QuanTAlib/lib/cycles/acp/acp.md

ACP: Ehlers Autocorrelation Periodogram

Autocorrelation periodogram scans every possible cycle length and ranks them by strength — a spectral fingerprint of the market.

Property	Value
Category	Cycle
Inputs	Source (close)
Parameters	minPeriod (default 8), maxPeriod (default 48), avgLength (default 3), enhance (default true)
Outputs	Single series (Acp)
Output range	Varies (see docs)
Warmup	maxPeriod * 2 bars
PineScript	acp.pine

• ACP 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.
• Similar: CG, HT_DCPeriod | Complementary: EBSW for trend/cycle classification | Trading note: Ehlers Autocorrelation Periodogram; identifies dominant cycle length adaptively.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

ACP 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

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

Performance Profile

Operation Count (Streaming Mode)

Operation	Count per bar	Notes
HP filter (2-pole IIR)	~8	Pre-processing trend removal
Super-Smoother (2-pole IIR)	~6	Anti-aliasing low-pass
Pearson autocorrelation	~5M	Mean, variance, cross-product over M samples per lag
Autocorrelation loop (N lags)	~5NM	Nested: N lags × M-sample windows
DFT cosine transform	~3NM	N periods × M cosine multiply-accumulates
Cosine evaluation	NM	Math.Cos calls (expensive transcendental)
Exponential smoothing	~2N	FMA per period bin
Cubic enhancement	~2N	Two multiplies per bin (when enabled)
AGC normalization	~2N	Max scan + N divides
Center-of-gravity	~3N	Weighted sum + division
Total (default N=41, M=48)	~16,000	Dominated by autocorrelation + DFT

Batch Mode (SIMD Analysis)

Aspect	Assessment
SIMD vectorizable	Partially: inner DFT cosine loops vectorizable; autocorrelation outer loop sequential
Bottleneck	Pearson autocorrelation: N×M multiply-accumulates with data-dependent means
Parallelism	DFT accumulation per period is independent; Vector<double> applicable to inner sums
Memory	O(N) power arrays + O(M) circular buffer for SSF history
Throughput	~100-200× slower than O(1) IIR indicators; most expensive cycle indicator

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.