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This is a basic question, but I was writing up some results and wasn't sure what terminology to use.

People seem to use the term "linear model" to describe what I would call "linear regression", $$Y_i = \beta_0 + \beta_1 x_i + \dots + \beta_p x_p + \epsilon_i.$$

I have the habit of referring to a linear model as anything linear in the parameters. This would be called a general linear model and would include anything of the form $\mathbf{Y = X B + U}$. This includes a variety of models that describe nonlinear relationships like polynomial regression, generalized additive models (GAMs), or regression with splines.

On the other hand, there are "nonlinear models" which are nonlinear functions of the parameters. For example, $\frac{\beta_1x}{\beta_2 + x}$.

Is there nomenclature that refers to models that describe nonlinear relationships but are still general linear models of the form $Y = XB + U$ which excludes models describing nonlinear relationships which are also nonlinear in the parameters?

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    $\begingroup$ I think you are confusing terms being linear in $x$ with linearity of terms. A polynomial regression is a linear model and is often taught a few weeks after introducing simple linear regression. Having terms like $x^4$ or $e^x$ is linear so long as these are multiplied by betas and added to get the predictor. $\endgroup$
    – kurtosis
    Jul 27, 2020 at 18:51

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Yes, Chris Bishop (machine-learning guru) has a chapter on these, called Linear Basis Function Models. Hyphenating properly it would be "linear basis-function models", to avoid any confusion -- the basis functions themselves are generally nonlinear.

In other words, these models are linear in the parameters, as you want, but the basis functions can be nonlinear. See Chapter 3 of his Pattern Recognition and Machine Learning, 2006.

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  • $\begingroup$ That's exactly the distinction I was making. Thank you. $\endgroup$
    – Eli
    Jul 27, 2020 at 20:11

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