linear vs nonlinear regression model

This is a question in Mathworktutorial of Machine learning. I do think it is wrong!

Quiz (Select two) :Which of these is not a linear regression model? please say your reasons for all of them .

A) y = a + bx + cx^2

B) y = ax1 + bx2 + cx3

C) y = ax1x2 + bx2x3 + cx1x3

D) y = a + bsin(x1) + csin(x2)

E) y = a + bsin(cx)

F) y = a + bx + x ^c

the answer is y = a + bsin(cx) and y = a + bx + x ^c ( why y = a + bx + cx^2 and y = a + bsin(x1) + csin(x2) are not not non-linear?

[John D'Errico]

No. It is not wrong. @the cyclist is of course correct.

y = a*x^2

is LINEAR in a. It is nonlinear in x. But that is not relevant to the question.

Answer 1

[John D'Errico]

@Mohammadreza

As I said in my comment, there are two different kinds of linearity. Consider the model:

y = a*exp(x)

Is y linear in x? A linear model has properties like:

y(k*x) == k*y(x)
y(x1+x2) = y(x1) + y(x2)

Is that the case here? NO!!!!!!!

An exponential function like that is not linear. y is clearly nonlinear in x. And that is why you are confused. You mistake one kind of nonlinearity for another.

That y is nonloinear in x, does not make the model nonlinear in a. It is truly linear in a. For example, we see that:

a1*exp(x) + a2*exp(x) = (a1 + a2)*exp(x)

Now, what were you asked?

"Which of these is not a linear regression model?"

So in a linear regression context versus a nonlinear regression context, which models are linear? Which models can have their paameters estimated using a LINEAR regression? That is all that matters here. It is not what you think that is at all relevant, since what you think is wrong. (That is why you are a student, after all.)

Answer 2

[the cyclist]

Whether a model is linear or not refers to the coefficients being solved for (a, b, c), not the data (x, x1, x2).

[Mohammadreza]

Thanks for answer. But I think It should be refer to the independent variables and that's (x, x1, x2).

[Torsten]

Then your thinking is not correct.

Linear or nonlinear always refers to the variables you solve for - and these are the fitting parameters.

[the cyclist]

You can of course think that's how it is, or how it should be. But that is not how experts refer to it. (This is a common misunderstanding that people new to mathematical modeling have.)

See, for example, the Linear regression models section of the Linear Model wikipedia page. It explicitly states that the function of the independent variables can be nonlinear.