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Miha Kralj
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- Remove 'C# Implementation Considerations' sections from 34 indicator .md files - Delete 29 temp PowerShell scripts (_fix_mojibake.ps1, _hex_scan.ps1, etc.) - Move test files into tests/ subdirectories for consistent project structure - Add trader-focused bullet points to indicator documentation
100 lines
4.0 KiB
Markdown
100 lines
4.0 KiB
Markdown
# PWMA: Parabolic Weighted Moving Average
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> *Linear weighting is for people who think the world is flat. PWMA squares the weights, because recent data isn't just more important—it's exponentially more important.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Trend (FIR MA) |
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| **Inputs** | Source (close) |
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| **Parameters** | `period` |
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| **Outputs** | Single series (Pwma) |
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| **Output range** | Tracks input |
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| **Warmup** | `period` bars |
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| **PineScript** | [pwma.pine](pwma.pine) |
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| **Signature** | [pwma_signature](pwma_signature.md) |
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- PWMA (Parabolic Weighted Moving Average) applies a parabolic ($i^2$) weighting scheme to the data window.
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- Parameterized by `period`.
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- Output range: Tracks input.
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- Requires `period` bars of warmup before first valid output (IsHot = true).
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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PWMA (Parabolic Weighted Moving Average) applies a parabolic ($i^2$) weighting scheme to the data window. This assigns massive importance to the most recent data points while still technically including the older data. It's like a WMA on steroids.
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## Historical Context
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While the WMA uses a linear triangle window ($1, 2, 3, \dots, n$), the PWMA uses a parabolic window ($1^2, 2^2, 3^2, \dots, n^2$). This was developed for traders who found the WMA too slow but the EMA too jittery. It provides a curve that turns faster than a WMA but is smoother than an EMA at the tail.
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## Architecture & Physics
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The "physics" is defined by the weight function $W_i = i^2$.
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This shifts the center of gravity of the filter heavily towards the right (recent data).
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## Mathematical Foundation
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$$ \text{PWMA} = \frac{\sum_{i=1}^{N} i^2 P_{t-N+i}}{\sum_{i=1}^{N} i^2} $$
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The O(1) update logic involves cascading the sums:
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$$ S1_{new} = S1_{old} - \text{Oldest} + \text{Newest} $$
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$$ S2_{new} = S2_{old} - S1_{old} + N \times \text{Newest} $$
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$$ S3_{new} = S3_{old} - 2 S2_{old} + S1_{old} + N^2 \times \text{Newest} $$
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## Performance Profile
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### Operation Count (Streaming Mode, Scalar)
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The O(1) algorithm uses triple cascading sums:
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| ADD/SUB | 9 | 1 | 9 |
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| MUL | 3 | 3 | 9 |
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| DIV | 1 | 15 | 15 |
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| **Total** | **13** | — | **~33 cycles** |
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**Hot path breakdown:**
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- S1 update: `S1_new = S1_old - oldest + newest` → 2 ADD/SUB
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- S2 update: `S2_new = S2_old - S1_old + N×newest` → 2 ADD/SUB + 1 MUL
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- S3 update: `S3_new = S3_old - 2×S2_old + S1_old + N²×newest` → 4 ADD/SUB + 2 MUL
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- Final: `PWMA = S3 / divisor` → 1 DIV (divisor precomputed)
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**Comparison with naive O(N) implementation:**
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| Mode | Complexity | Cycles (Period=100) |
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| :--- | :---: | :---: |
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| Naive (recalculate) | O(N) | ~700 cycles |
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| QuanTAlib O(1) | O(1) | ~33 cycles |
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| **Improvement** | **—** | **~21× faster** |
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### Batch Mode (SIMD)
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PWMA batch can vectorize prefix-sum cascades:
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| Operation | Scalar Ops (512 bars) | SIMD Ops (AVX2) | Speedup |
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| :--- | :---: | :---: | :---: |
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| S1 prefix sum | 512 | 64 | 8× |
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| S2 cascaded sum | 1024 | 128 | 8× |
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| S3 cascaded sum | 1536 | 192 | 8× |
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 10/10 | Matches mathematical definition exactly |
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| **Timeliness** | 9/10 | Very fast reaction to new data (heavy recent weighting) |
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| **Overshoot** | 3/10 | Parabolic weighting can cause overshoot |
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| **Smoothness** | 4/10 | Sensitive to recent noise |
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## Validation
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Validated against Ooples.
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| Library | Status | Notes |
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| :--- | :--- | :--- |
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| **QuanTAlib** | ✅ | Validated. |
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| **Ooples** | ✅ | Matches `CalculateParabolicWeightedMovingAverage` |
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| **Skender** | N/A | Not implemented |
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| **TA-Lib** | N/A | Not implemented |
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| **Tulip** | N/A | Not implemented. |
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