I derived a solution to a physics problem using a method of analysis that gives me an inversed relation, the independent variable $t$ expressed explicitly in terms of a nontrivial algebraic set of functions with the dependent variable $v$, namely $t= f(v, a,b,c,d)$,

where a,b,c,d, are parameters. From a physics standpoint it is variable $t$ that determines $v$, but obtaining such a solution of $v(t)$ it very complicated. My equation can be inverted using series functions but one sacrifices accuracy and some level of functionality. I need to determine the parameters a, b, c and d. For that purpose, I can use nonlinear regression vs. a reliable data set of ${(v_i,t_i)}$ values. The main sources of errors in the data are associated with variable $t$.

My question is if performing such a nonlinear regression for parameter estimation is a statistically valid approach. That is, the parameters and their confidence intervals could be considered accurate; having obtained very acceptable levels of significance in the statistical inferences of t statistics values, probability $p$, intrinsic curvature and parameter effect values, and well behaved standardized residuals

  • $\begingroup$ Just to be clear, it's only nonlinear regression if the coefficients you'r trying to find show up in nonlinear ways. So, for example, if you're trying to fit a quadratic $y=ax^2+bx+c$ to some data, it's linear regression because it's linear in the coefficients $a, b,$ and $c$. The fact that this relationship is nonlinear in $x$ is irrelevant to the terminology. Second, such a regression is definitely a statistically valid approach. You always want to look at your coefficient of determination, $R^2,$ to see how much of the variation in $t$ your fit explains. $\endgroup$ – Adrian Keister Jun 26 '18 at 13:38
  • $\begingroup$ Thank you for your remarks. I failed to mention that my solution is indeed nonlinear with respect to the parameters. It seems that the $R^2$ coefficient is not an applicable goodness-of-fit measure for nonlinear regression calculations, because it violates the basic premise about errors: SS_{ Regression} + SS_{ Error} = SS _{Total}.. $\endgroup$ – chavavic Jun 27 '18 at 15:00

It is a legitimate method. There is no close form solution for the nls, hence it is hard to tell something concrete without knowing nothing about $f(\cdot)$. For some types of non-linear models (e.g., the logistic model) - there are well established theoretical (inferential) results. For a general nls, assuming that the LS function is well behaving w.r.t the parameters and your initial values of them leads to the global minima - for large enough sample size you should be on the safe side.

  • $\begingroup$ Thank you, V. for your comment. You give me a sense of hope that I am doing something sensical. Would you clarify your statement regarding the "good behavior of the LS function w.r.t. the parameters"? Are you specifically referring to the curvature metrics for Max Intrinsic and Parameter effects or influences vouching (small enough below critical values) for ensuring linear behavior right around the converged parameter loci? $\endgroup$ – chavavic Jun 30 '18 at 1:04
  • $\begingroup$ Well, I'm not an expert in numerical methods. But the basic idea is that the function should be convex, smooth and continuously differentiable. Some rule of thumbs that I use is the "sensitivity" to the initial values and the optimization method., i.e., if by using different initial values and more than one optimization method you get +/- the same results, then it is a good sign. $\endgroup$ – V. Vancak Jun 30 '18 at 1:33
  • $\begingroup$ I see your point. Thank you. $\endgroup$ – chavavic Jul 1 '18 at 13:25

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