Laguerre
One exponential average, a chain of sections that only delay, and a weighted sum that does the rest of the work. Once you see which part does which, the gamma dial stops being mysterious.
Build one
Lag, noise gain, peak gain and leak are defined on the comparison page — the tool sits above the article deliberately, so the numbers arrive before the prose that explains them.
How it works
The chain has two different kinds of section, and the difference is the whole filter. The first is an ordinary one-pole low-pass: L0 = (1 − γ)·price + γ·L0[1], an exponential average by another name. Every section after it is an all-pass, which passes each frequency at exactly unit magnitude and changes only when things arrive — slow movement comes through almost immediately, fast movement is delayed more. So the chain attenuates once, at the front, and then only rearranges.
Most of the filtering is in the recombination. Take the section outputs, weight them 1:2:2:1 and average, and the frequencies whose delays now disagree cancel each other while the ones that arrived together survive. That is the whole mechanism, and it is why the weights are the interesting dial rather than an implementation detail.
gamma sets how much each section delays. Near zero the sections barely act and the output is nearly the input; near one they delay enormously and the filter becomes very slow.
What it costs
The cost is ordinary — gamma buys smoothness with lag, like everything else — but it is unusually hard to predict, because gamma is not a period and the relationship between them is not linear. A Laguerre at gamma 0.8 does not have a lookback of anything. You have to measure it.
What our measurements say
Which is what the table is for:
| Filter | Lag | Noise | Peak | Leak | vs an EMA of the same lag |
|---|---|---|---|---|---|
| Ehlers Laguerre (g=0.8) | 17.50 | 0.0309 | 1.000 | 0.00547 | leak 0.139× · noise 1.11× |
Only one row of the table is this construction: Ehlers Laguerre (g=0.8), 17.50 bars back and leaking 0.00547. Leak is measured over periods of 4.0 bars and shorter, the same band as every other page in this section, so the rows can be read against the full table.
Gamma is not a period
This matters more than it sounds. A comparison that puts a Laguerre beside an EMA of some round period is comparing two filters that sit different distances behind price, and some of what looks like a difference between the filters is that gap. The lag column above is what settles it: read it before believing any such comparison, including a favourable one.
The other constructions
- Windows — Choose a shape, normalise it, and the shape is the filter.
- Poles — Place the poles of a recursion and let it run.
- Cascades — Build something smooth and late, then subtract a deeper copy of it to cancel the lateness.
- Least squares — State the response you want, including the delay, and solve for it.
- Adaptive — Re-estimate the period every bar. The one construction with no fixed response, which is why it is absent from the table above.
- All of them on one ruler — the comparison this page is part of.