QuanTAlib/lib/trends_IIR/zltema/Zltema.md

ZLTEMA: Zero-Lag Triple Exponential Moving Average

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

TL;DR

• ZLTEMA takes a standard TEMA and feeds it a zero-lag signal: current price minus a lagged price.
• Parameterized by period.
• Output range: Tracks input.
• Requires Math.Max(lag + 1, EstimateWarmupPeriod(beta)) bars of warmup before first valid output (IsHot = true).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

"ZLTEMA combines the speed of zero-lag prediction with the smoothness of triple exponential averaging. You get the fastest response in the zero-lag family, with the best noise rejection from the TEMA cascade."

TEMA with lag compensation via a zero-lag signal

ZLTEMA takes a standard TEMA and feeds it a zero-lag signal: current price minus a lagged price. This produces a smoother that responds faster than TEMA without going fully raw. The triple EMA cascade provides maximum noise rejection in the exponential family while the zero-lag preprocessing maintains responsiveness.

Historical Context

ZLTEMA extends the zero-lag concept from ZLEMA to triple exponential moving averages. Where ZLEMA applies lag compensation to a single EMA and ZLDEMA to a double cascade, ZLTEMA applies it to a three-stage EMA cascade using the TEMA formula (3EMA1 - 3EMA2 + EMA3). This combination targets the extreme end: maximum smoothness with minimal lag.

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. Third EMA stage

\text{EMA3}_t = \text{EMA}(\text{EMA2}_t, \alpha)

6. TEMA output

\text{ZLTEMA}_t = 3 \cdot \text{EMA1}_t - 3 \cdot \text{EMA2}_t + \text{EMA3}_t

Warmup compensation

ZLTEMA uses EMA bias compensation during warmup on all three 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}}

TEMA formula:

\text{TEMA}_t = 3 \cdot \text{EMA1}_t - 3 \cdot \text{EMA2}_t + \text{EMA3}_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	6	4	24
MUL	3	3	9
Total	9		~33 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. EMA3 core: FMA(ema3Raw, beta, alpha * ema2) - 1 FMA + 1 MUL
5. TEMA output: FMA(3.0, ema1, FMA(-3.0, ema2, ema3)) - 2 FMA (nested)

Warmup path (with bias compensation):

Operation	Count	Cost (cycles)	Subtotal
FMA	6	4	24
MUL	5	3	15
DIV	1	15	15
CMP	2	1	2
Total	14		~56 cycles

Additional warmup operations:

• Decay tracking: e *= beta - 1 MUL
• Compensator calc: 1 / (1 - e) - 1 DIV
• Bias compensation: ema1Raw * compensator, ema2Raw * compensator, ema3Raw * compensator - 3 MUL
• Hot/compensated checks - 2 CMP

Batch Mode (SIMD Analysis)

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

Optimization	Benefit
FMA instructions	~33 cycles vs ~42 scalar
stackalloc buffer	Zero heap allocation for lag ≤256

Quality Metrics

Metric	Score	Notes
Accuracy	8/10	Matches PineScript reference
Timeliness	10/10	Fastest response in ZL family
Overshoot	4/10	Predictive signal plus TEMA amplification causes significant overshoot
Smoothness	8/10	Smoothest in ZL family due to triple EMA cascade

Validation

ZLTEMA is validated against a PineScript reference implementation.

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

Common Pitfalls

1. Maximum overshoot on turns

   The zero-lag signal is a forward estimate, and the TEMA formula (3EMA1 - 3EMA2 + EMA3) has the highest amplification in the exponential family. Expect more overshoot than ZLDEMA or ZLEMA when price reverses sharply.

2. Period semantics

   ZLTEMA uses EMA alpha; the lag term is derived from period but not equivalent to a window length. Do not compare ZLTEMA 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 triple EMA cascade requires longer warmup than ZLDEMA or ZLEMA.

4. Non-finite data

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

5. TEMA vs ZLTEMA

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

6. ZLDEMA vs ZLTEMA

   ZLTEMA adds a third EMA stage over ZLDEMA. This provides additional smoothing at the cost of more overshoot during reversals. Use ZLDEMA when overshoot is more concerning than noise; use ZLTEMA when maximum smoothness is required.