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hope you're doing well

I'm actually working with sets of parameters. For the sake of simplicity and replicability I'm formulating a general question altough it is related to this other question in which the idea was to find a solution for 5 parameters in a model where $f(x_1,x_2,x_3,x_4,x_5)$ is observed (known) once. For instance

-It could be the 3D position $(x_1,x_2,x_3)$, mass $x_4$ and hydrogen component $x_5$ of a galaxy, that allows the computation of a parameter $f$.

-It could be the (derived from a model) initial position of an object, its speed and mass, having measured its momentum.

Or anything else one could come up with. I wonder wether the parameters (solutions for the observable) do cluster in certain regions of the parameter space or not. I checked for FindClusters but I'm not sure if it's the right approach.

Question: Could you please suggest any readings/hints/ideas on the matter?

Idea

Also I'm attaching a simple sketch of what I'm thinking of (in $f:\mathbb{R}^2\to \mathbb{R}$). It would useful to study the correlations (because this parameters $x_1$, $x_2$ may be correlated) between groups of parameters that yield the same result (e.g., find different "types"/mechanisms that work for a model).

Description of the image: scattered data points $\{x_1,x_2, f(x_1,x_2)\}$ in which some (pink color) scatter in a "solution set 1" and others scatter in a "solution set 2".

nuwe
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    It is not clear to me. Do you have a couple of measurements {x1,x2,..f[x1,x2,..]} that belong to set S1. The same for S2 e.t.c. Now, a) do you want to fit different functions to different sets? Or b) do you want to fit one single function? a) seems obvious. For b) you could first calculate a suitable defined center of a set Si. E.g. the Mean or a weighted mean e.t.c. Finally you would then make a fit with the centers. – Daniel Huber Nov 03 '21 at 16:26
  • @DanielHuber I have a measurement $f(x_1,x_2,...)$ and I have calculated set of possibilities (according to a model) that satisfy $f(x_1,x_2,...)$. Another way to formulate the question may be: how could I know wether the parameters are or not close inside "a ball" in the parameter space and represent this in a plot? If I toggle one parameter inside the ball, do I end close to the measurement or the possible solutions are all scattered and uncorrelated? My intention is to constrain different types of solution (wishful thinking) but I don't really know how it's going to end up looking. – nuwe Nov 03 '21 at 18:16

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