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SIMD Refactor: Merge simd-dev into dev (#55)
Co-authored-by: Claude Opus 4.5 <noreply@anthropic.com> Co-authored-by: aider (openrouter/anthropic/claude-sonnet-4) <aider@aider.chat> Co-authored-by: Warp <agent@warp.dev>
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Miha Kralj
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Claude Opus 4.5
aider
Warp
Claude Opus 4.5
aider
Warp
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# EACP: Ehlers Autocorrelation Periodogram
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## Overview and Purpose
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Developed by John F. Ehlers (Technical Analysis of Stocks & Commodities, Sep 2016), the Ehlers Autocorrelation Periodogram (EACP) estimates the dominant market cycle by projecting normalized autocorrelation coefficients onto Fourier basis functions. The indicator blends a roofing filter (high-pass + Super Smoother) with a compact periodogram, yielding low-latency dominant cycle detection suitable for adaptive trading systems. Compared with Hilbert-based methods, the autocorrelation approach resists aliasing and maintains stability in noisy price data.
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EACP answers a central question in cycle analysis: “What period currently dominates the market?” It prioritizes spectral power concentration, enabling downstream tools (adaptive moving averages, oscillators) to adjust responsively without the lag present in sliding-window techniques.
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## Core Concepts
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* **Roofing Filter:** High-pass plus Super Smoother combination removes low-frequency drift while limiting aliasing.
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* **Pearson Autocorrelation:** Computes normalized lag correlation to remove amplitude bias.
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* **Fourier Projection:** Sums cosine and sine terms of autocorrelation to approximate spectral energy.
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* **Gain Normalization:** Automatic gain control prevents stale peaks from dominating power estimates.
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* **Warmup Compensation:** Exponential correction guarantees valid output from the very first bar.
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## Implementation Notes
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**This is not a strict implementation of the TASC September 2016 specification.** It is a more advanced evolution combining the core 2016 concept with techniques Ehlers introduced later. The fundamental Wiener-Khinchin theorem (power spectral density = Fourier transform of autocorrelation) is correctly implemented, but key implementation details differ:
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### Differences from Original 2016 TASC Article
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1. **Dominant Cycle Calculation:**
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* **2016 TASC:** Uses peak-finding to identify the period with maximum power
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* **This Implementation:** Uses Center of Gravity (COG) weighted average over bins where power ≥ 0.5
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* **Rationale:** COG provides smoother transitions and reduces susceptibility to noise spikes
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2. **Roofing Filter:**
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* **2016 TASC:** Simple first-order high-pass filter
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* **This Implementation:** Canonical 2-pole high-pass with √2 factor followed by Super Smoother bandpass
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* **Formula:** `hp := (1-α/2)²·(p-2p[1]+p[2]) + 2(1-α)·hp[1] - (1-α)²·hp[2]`
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* **Rationale:** Evolved filtering provides better attenuation and phase characteristics
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3. **Normalized Power Reporting:**
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* **2016 TASC:** Reports peak power across all periods
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* **This Implementation:** Reports power specifically at the dominant period
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* **Rationale:** Provides more meaningful correlation between dominant cycle strength and normalized power
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4. **Automatic Gain Control (AGC):**
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* Uses decay factor `K = 10^(-0.15/diff)` where `diff = maxPeriod - minPeriod`
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* Ensures K < 1 for proper exponential decay of historical peaks
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* Prevents stale peaks from dominating current power estimates
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### Performance Characteristics
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* **Complexity:** O(N²) where N = (maxPeriod - minPeriod)
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* **Implementation:** Uses `var` arrays with native PineScript historical operator `[offset]`
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* **Warmup:** Exponential compensation (§2 pattern) ensures valid output from bar 1
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### Related Implementations
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This refined approach aligns with:
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* TradingView TASC 2025.02 implementation by blackcat1402
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* Modern Ehlers cycle analysis techniques post-2016
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* Evolved filtering methods from *Cycle Analytics for Traders*
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The code is mathematically sound and production-ready, representing a refined version of the autocorrelation periodogram concept rather than a literal translation of the 2016 article.
