QuanTAlib/lib/trends_IIR/hema/Hema.md

HEMA: Hull Exponential Moving Average

HMA is a topology. HEMA keeps the topology and swaps the physics: windows to decay, with identical lag.

Property	Value
Category	Trend (IIR MA)
Inputs	Source (close)
Parameters	period
Outputs	Single series (Hema)
Output range	Tracks input
Warmup	EstimateWarmupPeriod() bars
PineScript	hema.pine
Signature	hema_signature

• HEMA is a Hull-style moving average built entirely from exponential smoothers.
• Similar: EMA, DEMA | Complementary: Trend following | Trading note: Hull-style EMA; applies Hulls lag-reduction technique to EMA.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

An EMA-domain analog of HMA with WMA-lag-matched alphas

HEMA is a Hull-style moving average built entirely from exponential smoothers. It preserves the classic HMA pipeline (fast minus slow, then smooth) but replaces WMA sub-filters with EMAs whose alphas are tuned to produce identical lag to the WMA stages they replace. At period N: HEMA(N) and HMA(N) have the same theoretical group delay, but HEMA has infinite memory and smoother transient behavior.

Historical Context

The Hull Moving Average was designed around weighted moving averages (WMA), which have finite memory and are parameterized by a window length. EMA-family filters have infinite memory and are parameterized by a decay rate. Mapping HMA to an EMA world is not "replace WMA with EMA and hope." You need a clear definition of what the period means in EMA terms, and a de-lag combiner that stays consistent when the underlying smoother is exponential.

Early implementations used a half-life mapping (\alpha = 1 - e^{-\ln 2 / N}), but this produces EMA lag \approx 1.44N instead of WMA lag (N-1)/3. The mismatch made HEMA(10) behave like HMA(30) in practice: roughly 4.6x more sluggish at every period. The current implementation uses a WMA-lag-matched alpha (\alpha = 3/(N+2)) that produces exactly the same lag as WMA(N), making period comparisons between HMA and HEMA meaningful.

Architecture and Physics

Topology (the pipeline)

Given input series x_t and user period N:

1. Slow smoother

s_t = \text{EMA}_{\alpha_s}(x_t), \quad \alpha_s = \frac{3}{N+2}

1. Fast smoother (integer floor sub-period, same as HMA)

f_t = \text{EMA}_{\alpha_f}(x_t), \quad \alpha_f = \frac{3}{\lfloor N/2 \rfloor + 2}

1. De-lag combiner (DC gain = 1)

d_t = \frac{f_t - r\,s_t}{1-r}

1. Final smoothing (integer floor sub-period, same as HMA)

\text{HEMA}_t = \text{EMA}_{\alpha_m}(d_t), \quad \alpha_m = \frac{3}{\lfloor\sqrt{N}\rfloor + 2}

This mirrors classic HMA:

\text{HMA}_N(x) = \text{WMA}_{\lfloor\sqrt{N}\rfloor}\!\left(2\,\text{WMA}_{\lfloor N/2\rfloor}(x)-\text{WMA}_N(x)\right)

The difference: HEMA's stages are exponential with infinite memory. The sub-periods use integer floor division to match HMA's behavior exactly.

WMA-lag-matched alpha (what "Period" actually means)

HEMA's Period = N means: the EMA has the same lag as WMA(N).

For a WMA of length N, the steady-state mean lag is:

\text{lag}_{\text{WMA}}(N) = \frac{N-1}{3}

For an EMA with smoothing constant \alpha, the steady-state mean lag is:

\text{lag}_{\text{EMA}}(\alpha) = \frac{1-\alpha}{\alpha}

Setting these equal and solving for \alpha:

\frac{1-\alpha}{\alpha} = \frac{N-1}{3} \implies \alpha = \frac{3}{N+2}

This makes "WMA-equivalent period" the primitive, and \alpha derived. At N=10: \alpha = 3/12 = 0.25, lag = 0.75/0.25 = 3.0 bars, exactly matching WMA(10) lag.

The de-lag ratio r: derived, not guessed

Classic HMA uses 2f - s. That implicitly assumes a particular lag relationship between the fast and slow smoothers.

In EMA space, the "correct" proportionality uses EMA's steady-state mean lag:

\text{lag}(\alpha)\approx \frac{1-\alpha}{\alpha}

Compute:

r = \frac{\text{lag}_\text{fast}}{\text{lag}_\text{slow}} = \frac{(1-\alpha_f)/\alpha_f}{(1-\alpha_s)/\alpha_s}

Then the combiner:

d_t = \frac{f_t - r\,s_t}{1-r}

Why this form?

• DC gain is exactly 1 (flat input stays flat).
• For "large" N (small \alpha), the ratio tends toward:

r \approx \frac{\alpha_s}{\alpha_f} \approx \frac{1}{2}

and the combiner approaches d_t \approx 2f_t - s_t, i.e., the classic HMA shape emerges as a limiting case.

Warmup: unbiased EMA from bar 1

Raw EMA recursion assumes the filter has run forever. Early outputs are biased toward zero (or the initial state). HEMA uses exact bias compensation during warmup by tracking each stage's decay:

If y_t is the raw EMA state and \beta = 1-\alpha, the bias-corrected output is:

y_t^{*} = \frac{y_t}{1-\beta^{t}}

HEMA performs this independently for slow stage, fast stage, and smooth stage, and exits warmup only when all three decays are negligible.

Practical implication: early samples converge fast to a meaningful value. Use IsHot (or WarmupPeriod) if you need "fully settled" behavior for signal generation.

