# TEMA: Triple Exponential Moving Average

> "Patrick Mulloy looked at the lag of an EMA and took it personally. TEMA is what happens when you apply algebra to impatience."

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.

## Historical Context

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.

## Architecture & Physics

TEMA is not just "EMA applied three times." That would be $EMA(EMA(EMA(x)))$. TEMA is a composite:
$$ TEMA = 3 \cdot EMA_1 - 3 \cdot EMA_2 + EMA_3 $$

This formula effectively projects the trend forward to compensate for the delay inherent in smoothing.

### Convergence Speed

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.

## Mathematical Foundation

### 1. The Cascade

$$ EMA_1 = EMA(Price) $$
$$ EMA_2 = EMA(EMA_1) $$
$$ EMA_3 = EMA(EMA_2) $$

### 2. The Combination

$$ TEMA = (3 \times EMA_1) - (3 \times EMA_2) + EMA_3 $$

## Performance Profile

| Metric | Score | Notes |
| :--- | :--- | :--- |
| **Throughput** | 9 | High; O(1) calculation with 3 EMA steps. |
| **Allocations** | 0 | Zero-allocation in hot paths. |
| **Complexity** | O(1) | Constant time regardless of period. |
| **Accuracy** | 10 | Matches TA-Lib exactly. |
| **Timeliness** | 10 | Extremely low lag; nearly zero-lag tracking. |
| **Overshoot** | 8 | Significant overshoot on sharp reversals. |
| **Smoothness** | 6 | Less smooth than SMA/EMA due to high responsiveness. |

## Validation

| Library | Status | Notes |
| :--- | :--- | :--- |
| **QuanTAlib** | ✅ | Validated. |
| **TA-Lib** | ✅ | Matches `TA_TEMA` exactly. |
| **Skender** | ✅ | Matches `GetTema` exactly. |
| **Tulip** | ✅ | Matches `tema` exactly. |
| **Ooples** | ❌ | Diverges significantly due to initialization logic. |

### Common Pitfalls

1. **Overshoot**: TEMA is so responsive it can overshoot price turns, creating a "whiplash" effect in volatile markets.
2. **Noise**: By reducing lag, TEMA sacrifices some noise suppression. It is "nervous" compared to an SMA.
3. **Identity Crisis**: Often confused with T3 (Tillson). T3 is a generalized version; TEMA is specifically T3 with $v=1$.
