QuanTAlib/lib/trends_IIR/zlema/Zlema.md

ZLEMA: Zero-Lag Exponential Moving Average

ZLEMA does not erase lag. It predicts just enough to act early, then pays the price in overshoot.

Property	Value
Category	Trend (IIR MA)
Inputs	Source (close)
Parameters	period
Outputs	Single series (Zlema)
Output range	Tracks input
Warmup	Math.Max(lag + 1, EstimateWarmupPeriod(beta)) bars
PineScript	zlema.pine
Signature	zlema_signature

• ZLEMA takes a standard EMA and feeds it a zero-lag signal: current price minus a lagged price.
• Similar: DEMA, HMA | Complementary: Momentum confirmation | Trading note: Zero-Lag EMA; pre-adjusts input to remove delay.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

EMA with lag compensation via a zero-lag signal

ZLEMA takes a standard EMA and feeds it a zero-lag signal: current price minus a lagged price. This produces a smoother that responds faster than EMA without going fully raw. It is not magic. It shifts some lag into controlled overshoot.

Historical Context

ZLEMA is a widely used variation on EMA intended to reduce delay without abandoning exponential smoothing. It appears in multiple technical analysis toolkits and is often described as a "predictive EMA." The prediction is simple: extrapolate the current price by subtracting a lagged value.

Architecture & Physics

Pipeline

1. Lag estimate

\text{lag} = \max(1, \text{round}((N-1)/2))

2. Zero-lag signal

s_t = 2 \cdot x_t - x_{t-\text{lag}}

3. EMA smoothing

\text{ZLEMA}_t = \text{EMA}(s_t, \alpha)

Warmup compensation

ZLEMA uses EMA bias compensation during warmup:

y_t^{*} = \frac{y_t}{1 - (1 - \alpha)^t}

This avoids the early-stage bias toward zero and makes the first values usable.

Math Foundation

EMA update:

y_t = y_{t-1} + \alpha (s_t - y_{t-1})

Zero-lag signal:

s_t = 2 \cdot x_t - x_{t-\text{lag}}

Alpha from period:

\alpha = \frac{2}{N + 1}

Performance Profile

Operation Count (Streaming Mode, Scalar)

Hot path (after warmup, compensation complete):

Operation	Count	Cost (cycles)	Subtotal
FMA	2	4	8
MUL	1	3	3
Total	3		~11 cycles

The hot path consists of:

1. Zero-lag signal: FMA(2.0, val, -lagged)  1 FMA
2. EMA core: FMA(zlemaRaw, beta, alpha * signal)  1 FMA + 1 MUL

Warmup path (with bias compensation):

Operation	Count	Cost (cycles)	Subtotal
FMA	2	4	8
MUL	2	3	6
DIV	1	15	15
CMP	2	1	2
Total	7		~31 cycles

Additional warmup operations:

• Decay tracking: e *= beta  1 MUL
• Bias compensation: zlemaRaw / (1 - e)  1 DIV
• Hot/compensated checks  2 CMP

Batch Mode (SIMD Analysis)

ZLEMA is an IIR filter with lag buffer dependency  not directly vectorizable across bars. However, within-bar operations use FMA intrinsics.

Optimization	Benefit
FMA instructions	~11 cycles vs ~14 scalar
stackalloc buffer	Zero heap allocation for lag d256

Quality Metrics

Metric	Score	Notes
Accuracy	8/10	Matches PineScript reference
Timeliness	8/10	Faster response than EMA
Overshoot	6/10	Predictive signal causes overshoot on reversals
Smoothness	7/10	Between EMA and raw price

Validation

ZLEMA is validated against a PineScript reference implementation.

Library	Status	Tolerance	Notes
TA-Lib	N/A	-	No ZLEMA in TA-Lib
Skender	N/A	-	No ZLEMA in Skender
Tulip	Partial	-	Tulip has zlema but not used here
Ooples	N/A	-	No ZLEMA in Ooples
PineScript	? Passed	1e-10	Matches lib/trends_IIR/zlema/zlema.pine

Common Pitfalls

1. Overshoot on turns

   The zero-lag signal is a forward estimate. It can overshoot when price reverses sharply. This is expected behavior.

2. Period semantics

   ZLEMA uses EMA alpha; the lag term is derived from period but not equivalent to a window length. Do not compare ZLEMA period directly to SMA window length.

3. Warmup discipline

   Use IsHot / WarmupPeriod before acting on signals. Early values are bias-corrected but still unstable.

4. Non-finite data

   NaN or Infinity is replaced with the last valid value. Before the first valid sample, output is NaN.