I have two variables and a single output for example:
$$x_1=\{0.2,0.4,0.6\} , x_2=\{0.2,0.6,1.9\} , y=\{10, 18, 30\}$$
I want to find an equation to obtain y from both $x_1$ and $x_2$: $y=f(x_1,x_2)$
What kind of equation is suitable for this problem and how to find the coefficients? I am not sure but some equations by googling is:
$$y=x_1^{\alpha}\cdot x_2^{\beta}$$
Find best alpha beta using dataset or maybe bivariate normal distribution
Thanks for your help in advance
The function $y=e^{\gamma}\cdot x_1^{\alpha}\cdot x_2^{\beta}$ fits the data quite well. $e^{\gamma}$ is a constant
To apply linear regression, you have to linearize the equation.
$ln(y)=ln(e^{\gamma}\cdot x_1^{\alpha}\cdot x_2^{\beta})$
$ln(y)=\gamma+\alpha \cdot ln(x_1)+\beta \cdot ln(x_2)$
First you have to transform the values of $y, x_1$ and $x_2$ in ln-values. I inserted the values in a table:
After applying linear regression the equation is:
$y = 0.3290563948\cdot x_1 + 0.327415062\cdot x_2 + 3.359135209$
And the regression line is $y_i=e^{3.359135209}\cdot x_{1i}^{0.3290563948}\cdot x_{2i}^{0.327415062}$
Inserting the values for i=1
$e^{3.359135209}\cdot 0.2^{0.3290563948}\cdot 0.2^{0.327415062}=10$
Inserting the values for i=2
$e^{3.359135209}\cdot 0.4^{0.3290563948}\cdot 0.6^{0.327415062}=18$
Thus there is more or less no difference between $y_i$ and the estimated $y_i$.