QuanTAlib/lib/trends_IIR/t3/T3.md

T3: Tillson T3 Moving Average

"If one EMA is good, six must be better. Tim Tillson's logic is impeccable, provided you hate noise more than you love latency."

The T3 Moving Average is a hyper-smooth, low-lag filter that cascades six Exponential Moving Averages (EMAs). Unlike standard cascading (which increases lag), T3 uses a "Volume Factor" (v) to weight the EMAs in a way that partially cancels out the lag, resulting in a curve that is smoother than an EMA but more responsive than an SMA.

Historical Context

Introduced by Tim Tillson in Technical Analysis of Stocks & Commodities (Jan 1998), "Smoothing Techniques for More Accurate Signals." Tillson sought to improve upon the DEMA (Double EMA) and TEMA (Triple EMA) concepts by generalizing the lag-reduction mathematics.

Architecture & Physics

T3 is essentially a filter of filters. It passes data through a chain of 6 EMAs: Input \to EMA_1 \to EMA_2 \to EMA_3 \to EMA_4 \to EMA_5 \to EMA_6

It then combines these outputs using coefficients derived from the Volume Factor (v).

The Volume Factor (v)

• $v = 0$: T3 becomes a standard EMA (actually, a triple EMA of EMAs).
• $v = 1$: T3 behaves like DEMA/TEMA with aggressive lag reduction (and potential overshoot).
• $v = 0.7$: The default. A "Goldilocks" zone of smoothness and responsiveness.

Mathematical Foundation

1. Coefficients

Given v (default 0.7):

c_1 = -v^3 c_2 = 3v^2 + 3v^3 c_3 = -6v^2 - 3v - 3v^3 c_4 = 1 + 3v + 3v^2 + v^3

2. The Formula

(Note: There are multiple variations of T3. QuanTAlib uses the standard Tillson formula).

T3 = c_1 e_6 + c_2 e_5 + c_3 e_4 + c_4 e_3

Where e_n is the output of the $n$-th EMA in the cascade.

Performance Profile

Operation Count (Streaming Mode)

T3 requires 6 cascaded EMA updates plus the weighted combination:

Operation	Count	Cost (cycles)	Subtotal
EMA update (×6)	6	7	42
MUL (c1×e6, c2×e5, c3×e4, c4×e3)	4	3	12
ADD (combination)	3	1	3
Total (hot)	13	—	~57 cycles

During warmup, each EMA stage has additional compensator overhead (~21 cycles × 6 = ~126 cycles).

Total during warmup: ~183 cycles/bar; Post-warmup: ~57 cycles/bar.

Batch Mode (SIMD Analysis)

T3 is inherently recursive due to 6 cascaded EMAs. SIMD parallelization across bars is not possible:

Optimization	Operations	Cycles Saved
FMA in each EMA stage	6 FMA vs 6×(MUL+ADD)	~12 cycles
FMA in coefficient combination	4 FMA ops	~8 cycles

Per-bar efficiency: ~57 cycles is 8× EMA cost, reflecting 6 EMA stages + 4-term combiner.

Quality Metrics

Metric	Score	Notes
Accuracy	10/10	Matches TA-Lib exactly
Timeliness	9/10	Very low lag due to volume factor cancellation
Overshoot	6/10	Can overshoot significantly if v > 1
Smoothness	10/10	Extremely smooth due to 6-pole filtering

Benchmark Results

Metric	Value	Notes
Throughput	~12 ns/bar	6× EMA overhead
Allocations	0 bytes	Zero-allocation in hot paths
Complexity	O(1)	Constant time regardless of period
State Size	192 bytes	Six EMA states (32 bytes each)

Validation

Library	Status	Notes
TA-Lib	✅	Matches TA_T3 exactly.
Skender	✅	Matches GetT3 exactly.
Tulip	N/A	Not implemented.
Ooples	✅	Matches CalculateTillsonT3MovingAverage.

Common Pitfalls

1. Warmup: Because it cascades 6 EMAs, T3 takes significantly longer to stabilize than a standard EMA. A T3(10) might need 60+ bars to converge.
2. Overshoot: With high v values (>1), T3 can overshoot price turns, creating false breakout signals.
3. Complexity: It is computationally heavier than SMA or EMA (approx 6x ops), though still negligible on modern CPUs.