Miha Kralj
aec3a64e4e
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
4.3 KiB
PTA: Ehlers Precision Trend Analysis
| Property | Value |
|---|---|
| Category | Dynamics |
| Author | John F. Ehlers |
| Source | TASC, September 2024 |
| Parameters | longPeriod (default 250), shortPeriod (default 40) |
| Output | Zero-centered trend indicator |
| Range | Unbounded (zero-centered) |
| Warmup | longPeriod bars |
Historical Context
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.
Architecture & Physics
Stage 1: Dual 2-Pole Butterworth Highpass Filters
Both filters use the standard Ehlers 2-pole Butterworth HP formulation:
a_1 = e^{-\sqrt{2} \cdot \pi / P}
b_1 = 2 \cdot a_1 \cdot \cos(\sqrt{2} \cdot \pi / P)
c_2 = b_1, \quad c_3 = -a_1^2, \quad c_1 = \frac{1 + c_2 - c_3}{4}
HP = c_1 \cdot (src - 2 \cdot src_1 + src_2) + c_2 \cdot HP_1 + c_3 \cdot HP_2
HP1 uses longPeriod (default 250), HP2 uses shortPeriod (default 40).
Stage 2: Bandpass via Subtraction
\text{Trend} = HP_1 - HP_2
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.
Key Properties
- Near-zero lag: Highpass filters inherently have minimal lag, unlike lowpass (MA-based) trend indicators.
- Positive = Uptrend: When PTA > 0, price trend is up.
- Negative = Downtrend: When PTA < 0, price trend is down.
- Zero crossings: Signal trend reversals.
Performance Profile
Operation Count (Streaming Mode, Scalar)
| Operation | Count |
|---|---|
| Subtractions | 3 |
| Multiplications | 4 |
| FMA | 4 |
| IIR state updates | 6 |
| Total | 17 FLOPs |
Batch Mode (SIMD Analysis)
No SIMD vectorization possible — serial IIR dependency chain on HP state. The batch path uses scalar FMA loop, O(1) per bar, zero allocation.
Quality Metrics
| Metric | Value |
|---|---|
| Lag | Near zero |
| Smoothness | High (IIR filtering) |
| Frequency range | shortPeriod–longPeriod |
| Allocations | 0 (hot path) |
Validation
Behavioral Test Summary
| Test | Description |
|---|---|
| ConstantInput → Zero | Constant price has zero 2nd-order difference → PTA = 0 |
| Uptrend → Positive | Steadily rising prices produce positive PTA |
| Downtrend → Negative | Steadily falling prices produce negative PTA |
| LongPeriod > ShortPeriod | Constructor enforces ordering constraint |
| Symmetry | Mirrored price produces mirrored PTA (negated) |
Common Pitfalls
- longPeriod must exceed shortPeriod — otherwise the bandpass is inverted. Constructor throws.
- IIR Bootstrap — First 2 bars output 0.0 while source history fills. Full convergence at ~longPeriod bars.
- Default 250 bars — Requires substantial history before the long HP stabilizes. Reduce for shorter timeframes.
- Not a price overlay — Output is zero-centered, plotted in separate window.
References
- Ehlers, J. F. "Precision Trend Analysis." Technical Analysis of Stocks & Commodities, September 2024.
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