wickra/docs/wiki/indicators/moving-averages/Indicator-T3.md

T3

Tillson T3 — a six-fold cascaded EMA recombined with a volume factor v to give a smooth, low-lag trend line.

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 required; v = 0.7 (Python default)
Warmup period	6·period − 5
Interpretation	Smooth trend line with less lag than a same-period EMA.

Formula

T3 is the generalised DEMA (GD) applied three times. Tim Tillson's expansion of GD(GD(GD(price))) over six chained EMAs — e1 … e6, each of the same period, where e2 = EMA(e1), e3 = EMA(e2), … — is:

v2 = v²,  v3 = v³
c1 = −v3
c2 = 3·v2 + 3·v3
c3 = −6·v2 − 3·v − 3·v3
c4 = 1 + 3·v + v3 + 3·v2
T3 = c1·e6 + c2·e5 + c3·e4 + c4·e3

The four coefficients always sum to 1, so a constant price series maps to itself. The volume factor v controls the lag/overshoot trade-off: v = 0 collapses T3 to the plain triple-cascaded EMA e3; the conventional v = 0.7 adds a corrective hump that sharpens turns.

Parameters

Name	Type	Default	Valid range	Description
period	usize	none	>= 1	Length of every EMA in the cascade. period = 0 errors with Error::PeriodZero.
v	f64	0.7 (Python)	[0.0, 1.0]	Volume factor. Non-finite or out-of-range values error with Error::InvalidPeriod.

The Python binding defaults v to 0.7 via #[pyo3(signature = (period, v=0.7))]; period is always explicit. The Node and WASM constructors take both arguments explicitly.

Inputs / Outputs

From crates/wickra-core/src/indicators/t3.rs:

impl Indicator for T3 {
    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

T3::new(period, v).warmup_period() == 6·period − 5. Each stage of the SMA-seeded EMA cascade adds period − 1 bars of delay: e1 seeds at input period, e2 at 2·period − 1, …, e6 at 6·period − 5. T3 emits its first value once e6 is ready, since the output formula needs e3 through e6.

Edge cases

• Constant series. Because c1 + c2 + c3 + c4 = 1 for any v, a flat input series produces a flat output equal to the constant (coefficients_sum_to_one and constant_series_yields_the_constant pin this).
• v = 0. The coefficients become c1 = c2 = c3 = 0, c4 = 1, so T3 is exactly the third stage of the EMA cascade (zero_volume_factor_collapses_to_triple_cascaded_ema pins this).
• NaN / infinity inputs. Non-finite inputs are silently dropped — the cascade is not advanced — and the previous valid value is returned.
• Reset. t3.reset() clears all six EMAs and the cached value.

Examples

Rust

use wickra::{BatchExt, Indicator, T3};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let prices: Vec<f64> = (1..=40).map(f64::from).collect();
    let mut t3 = T3::new(3, 0.7)?;
    let out = t3.batch(&prices);
    println!("warmup_period = {}", t3.warmup_period());
    println!("first ready index = {:?}", out.iter().position(Option::is_some));
    Ok(())
}

Output:

warmup_period = 13
first ready index = Some(12)

T3(3, 0.7) warms up after 6·3 − 5 = 13 inputs, so the first non-None output sits at index 12. On a pure ramp the output then tracks the input trend with a smooth, near-constant offset.

Python

import numpy as np
import wickra as ta

t3 = ta.T3(5)  # v defaults to 0.7
prices = np.linspace(100.0, 140.0, 60)
out = t3.batch(prices)
print("warmup_period =", t3.warmup_period())
print("ready values:", np.count_nonzero(~np.isnan(out)))

Output:

warmup_period = 25
ready values: 36

Node

const ta = require('wickra');
const t3 = new ta.T3(5, 0.7);
const prices = Array.from({ length: 60 }, (_, i) => 100 + i);
console.log('warmupPeriod:', t3.warmupPeriod());
console.log('last:', t3.batch(prices).at(-1));

Interpretation

T3 is a "best of both" trend line — close to Tema in lag reduction but visibly smoother, because the six-EMA cascade filters noise the three-EMA Tema lets through. Use it as a single trend filter or as the slow leg of a crossover where you want a clean line. Raise v toward 1 for sharper turns (more overshoot), lower it toward 0 for maximum smoothness (v = 0 is just a triple EMA).

Common pitfalls

• Treating v as optional outside Python. Only the Python binding defaults v to 0.7; the Rust, Node and WASM constructors require it.
• Underestimating warmup. 6·period − 5 grows fast — a T3(20) needs 115 bars before its first value.

References

Tim Tillson, "Better Moving Averages", Technical Analysis of Stocks & Commodities (1998). The six-EMA expansion and coefficient formulas here match Tillson's published derivation and TA-Lib's T3.

See also

• Indicator-Tema.md — the three-EMA relative.
• Indicator-Dema.md — the two-EMA relative.
• Indicator-Zlema.md — low-lag average via de-lagging.
• Indicators-Overview.md — the full taxonomy.