mirror of
https://github.com/mihakralj/QuanTAlib.git
synced 2026-08-16 09:38:05 +00:00
117 lines
4.9 KiB
Markdown
117 lines
4.9 KiB
Markdown
# 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.*
|
||
|
||
| Property | Value |
|
||
| ---------------- | -------------------------------- |
|
||
| **Category** | Trend (IIR MA) |
|
||
| **Inputs** | Source (close) |
|
||
| **Parameters** | `period`, `vfactor` (default 0.7) |
|
||
| **Outputs** | Single series (T3) |
|
||
| **Output range** | Tracks input |
|
||
| **Warmup** | `period * 6` bars |
|
||
| **PineScript** | [t3.pine](t3.pine) |
|
||
| **Signature** | [t3_signature](t3_signature.md) |
|
||
|
||
- The T3 Moving Average is a hyper-smooth, low-lag filter that cascades six Exponential Moving Averages (EMAs).
|
||
- **Similar:** [TEMA](../tema/tema.md), [DEMA](../dema/dema.md) | **Complementary:** Signal line crossover | **Trading note:** Tillsons T3; generalized DEMA with volume factor.
|
||
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
|
||
|
||
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. |