wickra/docs/wiki/indicators/moving-averages/Indicator-Zlema.md

ZLEMA

Zero-Lag Exponential Moving Average — an EMA fed a de-lagged price series so it tracks turns with almost no group delay.

Quick reference

Field	Value
Family	Moving Averages
Input type	f64 (single close)
Output type	f64
Output range	unbounded; tracks the input price scale
Default parameters	period is required (no default in either binding)
Warmup period	lag + period where lag = (period − 1) / 2
Interpretation	Low-lag trend line; crossings of price react far sooner than a plain EMA.

Formula

lag        = (period − 1) / 2          (integer division)
de_lagged_t = 2·price_t − price_{t−lag}
ZLEMA_t     = EMA_period(de_lagged)_t

The trick (Ehlers & Way, 2010): price_t − price_{t−lag} is a momentum term. Adding it to the current price over-shoots in the direction of the recent move by exactly enough to cancel the EMA's lag. The inner EMA then smooths that de-lagged series with the usual α = 2 / (period + 1).

Parameters

Name	Type	Default	Valid range	Description
period	usize	none	>= 1	EMA length. period = 0 errors with Error::PeriodZero. The lag offset is derived as (period − 1) / 2.

There is no Python #[pyo3(signature = …)] default for ZLEMA, so wickra.ZLEMA(period) requires the period explicitly. The derived lag is exposed as a read-only property.

Inputs / Outputs

From crates/wickra-core/src/indicators/zlema.rs:

impl Indicator for Zlema {
    type Input = f64;
    type Output = f64;
    // update(&mut self, input: f64) -> Option<f64>
}

A single f64 close in, an Option<f64> out. Python maps this to float | None / numpy.ndarray (NaN warmup); Node to number | null / Array<number> (NaN warmup).

Warmup

Zlema::new(period).warmup_period() == lag + period. The de-lagged series is undefined until lag prior inputs exist, so it produces its first value on input lag + 1; the inner EMA then needs period de-lagged values to seed. The first non-None output therefore lands on input lag + period.

Edge cases

• Constant series. De-lagging a constant gives the same constant (2c − c = c), so ZLEMA of a flat series is flat (constant_series_yields_the_constant pins this).
• NaN / infinity inputs. Non-finite inputs are silently dropped: the rolling lag buffer is not advanced and the inner EMA is not fed, so the previous valid value (if any) is returned.
• period = 1. lag = 0, the de-lagged series equals the raw price, and ZLEMA(1) degenerates to a pass-through.
• Reset. zlema.reset() clears the lag buffer and the inner EMA.

Examples

Rust

use wickra::{BatchExt, Indicator, Zlema};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let mut zlema = Zlema::new(3)?;
    let out: Vec<Option<f64>> = zlema.batch(&[1.0, 2.0, 3.0, 4.0, 5.0]);
    println!("{:?}", out);
    println!("lag = {}, warmup_period = {}", zlema.lag(), zlema.warmup_period());
    Ok(())
}

Output:

[None, None, None, Some(4.0), Some(5.0)]
lag = 1, warmup_period = 4

ZLEMA(3) has lag = 1. The de-lagged series of [1,2,3,4,5] is [_, 3, 4, 5, 6]; EMA(3) of that seeds at mean(3,4,5) = 4.0, then 0.5·6 + 0.5·4 = 5.0. This matches the reference_values test in crates/wickra-core/src/indicators/zlema.rs.

Python

import numpy as np
import wickra as ta

zlema = ta.ZLEMA(3)
print(zlema.batch(np.array([1.0, 2.0, 3.0, 4.0, 5.0])))
print("lag =", zlema.lag, "warmup_period =", zlema.warmup_period())

Output:

[nan nan nan  4.  5.]
lag = 1 warmup_period = 4

Node

const ta = require('wickra');
const zlema = new ta.ZLEMA(3);
console.log(zlema.batch([1, 2, 3, 4, 5]));
console.log('warmupPeriod:', zlema.warmupPeriod());

Output:

[ NaN, NaN, NaN, 4, 5 ]
warmupPeriod: 4

Interpretation

Zlema is a low-lag trend line. Use it where an Ema would lag too much into a reversal — for example as the fast leg of a crossover system, or as a trailing reference that should react quickly. The momentum injection that removes the lag also makes Zlema overshoot on sharp spikes, so it is noisier than the Ema it is built on; pair it with a slower filter if whipsaws are a concern.

Common pitfalls

• Expecting Ema-identical values. Zlema is deliberately not an Ema — it leads price. The two only coincide for period = 1.
• Forgetting the extra warmup. Warmup is lag + period, not period; budget (period − 1) / 2 extra bars before the first output.

References

John Ehlers and Ric Way, "Zero Lag (Well, Almost)", Technical Analysis of Stocks & Commodities (2010). The implementation here uses the standard lag = (period − 1) / 2 and an SMA-seeded inner EMA.

See also

• Indicator-Ema.md — the inner average ZLEMA de-lags.
• Indicator-Hma.md — another low-lag average, via WMAs.
• Indicator-T3.md — low-lag average via a six-EMA cascade.
• Indicators-Overview.md — the full taxonomy.