Triple Exponential Average (TRIX)

I. What is the TRIX Indicator

TRIX (Triple Exponential Average) is a momentum oscillator that measures price momentum by calculating the rate of change of a triple-smoothed Exponential Moving Average (EMA) of closing prices. Thanks to the layered filtering effect of the triple EMA, TRIX effectively eliminates short-term price noise and retains only the most significant trend change signals.

Historical Background

TRIX was invented by Jack Hutson in 1983 and published in Technical Analysis of Stocks and Commodities magazine — a publication Hutson himself founded and edited. He designed TRIX with the goal of creating an indicator that could filter out “meaningless price fluctuations” and only produce signals when a truly significant trend change occurs. The triple EMA design was inspired by the concept of multi-stage low-pass filters in signal processing.

Indicator Classification

• Type: Oscillator, displayed in a separate panel
• Category: Momentum
• Default Parameter: Period $n = 15$
• Value Range: Unbounded, oscillates around the zero line (unit: %)

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II. Mathematical Principles and Calculation

Core Formula

Step 1: Calculate triple EMA

$EMA1_t = EMA(Close, n)_t$

$EMA2_t = EMA(EMA1, n)_t$

$EMA3_t = EMA(EMA2, n)_t$

Step 2: Calculate TRIX (rate of change)

$TRIX_t = \frac{EMA3_t - EMA3_{t-1}}{EMA3_{t-1}} \times 100$

Optional: Signal line

$Signal_t = SMA(TRIX, s)_t$

where $s$ is typically 9.

EMA Recursive Formula

$EMA_t = \alpha \cdot X_t + (1 - \alpha) \cdot EMA_{t-1}, \quad \alpha = \frac{2}{n + 1}$

Step-by-Step Calculation Logic

1. Calculate first EMA: Apply $n$-period EMA to closing prices
2. Calculate second EMA: Apply $n$-period EMA to the first EMA
3. Calculate third EMA: Apply $n$-period EMA to the second EMA
4. TRIX = percentage rate of change of the triple EMA
5. Optional: Calculate 9-period SMA as signal line

Smoothing Effect of Triple EMA

For triple EMA with period $n = 15$:

• Equivalent smoothing period is approximately $3 \times (n-1) + 1 = 43$ periods
• This acts as a very smooth low-pass filter — only price movements sustained over several weeks can significantly affect the TRIX value

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III. Python Implementation

import numpy as np
import pandas as pd


def trix(close: pd.Series,
         period: int = 15,
         signal_period: int = 9) -> pd.DataFrame:
    """
    Calculate TRIX Indicator

    Parameters
    ----------
    close : pd.Series
        Close price series
    period : int
        EMA period, default 15
    signal_period : int
        Signal line SMA period, default 9

    Returns
    -------
    pd.DataFrame
        DataFrame with TRIX and Signal columns
    """
    # Triple EMA
    ema1 = close.ewm(span=period, adjust=False).mean()
    ema2 = ema1.ewm(span=period, adjust=False).mean()
    ema3 = ema2.ewm(span=period, adjust=False).mean()

    # TRIX = percentage change of triple EMA
    trix_values = (ema3 - ema3.shift(1)) / ema3.shift(1) * 100

    # Signal line
    signal_line = trix_values.rolling(window=signal_period, min_periods=signal_period).mean()

    return pd.DataFrame({
        "TRIX": trix_values,
        "Signal": signal_line
    }, index=close.index)


# ========== Usage Example ==========
if __name__ == "__main__":
    np.random.seed(42)
    dates = pd.date_range("2024-01-01", periods=150, freq="D")
    trend = np.linspace(0, 15, 150)
    noise = np.cumsum(np.random.randn(150) * 0.8)
    price = 100 + trend + noise

    df = pd.DataFrame({
        "date": dates,
        "open":  price + np.random.randn(150) * 0.3,
        "high":  price + np.abs(np.random.randn(150) * 0.6),
        "low":   price - np.abs(np.random.randn(150) * 0.6),
        "close": price,
        "volume": np.random.randint(1000, 10000, size=150),
    })
    df.set_index("date", inplace=True)