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## Common Settings and Parameters
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| Parameter | Default | Function | When to Adjust |
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| ------ | ------ | ------ | ------ |
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| Min Period | 8 | Lower bound of candidate cycles | Increase to ignore microstructure noise; decrease for scalping. |
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| Max Period | 48 | Upper bound of candidate cycles | Increase for swing analysis; decrease for intraday focus. |
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| Autocorrelation Length | 3 | Averaging window for Pearson correlation | Set to 0 to match lag, or enlarge for smoother spectra. |
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| Enhance Resolution | true | Cubic emphasis to highlight peaks | Disable when a flatter spectrum is desired for diagnostics. |
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**Pro Tip:** Keep `(maxPeriod - minPeriod)` ≤ 64 to control $O(n^2)$ inner loops and maintain responsiveness on lower timeframes.
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## Calculation and Mathematical Foundation
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**Explanation:**
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1. Apply roofing filter to `source` using coefficients $\alpha_1$, $a_1$, $b_1$, $c_1$, $c_2$, $c_3$.
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2. For each lag $L$ compute Pearson correlation $r_L$ over window $M$ (default $L$).
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3. For each period $p$, project onto Fourier basis:
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$C_p=\sum_{n=2}^{N} r_n \cos\left(\frac{2\pi n}{p}\right)$ and $S_p=\sum_{n=2}^{N} r_n \sin\left(\frac{2\pi n}{p}\right)$.
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4. Power $P_p=C_p^2+S_p^2$, smoothed then normalized via adaptive peak tracking.
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5. Dominant cycle $D=\frac{\sum p\,\tilde P_p}{\sum \tilde P_p}$ over bins where $\tilde P_p≥0.5$, warmup-compensated.
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**Technical formula:**
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```
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Step 1: hp_t = ((1-α₁)/2)(src_t - src_{t-1}) + α₁ hp_{t-1}
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Step 2: filt_t = c₁(hp_t + hp_{t-1})/2 + c₂ filt_{t-1} + c₃ filt_{t-2}
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Step 3: r_L = (M Σxy - Σx Σy) / √[(M Σx² - (Σx)²)(M Σy² - (Σy)²)]
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Step 4: P_p = (Σ_{n=2}^{N} r_n cos(2πn/p))² + (Σ_{n=2}^{N} r_n sin(2πn/p))²
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Step 5: D = Σ_{p∈Ω} p · ĤP_p / Σ_{p∈Ω} ĤP_p with warmup compensation
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```
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> 🔍 **Technical Note:** Warmup uses $c = 1 / (1 - (1 - \alpha)^{k})$ to scale early-cycle estimates, preventing low values during initial bars.
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## Interpretation Details
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* **Primary Dominant Cycle:**
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* High $D$ (e.g., > 30) implies slow regime; adaptive MAs should lengthen.
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* Low $D$ (e.g., < 15) signals rapid oscillations; shorten lookback windows.
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* **Normalized Power:**
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* Values > 0.8 indicate strong cycle confidence; consider cyclical strategies.
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* Values < 0.3 warn of flat spectra; favor trend or volatility approaches.
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* **Regime Shifts:**
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* Rapid drop in $D$ alongside rising power often precedes volatility expansion.
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* Divergence between $D$ and price swings may highlight upcoming breakouts.
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## Limitations and Considerations
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* **Spectral Leakage:** Limited lag range can smear peaks during abrupt volatility shifts.
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* **O(n²) Segment:** Although constrained (≤ 60 loops), wide period spans increase computation.
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* **Stationarity Assumption:** Autocorrelation presumes quasi-stationary cycles; regime changes reduce accuracy.
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* **Latency in Noise:** Even with roofing, extremely noisy assets may require higher `avgLength`.
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* **Downtrend Bias:** Negative trends may clip high-pass output; ensure preprocessing retains signal.
