Miha Kralj
a7b7207801
- Updated mathematical foundations and performance profiles where necessary to maintain clarity and coherence.
2.7 KiB
HMA: Hull Moving Average
"Alan Hull looked at the lag in moving averages and said, 'I can fix that.' And he did, by making the math do gymnastics."
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.
Historical Context
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.
Architecture & Physics
The HMA is built from three Weighted Moving Averages (WMAs):
- WMA(n/2): A fast WMA of half the period.
- WMA(n): A slow WMA of the full period.
- WMA(sqrt(n)): A smoothing WMA applied to the difference.
The core logic is: 2 \times \text{WMA}(n/2) - \text{WMA}(n).
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.
Mathematical Foundation
\text{Raw} = 2 \times \text{WMA}(P, \frac{N}{2}) - \text{WMA}(P, N)
\text{HMA} = \text{WMA}(\text{Raw}, \sqrt{N})
Where N is the period.
Performance Profile
HMA is computationally more intensive than a simple WMA due to the three passes, but our implementation optimizes the intermediate step.
| Metric | Complexity | Notes |
|---|---|---|
| Throughput | High | 3x WMA cost + vector math |
| Complexity | O(1) | Constant time update |
| Accuracy | 8/10 | Excellent at tracking price action |
| Timeliness | 9/10 | Very responsive, minimal lag |
| Overshoot | 5/10 | Prone to overshoot due to lag correction |
| Smoothness | 8/10 | Surprisingly smooth given its speed |
Validation
Validated against Alan Hull's original formula and standard library implementations.
| Provider | Error Tolerance | Notes |
|---|---|---|
| Fidelity | 10^{-9} |
Matches standard HMA |
| Skender | 10^{-9} |
Matches GetHma |
Common Pitfalls
- Overshoot: Like DEMA, HMA can overshoot price turns because of the lag correction.
- Period Sensitivity: The
\sqrt{N}smoothing is hardcoded into the definition. You can't easily tweak the smoothing independently of the lag correction without breaking the "Hull" definition. - Integer Math: The periods
N/2and\sqrt{N}are rounded to integers. This can cause slight discrepancies between implementations depending on rounding rules. Standard integer truncation is used in QuanTAlib.