QuanTAlib/lib/trends_FIR/wma/Wma.md

WMA: Weighted Moving Average

Because yesterday matters more than last Tuesday. WMA is the linear answer to the question: 'What have you done for me lately?'

Property	Value
Category	Trend (FIR MA)
Inputs	Source (close)
Parameters	period
Outputs	Single series (Wma)
Output range	Tracks input
Warmup	period bars
PineScript	wma.pine
Signature	wma_signature

• The Weighted Moving Average (WMA) assigns a linearly decreasing weight to data points.
• Similar: FWMA, TRIMA | Complementary: WMA crossover systems | Trading note: Linearly weighted MA; recent prices get higher weight, faster response than SMA.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

The Weighted Moving Average (WMA) assigns a linearly decreasing weight to data points. The most recent price gets weight N, the one before it N-1, down to 1. This makes it more responsive to recent price changes than an SMA, but without the infinite tail of an EMA.

Historical Context

WMA is the "finite impulse response" (FIR) counterpart to the EMA. It was developed to reduce the lag of the SMA while maintaining a finite window of influence.

Architecture & Physics

A naive WMA implementation is O(N), requiring a full loop over the history window for every update. QuanTAlib uses a dual running-sum algorithm to achieve O(1) complexity.

The O(1) Algorithm

Two sums are maintained:

1. Sum: The simple sum of values (like SMA).
2. WSum: The weighted sum.

WSum_{new} = WSum_{old} - Sum_{old} + (N \times Price_{new}) Sum_{new} = Sum_{old} - Price_{oldest} + Price_{new}

This allows calculating a WMA(1000) as fast as a WMA(10).

SIMD Optimization

For batch processing, Wma.Batch uses advanced vectorization (AVX2/AVX-512/Neon). It computes prefix sums and weighted updates in parallel, achieving throughputs that scalar code cannot touch.

Mathematical Foundation

1. The Formula

WMA = \frac{\sum_{i=0}^{N-1} (N-i) \times P_{t-i}}{\frac{N(N+1)}{2}}

The denominator is the sum of the weights (triangular number).

Performance Profile

Operation Count (Streaming Mode, Scalar)

The O(1) algorithm eliminates the O(N) weighted sum on each bar:

Operation	Count	Cost (cycles)	Subtotal
ADD/SUB	4	1	4
MUL	1	3	3
DIV	1	15	15
Total	6	—	~22 cycles

Hot path breakdown:

• WSum_new = WSum_old - Sum_old + (N × Price_new): 2 SUB + 1 MUL
• Sum_new = Sum_old - Price_oldest + Price_new: 2 SUB
• WMA = WSum / divisor: 1 DIV (divisor is precomputed constant)

Comparison with naive O(N) implementation:

Mode	Complexity	Cycles (Period=100)
Naive (recalculate)	O(N)	~400 cycles
QuanTAlib O(1)	O(1)	~22 cycles
Improvement	—	~18× faster

Batch Mode (SIMD/FMA)

WMA batch uses prefix sums for both Sum and WSum, enabling vectorization:

Operation	Scalar Ops (512 bars)	SIMD Ops (AVX2)	Speedup
Prefix sum (Sum)	512	64	8×
Weighted prefix sum	512	64	8×
Final divisions	512	64	8×

The batch path achieves near-linear scaling for large datasets.

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Matches TA-Lib, Skender, Tulip exactly
Timeliness	6/10	Linear weighting improves responsiveness over SMA
Overshoot	10/10	Never overshoots input data range (FIR property)
Smoothness	4/10	Less smooth than SMA; follows price closely

Validation

Library	Status	Notes
TA-Lib	✅	Matches TA_WMA exactly.
Skender	✅	Matches GetWma exactly.
Tulip	✅	Matches wma exactly.
Ooples	✅	Matches CalculateWeightedMovingAverage.