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I know that the first degree of the polynomial equation is considered as a linear function.

But, I found some things confusing in linear regression.

1. f(x)= w1 x1+ w2 x2 + W3 x3 --> linear function
2. f(x)= w1 x1+ w2 x2 + W3 x1 x3 --> is it linear? if not, then why?
3. f(x)= w1 x1+ w2 x2 + W3 W4 x3 --> is it linear? if not, then why?
4. f(x)= w1 x1+ w2 x2 + W3 x3 x3 --> it is not linear.    

Please help.

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  • $\begingroup$ To reduce any risk of confusion by others, this question and current answers deal with whether the function is linear in variables ($X$'s), not with whether these regression equations are linear in parameters, which all of them are. There is a separate question on what linear in parameters means: datascience.stackexchange.com/q/12274/69527. $\endgroup$
    – AlexK
    May 5, 2019 at 23:05

2 Answers 2

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A function $x = (x_1, \dots, x_n) \mapsto f(x_1, \dots, x_n) \in \mathbb{R}$ is linear if and only if for every $a,b \in \mathbb{R}$ it holds that $f(a (x_1, \dots, x_n) + b (\xi_1, \dots, \xi_n)) = a f(x_1, \dots, x_n) + b f(\xi_1, \dots,\xi_n)$.

In your second example try computing $f(a x + b \xi)$ to see that the last term $w_3 x_1 x_3$ will make $f(a x + b \xi) \neq a f(x) + b f(\xi)$.

The third example has a coefficient which is expressed as a product, but the function remains linear (just rename $w_3 w_4$ to $w$).

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Your third example is certainly linear as we can simply imagine $W3.W4$ to be some $W5$.

Regarding your other 2 examples, you need to clarify what your input is. If your input is $f(x1, x2, x3)$ then they are not linear. However, say in the second example if the input to your function is $f(x1,x2,x1x3)$ then it is still a linear function!

You can refer to @Miguel's answer to test it out.

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