    # Calculate TRIX
    trix_df = trix(df["close"], period=15, signal_period=9)
    df = pd.concat([df, trix_df], axis=1)

    print("=== TRIX Results (Last 15 Rows) ===")
    print(df[["close", "TRIX", "Signal"]].tail(15).to_string())

    # Zero-line crossover
    df["zero_cross_up"] = (df["TRIX"] > 0) & (df["TRIX"].shift(1) <= 0)
    df["zero_cross_down"] = (df["TRIX"] < 0) & (df["TRIX"].shift(1) >= 0)

    print("\n=== TRIX Crosses Above Zero ===")
    print(df[df["zero_cross_up"]][["close", "TRIX"]].to_string())
    print("\n=== TRIX Crosses Below Zero ===")
    print(df[df["zero_cross_down"]][["close", "TRIX"]].to_string())

    # Signal line crossover
    df["sig_cross_up"] = (df["TRIX"] > df["Signal"]) & \
                         (df["TRIX"].shift(1) <= df["Signal"].shift(1))
    df["sig_cross_down"] = (df["TRIX"] < df["Signal"]) & \
                           (df["TRIX"].shift(1) >= df["Signal"].shift(1))

    print("\n=== TRIX Crosses Above Signal Line ===")
    print(df[df["sig_cross_up"]][["close", "TRIX", "Signal"]].head(10).to_string())

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IV. Problems the Indicator Solves

1. Filtering Market Noise

TRIX’s core value lies in its triple smoothing effect. A large volume of short-term random fluctuations are filtered layer by layer through the EMAs — only truly sustained price movements can significantly influence the TRIX value.

2. Zero-line Crossover (Trend Change)

Signal	Condition	Meaning
Buy	TRIX crosses above zero	Triple-smoothed price begins rising -> sustained uptrend begins
Sell	TRIX crosses below zero	Triple-smoothed price begins falling -> sustained downtrend begins

3. Signal Line Crossover

• TRIX crosses above signal line -> Momentum strengthening, buy signal
• TRIX crosses below signal line -> Momentum weakening, sell signal

4. Divergence Signals

Because TRIX is highly smoothed, its divergence signals are more reliable:

• Bearish divergence: New price high + TRIX fails to make new high -> reliable signal of long-term momentum decline
• Bullish divergence: New price low + TRIX fails to make new low -> reliable signal of long-term momentum recovery

5. Trend Strength Assessment

• TRIX absolute value continuously increasing -> trend accelerating
• TRIX absolute value beginning to decrease -> trend decelerating
• TRIX approaching zero line -> trend about to change

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V. Advantages, Disadvantages, and Use Cases

Advantages

Advantage	Description
Exceptional noise filtering	Triple EMA almost completely eliminates short-term noise
Very few false signals	High smoothing means only genuine trend changes produce signals
Reliable signals	Zero-line and signal-line crossovers have relatively high accuracy
More reliable divergence	Divergence on smoothed data is more meaningful than on raw price

Disadvantages

Disadvantage	Description
Significant lag	Triple smoothing causes large signal delay, potentially missing early moves
Not suited for short-term	Excessive smoothing prevents response to short-term price changes
Parameter sensitive	Period selection has a significant impact on signal timing
Requires trending markets	In prolonged sideways markets, TRIX oscillates repeatedly around zero

Use Cases

• Medium-to-long-term trend trading: Capturing trends lasting weeks to months
• Position management: TRIX direction can serve as a reference for position direction
• Filter: Only consider long strategies when TRIX > 0
• Broad indices: Works well on indices with relatively smooth volatility

Comparison with Related Indicators

Comparison	TRIX	MACD	ROC
Smoothing degree	Very high (triple EMA)	Medium (dual EMA difference)	None
Lag	Large	Medium	Small
False signal rate	Very low	Low	High
Suited for	Long-term	Medium-term	Short-term
Signal line	Optional SMA(9)	EMA(9)	None