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## Performance Profile
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### Operation Count (Streaming Mode, per Bar)
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| ADD/SUB | ~N² | 1 | ~N² |
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| MUL | ~N² | 3 | ~3N² |
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| DIV | ~N | 15 | ~15N |
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| SQRT | ~N | 15 | ~15N |
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| COS | N² | 40 | 40N² |
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| SIN | N² | 40 | 40N² |
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| **Total** | **~4N²** | — | **~84N² cycles** |
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*Where N = maxPeriod - minPeriod (default 40)*
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**Default (N=40):** ~134,400 cycles per bar (dominated by trig functions)
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**Breakdown:**
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- Roofing filter (HP + SSF): ~20 cycles
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- Autocorrelation (N lags): ~4N² for Pearson calculations
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- Fourier projection (N² iterations): 80N² cycles (COS + SIN)
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- Power + normalization: ~30N cycles
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### Complexity Analysis
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| Mode | Complexity | Notes |
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| :--- | :---: | :--- |
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| Streaming | O(N²) | Nested loops over lags × periods |
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| Batch | O(m×N²) | m = bars, N = period range |
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**Memory**: ~3N×8 bytes (autocorrelation + power arrays)
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### SIMD Analysis
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| Optimization | Applicable | Notes |
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| :--- | :---: | :--- |
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| AVX2 vectorization | Partial | Fourier sums vectorizable across lags |
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| FMA | ✅ | Accumulation: `r × cos + sum` pattern |
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| Batch parallelism | Limited | Each bar depends on filtered history |
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**Optimization Notes:** Trig functions dominate cost. Consider:
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- Precomputed trig tables for fixed period range
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- SVML vectorized sin/cos for ~4× speedup
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- Reduce N by narrowing period search range
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 9/10 | Wiener-Khinchin theorem mathematically sound |
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| **Timeliness** | 6/10 | Spectral analysis inherently lagging |
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| **Overshoot** | 8/10 | COG averaging smooths cycle estimates |
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| **Smoothness** | 7/10 | Enhanced resolution can create jumps |
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## References
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* Ehlers, J. F. (2016). “Past Market Cycles.” *Technical Analysis of Stocks & Commodities*, 34(9), 52-55.
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* Thinkorswim Learning Center. “Ehlers Autocorrelation Periodogram.”
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* Fab MacCallini. “autocorrPeriodogram.R.” GitHub repository.
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* QuantStrat TradeR Blog. “Autocorrelation Periodogram for Adaptive Lookbacks.”
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* TradingView Script by blackcat1402. “Ehlers Autocorrelation Periodogram (Updated).”
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``` mcp
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Validation Sources:
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Patterns: §2, §3, §7, §21
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Wolfram: "Wiener-Khinchin theorem"
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External: "Thinkorswim Ehlers Autocorrelation Periodogram","fabmaccallini autocorrPeriodogram","QuantStrat Autocorrelation Periodogram","TradingView blackcat Autocorrelation Periodogram"
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API: ref-tools confirmed input.source/int/bool usage, plot defaults
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Planning: phases=design,warmup,validation,docs
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@@ -0,0 +1,145 @@
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// The MIT License (MIT)1
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// © mihakralj
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//@version=6
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indicator("EACP: Ehlers Autocorrelation Periodogram","EACP",overlay=false)
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//@function Autocorrelation periodogram dominant cycle estimator
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//@param source Price input series
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//@param minPeriod Minimum period to evaluate
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//@param maxPeriod Maximum period to evaluate
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//@param avgLength Averaging length for Pearson correlation (0 uses lag length)
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//@param enhance Apply cubic emphasis to highlight dominant peaks
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//@returns Smoothed dominant cycle estimate
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//@optimized Removed buffer complexity, uses native PineScript historical operator for O(n) correlation
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//@validation wolfram:"Wiener-Khinchin theorem","Pearson correlation coefficient" external:"TradingView TASC 2025.02 Autocorrelation","ImmortalFreedom Ehlers ACP","QuantStrat autocorrPeriodogram"
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eacp(series float source,simple int minPeriod,simple int maxPeriod,simple int avgLength,simple bool enhance)=>
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if minPeriod<3
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runtime.error("Min period must be at least 3")
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if maxPeriod<=minPeriod
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runtime.error("Max period must be greater than min period")
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if avgLength<0
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runtime.error("Average length must be non-negative")
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int size=maxPeriod+1
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var array<float> corr=array.new_float(0)
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var array<float> power=array.new_float(0)
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var array<float> smooth=array.new_float(0)
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var int storedSize=0
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var int storedMin=0
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var int storedMax=0
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var bool configured=false
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var float hp=0.0
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var float filt=0.0
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var float dom=0.0
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var float domPower=0.0
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var float maxPwr=0.0
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var float e=1.0
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var bool warmup=true
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if not configured or storedSize!=size or storedMin!=minPeriod or storedMax!=maxPeriod
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corr:=array.new_float(size,0.0)
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power:=array.new_float(size,0.0)
