Miha Kralj
4.1 KiB
PWMA: Parabolic Weighted Moving Average
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.
| Property | Value |
|---|---|
| Category | Trend (FIR MA) |
| Inputs | Source (close) |
| Parameters | period |
| Outputs | Single series (Pwma) |
| Output range | Tracks input |
| Warmup | period bars |
| PineScript | pwma.pine |
| Signature | pwma_signature |
- PWMA (Parabolic Weighted Moving Average) applies a parabolic (
i^2) weighting scheme to the data window. - Similar: FWMA, WMA | Complementary: Trend filters | Trading note: Pascal-Weighted MA; weights from Pascals triangle for smooth, symmetric kernel.
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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.
Historical Context
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.
Architecture & Physics
The "physics" is defined by the weight function W_i = i^2.
This shifts the center of gravity of the filter heavily towards the right (recent data).
Mathematical Foundation
\text{PWMA} = \frac{\sum_{i=1}^{N} i^2 P_{t-N+i}}{\sum_{i=1}^{N} i^2}
The O(1) update logic involves cascading the sums:
S1_{new} = S1_{old} - \text{Oldest} + \text{Newest}
S2_{new} = S2_{old} - S1_{old} + N \times \text{Newest}
S3_{new} = S3_{old} - 2 S2_{old} + S1_{old} + N^2 \times \text{Newest}
Performance Profile
Operation Count (Streaming Mode, Scalar)
The O(1) algorithm uses triple cascading sums:
| Operation | Count | Cost (cycles) | Subtotal |
|---|---|---|---|
| ADD/SUB | 9 | 1 | 9 |
| MUL | 3 | 3 | 9 |
| DIV | 1 | 15 | 15 |
| Total | 13 | — | ~33 cycles |
Hot path breakdown:
- S1 update:
S1_new = S1_old - oldest + newest→ 2 ADD/SUB - S2 update:
S2_new = S2_old - S1_old + N×newest→ 2 ADD/SUB + 1 MUL - S3 update:
S3_new = S3_old - 2×S2_old + S1_old + N²×newest→ 4 ADD/SUB + 2 MUL - Final:
PWMA = S3 / divisor→ 1 DIV (divisor precomputed)
Comparison with naive O(N) implementation:
| Mode | Complexity | Cycles (Period=100) |
|---|---|---|
| Naive (recalculate) | O(N) | ~700 cycles |
| QuanTAlib O(1) | O(1) | ~33 cycles |
| Improvement | — | ~21× faster |
Batch Mode (SIMD)
PWMA batch can vectorize prefix-sum cascades:
| Operation | Scalar Ops (512 bars) | SIMD Ops (AVX2) | Speedup |
|---|---|---|---|
| S1 prefix sum | 512 | 64 | 8× |
| S2 cascaded sum | 1024 | 128 | 8× |
| S3 cascaded sum | 1536 | 192 | 8× |
Quality Metrics
| Metric | Score | Notes |
|---|---|---|
| Accuracy | 10/10 | Matches mathematical definition exactly |
| Timeliness | 9/10 | Very fast reaction to new data (heavy recent weighting) |
| Overshoot | 3/10 | Parabolic weighting can cause overshoot |
| Smoothness | 4/10 | Sensitive to recent noise |
Validation
Validated against Ooples.
| Library | Status | Notes |
|---|---|---|
| QuanTAlib | ✅ | Validated. |
| Ooples | ✅ | Matches CalculateParabolicWeightedMovingAverage |
| Skender | N/A | Not implemented |
| TA-Lib | N/A | Not implemented |
| Tulip | N/A | Not implemented. |