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docs(pta): rewrite Pta.md to canonical template v3
Add tagline, full property table, 3-bullet summary, extended historical context, mathematical foundation, parameter mapping, frequency response analysis, phase lag discussion, quality metrics, validation table, and detailed common pitfalls.
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# 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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> *Most trend indicators smooth price and inherit lag as a tax. PTA sidesteps the toll entirely — two highpass filters, one subtraction, and the trend arrives before the moving average even notices it moved.*
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| Property | Value |
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| ---------------- | ------------------------------------------------------ |
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| **Category** | Dynamics |
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| **Inputs** | Source (close) |
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| **Parameters** | `longPeriod` (default 250), `shortPeriod` (default 40) |
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| **Outputs** | Single series (Pta) |
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| **Output range** | Unbounded (zero-centered) |
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| **Warmup** | `longPeriod` bars |
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| **PineScript** | [pta.pine](pta.pine) |
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- PTA (Precision Trend Analysis) applies two 2-pole Butterworth highpass filters with different cutoff periods to the same input, then subtracts: Trend = HP(longPeriod) − HP(shortPeriod). This preserves cyclic components between `shortPeriod` and `longPeriod` bars, producing a zero-centered trend indicator with near-zero phase lag.
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- **Similar:** [Decycler](../../trends_IIR/decycler/Decycler.md), [DECO](../../oscillators/deco/Deco.md) | **Complementary:** ADX for trend strength confirmation, SuperTrend for directional bias | **Trading note:** Positive = uptrend, negative = downtrend; zero crossings signal reversals. Not a price overlay — plot in separate window.
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- No external validation libraries implement PTA. Validated through self-consistency, behavioral testing, and PineScript reference comparison.
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PTA (Precision Trend Analysis) is John F. Ehlers' 2024 approach to extracting market trend with near-zero lag. Published in the September 2024 issue of *Technical Analysis of Stocks & Commodities*, the technique inverts the conventional wisdom: instead of smoothing price with a lowpass filter (which always introduces lag proportional to the filter order), PTA uses two highpass filters — which have almost no lag — and subtracts them to create a bandpass that isolates the trend-relevant frequency band. With default parameters of `longPeriod = 250` (~1 trading year) and `shortPeriod = 40` (~2 months), PTA captures intermediate-term market trends while rejecting both high-frequency noise and ultra-long-term drift. The output is zero-centered and unbounded: positive values indicate uptrend, negative values indicate downtrend, and zero crossings mark trend reversals. Each streaming bar requires only 17 floating-point operations — two IIR evaluations plus one subtraction — making PTA one of the cheapest trend indicators available.
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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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Ehlers' body of work on digital signal processing applied to financial markets spans three decades, with a consistent theme: treat price as a signal and apply engineering-grade filter design rather than ad hoc smoothing. His earlier contributions — the Decycler (2015), Super Smoother (2013), and various Hilbert Transform indicators — all apply specific filter topologies to extract actionable information from price series.
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The Decycler indicator, published in TASC in 2015, subtracts a highpass filter output from the original price to obtain a lowpass-filtered trend. PTA inverts this approach: instead of keeping what the highpass *removes*, PTA operates entirely within the highpass domain. By applying two highpass filters with different cutoff frequencies and subtracting, it creates a bandpass filter that preserves only the frequency band between the two cutoffs. This is the same principle as an analog bandpass filter built from differential highpass stages.
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The key innovation of PTA over the Decycler is the elimination of the lowpass path entirely. The Decycler's output tracks price closely (it is a lowpass of price), which makes it useful as a trend overlay but problematic for trend *magnitude* assessment. PTA's output is zero-centered and measures trend *energy* in the selected frequency band, making it a proper trend strength and direction indicator rather than a smoothed price estimate.
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The default parameters — `longPeriod = 250` and `shortPeriod = 40` — correspond to approximately one trading year and two trading months respectively. This isolates the intermediate-term trend band that most swing and position traders target. Shorter `shortPeriod` values (e.g., 10–20) capture faster trends; larger `longPeriod` values extend the analysis to secular trends.
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## Architecture & Physics
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### Stage 1: Dual 2-Pole Butterworth Highpass Filters
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### Stage 1: 2-Pole Butterworth Highpass Coefficients
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Both filters use the standard Ehlers 2-pole Butterworth HP formulation:
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Both highpass filters use the standard Ehlers 2-pole Butterworth formulation. For a given cutoff period $P$:
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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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$$\alpha = e^{-\sqrt{2} \cdot \pi / P}$$
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HP1 uses `longPeriod` (default 250), HP2 uses `shortPeriod` (default 40).
