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108 lines
4.7 KiB
Markdown
108 lines
4.7 KiB
Markdown
# HMA: Hull Moving Average
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> *Alan Hull looked at the lag in moving averages and said, 'I can fix that.' And he did, by making the math do gymnastics.*
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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 (Hma) |
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| **Output range** | Tracks input |
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| **Warmup** | `period + sqrtPeriod - 1` bars |
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| **PineScript** | [hma.pine](hma.pine) |
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| **Signature** | [hma_signature](hma_signature.md) |
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- HMA (Hull Moving Average) is a solution to the eternal struggle between smoothness and lag.
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- **Similar:** [DEMA](../../trends_IIR/dema/dema.md), [TEMA](../../trends_IIR/tema/tema.md) | **Complementary:** Signal line crossover | **Trading note:** Alan Hulls MA; cascades WMAs to nearly eliminate lag while maintaining smoothness.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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HMA (Hull Moving Average) is a solution to the eternal struggle between smoothness and lag. Most indicators force you to choose one; HMA gives you both. It achieves this by using weighted moving averages (WMAs) in a clever configuration that cancels out lag while maintaining the smoothing properties of the WMA.
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## Historical Context
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Developed by Alan Hull in 2005, the HMA was designed to be "responsive, accurate, and smooth." Hull realized that lag is essentially a function of the period, and by combining averages of different periods (specifically, a full period and a half period), he could mathematically offset the lag.
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## Architecture & Physics
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The HMA is built from three Weighted Moving Averages (WMAs):
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1. **WMA(n/2)**: A fast WMA of half the period.
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2. **WMA(n)**: A slow WMA of the full period.
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3. **WMA(sqrt(n))**: A smoothing WMA applied to the difference.
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The core logic is: $2 \times \text{WMA}(n/2) - \text{WMA}(n)$.
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This operation "over-weights" the recent data, pushing the average forward to align with the current price. The final WMA smooths out the resulting noise.
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## Mathematical Foundation
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$$ \text{Raw} = 2 \times \text{WMA}(P, \frac{N}{2}) - \text{WMA}(P, N) $$
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$$ \text{HMA} = \text{WMA}(\text{Raw}, \sqrt{N}) $$
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Where $N$ is the period.
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## Performance Profile
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### Operation Count (Streaming Mode, Scalar)
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HMA chains three WMA instances. Each WMA is O(1) with ~22 cycles (see WMA.md).
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| Component | Operations | Cost (cycles) |
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| :--- | :--- | :---: |
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| WMA(N/2) | 4 ADD/SUB, 1 MUL, 1 DIV | ~22 |
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| WMA(N) | 4 ADD/SUB, 1 MUL, 1 DIV | ~22 |
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| Combiner: 2×WMA₁ - WMA₂ | 1 MUL, 1 SUB | ~4 |
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| WMA(√N) | 4 ADD/SUB, 1 MUL, 1 DIV | ~22 |
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| **Total** | **~18 ops** | **~70 cycles** |
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**Hot path breakdown:**
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- `Raw = 2 × WMA(n/2) - WMA(n)`: 1 MUL + 1 SUB
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- Three independent WMA updates execute in sequence
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- Each WMA uses O(1) dual running-sum algorithm
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### Batch Mode (SIMD)
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Each WMA component benefits from SIMD prefix-sum optimization:
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| Component | Scalar (512 bars) | SIMD (AVX2) | Speedup |
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| :--- | :---: | :---: | :---: |
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| WMA(N/2) batch | ~11K cycles | ~3K cycles | ~4× |
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| WMA(N) batch | ~11K cycles | ~3K cycles | ~4× |
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| Combiner | ~2K cycles | ~250 cycles | ~8× |
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| WMA(√N) batch | ~11K cycles | ~3K cycles | ~4× |
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| **Total** | **~35K** | **~9K** | **~4×** |
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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 Skender, Tulip exactly |
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| **Timeliness** | 9/10 | Lag-compensated design; very responsive |
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| **Overshoot** | 4/10 | Can overshoot on sharp reversals (algebraic correction side effect) |
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| **Smoothness** | 6/10 | Final √N smoothing moderates noise |
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### Zero-Allocation Design
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HMA is implemented by chaining three `Wma` instances. Since `Wma` is zero-allocation, HMA inherits this property.
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## Validation
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Validated against Skender, Tulip, and Ooples.
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| Library | Status | Notes |
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| :--- | :--- | :--- |
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| **Skender** | ✅ | Matches `GetHma`. |
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| **Tulip** | ✅ | Matches `hma`. |
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| **Ooples** | ✅ | Matches `CalculateHullMovingAverage` (with rounding caveats). |
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| **TA-Lib** | ❌ | Not implemented. |
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### External Library Discrepancies
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**OoplesFinance.StockIndicators**:
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Discrepancies exist due to different rounding methods for integer periods.
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* **QuanTAlib**: Uses integer truncation (floor) for $N/2$ and $\sqrt{N}$.
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* **Ooples**: Uses `Math.Round` (nearest integer).
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This results in different effective periods for $N=14$ ($\sqrt{14} \approx 3.74 \to 3$ vs $4$) and others where the fractional part $\ge 0.5$. Validation tests match exactly for periods where rounding logic aligns (e.g., $N=9, 20, 50$). |