Laguerre Filter [BackQuant]Laguerre Filter
Overview
The Laguerre Filter is a powerful trend-following tool designed to smooth price action while maintaining responsiveness to market changes. It is based on the Laguerre recursive filter, which is a type of signal processing filter that adapts to both the current price dynamics and the underlying trend. The Laguerre Filter can be seen as a method to reduce market noise, enabling traders to more easily identify the strength and direction of trends while minimizing lag.
The Laguerre Filter is well-suited for markets with varying volatility levels, offering a smoother representation of price action without the delay associated with traditional moving averages. By dynamically adjusting to price movements, the Laguerre Filter provides a more adaptive and reliable signal compared to simpler smoothing techniques.
What is the Laguerre Filter?
The Laguerre Filter is derived from the Laguerre polynomial, which is used in signal processing for smooth filtering of data. The Laguerre filter is a recursive filter, meaning that each new value is calculated based on both the current price data and previous values, with a weighting system that allows it to adapt to market conditions. This recursive nature helps reduce the impact of short-term fluctuations, enabling the filter to focus on the underlying trend.
The Laguerre filter uses a feedback mechanism, where the input signal (price data) is smoothed iteratively. This iterative process helps avoid the lag that is typically associated with traditional moving averages while still capturing the overall trend direction.
The filter is designed to have:
Adaptive behavior: It reacts quickly to significant price changes while ignoring minor fluctuations.
Reduced noise: By filtering out random short-term price movements, it provides a clearer view of the underlying trend.
Customizability: Traders can adjust the filter’s sensitivity through user inputs, making it adaptable to different market conditions.
Core Calculation Methodology
The core of the Laguerre Filter lies in its recursive calculation:
Each new value is calculated using the previous value along with the current price input.
The recursive formula is governed by two key parameters: the damping factor (gamma) and the order of the filter (number of Laguerre elements).
The damping factor controls how responsive the filter is to changes in price. A higher gamma value makes the filter smoother but introduces more lag, while a lower gamma value makes it more reactive to price changes but can introduce more noise.
The order defines how many Laguerre elements are used in the calculation. A higher order results in a smoother output but with more delay, while a lower order provides a faster response but less smoothing.
The filter works by weighting previous values with a binomial weighting system, which assigns more weight to recent values and less weight to older values. This creates a dynamic smoothing effect that adapts to price volatility, ensuring that the filter is neither too slow nor too noisy.
Signal Logic and Trend Detection
The Laguerre Filter continuously evaluates the strength and direction of the trend by comparing the current smoothed value to the previous value:
If the current value is greater than the previous value, the trend is considered bullish, and the filter will signal a long condition.
If the current value is less than the previous value, the trend is considered bearish, and the filter will signal a short condition.
The trend detection logic is based on the recursive nature of the filter, which smooths price movements over time. This allows the filter to capture the broader trend while minimizing the influence of short-term price fluctuations.
The trend state is also visually represented by color-coding:
Green color represents an uptrend (bullish condition).
Red color represents a downtrend (bearish condition).
Neutral (white) indicates no clear trend direction.
This color-coding helps traders easily identify the prevailing trend and decide whether to enter or exit trades based on the trend's strength.
Laguerre Filter Behavior and Performance
The performance of the Laguerre Filter can be influenced by several factors:
Gamma (Damping Factor): A higher gamma value results in a smoother filter but increases lag. A lower gamma value allows for a faster response but may introduce more noise, making it more reactive to smaller price changes.
Filter Order: The order determines how many Laguerre elements are used in the filter calculation. A higher order provides more smoothing but increases lag, while a lower order results in a quicker response but less smoothing.
The sweet spot for gamma is typically between 0.7 and 0.85, where the filter offers a good balance between smoothness and responsiveness. The filter order is usually set to 4 for classic Laguerre filtering, but higher orders can be used for more smoothing if needed.
The Laguerre Filter’s performance shines in markets with sustained trends, where the filter can effectively capture and represent the underlying direction without excessive lag. It is particularly useful in volatile markets, as it helps smooth out noise while providing a clear picture of the trend.
