# Difference between Non linear regression vs Polynomial regression

I have been reading a couple of articles regarding polynomial regression vs non-linear regression, but they say that both are a different concept. I mean when you say polynomial regression, in fact, it implies that its Nonlinear right. Then why there is a difference in the Data Science world regarding both the concepts?

the difference is probably easily seen with an example. Linear regression assumes a form $$f(x, \beta) = \beta_0 + \beta_1 x_1 + \cdots + \beta_n x_n$$ with some covariates $$x_i$$ and some parameters to estimate $$\beta_i$$. An example of non-linear regression would be something like $$f(x,\beta) = \frac{\beta_1 x_1 + \beta_2 x_2}{\beta_3 x_3 + \cdots + \beta_n x_n}.$$ Essentially you are assuming your model to be of a nonlinear form. Polynomial regression on the other hand is a fixed type of regression where the model follows a fixed form $$f(x, \beta) = \beta_0 + \beta_1 x + \beta_2 x^2 + \cdots + \beta_n x^n$$ which is a nonlinear function, however it is still linear in the parameters $$\beta$$ you are trying to estimate.

That is to say,

Polynomial regression is non-linear in the way that $$x$$ is not linearly correlated with $$f(x, \beta)$$; the equation itself is still linear.

In the other hand, non-linear regression is both non-linear in equation and $$x$$ not linearly correlated with $$f(x, \beta)$$.

• Nice answer, could you also tell if it's preferable to use non linear regression using RF or Boosting etc. than polynomial regression due to difficulty in finding right polynomial degree. Oct 10, 2019 at 21:00
• can we say that polynomial is a special case (form) of non-linear regression? Dec 22, 2021 at 4:43
• @AbdulkarimKanaan Linearity refers to linearity in the parameters: a linear combination of features. Since each polynomial term is its own feature, polynomial regression is a linear regression.
– Dave
Dec 3, 2022 at 19:59

Well elaborated by Marvin,To add on that-We define linear model as "A model that expresses the target output value in terms of a sum of weighted input variables" taking this definition into consideration for polymonial regression- the features are just numbers within a weighted sum