QuanTAlib/lib/trends_IIR/zlema/Zlema.md

ZLEMA: Zero-Lag Exponential Moving Average

EMA with lag compensation via a zero-lag signal

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

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.