Visual Presentation
The Laguerre Filter provides a dynamic, color-coded line that follows the trend direction. This line can be displayed alongside price data to visually highlight the market trend. In addition to the main Laguerre line, several visual enhancements can be applied:
Gradient fill between the price and the Laguerre Filter line, providing a visual cue for bullish or bearish market conditions.
Candle coloring to reflect the current trend, making it easier to spot trend reversals or confirmations directly on the chart.
Background shading to visually highlight areas of strong trend or consolidation.
Edge glow effect that highlights trend boundaries, making it easy to spot key levels of support or resistance.
These visual elements enhance the usability of the Laguerre Filter, allowing traders to quickly assess the market trend and make informed decisions.
Practical Use Cases
1) Trend Following
The Laguerre Filter is ideal for trend-following strategies. By using the filter to identify the prevailing trend, traders can:
Enter long positions when the Laguerre Filter turns bullish (green).
Enter short positions when the Laguerre Filter turns bearish (red).
By aligning trades with the dominant trend, traders can improve their chances of success.
2) Trend Strength Assessment
The Laguerre Filter can also be used to assess the strength of the trend:
A rising Laguerre value indicates a strengthening uptrend.
A falling Laguerre value indicates a strengthening downtrend.
A flattening Laguerre value signals weakening momentum or consolidation.
This information can be used to adjust position sizing or to decide when to enter or exit a trade.
3) Trade Management
The Laguerre Filter can also assist in trade management:
Use the Laguerre line as a trailing stop for long positions in an uptrend.
Scale out of positions as the Laguerre value begins to flatten or reverse.
Use the Laguerre Filter to avoid trades when the market is in consolidation or lacks a clear trend.
Tuning Guidelines
The Laguerre Filter can be adjusted for different market conditions using the following parameters:
Gamma (Damping Factor): Adjust for the desired level of responsiveness versus smoothness. Typical values range from 0.7 to 0.85.
Filter Order: Adjust to control the level of smoothing. The default value of 4 is a good starting point, but higher orders can be used for smoother filters.
Summary
The Laguerre Filter is a versatile and adaptive trend-following indicator that smooths price data and reduces noise, making it easier to identify and follow trends. By using recursive smoothing techniques and adjustable parameters, the Laguerre Filter provides an accurate representation of market conditions with minimal lag. It is especially useful in volatile markets where traditional moving averages may fail to capture the underlying trend. With its color-coded trend detection, gradient fills, and customizable settings, the Laguerre Filter is a powerful tool for traders looking to stay aligned with the prevailing market direction.
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[Pandora] Laguerre Ultimate Explorations MulticatorIt's time to begin demonstrations differentiating the difference between known and actual feasibility beyond imagination... Welcome to my algorithmic twilight zone .
INTRODUCTION:
Hot off my press, I present this Laguerre multicator employing PSv6.0, originally formulated by John Ehlers for TASC - July 2025 Traders Tips. Basically I transcended Ehlers' notions of transversal filtration with an overhaul of his Laguerre design with my "what if" Pandora notions included. Striving beyond John Ehlers' original intended design. This action packed indicator is a radically revamped version of his original filter using novel techniques. My aim was to explore whether providing even more enhanced responsiveness and lesser lag is possible and how. Presented here is my mind warping results to witness.
EHLERS' LAGUERRE EXPLAINED:
First and foremost, the concept of Ehlers' Laguerre-izing method deserves a comprehensive deep dive. Ehlers' Laguerre filter design, as it functions originally, begins with his Ultimate Smoother (US) followed by a gang of four LERP (jargon for Linear intERPolation) filters. Following a myriad of cascading LERPs is a window-like FIR filter tapped into the LERP delay values to provide extra smoothness via the output.
On a side note, damping factor controlled LERP filters resemble EMAs indeed, but aren't exactly "periodic" filters that would have a period/length parameter and their subsequent calculations. I won't go into fine-grained relationship details, but EMA and LERP are indeed related in approach, being cousins of similar pedigree.
