Arnaud Legoux Moving Average (ALMA)

I. What is ALMA

The Arnaud Legoux Moving Average (ALMA) is an advanced moving average that uses a Gaussian distribution (normal distribution) as its weight function. It was co-invented by French engineer Arnaud Legoux and Greek mathematician Dimitris Kouzis-Loukas, and was first published around 2009.

Historical Background

Legoux and Kouzis-Loukas drew inspiration from the signal processing domain, arguing that the ideal moving average weight function should be a bell curve (Gaussian function), rather than a linear decay (WMA) or exponential decay (EMA). Gaussian weights offer the following advantages:

1. The center region carries the highest weight, with rapid decay on both sides
2. The weight curve is extremely smooth, with no hard cutoff
3. By adjusting the offset parameter, the weight center can be positioned anywhere within the window

ALMA allows users to fine-tune the weight distribution through offset and sigma parameters, achieving flexible control over the trade-off between lag and smoothness.

Indicator Classification

• Type: Overlay indicator, plotted on the price chart
• Category: Moving Averages
• Default Parameters: Period $n = 20$, offset $= 0.85$, sigma $= 6$, data source is closing price (Close)

---

II. Mathematical Principles and Calculation

Core Formula

ALMA weights are based on the Gaussian function:

$w_i = \exp\left(-\frac{(i - m)^2}{2s^2}\right)$

Where:

• $i = 0, 1, 2, \ldots, n-1$ (index within the window, 0 being the oldest data)
• $m = offset \times (n - 1)$ (center position of the Gaussian bell curve)
• $s = \frac{n}{sigma}$ (standard deviation of the Gaussian function, controlling the bell width)

The ALMA calculation formula is:

$ALMA_t = \frac{\sum_{i=0}^{n-1} w_i \cdot P_{t-n+1+i}}{\sum_{i=0}^{n-1} w_i}$

Parameter Details

offset (range 0 ~ 1, default 0.85)

Controls the position of the Gaussian bell center within the window:

• $offset = 0.5$: Weight center at the window midpoint; maximum lag but smoothest
• $offset = 1.0$: Weight center at the newest data point; minimum lag but less smooth
• $offset = 0.85$ (default): Biased toward recent data; a good balance between lag and smoothness

sigma (width parameter, default 6)

Controls the width of the Gaussian bell:

• Smaller $sigma$: Narrower bell, weights more concentrated near the center, behaves like a short-period average
• Larger $sigma$: Wider bell, more uniformly distributed weights, behaves like SMA

Numerical Example

For $n = 5$, $offset = 0.85$, $sigma = 6$:

• $m = 0.85 \times (5 - 1) = 3.4$
• $s = 5 / 6 \approx 0.833$

Weights for each index:

$i$	$(i - m)^2 / (2s^2)$	$w_i$
0	$(0 - 3.4)^2 / 1.389 = 8.324$	0.000
1	$(1 - 3.4)^2 / 1.389 = 4.147$	0.016
2	$(2 - 3.4)^2 / 1.389 = 1.411$	0.244
3	$(3 - 3.4)^2 / 1.389 = 0.115$	0.891
4	$(4 - 3.4)^2 / 1.389 = 0.259$	0.772

As shown, weight is concentrated at $i = 3$ and $i = 4$ (the two most recent days), consistent with the offset=0.85 bias.

---

III. Python Implementation

import numpy as np
import pandas as pd

def alma(close: pd.Series, period: int = 20,
         offset: float = 0.85, sigma: float = 6.0) -> pd.Series:
    """
    Calculate the Arnaud Legoux Moving Average (ALMA)

    Parameters
    ----------
    close : pd.Series
        Closing price series
    period : int
        Calculation period, default is 20
    offset : float
        Gaussian weight center offset (0~1), default is 0.85
    sigma : float
        Gaussian width parameter, default is 6

    Returns
    -------
    pd.Series
        ALMA value series
    """
    m = offset * (period - 1)
    s = period / sigma

