kingchenc
87b3f383d6
A5 changed Keltner and HMA to feed their sibling sub-indicators unconditionally, so warmup_period() is now the exact first-emission index for every indicator. The wiki still described the old ?-starvation behavior as correct. - Indicator-Keltner.md: the Warmup section, the worked example output (first emission now at i=2, not i=4), the summary table row, and the "reported warmup understates" pitfall now state that warmup_period() is exact. Example output regenerated by running the code. - Indicator-Hma.md: the Warmup section, all three language examples (first Some at index 10, not 13), the table row, and the chaining pitfall corrected. Outputs regenerated. - Indicators-Overview.md: dropped the claim that Hma and Kama lag their reported warmup — both were verified exact.
230 lines
7.7 KiB
Markdown
230 lines
7.7 KiB
Markdown
# HMA
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> Hull Moving Average — Alan Hull's
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> `WMA(2·WMA(n/2) − WMA(n), √n)`, a near-lag-free trend filter that
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> combines a fast `Wma(n/2)`, a slow `Wma(n)`, and a final smoothing pass.
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## Quick reference
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| Field | Value |
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|-------|-------|
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| Family | Trend |
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| Sub-category | Adaptive & hybrid |
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| Input type | `f64` (single close) |
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| Output type | `f64` |
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| Output range | unbounded; tracks the input price scale |
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| Default parameters | `period` is required (no default in either binding) |
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| Warmup period (`warmup_period()`) | `period + round(√period).max(1) − 1` — exact first-emission index |
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| Interpretation | Near-zero-lag trend line with an inherent smoothing step. |
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## Formula
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```
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half = max(period / 2, 1) // integer division
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smooth = max(round(sqrt(period)), 1) // nearest integer, floor at 1
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raw_t = 2 * WMA(price, half)_t - WMA(price, period)_t
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HMA_t = WMA(raw, smooth)_t
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```
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The "magic" is the `2·WMA(n/2) − WMA(n)` step: the fast WMA leads the
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slow WMA on a trend, so doubling the fast and subtracting the slow
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produces a series that is *ahead* of the input by roughly the WMA lag.
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The final `WMA(…, √n)` then smooths the resulting overshoot back down
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to a clean line. For `period = 9` this gives `half = 4`, `smooth = 3`;
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for `period = 14`, `half = 7`, `smooth = 4`.
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## Parameters
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| Name | Type | Default | Valid range | Description |
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|----------|---------|---------|-------------|-------------|
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| `period` | `usize` | none | `>= 1` | Top-level lookback. The inner WMA periods are derived from it. `period = 0` errors with `Error::PeriodZero`. |
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(Python class `wickra.HMA(period)` has no `#[pyo3(signature)]` default;
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pass `period` explicitly.)
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## Inputs / Outputs
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From `crates/wickra-core/src/indicators/hma.rs`:
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```rust
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impl Indicator for Hma {
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type Input = f64;
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type Output = f64;
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// update(&mut self, input: f64) -> Option<f64>
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}
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```
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Python returns `float | None` (streaming) / `numpy.ndarray` (batch,
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`NaN` for warmup). Node returns `number | null` / `Array<number>` with
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`NaN`.
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## Warmup
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`warmup_period()` returns:
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```
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period + round(sqrt(period)).max(1) - 1
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```
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which gives `11` for `Hma::new(9)`, `17` for `Hma::new(14)`,
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`19` for `Hma::new(16)`. This figure is **exact**: the first non-`None`
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output lands on input `warmup_period()` (index `warmup_period() - 1`).
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The number reflects how the three inner WMAs warm up *in parallel*: the
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slow `WMA(period)` emits at input `period`, then the smoothing
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`WMA(√period)` needs `√period − 1` more inputs on top.
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```rust
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fn update(&mut self, input: f64) -> Option<f64> {
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// Both raw WMAs are fed unconditionally so neither delays the other.
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let h = self.half_wma.update(input);
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let f = self.full_wma.update(input);
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match (h, f) {
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(Some(h), Some(f)) => self.smooth_wma.update(2.0 * h - f),
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_ => None,
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}
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}
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```
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`half_wma` and `full_wma` receive every input, so `full_wma` emits at
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input `period` (not later). The `2·half − full` diff then flows into
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`smooth_wma`, which needs `round(√period)` of those — giving a first
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emission at exactly `period + round(√period) − 1`.
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| `period` | `round(√period)` | `warmup_period()` | First emission (input #) |
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|----------|------------------|-------------------|--------------------------|
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| 9 | 3 | 11 | 11 |
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| 14 | 4 | 17 | 17 |
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| 16 | 4 | 19 | 19 |
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This is pinned by the `first_emission_matches_warmup_period` test in
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`hma.rs`: the first call that returns `Some` is exactly at
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`warmup_period() - 1` (0-indexed).
