What is curve linear?
In linear regression, we are fitting a polynomial to a set of data points. In Bishop's book of Pattern Recognition & Machine Learning, there are a few examples where the fit is a curve or a straight line. I am a bit confused if a curve is linear or not.
The term linear means that the fit should be a linear function or a polynomial of degree 1 i.e., a straight line.
But in many resources, examples are shown where the fit can be a polynomial of degree 3,9 etc. So, are these higher order polynomials linear?
Linear in linear regression means linear in parameters. It refers to the relationship between the parameters that you are estimating (e.g., β) and the dependent variable (e.g., yi). Hence, y=exβ+ϵ is linear, but y=eβx+ϵ is not. This has nothing to do with the powers of the independent variables. There are a few examples where the fit is a curve or a straight line. The fit can be a curve linear and can incorporate higher powers of independent variables and be linear in parameters -- the betas.