Pseudo Polynomial Channel

Introduction

Back when i started using pine i made a script called periodic channel who aimed to rescale an average correlated sine wave to the price...don't worked very well. So i tried to fix problems induced by the indicator without much success, i had to redo it from scratch while abandoning the idea of rescaling correlated smooth functions to the price, at that time i also received requests regarding polynomial channel, some plateformes included this indicator, this led me to the idea to estimate it in order to both respond to the periodic channel problems and the requests i received, i have tried many many things and recently i tweaked a linear extrapolation to have an approximation.

Linear Extrapolation To Pseudo Polynomial Regression

I could be wrong but a polynomial regression must use constant parameters in order to provide a really smooth output, at least constant for a set of time. The moving averages forms (Savitzky-Golay moving average) who smooth polynomials across a window to the data don't have such smoothness, so how to estimate a polynomial regression while having a parameter providing control over the smoothness, a response to this is by using a recursive linear extrapolation. I posted a linear extrapolation indicator long ago, i used the same formula while adding a function to morph the output and the input in the form of :

morph * output + (1-morph) * input

How can this provide an estimate of a polynomial regression ? Well i'm not even sure myself but if you use the output as input (morph = 1) for the linear extrapolation function you should get a rough estimate of a line, this is what i thought at first and it proved to be right


Based on this observation i thought that it would be possible to get polynomial results by lowering morph, and as expected it worked well but showed a periodic pattern, this is why i smooth k in line 10.

0.9 for morph work well, higher values create sometimes smoother results but damage heavily the estimation.

Parameters

Morph have been introduced earlier, it control the amount of output and input the linear extrapolation should process, lower values create rougher but more stables results, if you see that the estimation is going nuts lower morph or change length, also lower length if you increase morph.


High overshoot, morph to 0.8 can help have a better estimation at the cost of less smoothness.

Length control the indicator smoothing, this parameter differ heavily from other filters, therefore low values can create mid/long term smoothing, it can also depend on which market instrument you are applying it, so there are no fixed optimal length.

Mult control how spread the bands are, to do so mult multiply the cumulative mean error, you can change this error measurement by anything you want like standard deviation/atr/range but take into account that you may create a separate parameter to control the error instead of length. Mult can be a float and like length can have different optimal values depending on the market the indicator is applied to.

Flatten do exactly what is name imply, it flatten the overall output to have a better estimation, can be a float. The result is less smooth.


Flatten = 2

More Exemples


BTCUSD length = 25 and mult = 4


XPDUSD length = 25 and mult = 1


ALPHABET length = 6 and morph = 0.99

Conclusion

I tried to estimate a polynomial channel by using recursion in the linear extrapolation function. This build is way more stable than the periodic channel but its still a bit inaccurate in my opinion. I hope this code can still help someone build something really nice, if so share your results :)

I apologize for those expecting a legit polynomial channel build but i really don't know how to do that, as i said parameters for the regression must be constants, i hope it still fine :)

Thanks for reading !