QuanTAlib/lib/trends/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?'"

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

Metric	Score	Notes
Throughput	10	High; O(1) calculation via dual running sums.
Allocations	0	Zero-allocation in hot paths.
Complexity	O(1)	Constant time regardless of period.
Accuracy	10	Matches TA-Lib exactly.
Timeliness	6	More responsive than SMA due to linear weighting.
Overshoot	0	Never overshoots the input data range.
Smoothness	4	Less smooth than SMA; follows price more closely.

Validation

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

Common Pitfalls

1. Drift: Like SMA, the O(1) algorithm is susceptible to floating-point drift. QuanTAlib resets the sums every 10,000 ticks to guarantee accuracy.
2. Aggressiveness: WMA reacts faster than SMA but can be "twitchy." It is often used as a component in other indicators (e.g., HMA) rather than a standalone trend filter.
3. Weights: Users sometimes confuse WMA (linear weights) with EMA (exponential weights) or VWAP (volume weights).