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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 $y = ... + a_5 * P_5$

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  1. You can calculate pi and run linear regression. You should not be confused about the term "polynomial regression". A regression on polynomial basis expansion (even some of the terms do not exists) can be called polynomial regression.

  2. Drop the dependent variables or add regularization

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  • $\begingroup$ Another point I forgot to ask... If already know that some coefficient are between such and such values, in python is there any way to express it ? $\endgroup$
    – Yassine
    Mar 29, 2019 at 4:56

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