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Miha Kralj
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TASC Sep 2024. Dual 2-pole Butterworth highpass bandpass for near-zero-lag trend extraction. HP(long) - HP(short) preserves cycles between shortPeriod and longPeriod. - Core: Pta.cs with O(1) streaming, Span batch, state rollback - Quantower: PtaIndicator adapter with LineSeries + SetValue - Tests: 31 lib + 11 Quantower (all passing) - Pine: pta.pine PineScript v6 reference - Docs: Pta.md canonical template v3 - Python: Exports.Generated.cs + _bridge.py + dynamics.py - Indexes: _sidebar.md, lib/_index.md, dynamics/_index.md, docs/indicators.md, docs/pinescript.md
92 lines
4.3 KiB
Markdown
92 lines
4.3 KiB
Markdown
# PTA: Ehlers Precision Trend Analysis
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| Property | Value |
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| :------------- | :----------------------------------------------- |
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| **Category** | Dynamics |
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| **Author** | John F. Ehlers |
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| **Source** | TASC, September 2024 |
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| **Parameters** | longPeriod (default 250), shortPeriod (default 40)|
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| **Output** | Zero-centered trend indicator |
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| **Range** | Unbounded (zero-centered) |
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| **Warmup** | longPeriod bars |
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## Historical Context
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Traditional trend-following indicators like moving averages are lowpass filters with unavoidable lag. Ehlers' insight is to use highpass filters instead — they have nearly zero lag. By applying two highpass filters with different cutoff periods and subtracting, PTA creates a bandpass that preserves cyclic components between the short and long periods while eliminating noise and very long-term drift.
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## Architecture & Physics
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### Stage 1: Dual 2-Pole Butterworth Highpass Filters
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Both filters use the standard Ehlers 2-pole Butterworth HP formulation:
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$$a_1 = e^{-\sqrt{2} \cdot \pi / P}$$
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$$b_1 = 2 \cdot a_1 \cdot \cos(\sqrt{2} \cdot \pi / P)$$
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$$c_2 = b_1, \quad c_3 = -a_1^2, \quad c_1 = \frac{1 + c_2 - c_3}{4}$$
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$$HP = c_1 \cdot (src - 2 \cdot src_1 + src_2) + c_2 \cdot HP_1 + c_3 \cdot HP_2$$
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HP1 uses `longPeriod` (default 250), HP2 uses `shortPeriod` (default 40).
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### Stage 2: Bandpass via Subtraction
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$$\text{Trend} = HP_1 - HP_2$$
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HP1 passes frequencies above 1/longPeriod. HP2 passes frequencies above 1/shortPeriod. The difference preserves only the band between shortPeriod and longPeriod — the trend-relevant frequencies.
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### Key Properties
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- **Near-zero lag**: Highpass filters inherently have minimal lag, unlike lowpass (MA-based) trend indicators.
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- **Positive = Uptrend**: When PTA > 0, price trend is up.
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- **Negative = Downtrend**: When PTA < 0, price trend is down.
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- **Zero crossings**: Signal trend reversals.
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## Performance Profile
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### Operation Count (Streaming Mode, Scalar)
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| Operation | Count |
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| :----------------- | :---- |
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| Subtractions | 3 |
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| Multiplications | 4 |
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| FMA | 4 |
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| IIR state updates | 6 |
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| **Total** | **17 FLOPs** |
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### Batch Mode (SIMD Analysis)
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No SIMD vectorization possible — serial IIR dependency chain on HP state. The batch path uses scalar FMA loop, O(1) per bar, zero allocation.
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### Quality Metrics
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| Metric | Value |
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| :---------------- | :------------------- |
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| Lag | Near zero |
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| Smoothness | High (IIR filtering) |
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| Frequency range | shortPeriod–longPeriod |
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| Allocations | 0 (hot path) |
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## Validation
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### Behavioral Test Summary
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| Test | Description |
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| :------------------------ | :----------------------------------------------------------- |
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| ConstantInput → Zero | Constant price has zero 2nd-order difference → PTA = 0 |
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| Uptrend → Positive | Steadily rising prices produce positive PTA |
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| Downtrend → Negative | Steadily falling prices produce negative PTA |
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| LongPeriod > ShortPeriod | Constructor enforces ordering constraint |
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| Symmetry | Mirrored price produces mirrored PTA (negated) |
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## Common Pitfalls
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1. **longPeriod must exceed shortPeriod** — otherwise the bandpass is inverted. Constructor throws.
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2. **IIR Bootstrap** — First 2 bars output 0.0 while source history fills. Full convergence at ~longPeriod bars.
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3. **Default 250 bars** — Requires substantial history before the long HP stabilizes. Reduce for shorter timeframes.
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4. **Not a price overlay** — Output is zero-centered, plotted in separate window.
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## References
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- Ehlers, J. F. "Precision Trend Analysis." *Technical Analysis of Stocks & Commodities*, September 2024.
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- [TradingView Implementation](https://www.tradingview.com/script/XxSVTg0v-TASC-2024-09-Precision-Trend-Analysis/)
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- [Financial Hacker Analysis](https://financial-hacker.com/ehlers-precision-trend-analysis/)
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