← PearsonAutocorrelation(ssfBuf, M)

        // DFT: convert autocorrelation to power spectrum
        for p = minPeriod to maxPeriod:
            cosPower ← 0
            for k = 0 to M-1:
                cosPower += corr[k] * cos(2π * k / p)
            power[p] ← cosPower²

        // Exponential smoothing of spectrum
        for p = minPeriod to maxPeriod:
            smoothPower[p] ← 0.2 * power[p] + 0.8 * smoothPower[p]

        // Optional cubic enhancement
        if enhance:
            for p: smoothPower[p] ← smoothPower[p]³

        // AGC normalization
        maxPow ← max(smoothPower)
        for p: smoothPower[p] /= maxPow   // normalize to [0, 1]

        // Center-of-gravity dominant cycle
        num ← 0; den ← 0
        for p = minPeriod to maxPeriod:
            num += smoothPower[p] * p
            den += smoothPower[p]
        dominantCycle ← (den > 0) ? num / den : (minPeriod + maxPeriod) / 2

        emit dominantCycle

Output Interpretation

Output	Meaning
dominantCycle	Estimated dominant period in bars (use to tune other indicators)
Stable value	Market exhibiting regular cyclical behavior
Rapidly changing value	Market transitioning between regimes
Pegged at maxPeriod	No clear cycle detected; likely trending

Resources

• Ehlers, J.F. Cycle Analytics for Traders. Wiley, 2013.
• Ehlers, J.F. Cybernetic Analysis for Stocks and Futures. Wiley, 2004.
• Wiener, N. "Generalized Harmonic Analysis." Acta Mathematica, 55(1), 1930.
• Khinchin, A. "Korrelationstheorie der stationären stochastischen Prozesse." Mathematische Annalen, 109(1), 1934.