# Laguerre Filter

A Trend concept (Digital filters & smoothers) in the LuxAlgo Library, with 1 indicator implementation.

## What is a Laguerre Filter?

The Laguerre filter is a smoother John Ehlers built from the Laguerre transform, a mathematical basis that lets a filter spend its memory unevenly across time. Structurally it is a cascade of four one-pole stages sharing a single damping factor, gamma, between 0 and 1; the plotted line is a fixed 1-2-2-1 weighted blend of the four stage outputs. The warped time base samples recent bars finely and compresses older ones, so the design targets heavy smoothing from only four internal elements, aiming for less lag than a conventional average of similar smoothness.

Gamma is the only knob. At 0 the cascade collapses to a fixed weighted smoother of the last four bars; raising it stretches the effective lookback, adding smoothness and lag without adding stages. The same four-stage skeleton powers [Laguerre RSI](https://www.luxalgo.com/library/concept/laguerre-rsi/), which builds an RSI-style ratio from the differences between adjacent stages instead of plotting the blend.

Ehlers published the design in the early 2000s, in a paper and book chapter memorably titled Time Warp, Without Space Travel, and the name honors Edmond Laguerre, the nineteenth-century French mathematician whose orthogonal polynomials underlie the transform. The mechanics are compact: the first stage is an ordinary one-pole smoother of price, and each later stage is an all-pass section that receives the previous stage's output and delays it in the warped time base. Low-numbered stages carry the recent past in fine detail while high-numbered stages hold a compressed summary of older history, which is exactly the memory allocation a plain moving average cannot make.

On the Library the lineage is visible in TheLark's Laguerre Moving Average and Laguerre RSI ports and in ChrisMoody's Laguerre PPO PercentileRank study, which subtracts a slow Laguerre line from a fast one, then ranks the spread by percentile to flag potential tops and bottoms. The honest framing for all of them: the filter is a well-engineered smoother, not a forecaster. It still lags genuine turns, gamma still trades speed against stability, and the warped memory that makes it elegant also makes its effective lookback harder to reason about than a simple length.

## How to identify a Laguerre filter on a chart

The line looks like a smooth moving average; the giveaways are the parameter it exposes and how it behaves relative to averages of similar smoothness.

1. Look at the input: a single gamma between 0 and 1 instead of a bar-count length is the Laguerre signature.
2. Check the construction if source is available: four cascaded stages sharing gamma, blended with 1-2-2-1 weights and divided by 6.
3. Compare against an [EMA](https://www.luxalgo.com/library/concept/ema/) tuned to similar smoothness: the Laguerre line should track turns with visibly less delay, which is the design's whole claim.
4. Sweep gamma: near 0 the line hugs price and reduces to a four-bar weighted smoother, near 1 it flattens dramatically; the compounding sensitivity of one shared parameter is characteristic.
5. Note the derivatives: an oscillator built from differences between the stages is the Laguerre RSI, and a fast-minus-slow pair of these filters is the Laguerre PPO.

## How it's calculated

A recursive four-stage smoothing filter by John Ehlers that delivers heavy smoothing with comparatively little lag.

```
L0_t = (1 - gamma) × P_t + gamma × L0_(t-1)
L1_t = -gamma × L0_t + L0_(t-1) + gamma × L1_(t-1)
L2_t = -gamma × L1_t + L1_(t-1) + gamma × L2_(t-1)
L3_t = -gamma × L2_t + L2_(t-1) + gamma × L3_(t-1)
Laguerre_t = (L0_t + 2 × L1_t + 2 × L2_t + L3_t) / 6

  P_t: source price at bar t (commonly close or (high + low) / 2)
  t: bar index
  gamma: damping factor between 0 and 1 (commonly 0.8)
  L0_t: first Laguerre ladder stage at bar t
  L1_t: second Laguerre ladder stage at bar t
  L2_t: third Laguerre ladder stage at bar t
  L3_t: fourth Laguerre ladder stage at bar t
  Laguerre_t: filter output at bar t
```

Published by John Ehlers in Time Warp Without Space Travel, with gamma = 0.8 in his filter example.

Higher gamma means heavier smoothing and more lag; gamma = 0 collapses the filter to a 4-bar weighted average with weights 1, 2, 2, 1.

The Laguerre RSI is built from the up and down differences between these same four stages, commonly with gamma = 0.5.

## How traders use it

- As a low-lag trend line: price above a rising filter reads as bullish bias, with crossovers traded the way [EMA](https://www.luxalgo.com/library/concept/ema/) crosses are, accepting that the line still lags genuine turns.
- As the smoothing stage inside other studies, where its short internal length keeps derived oscillators responsive while stripping bar-to-bar noise.
- As a regime gate: a flat filter with price whipping across it marks chop where trend entries tend to fail, used much like an [MA slope filter](https://www.luxalgo.com/library/concept/ma-slope-filter/).
- As a crossover pair: two gammas give a fast and slow line for [moving average crossover](https://www.luxalgo.com/library/concept/moving-average-crossovers/) systems, or their spread becomes a Laguerre PPO ranked by percentile for top and bottom flags.
- As dynamic support and resistance: a mid-gamma filter serves the same role as [dynamic S/R via MA](https://www.luxalgo.com/library/concept/dynamic-s-r-via-ma/), a moving reference that pullbacks in a trend repeatedly test.

