# 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](pwma.pine)                       |
| **Signature**    | [pwma_signature](pwma_signature.md) |

- PWMA (Parabolic Weighted Moving Average) applies a parabolic ($i^2$) weighting scheme to the data window.
- Parameterized by `period`.
- Output range: Tracks input.
- Requires `period` bars of warmup before first valid output (IsHot = true).
- 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. |

## C# Implementation Considerations

QuanTAlib's PWMA uses triple cascading sums to achieve O(1) streaming updates. The implementation demonstrates several high-performance patterns:

### State Management

```csharp
[StructLayout(LayoutKind.Auto)]
private record struct State(
    double Sum,      // Running sum (S1)
    double WSum,     // Weighted sum (S2)
    double PSum,     // Parabolic sum (S3)
    double LastInput,
    double LastValidValue,
    int TickCount)
```

The state captures all three cascading sums needed for the O(1) update formula. `TickCount` tracks iterations for periodic resync.

### Key Optimizations

| Technique | Implementation | Benefit |
| :--- | :--- | :--- |
| **Precomputed divisor** | `_divisor = period * (period + 1.0) * (2.0 * period + 1.0) / 6.0` | Eliminates division in hot path |
| **FMA cascade** | All three sum updates use `FusedMultiplyAdd` | Hardware-accelerated multiply-add |
| **Dual buffer** | `_buffer` + `_p_buffer` for bar correction | O(1) state restoration on `isNew=false` |
| **Periodic resync** | Full recalculation every 1000 ticks | Bounds floating-point drift |
| **stackalloc** | Batch `Calculate` uses stack for period ≤ 512 | Zero heap allocation |

### FMA in Cascade Updates

The cascading sum formulas map directly to FMA operations:

```csharp
// S1 update (simple running sum)
double newSum = _state.Sum - oldest + newest;

// S2 update: S2_new = S2_old - S1_old + N×newest
double newWSum = Math.FusedMultiplyAdd(period, newest, _state.WSum - _state.Sum);

// S3 update: S3_new = S3_old - 2×S2_old + S1_old + N²×newest
double newPSum = Math.FusedMultiplyAdd(period * period, newest,
    _state.PSum - 2 * oldWSum + oldSum);
```

### Memory Layout

| Field | Type | Size | Purpose |
| :--- | :--- | :---: | :--- |
| `Sum` | double | 8 bytes | Running sum S1 |
| `WSum` | double | 8 bytes | Weighted sum S2 |
| `PSum` | double | 8 bytes | Parabolic sum S3 |
| `LastInput` | double | 8 bytes | Previous input value |
| `LastValidValue` | double | 8 bytes | NaN substitution |
| `TickCount` | int | 4 bytes | Resync counter |
| **State total** | | **44 bytes** | Compiler-aligned |
| `_buffer` | RingBuffer | 24 + 8N | Sliding window |
| `_p_buffer` | RingBuffer | 24 + 8N | Bar correction backup |

### Bar Correction Pattern

```csharp
if (isNew)
{
    _p_state = _state;           // Snapshot for rollback
    _p_buffer.CopyFrom(_buffer); // Buffer snapshot
}
else
{
    _state = _p_state;           // Restore previous state
    _buffer.CopyFrom(_p_buffer); // Restore buffer
}
```

### Periodic Resync

Triple cascading sums accumulate floating-point errors faster than simple running sums. The implementation resyncs every 1000 ticks:

```csharp
if (_state.TickCount >= 1000)
{
    // Full O(N) recalculation to reset drift
    RecalculateFromBuffer();
    _state = _state with { TickCount = 0 };
}
```

### Common Pitfalls

1. **Resync**: Because triple running sums are used, floating-point errors can accumulate faster than in a simple SMA. The implementation automatically resyncs every 1000 ticks to maintain precision.
2. **Sensitivity**: This indicator is very sensitive to the most recent bar. It can "repaint" visually if used on an open bar (though the math is consistent).