# Multivariate (Polynomial?) Regression

I'm trying to solve one problem, I'm not sure how to do it...

Here is my problem (related to Heat transfer/Fluid mechanics) that I tried to simplify :

• I have a few independents measurements $$x_1, x_2, x_3, x_4$$ and $$y$$
• These datas give me some parameters, using some constants :
• $$P_1 = x_1 + i$$
• $$P_2 = j * x_1*x_2$$
• $$P_3 = x_3^\frac{1}{2} + k$$
• $$P_4 = x_4^4 - \frac{x_3^4}{l}$$
• Then I have : $$y = a_1 * P_1 + a_2* P_2 + a_3 * P_3 + a_4 * P_4$$

My questions are :

1. How can I find the coefficients $$a_1, a_2, a_3$$ and $$a_4$$ (using python) ?

• Shall calculate all $$P_i$$ and make a linear regression on the formula ?
• Shall expand everything and make a polynomial regression on $$x_i$$ ?
• Shall i do something else than regression (ANN, etc.) ?
• Am I missing something here ?
2. What should I do in case of one $$P_i$$ is dependent ? For example : $$P_5 = x_1(x_2+1)$$ (which can be extracted from $$P_1$$ and $$P_2$$) with $$y = ... + a_5 * P_5$$