EXAMINING LAGUERRE:
I focused firstly on US initialization obstacles at Pine's bar_index==0 with nz() in abundance. The next primary notion of intrigue I mostly wondered about was, why are there four LERP elements instead of fewer or more. Why not three or why not two LERPs, etc... 1-4-6-4-1, I remember seeing those coefficients before in high pass filters.
Gathering my thoughts from that highpass knowledge base, I devised other tapped configuration modes to inspect their behavior out of curiosity. Eureka! There is actually more to Laguerre than Ehlers' mind provided, now that I had formulated additional modes. Each mode exhibits it's own lag/smoothness characteristics better than the quad LERPed version. I narrowed it down to a total of 5 modes for exploration. Mode 0 is just the raw US by itself.
ANALYZING FILTER BEHAVIORS:
Which option might be possibly superior, and how may I determine that? Fortunately, I have a custom-built analyzer allowing me to thoroughly examine transient responses across multiple periodicities simultaneously, providing remarkable visual insights.
While Ehlers has meagerly touched upon presenting general frequency responses in his books, I have excelled far beyond that. This robust filter analysis capability enables me to observe finer aspects hidden to others, ultimately leading to the deprecation of numerous existing filters. Not only this, but inventing entirely new species of filtration whether lowpass, highpass, or bandpass is already possible with a thorough comprehensive evaluation.
Revealing what's quirky with each filter and having the ability to discover what filters may be lacking in performance, is one of it's implications. I'm just going to explain this: For example US has a little too much overshoot to my liking, along with nonconformant cutoff frequency compliance with the period parameter. Perhaps Ehlers should inspect US coefficients a bit closer... I hope stating this is not received in an ill manner, as it's not my intention here.
What this technically eludes to is that UltimateSmoother can be further improved, analogous to my Laguerre alterations described above. I will also state Laguerre can indeed be reformulated to an even greater extent concerning group delay, from what I have already discussed. Another exciting time though... More investigative research is warranted.
LAGUERRE CONCLUSIONS:
After analyzing Laguerre's frequency compliance, transient responses, amplitudes, lag, symmetry across periodicities, noise rejection, and smoothness... I favor mode 3 for a multitude of reasons over the mode 4 configuration, but mostly superb smoothing with less lag, AND I also appreciated mode 1 & 2 for it's lower lag performance options.
Each mode and lag (phase shift) damping value has it's own unique characteristics at extremes, yet they demonstrate additional finesse in it's new hybrid form without adding too much more complexity. This multicator has a bunch of Laguerre filters in the overlay chart over many periodicities so you can easily witness it's differing periodic symmetries on an input signal while adjusting lag and mode.
LAGUERRE OSCILLATOR:
The oscillator is integrated into the laguerreMulti() function for the intention of posterity only. I performed no evaluation on it, only providing the code in Pine. That wasn't part of my intended exploration adventure, as I'm more TREND oriented for the time being, focusing my efforts there.
Market analysis has two primary aspects in my observations, one cyclic while the other is trending dynamics... There's endless oscillators, but my expectations for trend analysis seems a little lesser explored in my opinion, hence my laborious trend endeavors. Ehlers provided both indicator facets this time around, and I hope you find the filtration aspect more intriguing after absorption of this reading.
FUNCTION MODULES EXPLAINED:
The Ultimate Smoother is an advanced IIR lowpass smoothing filter intended to minimize noise in time series data with minimal group delay, similar to a traditional biquad filter. This calculation helps to create a smoother version of the original signal without the distortions of short-term fluctuations and with minimal lag, adjustable by period.
The Modified Laguerre Lowpass Filter (MLLF) enhances the functionality of US by introducing a Laguerre mode parameter along side the lag parameter to refine control over the amount of additional smoothing/lag applied to the signal. By tethering US with this LERPed lag mechanism, MLLF achieves an effective balance between responsiveness and smoothness, allowing for customizable lag adjustments via multiple inputs. This filter ends with selecting from a choice of weighted averages derived from a gang of up to four cascading LERP calculations, resulting with smoother representations of the data.