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$$c_2 = 2 \alpha \cos\!\left(\frac{\sqrt{2} \cdot \pi}{P}\right)$$
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### Stage 2: Bandpass via Subtraction
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$$c_3 = -\alpha^2$$
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$$\text{Trend} = HP_1 - HP_2$$
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$$c_1 = \frac{1 + c_2 - c_3}{4}$$
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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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The coefficients are precomputed once in the constructor. Two independent sets are stored: $\{c_{1L}, c_{2L}, c_{3L}\}$ for `longPeriod` and $\{c_{1S}, c_{2S}, c_{3S}\}$ for `shortPeriod`.
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### Key Properties
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### Stage 2: Highpass Filter Recurrence
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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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Each filter applies the same 2nd-order IIR recurrence per bar:
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$$HP_n = c_1 \cdot (x_n - 2x_{n-1} + x_{n-2}) + c_2 \cdot HP_{n-1} + c_3 \cdot HP_{n-2}$$
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where $x_n$ is the current source value. The term $(x_n - 2x_{n-1} + x_{n-2})$ is the discrete second difference — it is shared between both filters since they process the same input, saving 2 operations.
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HP1 (long-period) removes only the very lowest frequencies (below $1/\text{longPeriod}$), passing everything above. HP2 (short-period) removes a wider band of low frequencies (below $1/\text{shortPeriod}$), passing only higher frequencies.
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### Stage 3: Bandpass via Subtraction
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$$\text{PTA} = HP_1 - HP_2$$
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HP1 passes frequencies above $f_L = 1/\text{longPeriod}$. HP2 passes frequencies above $f_S = 1/\text{shortPeriod}$ (where $f_S > f_L$). The subtraction cancels the high-frequency components that both filters pass, leaving only the band between $f_L$ and $f_S$ — the trend-relevant frequencies.
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### State Management
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The indicator maintains a `State` record struct containing:
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- `Hp1`, `Hp1_1`: Current and previous HP1 values (long-period filter)
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- `Hp2`, `Hp2_1`: Current and previous HP2 values (short-period filter)
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- `Src1`, `Src2`: Previous two source values (shared across both filters)
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- `Count`: Bar counter for warmup tracking
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A shadow state (`_p_state`) enables bar correction — when `isNew = false`, the previous state is restored before recomputing.
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## Mathematical Foundation
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### Frequency Response
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The 2-pole Butterworth highpass has a frequency response magnitude:
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$$|H(f)|^2 = \frac{1}{1 + \left(\frac{f_c}{f}\right)^{2n}}$$
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where $f_c$ is the cutoff frequency and $n = 2$ (two poles). The −3 dB point occurs at $f = f_c = 1/P$.
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The PTA bandpass response is:
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$$|H_{\text{PTA}}(f)| = |H_1(f)| - |H_2(f)|$$
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This creates a passband centered between $f_L$ and $f_S$ with smooth rolloff determined by the Butterworth characteristic — maximally flat in the passband with no ripple.
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### Parameter Mapping
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| Symbol | Parameter | Default | Constraint |
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|--------|-----------|---------|------------|
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| $P_L$ | longPeriod | 250 | $P_L \geq 3$ |
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| $P_S$ | shortPeriod | 40 | $P_S \geq 2$, $P_S < P_L$ |
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| Configuration | Passband | Use Case |
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|---------------|----------|----------|
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| 250 / 40 | 40–250 bars | Position trading, daily charts |
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| 125 / 20 | 20–125 bars | Swing trading |
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| 60 / 10 | 10–60 bars | Active trading, 4H charts |
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| 500 / 100 | 100–500 bars | Secular trend analysis |
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### Phase Lag Analysis
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Highpass filters have near-zero phase lag for frequencies well above the cutoff. Since PTA operates by subtracting two highpass outputs, the lag of the combined bandpass is dominated by the slower (long-period) filter near its cutoff frequency, but remains negligible for the center of the passband. This is in stark contrast to lowpass-based trend indicators (SMAs, EMAs), which accumulate phase lag proportional to the filter order and period.
