[Pandora] Laguerre Ultimate Explorations Multicator

It'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.