I am taking a machine learning course and today we were given an example of regression, with two attributes $x_,x_2$ and $y$ being the real valued outcome.
$y$ is a quadratic function of $x_1,x_2$, given as
$y=\alpha_1 x_1^2+\alpha_2 x_2^2+\alpha_3 x_1x_2+\alpha_4x_1+\alpha_5x_2+c$ Now my teacher told that we can view this function as a linear function of $\lt x_1^2,x_2^2,x_1x_2,x_1,x_2 \gt$.
But I can't really visualize how can a curved function look like a straight line.
Please some one explain it or give some good reference.
Quadratic function as a linear function
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0http://en.wikipedia.org/wiki/Linear_combination – 2012-07-30
1 Answers
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It would seem to me that your teacher has completely botched his/her explanation. What s/he probably intended to say is that your model function ($y$ as a function of the parameters $\alpha_k$) is what's termed as a linear model (and thus, the kind of regression you need to do here is called a linear regression).
Here, you can think of the adjective "linear" like so: if you take the partial derivative of $y$ with respect to any of the $\alpha_j$, you will obtain an expression that is free of any of the $\alpha_j$ (in fact, the result will be whatever was multiplied with the $\alpha_j$ under consideration). As an example,
$\frac{\partial y}{\partial \alpha_4}=x_1$
See also this answer in CV.