## Laguerre filter vs other smoothers

- **EMA** (https://www.luxalgo.com/library/concept/ema/): An EMA is a single one-pole stage; the Laguerre filter cascades four and blends them, spending memory unevenly so it can smooth harder per unit of lag. The price is interpretability: an EMA length maps to intuition, gamma does not.
- **Ehlers SuperSmoother** (https://www.luxalgo.com/library/concept/ehlers-supersmoother/): Both are Ehlers designs with different goals. The SuperSmoother is a two-pole filter engineered to reject high-frequency noise near the bar-sampling limit; the Laguerre filter is about warped memory allocation. They are frequently combined, SuperSmoother first, Laguerre logic after.
- **Adaptive-lookback MA** (https://www.luxalgo.com/library/concept/adaptive-lookback-ma/): Adaptive averages change their effective length as conditions change; the classic Laguerre filter keeps gamma fixed. Adaptive Laguerre variants exist that drive gamma from measured volatility or efficiency, merging the two ideas at the cost of another estimation layer.

## FAQ

### What does gamma control in a Laguerre filter?

Gamma is the damping factor shared by the four internal stages, running from 0 to 1. At 0 the filter reduces to a fixed smoother of the last four bars; raising gamma stretches the effective memory, so the line gets smoother and lag grows. Because gamma feeds every stage, small changes compound, and settings should be adjusted gradually and tested.

### Is a Laguerre filter better than an EMA?

It is a different trade-off, not a strict upgrade. The design goal is more smoothing per unit of lag than a conventional average, and the line often looks steadier at comparable responsiveness. It still lags turns, still whipsaws in ranges, and its parameter is less intuitive than a length. Whether it improves a specific strategy is an empirical question for backtesting.

### Where does the Laguerre filter get its name?

From Edmond Laguerre, the nineteenth-century French mathematician whose orthogonal polynomials define the transform Ehlers borrowed. Laguerre polynomials are naturally weighted by a decaying exponential over an infinite horizon, which is what lets a four-term filter carry a long, unevenly allocated memory. Ehlers introduced the trading application in the early 2000s under the title Time Warp, Without Space Travel.

### How is the Laguerre filter computed internally?

Stage zero is a one-pole smoother of price with coefficient 1 minus gamma. Each subsequent stage is an all-pass section: it combines the current and previous values of the stage above with its own previous value, scaled by gamma. The plotted filter is the blend of the four stages with weights 1, 2, 2, 1, divided by 6. Four small recursions, one shared parameter.

### What is the difference between the Laguerre filter and the Laguerre RSI?

Same ladder, different readout. The filter plots the weighted blend of the four stages as a price-scale line. The [Laguerre RSI](https://www.luxalgo.com/library/concept/laguerre-rsi/) instead compares adjacent stages, summing the positive differences as up-pressure and negative ones as down-pressure, and plots the ratio from 0 to 1 as an oscillator. One skeleton feeds both a trend tool and a momentum tool.

### Does the Laguerre filter repaint?

No. It is a causal recursion over confirmed prices: each value uses only current and past data, so history never redraws. Its limitations are the ordinary ones, lag on sharp turns and whipsaw in congestion, not lookahead. On a live bar the current value updates tick by tick until the bar closes, which is standard for any indicator computed on the developing bar.

## Implementations in the Library

- Laguerre Filter (LuxAlgo): https://www.luxalgo.com/library/indicator/laguerre-filter/

## Related concepts

- Gaussian Filter: https://www.luxalgo.com/library/concept/gaussian-filter/
- Butterworth Filter: https://www.luxalgo.com/library/concept/butterworth-filter/
- Chebyshev Filters: https://www.luxalgo.com/library/concept/chebyshev-filters/
- Elliptic Filter: https://www.luxalgo.com/library/concept/elliptic-filter/
- Bessel Filter: https://www.luxalgo.com/library/concept/bessel-filter/
- Ehlers SuperSmoother: https://www.luxalgo.com/library/concept/ehlers-supersmoother/
- Ehlers Instantaneous Trendline: https://www.luxalgo.com/library/concept/ehlers-instantaneous-trendline/
- Ultimate Smoother: https://www.luxalgo.com/library/concept/ultimate-smoother/
- Swiss Army Knife Filter: https://www.luxalgo.com/library/concept/swiss-army-knife-filter/
- Windowed FIR Smoothing: https://www.luxalgo.com/library/concept/windowed-fir-smoothing/

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Source: https://www.luxalgo.com/library/concept/laguerre-filter/ (LuxAlgo Library, the encyclopedia of trading & technical analysis). Free to use with attribution: https://www.luxalgo.com/library/license/