Laguerre Filter EMA by Ehlers comparison with EMA

• 10/14/2025 at 2:45 PM #252497
  elekin.

  Hello, I have a question about the Laguerre Filter Indicator by John F. Ehlers, the first version that appeared in the book “Cybernetic Analysis for stocks and futures” at the Chapter 14th.

  This version is built using an Exponential Moving Average as the first level and three successive levels, with a total of four. I’d like to compare the Laguerre Filter with an EMA, but I don’t know what’s the formula to apply to find the actual EMA’s Period considering a certain value of gamma.
  I found this  formula: Nb of levels * (2 / (1 – gamma) -1), but, as you can see in the picture attached, the EMA (16) is much rapid than the Laguerre Filter (gamma = 0,6), while we know by Ehlers that this indicator is built just for reducing the lag of traditional moving averages.
  Do you have a clue to find the right conversion formula?

  The following is the code of the indicator:

  ONCE L0 = 0
  ONCE L1 = 0
  ONCE L2 = 0
  ONCE L3 = 0

  //Levels

  IF BarIndex > 0 THEN

  L0 = (1 – gamma) * Close + gamma * L0[1]

  L1 = – gamma * L0 + L0[1] + gamma * L1[1]

  L2 = – gamma * L1 + L1[1] + gamma * L2[1]

  L3 = – gamma * L2 + L2[1] + gamma * L3[1]

  ENDIF

  //Final Output

  LF = (L0 + 2*L1 + 2*L2 + L3)/6

  Return LF as “Laguerre Filter EMA”

  

  elekin.

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  10/14/2025 at 3:00 PM #252608
  Nicolas

  The Laguerre Filter as defined by John F. Ehlers indeed differs significantly from a classic Exponential Moving Average (EMA), as its recursive multi-level structure (L0–L3) provides both smoothing and phase advancement that do not correspond linearly to an EMA period.

  The formula you mentioned :
  Nb of levels × (2 / (1 – γ) – 1) —
  is only a rough empirical approximation. It cannot precisely reproduce the dynamic response of the Laguerre Filter because each level in the recursive computation interacts nonlinearly with γ (gamma).

  There is no exact “conversion formula” between γ and an equivalent EMA period. However, practical testing shows that:

  • For γ = 0.5, the filter roughly resembles an EMA with a period around 8–10.

  • For γ = 0.6, the equivalent EMA is closer to 12–16.

  • For γ = 0.8, it behaves somewhat like an EMA with a period near 30–35.

  These estimates depend on the number of levels (4 in your example) and the data’s sampling rate.
  If you wish to compare directly with an EMA, the best approach is to use empirical calibration — i.e., adjust the EMA period until its smoothness visually matches the Laguerre Filter output for your chosen γ.

  In summary:

  • The Laguerre Filter reduces lag while maintaining smoother transitions than an EMA of similar responsiveness.

  • Its γ parameter controls both lag and smoothing simultaneously, but not in a way that maps directly to a single EMA period.

  I hope this clarifies the relationship for your comparison tests.

  

  Nicolas

  Keymaster

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  10/14/2025 at 3:04 PM #252609
  Nicolas

  It’s also worth emphasizing that one of the main strengths of the Laguerre Filter is precisely that it has no fixed period. Unlike moving averages (EMA, SMA, WMA, etc.) where the smoothing is tied to a specific lookback window, the Laguerre Filter’s responsiveness is governed solely by the gamma (γ) parameter, which dynamically adapts the weighting of recent versus past data through its recursive structure.

  This means the Laguerre Filter doesn’t “wait” for a fixed number of bars to react, it continuously adjusts, producing smoother and faster signals without introducing the same degree of lag typical of long-period moving averages. That’s why Ehlers designed it: to achieve low lag with strong smoothing, something that’s not possible with a simple EMA formula.

  In short:
  There’s no “period” to translate and that’s exactly what makes the Laguerre Filter special!

  

  Nicolas

  Keymaster

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