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58 lines
2.6 KiB
Markdown
58 lines
2.6 KiB
Markdown
# TEMA: Triple Exponential Moving Average
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> "Patrick Mulloy looked at the lag of an EMA and took it personally. TEMA is what happens when you apply algebra to impatience."
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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.
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## Historical Context
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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.
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## Architecture & Physics
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TEMA is not just "EMA applied three times." That would be $EMA(EMA(EMA(x)))$. TEMA is a composite:
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$$ TEMA = 3 \cdot EMA_1 - 3 \cdot EMA_2 + EMA_3 $$
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This formula effectively projects the trend forward to compensate for the delay inherent in smoothing.
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### Convergence Speed
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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.
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## Mathematical Foundation
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### 1. The Cascade
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$$ EMA_1 = EMA(Price) $$
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$$ EMA_2 = EMA(EMA_1) $$
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$$ EMA_3 = EMA(EMA_2) $$
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### 2. The Combination
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$$ TEMA = (3 \times EMA_1) - (3 \times EMA_2) + EMA_3 $$
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## Performance Profile
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### Zero-Allocation Design
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QuanTAlib's `Tema` implementation does not create three separate `Ema` objects. Instead, it maintains three lightweight `EmaState` structs within the main class. This ensures zero heap allocations during updates and keeps the memory footprint minimal.
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| Metric | Score | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | High | 3 EMAs |
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| **Complexity** | O(1) | Constant time update |
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| **Accuracy** | 8/10 | Extremely responsive to turns |
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| **Timeliness** | 9/10 | Near-zero lag (Lag ≈ 0) |
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| **Overshoot** | 4/10 | Significant overshoot on reversals |
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| **Smoothness** | 7/10 | Smoother than DEMA, less than T3 |
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## Validation
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Validated against TA-Lib (`TA_TEMA`) and Skender.Stock.Indicators.
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### Common Pitfalls
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1. **Overshoot**: TEMA is so responsive it can overshoot price turns, creating a "whiplash" effect in volatile markets.
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2. **Noise**: By reducing lag, TEMA sacrifices some noise suppression. It is "nervous" compared to an SMA.
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3. **Identity Crisis**: Often confused with T3 (Tillson). T3 is a generalized version; TEMA is specifically T3 with $v=1$.
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