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Polynomial regression is a common way of doing curvilinear regression. It is common to also use the inverse transform x^-1 (http://pareonline.net/getvn.asp?v=8&n=6).

One can extend the concept of the inverse transform by thinking about Laurent polynomials.

My question is do some practitioners use Laurent polynomial regression? What are the general "rules" around them? With regular polynomials one tends to get a hill or a valley for each additional power.

I read this post which got me thinking about it: https://math.stackexchange.com/questions/231357/the-degree-of-a-polynomial-which-also-has-negative-exponents

https://en.wikipedia.org/wiki/Laurent_polynomial

Example R code:

x=runif(100,2,10)/2
y=jitter(1+abs(cos(x)))
df=data.frame(x=x,y=y)
df=df[order(df$x),]
plot(df$x,df$y)
lines(df$x,predict(lm(y~x+I(x^2)+I(x^3),df)))
title("3rd degree poly");

plot(df$x,df$y)
lines(df$x,predict(lm(y~x+I(x^-1)+I(x^-2),df)))
title("Laurent poly");

I wonder if some regular polynomial regression could benefit by having negative powers in addition to the positive?

Thoughts?

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