TEMA: Triple Exponential Moving Average

"Patrick Mulloy looked at the lag of an EMA and took it personally. TEMA is what happens when you apply algebra to impatience."

The Triple Exponential Moving Average (TEMA) is a lag-reducing filter that combines a single, double, and triple EMA. Unlike a simple triple smoothing (which would be incredibly slow), TEMA uses a weighted combination of the three to cancel out the lag, resulting in an indicator that hugs price action tighter than a spandex cycling short.

Historical Context

Introduced by Patrick Mulloy in Technical Analysis of Stocks & Commodities (Jan 1994), "Smoothing Data With Less Lag." Mulloy's goal was to replace the standard moving averages in MACD and other indicators to reduce the delay in signal generation.

Architecture & Physics

TEMA is not just "EMA applied three times." That would be EMA(EMA(EMA(x))). TEMA is a composite:

TEMA = 3 \cdot EMA_1 - 3 \cdot EMA_2 + EMA_3

This formula effectively projects the trend forward to compensate for the delay inherent in smoothing.

Convergence Speed

Because of the aggressive weighting, TEMA converges (warms up) faster than a standard EMA. While an EMA takes \approx 3.45(N+1) steps to converge to 99.9%, TEMA stabilizes quicker due to the subtraction terms canceling out the initial error.

Mathematical Foundation

1. The Cascade

EMA_1 = EMA(Price) EMA_2 = EMA(EMA_1) EMA_3 = EMA(EMA_2)

2. The Combination

TEMA = (3 \times EMA_1) - (3 \times EMA_2) + EMA_3

Performance Profile

Metric	Score	Notes
Throughput	9	High; O(1) calculation with 3 EMA steps.
Allocations	0	Zero-allocation in hot paths.
Complexity	O(1)	Constant time regardless of period.
Accuracy	10	Matches TA-Lib exactly.
Timeliness	10	Extremely low lag; nearly zero-lag tracking.
Overshoot	8	Significant overshoot on sharp reversals.
Smoothness	6	Less smooth than SMA/EMA due to high responsiveness.

Validation

Library	Status	Notes
QuanTAlib	✅	Validated.
TA-Lib	✅	Matches `TA_TEMA` exactly.
Skender	✅	Matches `GetTema` exactly.
Tulip	✅	Matches `tema` exactly.
Ooples	❌	Diverges significantly due to initialization logic.

Common Pitfalls

Overshoot: TEMA is so responsive it can overshoot price turns, creating a "whiplash" effect in volatile markets.
Noise: By reducing lag, TEMA sacrifices some noise suppression. It is "nervous" compared to an SMA.
Identity Crisis: Often confused with T3 (Tillson). T3 is a generalized version; TEMA is specifically T3 with v=1.

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