The Laguerre Oscillator is a momentum-like indicator derived from the output of US and a singular LERPed lowpass filter. It calculates the difference between the US data and Laguerre filter data, normalizing it by the root mean square (RMS). This quasi-normalization technique helps to assess the intensity of the momentum on any timeframe within an expected bound range centered around 0.0. When the Laguerre Oscillator is positive, it suggests that the smoothed data is trending upward, while a negative value indicates a downward trend. Adjustability is controlled with period, lag, Laguerre mode, and RMS period.
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TASC 2025.07 Laguerre Filters█ OVERVIEW
This script implements the Laguerre filter and oscillator described by John F. Ehlers in the article "A Tool For Trend Trading, Laguerre Filters" from the July 2025 edition of TASC's Traders' Tips . The new Laguerre filter utilizes the UltimateSmoother filter in place of an exponential moving average (EMA) in its calculation, offering improved responsiveness and reduced lag.
█ CONCEPTS
As Ehlers explains in his article, the Laguerre filter is a form of transversal filter . A transversal filter calculates an output signal using a tapped delay line . It creates multiple delayed versions of an input signal, applies weight to each delay, and then calculates their sum to generate the filtered result.
The Laguerre filter's structure relies on Laguerre polynomials — solutions to a differential equation solved by Edmond Laguerre in the 1800s. When Ehlers analyzed the formula for these polynomials on discrete systems (e.g., financial time series), he found that the first term's expression corresponds to an EMA response, and all subsequent terms correspond to an all-pass response. In contrast to other filter types, an all-pass filter produces phase shift (i.e., delay) in an input signal's components without affecting its amplitude.
Ehlers observed that these characteristics of Laguerre polynomials make them suitable for use in a transversal filter structure, and thus the Laguerre filter was born. However, he notes that EMAs are not great filters in general. As such, to improve on the Laguerre filter's design, Ehlers modified it by replacing the EMA term with his UltimateSmoother filter. The resulting Laguerre filter has significantly reduced lag, achieving a tighter response to market fluctuations while maintaining smoothness. Ehlers suggests that traders can analyze crossings between the UltimateSmoother and this Laguerre filter, or those between two Laguerre filters of different order, for helpful buy and sell signals.
In addition to the Laguerre filter, Ehlers derived a smooth, low-lag oscillator based on the difference between the first and second terms in the modified filter structure, scaled by the root mean square (RMS). The resulting oscillator provides an alternative filtered representation of market data, which can help traders identify swing and mean-reversion signals.
█ USAGE
This indicator calculates both the Laguerre filter and the Laguerre oscillator described in Ehlers' article. It displays the Laguerre filter on the main chart pane and the oscillator in a separate pane.
Users can control the behavior of the filter and oscillator with the inputs in the "Settings/Inputs" tab:
The "Period" input defines the critical period of the UltimateSmoother used in the Laguerre filter and oscillator calculations. Its default value is 30.
The "Gamma" input determines the weighting behavior of the Laguerre filter and oscillator. It accepts a positive value between 0 and 1. Use a lower value for quicker responsiveness to market changes, and a higher value for trends. The default value is 0.5.
The "RMS length" input determines the length of the RMS calculation for oscillator normalization. The default value is 100 bars.
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Relative Momentum Index with Laguerre FilterThe Relative Momentum Index
The Relative Momentum Index (RMI) is an oscillator that is a variation of the Relative Strength Index (RSI), but incorporates momentum over a variable lookback period rather than just consecutive price changes, which can help identify reversals and filter out noise.
It measures the momentum of price changes over a specified period, rather than just the magnitude of price changes like the RSI does.
It counts up and down days from the current closing price relative to the closing price a certain number of days ago (e.g. 5 days ago), instead of just comparing consecutive daily closes like the RSI
It is calculated by taking the ratio of the average upward price changes to the average downward price changes over a given period, where each change is measured from the close X days ago (X is the “momentum” period)
Like the RSI, the RMI oscillates between 0 and 100, with readings above 70 considered overbought and below 30 oversold.
