I have to obtain an empirical equation of efficiency (E) in percentage. After applying the Buckingham-Pi theorem to the variables on which efficiency depends, I got the following non-dimensional functional relationship:
E = f(X1, X2, X3, X4, X5)
Where X1......X5 are non-dimensional independent variables and E is an efficiency.
Now, I have to develop an empirical relation between efficiency (E) and non-dimensional independent variables (X1, X2, X3, X4, X5) using non-linear multiple regression analysis.
I want to use the power function: a(X^b)
Hence, E = C1.(X1^n1).(X2^n2).(X3^n3).(X4^n4).(X5^n5)
where C1 is constant and n1.....n5 are exponents.
Please suggest to me an easy method/software to do this task. I would appreciate your help.
The 1-st approach is to make log of your E function that will convert it into a linear regression for lnE vs lnXi, i.e. lnE=lnC+n1*lnx1+n2*lnx2+....
If you convert it back after estimation lnC and n(i) parameters using any standard linear regression you will get the needed function E=f(Xi).
However due to non-linear transformation the values of the parameters will be biased, i.e., they will not be equal to their values vs obtained directly from non-linear estimation of the original non-transformed function.
Depending on your final goal these biased parameters could be acceptable or not.
If you goal is just getting an interpolation function for E these biased parameters could suffice.
If your goal is making sense of the specific numerical values of the parameters, then they must be estimated from the original non-linear function. To do so, you may try Excel solver with non-linear minimization option, or any other package with non-linear optimization option which gives the likely global minimum vs a set of possible local minimums. There is no "easy" method for minimization of a highly non-linear function, but some numerical approaches do exist. See, for example, Numerical Recipes. The art of scientific computing. by W. Press et al. or other relevant sources on non-linear numerical optimization methods.
Attached is an introduction for you. Adaptive LASSO works on most dependant variables and I have attached some software that you should be able to use. best wishes David Booth
BTW a linear statistical model is one that is linear in the regression coefficients not necessarily in the predictor variables. Best wishes David Booth
David Eugene Booth The original function in question is an example of non-linear regression because the relationship E = C1(X1^n1).(X2^n2).(X3^n3).(X4^n4).(X5^n5) is the product of the power functions of predictor variables that are non-linear ones with respect to parameters n1, ..., n5.
The log-transformation makes E=f(Xi, ni) linear one with respect to parameters lnC and n1,...n5 but results in biasing of the estimated numerical values.
Glmnet is a package for linear regression with respect to parameters.
Therefore, it is not suitable for this particular problem at all, unless E=f(Xi, ni) is log-transformed.
Thank you Professor Kolker for your comment but I refer you to any competent linear statistical model book. In addition please note that I made no comment at all about what you said but I attempted to clarify and give a method that would would work and you might also note that my paper is an example of a logistic regression which can be transformed quite like yours. David Booth