QuanTAlib/lib/trends_IIR/tema/Tema.md

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

Operation Count (Streaming Mode)

TEMA requires 3 cascaded EMA updates plus the combination formula:

Operation	Count	Cost (cycles)	Subtotal
EMA update (×3)	3	7	21
MUL (3×e1, 3×e2)	2	3	6
SUB (3×e1 - 3×e2)	1	1	1
ADD (+ e3)	1	1	1
Total (hot)	7	—	~29 cycles

During warmup, each EMA stage has additional compensator overhead (~21 cycles × 3 = ~63 cycles).

Total during warmup: ~92 cycles/bar; Post-warmup: ~29 cycles/bar.

Batch Mode (SIMD Analysis)

TEMA is inherently recursive due to cascaded EMAs. SIMD parallelization across bars is not possible. Each EMA stage must complete before feeding the next:

Optimization	Operations	Cycles Saved
FMA in each EMA stage	3 FMA vs 3×(MUL+ADD)	~6 cycles
Inline combination	Avoid intermediate stores	~2 cycles

Per-bar efficiency: ~29 cycles is 4× EMA cost, as expected for 3 EMA stages + combiner.

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Matches TA-Lib exactly
Timeliness	10/10	Extremely low lag; nearly zero-lag tracking
Overshoot	5/10	Significant overshoot on sharp reversals
Smoothness	6/10	Less smooth than SMA/EMA due to high responsiveness

Benchmark Results

Metric	Value	Notes
Throughput	~6 ns/bar	3× EMA overhead
Allocations	0 bytes	Zero-allocation in hot paths
Complexity	O(1)	Constant time regardless of period
State Size	96 bytes	Three EMA states (32 bytes each)

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.

C# Implementation Considerations

State Management

TEMA maintains six EmaState instances—three current, three previous—enabling atomic rollback on bar corrections:

private record struct EmaState(double Ema, double E, bool IsHot, bool IsCompensated);

private EmaState _state1, _state2, _state3;
private EmaState _p_state1, _p_state2, _p_state3;

The E field tracks bias compensation factor for each EMA stage independently. Each state auto-transitions via IsCompensated flag when bias becomes negligible.

Precomputed Constants

Constructor calculates smoothing constants once:

_alpha = 2.0 / (period + 1);
_decay = 1 - _alpha;

These constants are reused across all three EMA stages, avoiding repeated division.

FMA Usage

Each EMA update uses FusedMultiplyAdd for the standard EMA formula:

double newEma = Math.FusedMultiplyAdd(state.Ema, _decay, _alpha * input);

The final TEMA combination 3*e1 - 3*e2 + e3 could use FMA but the coefficients (3, -3, 1) make chained FMA marginal; current implementation uses direct arithmetic.

Bar Correction Pattern

TEMA's cascaded structure requires coordinated state rollback:

if (isNew)
{
    _p_state1 = _state1;
    _p_state2 = _state2;
    _p_state3 = _state3;
}
else
{
    _state1 = _p_state1;
    _state2 = _p_state2;
    _state3 = _p_state3;
}

All three stages rollback atomically, ensuring consistent cascade state when isNew=false.

Memory Layout

Field	Type	Size	Purpose
_alpha	double	8B	Smoothing constant
_decay	double	8B	1 - alpha
_state1	EmaState	24B	First EMA state
_state2	EmaState	24B	Second EMA state
_state3	EmaState	24B	Third EMA state
_p_state1	EmaState	24B	Previous state 1
_p_state2	EmaState	24B	Previous state 2
_p_state3	EmaState	24B	Previous state 3
Total		160B	Per indicator instance

Each EmaState contains: Ema (8B), E (8B), IsHot (1B), IsCompensated (1B) + padding (~6B) = ~24B.

Common Pitfalls

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