QuanTAlib/lib/trends_IIR/zldema/Zldema.md

ZLDEMA: Zero-Lag Double Exponential Moving Average

ZLDEMA combines the speed of zero-lag prediction with the smoothness of double exponential averaging. You get faster response than ZLEMA, with better trend-following than DEMA.

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

• ZLDEMA takes a standard DEMA and feeds it a zero-lag signal: current price minus a lagged price.
• Similar: ZLEMA, DEMA | Complementary: Signal line crossovers | Trading note: Zero-Lag DEMA; combines zero-lag with double-exponential.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

DEMA with lag compensation via a zero-lag signal

ZLDEMA takes a standard DEMA and feeds it a zero-lag signal: current price minus a lagged price. This produces a smoother that responds faster than DEMA without going fully raw. The dual EMA cascade provides additional noise rejection while the zero-lag preprocessing maintains responsiveness.

Historical Context

ZLDEMA extends the zero-lag concept from ZLEMA to double exponential moving averages. Where ZLEMA applies lag compensation to a single EMA, ZLDEMA applies it to a two-stage EMA cascade using the DEMA formula (2*EMA1 - EMA2). This combination targets the middle ground between ZLEMA's speed and TEMA's smoothness.

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. First EMA stage

\text{EMA1}_t = \text{EMA}(s_t, \alpha)

4. Second EMA stage

\text{EMA2}_t = \text{EMA}(\text{EMA1}_t, \alpha)

5. DEMA output

\text{ZLDEMA}_t = 2 \cdot \text{EMA1}_t - \text{EMA2}_t

Warmup compensation

ZLDEMA uses EMA bias compensation during warmup on both EMA stages:

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}}

DEMA formula:

\text{DEMA}_t = 2 \cdot \text{EMA1}_t - \text{EMA2}_t

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	4	4	16
MUL	2	3	6
Total	6		~22 cycles

The hot path consists of:

1. Zero-lag signal: FMA(2.0, val, -lagged) - 1 FMA
2. EMA1 core: FMA(ema1Raw, beta, alpha * signal) - 1 FMA + 1 MUL
3. EMA2 core: FMA(ema2Raw, beta, alpha * ema1) - 1 FMA + 1 MUL
4. DEMA output: FMA(2.0, ema1, -ema2) - 1 FMA

Warmup path (with bias compensation):

Operation	Count	Cost (cycles)	Subtotal
FMA	4	4	16
MUL	4	3	12
DIV	1	15	15
CMP	2	1	2
Total	11		~45 cycles

Additional warmup operations:

• Decay tracking: e *= beta - 1 MUL
• Compensator calc: 1 / (1 - e) - 1 DIV
• Bias compensation: ema1Raw * compensator, ema2Raw * compensator - 2 MUL
• Hot/compensated checks - 2 CMP

Batch Mode (SIMD Analysis)

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

Optimization	Benefit
FMA instructions	~22 cycles vs ~28 scalar
stackalloc buffer	Zero heap allocation for lag ≤256

Quality Metrics

Metric	Score	Notes
Accuracy	8/10	Matches PineScript reference
Timeliness	9/10	Faster response than DEMA, comparable to ZLEMA
Overshoot	5/10	Predictive signal plus DEMA amplification causes overshoot
Smoothness	7/10	Smoother than ZLEMA due to dual EMA cascade

Validation

ZLDEMA is validated against a PineScript reference implementation.

Library	Status	Tolerance	Notes
TA-Lib	N/A	-	No ZLDEMA in TA-Lib
Skender	N/A	-	No ZLDEMA in Skender
Tulip	N/A	-	No ZLDEMA in Tulip
Ooples	N/A	-	No ZLDEMA in Ooples
PineScript	✓ Passed	1e-10	Matches lib/trends_IIR/zldema/zldema.pine

Common Pitfalls

1. Increased overshoot on turns

   The zero-lag signal is a forward estimate, and the DEMA formula (2*EMA1 - EMA2) further amplifies deviations. Expect more overshoot than ZLEMA when price reverses sharply.

2. Period semantics

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

3. Warmup discipline

   Use IsHot / WarmupPeriod before acting on signals. Early values are bias-corrected but still unstable. The dual EMA cascade requires longer warmup than single-stage ZLEMA.

4. Non-finite data

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

5. DEMA vs ZLDEMA

   ZLDEMA is not simply DEMA with a different alpha. The zero-lag preprocessing fundamentally changes the input signal, making ZLDEMA more responsive but also more prone to overshoot than standard DEMA.