    # Calculate Gaussian weights
    weights = np.array([
        np.exp(-((i - m) ** 2) / (2 * s * s))
        for i in range(period)
    ])
    weights /= weights.sum()  # Normalize

    def _alma(window):
        return np.dot(window, weights)

    return close.rolling(window=period, min_periods=period).apply(_alma, raw=True)


def alma_numpy(close: np.ndarray, period: int = 20,
               offset: float = 0.85, sigma: float = 6.0) -> np.ndarray:
    """
    Manual ALMA implementation using numpy
    """
    n = len(close)
    result = np.full(n, np.nan)

    m = offset * (period - 1)
    s = period / sigma

    # Construct Gaussian weights
    idx = np.arange(period)
    weights = np.exp(-((idx - m) ** 2) / (2 * s * s))
    weights /= weights.sum()

    for i in range(period - 1, n):
        window = close[i - period + 1 : i + 1]
        result[i] = np.dot(window, weights)

    return result


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

    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)

    # Compare ALMA with different offsets
    df["EMA_20"]  = df["close"].ewm(span=20, adjust=False).mean()
    df["ALMA_default"] = alma(df["close"], 20, offset=0.85, sigma=6)
    df["ALMA_fast"]    = alma(df["close"], 20, offset=0.95, sigma=6)
    df["ALMA_smooth"]  = alma(df["close"], 20, offset=0.50, sigma=6)

    print("=== ALMA Comparison with Different Offsets ===")
    print(df[["close", "EMA_20", "ALMA_default", "ALMA_fast", "ALMA_smooth"]].tail(10))

    # Compare different sigma values
    df["ALMA_s3"] = alma(df["close"], 20, offset=0.85, sigma=3)
    df["ALMA_s6"] = alma(df["close"], 20, offset=0.85, sigma=6)
    df["ALMA_s12"] = alma(df["close"], 20, offset=0.85, sigma=12)

    print("\n=== ALMA Comparison with Different Sigma Values ===")
    print(df[["close", "ALMA_s3", "ALMA_s6", "ALMA_s12"]].tail(10))

    # Crossover signal
    df["signal"] = np.where(df["close"] > df["ALMA_default"], 1, -1)
    df["cross"] = df["signal"].diff().abs() > 0
    print("\n=== ALMA Crossover Signals ===")
    print(df.loc[df["cross"], ["close", "ALMA_default", "signal"]])

---

IV. Problems the Indicator Solves

1. Flexible Lag-Smoothness Trade-off

ALMA’s most unique capability is the fine-grained control over lag and smoothness through the offset and sigma parameters. This is something other moving averages cannot achieve:

• Increase offset -> Reduce lag
• Increase sigma -> Increase smoothness

Traders can freely adjust these according to different market conditions and strategy requirements.

2. Eliminating Window Boundary Effects

Gaussian weights naturally decay to near zero at both ends of the window, avoiding the hard cutoff problem seen in SMA and WMA. When price data exits the window, ALMA’s value does not experience abrupt jumps.

3. Signal-Processing-Grade Smoothing

ALMA’s Gaussian kernel function is directly related to low-pass filters in signal processing. Its suppression of high-frequency noise in the frequency domain is superior to linear or exponential weighting schemes.

4. Reducing False Crossovers

Due to the smooth nature of Gaussian weights, ALMA produces fewer false crossover signals than same-period EMA, performing more stably especially in ranging markets.

---

V. Advantages, Disadvantages, and Use Cases

Advantages

Advantage	Description
Flexible parameters	offset and sigma provide fine-grained lag/smoothness control
No hard cutoff	Gaussian weights decay naturally; no window boundary jumps
High smoothness	Gaussian kernel smoothing outperforms linear/exponential weights
Theoretically rigorous	Based on well-established Gaussian filtering theory from signal processing

Disadvantages

Disadvantage	Description
Too many parameters	Three parameters (period, offset, sigma) increase tuning complexity
Higher computation	Each calculation requires a weighted sum over the entire window
High learning barrier	The Gaussian distribution concept is not intuitive for non-mathematical traders
Non-mainstream	Uncommon in traditional technical analysis textbooks; may lack shared understanding

Use Cases

• Quantitative strategy development: Research requiring fine control over moving average behavior
• Signal-processing-oriented analysis: Optimizing price filtering from a frequency-domain perspective
• Multi-market adaptation: Using different offset/sigma parameters for different markets

Comparison with Similar Indicators

Feature	SMA	EMA	WMA	ALMA
Weight function	Constant	Exponential	Linear	Gaussian
Adjustable parameters	1	1	1	3
Boundary effects	Severe	None	Moderate	None
Smoothness	Medium	Medium	Medium	High
Flexibility	Low	Low	Low	High