QuanTAlib/lib/trends_IIR/zltema/Zltema.md

ZLTEMA: Zero-Lag Triple Exponential Moving Average

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.

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
PineScript	zltema.pine
Signature	zltema_signature

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

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.