QuanTAlib/lib/trends_FIR/pwma/Pwma.md

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.
• 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

[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:

// 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

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:

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).