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Co-authored-by: Claude Opus 4.5 <noreply@anthropic.com> Co-authored-by: aider (openrouter/anthropic/claude-sonnet-4) <aider@aider.chat> Co-authored-by: Warp <agent@warp.dev>
102 lines
4.0 KiB
Markdown
102 lines
4.0 KiB
Markdown
# T3: Tillson T3 Moving Average
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> "If one EMA is good, six must be better. Tim Tillson's logic is impeccable, provided you hate noise more than you love latency."
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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.
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## Historical Context
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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.
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## Architecture & Physics
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T3 is essentially a filter of filters. It passes data through a chain of 6 EMAs:
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$Input \to EMA_1 \to EMA_2 \to EMA_3 \to EMA_4 \to EMA_5 \to EMA_6$
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It then combines these outputs using coefficients derived from the Volume Factor ($v$).
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### The Volume Factor ($v$)
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* **$v = 0$**: T3 becomes a standard EMA (actually, a triple EMA of EMAs).
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* **$v = 1$**: T3 behaves like DEMA/TEMA with aggressive lag reduction (and potential overshoot).
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* **$v = 0.7$**: The default. A "Goldilocks" zone of smoothness and responsiveness.
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## Mathematical Foundation
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### 1. Coefficients
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Given $v$ (default 0.7):
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$$ c_1 = -v^3 $$
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$$ c_2 = 3v^2 + 3v^3 $$
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$$ c_3 = -6v^2 - 3v - 3v^3 $$
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$$ c_4 = 1 + 3v + 3v^2 + v^3 $$
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### 2. The Formula
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(Note: There are multiple variations of T3. QuanTAlib uses the standard Tillson formula).
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$$ T3 = c_1 e_6 + c_2 e_5 + c_3 e_4 + c_4 e_3 $$
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Where $e_n$ is the output of the $n$-th EMA in the cascade.
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## Performance Profile
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### Operation Count (Streaming Mode)
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T3 requires 6 cascaded EMA updates plus the weighted combination:
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| EMA update (×6) | 6 | 7 | 42 |
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| MUL (c1×e6, c2×e5, c3×e4, c4×e3) | 4 | 3 | 12 |
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| ADD (combination) | 3 | 1 | 3 |
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| **Total (hot)** | **13** | — | **~57 cycles** |
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During warmup, each EMA stage has additional compensator overhead (~21 cycles × 6 = ~126 cycles).
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**Total during warmup:** ~183 cycles/bar; **Post-warmup:** ~57 cycles/bar.
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### Batch Mode (SIMD Analysis)
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T3 is inherently recursive due to 6 cascaded EMAs. SIMD parallelization across bars is not possible:
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| Optimization | Operations | Cycles Saved |
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| :--- | :---: | :---: |
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| FMA in each EMA stage | 6 FMA vs 6×(MUL+ADD) | ~12 cycles |
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| FMA in coefficient combination | 4 FMA ops | ~8 cycles |
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**Per-bar efficiency:** ~57 cycles is 8× EMA cost, reflecting 6 EMA stages + 4-term combiner.
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### Quality Metrics
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| Metric | Score | Notes |
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| :--- | :---: | :--- |
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| **Accuracy** | 10/10 | Matches TA-Lib exactly |
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| **Timeliness** | 9/10 | Very low lag due to volume factor cancellation |
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| **Overshoot** | 6/10 | Can overshoot significantly if $v > 1$ |
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| **Smoothness** | 10/10 | Extremely smooth due to 6-pole filtering |
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### Benchmark Results
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| Metric | Value | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | ~12 ns/bar | 6× EMA overhead |
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| **Allocations** | 0 bytes | Zero-allocation in hot paths |
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| **Complexity** | O(1) | Constant time regardless of period |
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| **State Size** | 192 bytes | Six EMA states (32 bytes each) |
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## Validation
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| Library | Status | Notes |
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| :--- | :--- | :--- |
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| **TA-Lib** | ✅ | Matches `TA_T3` exactly. |
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| **Skender** | ✅ | Matches `GetT3` exactly. |
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| **Tulip** | N/A | Not implemented. |
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| **Ooples** | ✅ | Matches `CalculateTillsonT3MovingAverage`. |
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### Common Pitfalls
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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.
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2. **Overshoot**: With high $v$ values ($>1$), T3 can overshoot price turns, creating false breakout signals.
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3. **Complexity**: It is computationally heavier than SMA or EMA (approx 6x ops), though still negligible on modern CPUs. |