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Miha Kralj
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- Remove 'C# Implementation Considerations' sections from 34 indicator .md files - Delete 29 temp PowerShell scripts (_fix_mojibake.ps1, _hex_scan.ps1, etc.) - Move test files into tests/ subdirectories for consistent project structure - Add trader-focused bullet points to indicator documentation
111 lines
4.2 KiB
Markdown
111 lines
4.2 KiB
Markdown
# WMA: Weighted Moving Average
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> *Because yesterday matters more than last Tuesday. WMA is the linear answer to the question: 'What have you done for me lately?'*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Trend (FIR MA) |
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| **Inputs** | Source (close) |
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| **Parameters** | `period` |
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| **Outputs** | Single series (Wma) |
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| **Output range** | Tracks input |
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| **Warmup** | `period` bars |
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| **PineScript** | [wma.pine](wma.pine) |
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| **Signature** | [wma_signature](wma_signature.md) |
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- The Weighted Moving Average (WMA) assigns a linearly decreasing weight to data points.
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- Parameterized by `period`.
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- Output range: Tracks input.
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- Requires `period` bars of warmup before first valid output (IsHot = true).
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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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.
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## Historical Context
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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.
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## Architecture & Physics
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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.
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### The O(1) Algorithm
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Two sums are maintained:
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1. `Sum`: The simple sum of values (like SMA).
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2. `WSum`: The weighted sum.
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$$ WSum_{new} = WSum_{old} - Sum_{old} + (N \times Price_{new}) $$
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$$ Sum_{new} = Sum_{old} - Price_{oldest} + Price_{new} $$
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This allows calculating a WMA(1000) as fast as a WMA(10).
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### SIMD Optimization
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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.
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## Mathematical Foundation
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### 1. The Formula
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$$ WMA = \frac{\sum_{i=0}^{N-1} (N-i) \times P_{t-i}}{\frac{N(N+1)}{2}} $$
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The denominator is the sum of the weights (triangular number).
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## Performance Profile
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### Operation Count (Streaming Mode, Scalar)
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The O(1) algorithm eliminates the $O(N)$ weighted sum on each bar:
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| ADD/SUB | 4 | 1 | 4 |
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| MUL | 1 | 3 | 3 |
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| DIV | 1 | 15 | 15 |
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| **Total** | **6** | — | **~22 cycles** |
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**Hot path breakdown:**
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- `WSum_new = WSum_old - Sum_old + (N × Price_new)`: 2 SUB + 1 MUL
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- `Sum_new = Sum_old - Price_oldest + Price_new`: 2 SUB
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- `WMA = WSum / divisor`: 1 DIV (divisor is precomputed constant)
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**Comparison with naive O(N) implementation:**
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| Mode | Complexity | Cycles (Period=100) |
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| :--- | :---: | :---: |
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| Naive (recalculate) | O(N) | ~400 cycles |
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| QuanTAlib O(1) | O(1) | ~22 cycles |
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| **Improvement** | **—** | **~18× faster** |
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### Batch Mode (SIMD/FMA)
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WMA batch uses prefix sums for both `Sum` and `WSum`, enabling vectorization:
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| Operation | Scalar Ops (512 bars) | SIMD Ops (AVX2) | Speedup |
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| :--- | :---: | :---: | :---: |
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| Prefix sum (Sum) | 512 | 64 | 8× |
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| Weighted prefix sum | 512 | 64 | 8× |
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| Final divisions | 512 | 64 | 8× |
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The batch path achieves near-linear scaling for large datasets.
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 10/10 | Matches TA-Lib, Skender, Tulip exactly |
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| **Timeliness** | 6/10 | Linear weighting improves responsiveness over SMA |
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| **Overshoot** | 10/10 | Never overshoots input data range (FIR property) |
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| **Smoothness** | 4/10 | Less smooth than SMA; follows price closely |
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## Validation
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| Library | Status | Notes |
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| :--- | :--- | :--- |
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| **TA-Lib** | ✅ | Matches `TA_WMA` exactly. |
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| **Skender** | ✅ | Matches `GetWma` exactly. |
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| **Tulip** | ✅ | Matches `wma` exactly. |
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| **Ooples** | ✅ | Matches `CalculateWeightedMovingAverage`. |
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