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Technical Details — Arnaud Legoux Moving Average (ALMA)

Section titled “Technical Details — Arnaud Legoux Moving Average (ALMA)”

Overview

Section titled “Overview”

The Arnaud Legoux Moving Average (ALMA) applies Gaussian-distribution weights to prices within the lookback window. The offset parameter (0-1) controls where the peak weight falls (0 = oldest, 1 = newest, 0.85 = default), and sigma (1-10) controls how concentrated the weights are.

Mathematical Derivation

Step 1 — Calculate Weight Position

Section titled “Step 1 — Calculate Weight Position”

Purpose: Determine the center of the Gaussian bell curve.

$$m = offset \times (n - 1)$$

Where:

  • $$offset$$ = Weight position parameter (0 to 1, default 0.85)
  • $$n$$ = Period

What This Measures: Where in the window the highest weight is applied

Step 2 — Calculate Weight Spread

Section titled “Step 2 — Calculate Weight Spread”

Purpose: Determine how concentrated the weights are.

$$s = \frac{n}{\sigma}$$

Where:

  • $$\sigma$$ = Sigma parameter (1 to 10, default 6)

What This Measures: How quickly weights decay away from the peak

Step 3 — Calculate Gaussian Weights

Section titled “Step 3 — Calculate Gaussian Weights”

Purpose: Generate bell-curve weights for each bar.

$$w[i] = e^{-\frac{(i - m)^2}{2s^2}}$$

Where:

  • $$i$$ = Bar index within the window
  • $$m$$ = Peak position
  • $$s$$ = Spread

What This Measures: The relative importance of each bar in the calculation

Step 4 — Apply Weighted Average

Section titled “Step 4 — Apply Weighted Average”

Purpose: Calculate the final ALMA value.

$$ALMA[t] = \frac{\sum_{i=0}^{n-1} w[i] \times Close[t-n+1+i]}{\sum_{i=0}^{n-1} w[i]}$$

What This Measures: Gaussian-weighted average price

Compact Formula Summary

$$m = offset \times (n-1)$$ $$s = n / \sigma$$ $$w[i] = e^{-(i-m)^2 / 2s^2}$$ $$ALMA[t] = \frac{\sum w[i] \times Close[i]}{\sum w[i]}$$

Default Parameters: Period = 14, Offset = 0.85, Sigma = 6

Complete Calculation Example

With Period = 5, Offset = 0.85, Sigma = 6:

  • m = 0.85 × 4 = 3.4 (weight peaks near the most recent bar)
  • Weights concentrate around bar index 3-4, giving most influence to recent prices
  • Result: smooth average that responds quickly to price changes
Key Takeaways from the Example
  1. Gaussian Weighting: Unlike linear weights (WMA) or exponential (EMA), ALMA uses a bell curve for the most mathematically elegant smoothing
  2. Three Parameters: Period, Offset, and Sigma give fine-grained control over responsiveness
  3. Offset Control: Offset = 0.85 means 85% of the weight window is behind the peak — emphasizing recent data
  4. Sigma Control: Lower sigma = more concentrated weights (responsive), higher sigma = more spread out (smooth)