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I have a function $f_\theta$ that is generic. More particularly, it is the function that describes the conversion of pressure to altitude.

$$ y_i=\frac{\mathrm{T}_{0}}{\mathrm{~L}}\left(\left(\frac{x_i}{\mathrm{P}_{0}}\right)^{-\mathrm{L} \mathrm{R} / \mathrm{g}}-1\right) $$

The problem is that there are a lot of assumptions about these constants so I do not get accurate results. I am interested in finding $\mathrm{T_0}$ and $\mathrm{P_0}$, but if it is not so complicated to extend it for all parameters, then I would be interested in finding all parameters.

What I do have is $n$ pairs $(x_i,y_i)$ where $y_i = f_{\theta^{\star}}(x_i)$ where $\theta^\star$ are the true parameters that I need to find.

What is a method to find $\theta^\star$ programmatically with a unique solution?

truvaking
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  • I don't see how is this an optimization problem. What function do you want to optimize? What is $\theta^*$?Also your problem looks like a maximum likelihood problem : https://en.wikipedia.org/wiki/Maximum_likelihood_estimation. Does this answer your question? – blamethelag Mar 18 '21 at 19:57
  • Sorry I am not very well acquainted with the terminology. It is like maximum likelihood function but instead of maximizing the likelihood function I want to minimize the difference between the sampled truth value and computed values. And instead of probability distribution I have a generic function. $\theta^\star$ are my constants here, but I treat them as parameters. – truvaking Mar 18 '21 at 20:01
  • If you find some open set $\Omega \subset \mathbb{R}^4$ containing the parameters you'd like to consider your problem can be put in the form $\underset{\theta \in \Omega}{\mathrm{argmin}} ~||(y_1,...,y_n)-(f_{\theta}(x_1),...,f_{\theta}(x_n))||$ that is an optimization problem with a $C^{\infty}$ function over an open set. I guess you could apply an optimization method like Newton or gradient descent but I am not that confident to specify which one to use. Those would converge if you start fairly close to the optimum, I guess you could try some random points or some you suspect to be correct. – blamethelag Mar 18 '21 at 20:31
  • You can formulate and solve this as a nonlinear least squares problem, using an off-the-shelf solver. The formulation can include constraints on the parameters, such as nonnegativity constraints, among possibly others. – Mark L. Stone Mar 19 '21 at 14:03

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