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## Edge cases
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- **Constant series.** Feeding `[10.0; n]` produces `Some(10.0)` once
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the chain is warm. All three WMAs converge to `10`, so
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`raw = 2·10 − 10 = 10`, then `WMA(10, smooth) = 10`. The unit test
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`constant_series_yields_constant_hma` pins this with `Hma::new(9)`
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over 80 constants.
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- **NaN / infinity inputs.** Inherited from the inner `Wma`: non-finite
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inputs are silently dropped at the half/full WMA boundary and never
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reach the `2·h − f` arithmetic.
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- **Reset.** `hma.reset()` resets all three internal WMAs; the next
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`update` starts a full warmup countdown.
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## Examples
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### Rust
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```rust
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use wickra::{BatchExt, Hma, Indicator};
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fn main() -> Result<(), Box<dyn std::error::Error>> {
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let mut hma = Hma::new(9)?;
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let prices: Vec<f64> = (1..=20).map(f64::from).collect();
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let out: Vec<Option<f64>> = hma.batch(&prices);
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println!("warmup_period = {}", hma.warmup_period());
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println!("{:?}", out);
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Ok(())
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}
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```
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Output:
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```
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warmup_period = 11
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[None, None, None, None, None, None, None, None, None, None, Some(11.0), Some(12.0), Some(13.0), Some(14.0), Some(15.0), Some(16.0), Some(17.0), Some(18.0), Some(19.0), Some(20.0)]
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```
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The first `Some` lands at index 10 (the 11th input) — exactly
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`warmup_period() - 1`, as the [Warmup](#warmup) section explains. On the
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linear ramp `1, 2, …, 20`, HMA tracks price exactly with no visible lag.
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### Python
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```python
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import numpy as np
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import wickra as ta
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hma = ta.HMA(9)
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out = hma.batch(np.arange(1.0, 21.0))
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print("warmup_period =", hma.warmup_period())
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print(out)
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```
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Output:
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```
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warmup_period = 11
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[nan nan nan nan nan nan nan nan nan nan 11. 12. 13. 14. 15. 16. 17. 18.
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19. 20.]
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```
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### Node
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```javascript
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const ta = require('wickra');
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const hma = new ta.HMA(9);
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const prices = Array.from({ length: 20 }, (_, i) => i + 1);
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console.log(hma.batch(prices));
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console.log('warmupPeriod:', hma.warmupPeriod());
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```
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Output:
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```
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[
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NaN, NaN, NaN, NaN, NaN, NaN,
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NaN, NaN, NaN, NaN, 11, 12,
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13, 14, 15, 16, 17, 18,
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19, 20
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]
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warmupPeriod: 11
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```
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## Interpretation
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`Hma` is the lag-reduction trend filter that does *not* require you to
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choose between responsiveness and noise: the final `WMA(√period)` pass
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is a built-in smoothing step that prevents the kind of whipsaw a `Tema`
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of the same period would produce on noisy data. On clean trending data
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it sits effectively on top of price; on choppy data the smoothing pass
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keeps the line readable.
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The textbook signal is colour-coded slope: HMA turning up = uptrend,
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turning down = downtrend. Crossover patterns (`Hma(9)` vs `Hma(20)`)
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also work and tend to be cleaner than the equivalent EMA pair.
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Prefer `Hma` over `Dema` / `Tema` when your data is noisy enough that
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the lag-reduction in those would manifest as whipsaws. Prefer `Tema` /
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`Dema` on cleaner data where you want one fewer smoothing step.
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## Common pitfalls
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- **Mis-reading the warmup as a lag.** `warmup_period()` is the exact
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first-emission index (`Hma::new(9).warmup_period() == 11`, first
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`Some` at the 11th input), so it can be used directly for `Chain`
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alignment. The leading `None`/`NaN` values are warmup, not lag — once
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HMA emits it tracks price with near-zero lag.
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- **Picking `period = 2` or `3`.** The inner `half = period / 2` is an
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integer division floored at 1. For `period = 2`, `half = 1`,
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`smooth = 1`, and you essentially end up with `Wma(2·price − WMA(2))`
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which is a sharp, noisy line. HMA is designed for `period >= 9` or so;
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for shorter lookbacks reach for `Ema(period)` or `Wma(period)` instead.
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## References
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Alan Hull, *"How to Reduce Lag in a Moving Average"*, 2005 — the
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original HMA derivation, hosted on Hull's site at
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<https://alanhull.com/hull-moving-average>.
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## See also
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- [Indicator-Wma.md](Indicator-Wma.md) — the building block.
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- [Indicator-Tema.md](Indicator-Tema.md) — same lag-reduction goal, EMA-based.
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- [Indicator-Kama.md](Indicator-Kama.md) — adaptive smoothing instead of fixed.
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- [Indicators-Overview.md](../../Indicators-Overview.md) — the full taxonomy.
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