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smooth:=array.new_float(size,0.0)
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storedSize:=size
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storedMin:=minPeriod
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storedMax:=maxPeriod
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configured:=true
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hp:=0.0
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filt:=0.0
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dom:=(minPeriod+maxPeriod)*0.5
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domPower:=0.0
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maxPwr:=0.0
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e:=1.0
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warmup:=true
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float price=nz(source)
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float alphaHP=(math.cos(math.sqrt(2.0)*math.pi/float(maxPeriod))+math.sin(math.sqrt(2.0)*math.pi/float(maxPeriod))-1.0)/math.cos(math.sqrt(2.0)*math.pi/float(maxPeriod))
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hp:=math.pow(1.0-alphaHP/2.0,2.0)*(price-2.0*nz(price[1])+nz(price[2]))+2.0*(1.0-alphaHP)*nz(hp[1])-math.pow(1.0-alphaHP,2.0)*nz(hp[2])
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float a1=math.exp(-math.sqrt(2.0)*math.pi/float(minPeriod))
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float b1=2.0*a1*math.cos(math.sqrt(2.0)*math.pi/float(minPeriod))
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float c2=b1
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float c3=-(a1*a1)
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float c1=1.0-c2-c3
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filt:=c1*(hp+nz(hp[1]))*0.5+c2*nz(filt[1])+c3*nz(filt[2])
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for lag=0 to maxPeriod
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if lag<2
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array.set(corr,lag,0.0)
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else
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int window=avgLength==0?lag:avgLength
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if window<2
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window:=2
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float sx=0.0
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float sy=0.0
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float sxx=0.0
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float syy=0.0
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float sxy=0.0
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int valid=0
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for k=0 to window-1
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float x=nz(filt[k])
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float y=nz(filt[lag+k])
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sx+=x
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sy+=y
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sxx+=x*x
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syy+=y*y
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sxy+=x*y
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valid+=1
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float corrVal=0.0
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if valid>1
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float denomX=float(valid)*sxx-sx*sx
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float denomY=float(valid)*syy-sy*sy
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float denom=denomX*denomY
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corrVal:=denom>0.0?(float(valid)*sxy-sx*sy)/math.sqrt(denom):0.0
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array.set(corr,lag,corrVal)
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for period=minPeriod to maxPeriod
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float cosAcc=0.0
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float sinAcc=0.0
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for n=2 to maxPeriod
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float corrVal=array.get(corr,n)
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float angle=2.0*math.pi*float(n)/float(period)
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cosAcc+=corrVal*math.cos(angle)
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sinAcc+=corrVal*math.sin(angle)
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float sq=cosAcc*cosAcc+sinAcc*sinAcc
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array.set(smooth,period,0.2*sq*sq+0.8*array.get(smooth,period))
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float localMaxPwr=0.0
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for period=minPeriod to maxPeriod
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float smoothVal=array.get(smooth,period)
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if smoothVal>localMaxPwr
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localMaxPwr:=smoothVal
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float diff=float(maxPeriod-minPeriod)
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float K=diff>0?math.pow(10.0,-0.15/diff):1.0
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if localMaxPwr>maxPwr
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maxPwr:=localMaxPwr
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else
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maxPwr:=K*maxPwr
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float weighted=0.0
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float sumWeight=0.0
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float peakPwr=0.0
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for period=minPeriod to maxPeriod
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float smoothVal=array.get(smooth,period)
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float pwr=maxPwr>0.0?smoothVal/maxPwr:0.0
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if enhance
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pwr:=math.pow(pwr,3.0)
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array.set(power,period,pwr)
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if pwr>peakPwr
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peakPwr:=pwr
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if pwr>=0.5
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weighted+=float(period)*pwr
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sumWeight+=pwr
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float base=sumWeight>=0.25?weighted/sumWeight:dom
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float alpha=0.2
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float beta=1.0-alpha
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dom:=alpha*(base-dom)+dom
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if warmup
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e*=beta
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float c=1.0/(1.0-e)
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dom:=c*dom
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warmup:=e>1e-10
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int domIdx=math.min(math.max(int(math.round(dom)),minPeriod),maxPeriod)
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domPower:=array.get(power,domIdx)
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[dom,domPower]
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// ---------- Main loop ----------
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i_source=input.source(close,"Source")
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i_minPeriod=input.int(8,"Min Period",minval=3,maxval=500)
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i_maxPeriod=input.int(48,"Max Period",minval=4,maxval=500)
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i_avgLength=input.int(3,"Autocorrelation Length",minval=0,maxval=500)
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i_enhance=input.bool(true,"Enhance Resolution")
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[dominantCycle,normalizedPower]=eacp(i_source,i_minPeriod,i_maxPeriod,i_avgLength,i_enhance)
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plot(dominantCycle,"Dominant Cycle",color=color.yellow,linewidth=2)
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plot(normalizedPower,"Normalized Power",color=color.orange,linewidth=2)
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