For quadratic least squares statistical regression equations, the following is utilized for the calculation of coefficients $(a,b,c)$:
$a = \cfrac{∑(x^2y) * ∑(xx)-∑(xy) * ∑(xx^2)}{∑(xx) * ∑(x^2x^2)-∑(xx^2)^2}$
$b = \cfrac{ ∑(xy) * ∑(x^2x^2) - ∑(x^2y) * ∑(xx^2)}{∑(xx) * ∑(x^2x^2)- ∑(xx^2)^2}$
$c = \text{mean}(y) - b * \text{mean}(x) - a * \text{mean}(x^2)$
where,
$∑(xx) = ∑(x^2) - \cfrac{∑(x)^2}k$
$∑(xy) = ∑(xy) - \cfrac{∑(x)*∑(y)}k$
$∑(xx^2) = ∑(x^3) - \cfrac{∑(x)*∑(x^2)}k$
$∑(x^2y) = ∑(x^2y) - \cfrac{∑(x^2)*∑(y)}k$
$∑(x^2x^2) = ∑(x^4) - \cfrac{∑(x^2)^2}k$
Can anyone help me identify this same method for finding the coefficients $(a,b,c,d)$ with regard to cubic regression statistical equations?
Like this:
$a = ???$
$b = ???$
$c = ???$
$d = \text{mean}(y) - c * \text{mean}(x) - b * \text{mean}(x^2) - a * \text{mean}(x^3)$
where, the same identities as above, but possibly including $∑(x^3y)$, $∑(x^2x^3)$ and $∑(x^3x^3)$?