Mathematical Foundation

WMA-lag-matched alpha:

\alpha = \frac{3}{N+2}

where N is the period parameter (minimum 2). This produces EMA lag = (N-1)/3 = WMA(N) lag.

Sub-period alphas (integer floor, matching HMA):

\alpha_{\text{slow}} = \frac{3}{N+2}, \quad \alpha_{\text{fast}} = \frac{3}{\lfloor N/2 \rfloor+2}, \quad \alpha_{\text{smooth}} = \frac{3}{\lfloor\sqrt{N}\rfloor+2}

EMA recursion:

\text{EMA}_{t} = \alpha \cdot x_t + (1 - \alpha) \cdot \text{EMA}_{t-1}

Bias-compensated EMA:

\text{EMA}_{t}^{*} = \frac{\text{EMA}_{t}}{1 - (1-\alpha)^{t}}

De-lag combiner:

d_t = \frac{f_t - r \cdot s_t}{1 - r}

where:

r = \frac{(1-\alpha_f)/\alpha_f}{(1-\alpha_s)/\alpha_s}

Final output:

\text{HEMA}_t = \text{EMA}_{\text{smooth}}(d_t)

Performance Profile

Operation Count (Streaming Mode, Scalar)

Hot Path (Post-Warmup):

Operation	Count	Cost (cycles)	Subtotal
Stage 1: EMA Slow
FMA (emaSlowRaw x betaSlow + alphaSlow x input)	1	4	4
MUL (alphaSlow x input)	1	3	3
Stage 2: EMA Fast
FMA (emaFastRaw x betaFast + alphaFast x input)	1	4	4
MUL (alphaFast x input)	1	3	3
Stage 3: De-Lag Combiner
FMA (-ratio x emaSlow + emaFast)	1	4	4
MUL (x invOneMinusRatio)	1	3	3
Stage 4: Final EMA Smooth
FMA (emaSmoothRaw x betaSmooth + alphaSmooth x deLag)	1	4	4
MUL (alphaSmooth x deLag)	1	3	3
Total (Hot Path)			~28 cycles

Warmup Path (Additional Operations):

Operation	Count	Cost (cycles)	Subtotal
MUL (decay x beta)	3	3	9
DIV (1 / (1 - decay))	3	15	45
MUL (raw x invDecay)	3	3	9
CMP/MAX (decay comparisons)	3	1	3
Total (Warmup)			~66 cycles

Warmup total: ~94 cycles | Hot path total: ~28 cycles

Batch Mode (SIMD Analysis)

HEMA is not SIMD-parallelizable across bars due to:

1. All three EMA stages are recursive IIR filters (output[t] depends on output[t-1])
2. De-lag combiner depends on current slow/fast EMA values
3. Final smoother depends on de-lagged series

FMA optimization (already applied): All EMA updates use Math.FusedMultiplyAdd for single-rounding precision.

Quality Metrics

Metric	Score	Notes
Accuracy	9/10	WMA-lag-matched alphas produce identical theoretical lag to HMA
Timeliness	8/10	Faster response than plain EMA via de-lag combiner
Overshoot	6/10	De-lag combiner can overshoot during sharp reversals
Smoothness	7/10	Smoother than DEMA, less smooth than T3

Benchmark environment: .NET 10, Release build, no SIMD (stateful recursion). Measured via BenchmarkDotNet on synthetic GBM data (mu=0.0001, sigma=0.02, 10K bars).

Validation

HEMA is not commonly available in mainstream TA libraries. Validation uses a reference implementation.

Library	Status	Tolerance	Notes
TA-Lib	N/A	-	Not implemented
Skender	N/A	-	Not implemented
Tulip	N/A	-	Not implemented
Ooples	N/A	-	Not implemented
PineScript	Passed	1e-10	Matches lib/trends_IIR/hema/hema.pine

Validation strategy:

• PineScript reference is authoritative (included in repo).
• Cross-check via invariant tests: DC gain, step response monotonicity, no NaN propagation after first finite sample.
• Streaming vs batch vs span consistency verified in unit tests.

Common Pitfalls

1. Period semantics are now WMA-lag-matched

   Period = N means "same lag as WMA(N)." HEMA(10) and HMA(10) have the same theoretical group delay. Earlier versions used half-life semantics where HEMA(10) was roughly equivalent to HMA(30). If you are upgrading from the half-life version, expect HEMA to now be noticeably more responsive at the same period.

2. Warmup assumptions

   Early values are bias-corrected, but "fully settled" still takes time. Use IsHot / WarmupPeriod before acting on signals. Expect roughly 3\sqrt{N} bars for all three stages to stabilize.

3. Overshoot on reversals

   De-lag can overshoot. This is the price of reduced lag, same tradeoff as the DEMA/ZLEMA family. If overshoot is unacceptable, prefer a slower final smoother or reduce de-lag strength (requires custom variant).

4. Non-finite data handling

   Non-finite values are substituted with last valid value. Before the first valid input, output is NaN. If your upstream data source produces frequent gaps, consider pre-filtering or using a different indicator.

5. Bar correction discipline

   Use isNew=false when correcting the last bar (same timestamp, revised OHLC). Failing to do so causes state drift and inconsistent results across runs.

6. Integer floor sub-periods

   Sub-periods use integer floor division (period / 2, (int)Math.Sqrt(period)) to match HMA behavior exactly. This means HEMA(5) uses halfPeriod=2 and sqrtPeriod=2, not 2.5 and 2.236.

References

• Hull, A. "Hull Moving Average." Technical analysis methodology using WMA lag cancellation.
• Wolfram Alpha verification: EMA lag with alpha=3/(N+2) equals (N-1)/3, matching WMA(N) lag exactly.