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

TEMA

Triple Exponential Moving Average — Mulloy's 3·EMA1 − 3·EMA2 + EMA3 (where each EMA is fed from the previous one), the second-order lag-reduction sibling of DEMA.

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	3·period − 2
Interpretation	Even less lag than Dema, at the cost of more noise sensitivity.

Formula

Let EMA1 = EMA(price, period), EMA2 = EMA(EMA1, period), EMA3 = EMA(EMA2, period). Then:

TEMA_t = 3 * EMA1_t - 3 * EMA2_t + EMA3_t

All three EMAs share the same period, hence the same α = 2 / (period + 1). The coefficients (3, −3, 1) are the second-order finite-difference correction that removes both the first-order and second-order EMA lag terms — they come from expanding (1 − L)^{-3} where L is the lag operator.

Parameters

Name	Type	Default	Valid range	Description
period	usize	none	>= 1	Period shared by all three internal EMAs. period = 0 errors with Error::PeriodZero.

(Python class wickra.TEMA(period) has no #[pyo3(signature)] default; pass period explicitly.)

Inputs / Outputs

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

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

Python update returns float | None, batch returns a 1-D numpy.ndarray (float64, NaN for warmup). Node update returns number | null, batch returns Array<number> with NaN placeholders.

Warmup

Tema::new(period).warmup_period() == 3 * period - 2. Each stacked EMA adds period − 1 more inputs to the warmup count:

• ema1 emits first at input period.
• ema2, fed from ema1, emits first at input period + (period − 1) = 2·period − 1.
• ema3, fed from ema2, emits first at input (2·period − 1) + (period − 1) = 3·period − 2.

For Tema::new(14) this gives 40 (matches the table in Warmup Periods); for Tema::new(5) (the example below) it gives 13. The implementation uses ? short-circuit on every stage, so each inner EMA is only fed once the previous one emits.

Edge cases

• Constant series. Feeding [42.0; n] produces Some(42.0) once all three EMAs have converged: 3·42 − 3·42 + 42 = 42. The unit test constant_series_yields_constant_tema pins this with Tema::new(5) over 80 constants.
• NaN / infinity inputs. Inherited from the inner Ema: non-finite inputs are silently dropped at the ema1 boundary and never reach the 3·EMA1 − 3·EMA2 + EMA3 arithmetic.
• Reset. tema.reset() resets all three internal EMAs; the next update starts a full 3·period − 2 warmup countdown.

Examples

Rust

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

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let mut tema = Tema::new(5)?;
    let prices: Vec<f64> = (1..=20).map(f64::from).collect();
    let out: Vec<Option<f64>> = tema.batch(&prices);
    println!("warmup_period = {}", tema.warmup_period());
    println!("{:?}", out);
    Ok(())
}

Output:

warmup_period = 13
[None, None, None, None, None, None, None, None, None, None, None, None, Some(13.0), Some(14.0), Some(15.000000000000002), Some(16.000000000000004), Some(17.000000000000007), Some(18.000000000000007), Some(19.000000000000007), Some(20.0)]

The first Some lands at index 12 (the 13th input), matching 3·5 − 2 = 13. On the linear ramp 1, 2, …, 20, TEMA tracks the input ramp essentially exactly because both first- and second-order lag have been cancelled; the floating-point tail (15.000000000000002, 16.000000000000004, …) is ordinary IEEE-754 drift from the recursive subtractions.

Python

import numpy as np
import wickra as ta

tema = ta.TEMA(5)
out = tema.batch(np.arange(1.0, 21.0))
print("warmup_period =", tema.warmup_period())
print(out)

Output:

warmup_period = 13
[nan nan nan nan nan nan nan nan nan nan nan nan 13. 14. 15. 16. 17. 18.
 19. 20.]

Node

const ta = require('wickra');
const tema = new ta.TEMA(5);
const prices = Array.from({ length: 20 }, (_, i) => i + 1);
console.log(tema.batch(prices));
console.log('warmupPeriod:', tema.warmupPeriod());

Output:

[
                 NaN,                NaN,
                 NaN,                NaN,
                 NaN,                NaN,
                 NaN,                NaN,
                 NaN,                NaN,
                 NaN,                NaN,
                  13,                 14,
  15.000000000000002, 16.000000000000004,
  17.000000000000007, 18.000000000000007,
  19.000000000000007,                 20
]
warmupPeriod: 13

Interpretation

Tema removes more lag than Dema and noticeably more than Ema. On a clean trending series the line stays glued to price; on a noisy or sideways series the same lag-cancellation amplifies the noise — TEMA overshoots and reverses faster than DEMA, and very much faster than EMA.

The signals are the same crossover patterns: price-vs-TEMA and fast-TEMA-vs-slow-TEMA. The (3, −3, 1) coefficient pattern is also what makes Trix (also in this family) work — Trix is the percentage change of EMA3, the triple-smoothed series.

Prefer Tema when Dema still feels too laggy and your data is clean enough to tolerate the extra noise sensitivity. Prefer Hma if you want a similar lag profile but with a built-in smoothing step (WMA chain instead of EMA chain), which behaves more gracefully on noisy data.

Common pitfalls

• Forgetting the 3·period − 2 warmup. Tema::new(50) will not emit until input 148. That is a significant chunk of any short-term backtest. If you are running a side-by-side panel of indicators with different warmups, filter rows on ~np.isnan(...) (Python) / is_some() (Rust) per indicator rather than picking one global warmup cutoff.
• Using TEMA for noisy intraday data without a smoothing step. The same lag-cancellation that makes TEMA attractive on clean data turns into whipsaws on tick-by-tick feeds. Either raise period materially or switch to Hma, which has a final WMA smoothing pass built in.

References

Patrick G. Mulloy, "Smoothing Data with Less Lag", Technical Analysis of Stocks & Commodities, February 1994 (TEMA). The coefficient pattern (3, −3, 1) for cancelling first- and second-order EMA lag is derived in the same article.

See also

• Indicator-Ema.md — the building block.
• Indicator-Dema.md — second-order's sibling.
• Indicator-Hma.md — similar lag profile, built on WMAs.
• Indicators-Overview.md — the full taxonomy.