In trending markets, the RMI tends to remain in overbought or oversold territory for extended periods. In trading ranges, it oscillates more predictably between the overbought and oversold levels.
The RMI is generally considered better than the RSI at identifying potential reversal points, as it incorporates a momentum factor rather than just strength.
It can be used in a similar way to the RSI for trade signals, such as buying when it rises above 30 from below, or selling when it falls below 70 from above
The Laguerre filter
A Laguerre filter is a type of infinite impulse response (IIR) filter used for smoothing signals or data. The Laguerre filter provides a way to apply variable smoothing to a signal by adjusting its pole position, allowing you to control the balance between smoothness and lag based on your preferences. It is an alternative to simple moving averages that can better preserve the shape of the original signal. Indicator
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Jurik-Filtered, Adaptive Laguerre PPO [Loxx]Jurik-Filtered, Adaptive Laguerre PPO is an indicator used to find reversals. Smoothing with a Jurik Filter reduces noise and better identifies reversal points.
What is Laguerre Filter?
The Adaptive Laguerre is based on the Laguerre filter, described by John Ehlers in his paper “Time Warp – Without Space Travel”. It applies a variable gamma factor, based on how well the filter is tracking previous price movement. As with other adaptive moving averages, the Adaptive Laguerre tracks trending markets closely but will see less changes in range-bound markets.
The Adaptive Laguerre filter allows for an adjustment of the simple Laguerre filter. When price moves away from the filter, it becomes faster. When price moves sideward, the filter gets slower. Accordingly, this indicator belongs to the same class of moving average as the Kaufman Adaptive Moving Average (KAMA). It similar to the Volatility Index Dynamic Average (VIDYA) developed by Tushar Chande. The Adaptive Laguerre filter is smoother than the VIDYA and will adjust slower to price action after consolidations.
What is Jurik Volty?
One of the lesser known qualities of Juirk smoothing is that the Jurik smoothing process is adaptive. "Jurik Volty" (a sort of market volatility ) is what makes Jurik smoothing adaptive. The Jurik Volty calculation can be used as both a standalone indicator and to smooth other indicators that you wish to make adaptive.
What is the Jurik Moving Average?
Have you noticed how moving averages add some lag (delay) to your signals? ... especially when price gaps up or down in a big move, and you are waiting for your moving average to catch up? Wait no more! JMA eliminates this problem forever and gives you the best of both worlds: low lag and smooth lines.
Ideally, you would like a filtered signal to be both smooth and lag-free. Lag causes delays in your trades, and increasing lag in your indicators typically result in lower profits. In other words, late comers get what's left on the table after the feast has already begun.
Included:
-Toggle on/off bar coloring Indicator
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Laguerre FilterThis is the Laguerre Filter by John F. Ehlers. He published this filter design in the article Time Wrap - Without Space Travel.
This study open gamma and weight of each filter for to customize the filter design.
For example, setting gamma to 0.2, 0.7, 0.8,0.8 and weight 1,2,2,1, the curve is a very nice simulation to ALMZ(50),
setting gamma to 0.3, 0.3, 0.8, 0.9 and weight of 1,3,3,1 will produce a curve close to EMA(20),
basically, the Laguerre filter can simulate various type of moving averages.
The beauty of this Filter is it use only one data point and 3 filters output to reproduce various moving average, it is
an innovative approach to develop smoothing lines, and to avoid whipsaws on the price.
Each filter can be plotted for evaluation too.
Data of the first filter allows to input any bar, eg. 0 for current bar, 1 for one previous, for the purpose to evaluate forecasting idea.
Follows Ehlers' paper, this study allows to plot the weighted average line with all filter off.
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[blackcat] L2 Ehlers Laguerre FilterLevel: 2
Background
John F. Ehlers introuced Laguerre Filter in his "Cybernetic Analysis for Stocks and Futures" chapter 14 on 2004.
