Question

y_1 ~ ax_1^2+bx_1+c


whats the linear equation?

Answer 1

You wrote y1 ~ a x1^2 + b x1 + c. That is a quadratic function of x but it is linear in the parameters a, b, c. Written as a linear (regression) equation:

- For one observation i:
y_i = a x_i^2 + b x_i + c

- As a dot product (feature form) for a given x:
y = [x^2 x 1] · [a b c]^T

- For n observations in matrix form:
y = X β, where
X = [[x1^2 x1 1],
[x2^2 x2 1],
...
[xn^2 xn 1]],
β = [a b c]^T,
y = [y1 y2 ... yn]^T.

If you want the least-squares estimate of β:
β̂ = (X^T X)^{-1} X^T y.

If you meant something else by “linear equation,” tell me and I’ll adapt.