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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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| Operation | Count | Notes |
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| :----------------- | :---- | :------------------------------ |
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| Subtraction (2nd diff) | 2 | $x_n - 2x_{n-1} + x_{n-2}$ |
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| Multiplication | 1 | $2 \cdot x_{n-1}$ |
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| FMA (HP1) | 2 | $c_{1L} \cdot d + c_{2L} \cdot HP_1$ and $c_{3L} \cdot HP_{1,prev}$ |
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| FMA (HP2) | 2 | $c_{1S} \cdot d + c_{2S} \cdot HP_2$ and $c_{3S} \cdot HP_{2,prev}$ |
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| Subtraction (PTA) | 1 | $HP_1 - HP_2$ |
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| State updates | 6 | Hp1, Hp1_1, Hp2, Hp2_1, Src1, Src2 |
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| **Total** | **~14 FLOPs + 6 stores** | |
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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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| Operation | Vectorizable? | Notes |
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|:----------|:-------------:|:------|
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| Second difference | Yes | Independent per bar |
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| HP IIR recurrence | **No** | Serial dependency: $HP_n$ depends on $HP_{n-1}$ and $HP_{n-2}$ |
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| Final subtraction | Yes | Independent per bar |
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The IIR dependency chain prevents SIMD vectorization of the core computation. The batch path uses a scalar FMA loop with 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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| Metric | Score | Notes |
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|:-------|:-----:|:------|
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| **Accuracy** | 10/10 | Exact IIR arithmetic, FMA precision |
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| **Timeliness** | 9/10 | Near-zero phase lag; fastest trend indicator in the library |
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| **Smoothness** | 8/10 | Butterworth maximally-flat characteristic |
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| **Noise Rejection** | 8/10 | Dual-filter bandpass rejects both HF noise and LF drift |
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| **Interpretability** | 7/10 | Zero-centered; positive/negative intuitive, but unbounded magnitude requires context |
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## Validation
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| Library | Status | Notes |
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|:--------|:------:|:------|
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| **TA-Lib** | N/A | Not implemented |
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| **Skender** | N/A | Not implemented |
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| **Tulip** | N/A | Not implemented |
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| **TradingView** | Reference | Community scripts match the TASC article formula |
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| **PineScript** | ✓ | [pta.pine](pta.pine) reference validates algorithm |
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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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| Test | Expected |
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|------|----------|
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| Constant input | PTA = 0 (zero second difference) |
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| Monotonic uptrend | PTA > 0 after warmup |
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| Monotonic downtrend | PTA < 0 after warmup |
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| Mirrored series | PTA negated (symmetry) |
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| LongPeriod ≤ ShortPeriod | Constructor throws `ArgumentOutOfRangeException` |
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| ShortPeriod < 2 | Constructor throws `ArgumentOutOfRangeException` |
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| LongPeriod < 3 | Constructor throws `ArgumentOutOfRangeException` |
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| Bar correction | State rollback produces different result on modified input |
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| 4-API consistency | Streaming, batch TSeries, batch Span, Calculate all match |
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| Warmup period | `IsHot` transitions to `true` at bar `longPeriod` |
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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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1. **longPeriod must exceed shortPeriod.** The bandpass is defined as HP(longPeriod) − HP(shortPeriod). If longPeriod ≤ shortPeriod, the bandpass inverts and the output becomes meaningless. The constructor throws `ArgumentOutOfRangeException`.
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2. **IIR bootstrap phase.** The first 2 bars always output 0.0 because the second-order difference $(x_n - 2x_{n-1} + x_{n-2})$ requires 3 source values. Full convergence occurs at approximately `longPeriod` bars. The `IsHot` flag indicates when the warmup is complete.
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3. **Default 250-bar longPeriod requires substantial history.** For intraday or short-term applications, reduce `longPeriod` to match the analysis horizon. Using 250 on 5-minute bars means the long HP filter doesn't stabilize until ~21 trading hours of data.
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4. **Not a price overlay.** PTA output is zero-centered and unbounded. It measures trend energy, not price level. Always plot in a separate window. Overlaying on price produces a visually meaningless flat line near zero.
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5. **Magnitude is not normalized.** Unlike bounded oscillators (RSI, USI), PTA's amplitude scales with price volatility. A ±5 reading on a $10 stock is very different from ±5 on a $500 stock. Consider normalizing by ATR or price if comparing across instruments.
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6. **Zero crossings can whipsaw.** In ranging markets, PTA oscillates around zero and produces frequent false reversal signals. Filter zero crossings with a dead zone (e.g., PTA must exceed ±threshold before triggering) or confirm with a trend strength indicator like ADX or VHF.
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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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1. Ehlers, J. F. "Precision Trend Analysis." *Technical Analysis of Stocks & Commodities*, September 2024.
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2. Ehlers, J. F. "The Decycler." *Technical Analysis of Stocks & Commodities*, September 2015. (Predecessor using HP subtraction from price.)
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3. Ehlers, J. F. *Cycle Analytics for Traders*. Wiley, 2013. ISBN: 978-1118728512. (Butterworth filter design for financial signals.)
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4. PineScript reference: [pta.pine](pta.pine)
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