Difference between Non linear regression vs Polynomial regression

Question

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, it, in fact, implies that its Nonlinear right. They why there is a difference in the Data Science world regarding both the concepts?

Answer 1

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 $f(x, \beta)$.