Function
The Laguerre transform can be represented as an exponential moving average (EMA) low-pass filter (the first term) followed by a succession of allpass elements instead of unit delays (the k − 1 terms). All terms have exactly the same damping factor γ (gamma). We see that these are all pass networks by examining the frequency response. Dr. Ehlers made a filter using the Laguerre elements instead of the unit delay, whose coefficients are also /6 as with the FIR filter. The difference is that we have warped the time between the delay line
taps.
Key Signal
FIR ---> Laguerre Filter fast line
Filt ---> Laguerre Filter slow line
Pros and Cons
100% John F. Ehlers definition translation of original work, even variable names are the same. This help readers who would like to use pine to read his book. If you had read his works, then you will be quite familiar with my code style.
Remarks
The 33th script for Blackcat1402 John F. Ehlers Week publication.
Readme
In real life, I am a prolific inventor. I have successfully applied for more than 60 international and regional patents in the past 12 years. But in the past two years or so, I have tried to transfer my creativity to the development of trading strategies. Tradingview is the ideal platform for me. I am selecting and contributing some of the hundreds of scripts to publish in Tradingview community. Welcome everyone to interact with me to discuss these interesting pine scripts.
The scripts posted are categorized into 5 levels according to my efforts or manhours put into these works.
Level 1 : interesting script snippets or distinctive improvement from classic indicators or strategy. Level 1 scripts can usually appear in more complex indicators as a function module or element.
Level 2 : composite indicator/strategy. By selecting or combining several independent or dependent functions or sub indicators in proper way, the composite script exhibits a resonance phenomenon which can filter out noise or fake trading signal to enhance trading confidence level.
Level 3 : comprehensive indicator/strategy. They are simple trading systems based on my strategies. They are commonly containing several or all of entry signal, close signal, stop loss, take profit, re-entry, risk management, and position sizing techniques. Even some interesting fundamental and mass psychological aspects are incorporated.
Level 4 : script snippets or functions that do not disclose source code. Interesting element that can reveal market laws and work as raw material for indicators and strategies. If you find Level 1~2 scripts are helpful, Level 4 is a private version that took me far more efforts to develop.
Level 5 : indicator/strategy that do not disclose source code. private version of Level 3 script with my accumulated script processing skills or a large number of custom functions. I had a private function library built in past two years. Level 5 scripts use many of them to achieve private trading strategy. Indicator
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Trend-Following Combo-SuperTrend, EMA, Aroon, DMI, Laguerre RSIThis is a trend-following indicator which condenses two SuperTrend indicators -- one based on analysis over a shorter period of time (1.5, 7), and one based on analysis over a longer period of time (1.65, 100) -- into a single indicator which appears on your chart only when both the shorter- and longer-term analysis indicates a "SuperTrend" in the same direction.
Additionally, potential trade entry indicators are displayed in the form of up and down arrows when (by default) three of the following five indicators suggest that the market is trending in the same direction as both the shorter- and longer-term SuperTrend indicators:
EMA Crossover (8, 15)
Aroon Indicator (8)
Aroon Oscillator (8)
Directional Movement Index (DI +/-) (8)
Laguerre RSI (13)
You may update the parameters of any of the indicators to match your own preferences.
Additionally, you may also adjust the "Threshold" of indicators that must be in agreement with the SuperTrend to show a potential trade entry arrow. Bear in mind that if you set the Indicator Threshold too low, you will see more frequent trade entry arrows, many of which will not be profitable if taken. Similarly, set this value too high, and you will see fewer trade entry arrows that may not appear until after most of the "juice" in the trend has evaporated. Ideal values for the threshold seem to be between 2-4, depending on the symbol you are trading.
The following image shows all of the indicators referenced above on a 5-minute chart of the SPY during a single trading day:
And, here is the same period of time showing only the Trend-Following Combo indicator with default settings:
This indicator would not have been possible save for work contributed by the following:
SuperTrend by Rajandran R
Aroon w/ crossovers highlighted by seiglerj
Aroon Oscillator by jcrewolinsky
Directional Movement Index by PulseWire
Laguerre RSI (Self Adjusting Alpha with Fractals Energy) by everget
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