# 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.