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I have some experimental ice polarization data, and simulated the polarizations of possible size/shape particles that could be present in the system. I want to use my polarization data to quantitatively narrow down which shapes are impossible. Ideally, there would be a number like R^2 that compares the curve to the data, and some curves will have a terrible number and I'll know to discount those as possibilities. I have hundreds of possible combinations of sizes and shapes, so I need some way to remove sizes/shapes from my list of possibilities after checking their polarizations.

I only know basic statistical ways of comparing models to data and I don't think any of them are relevant here. I don't think R^2 applies because of the non-linearity, and I thought about cross-correlation, but I don't care about the shift of the data, only the behavior at each angle. I'm also unsure of how to treat the error bars: should I only use the bare data, or maybe do two separate analyses for the minimums and maximums? Thanks!

I've attached a photo of one of my plots: here I have the simulated polarization of particles with three different aspect ratios and want to know if any of them fit the data better than the other two. enter image description here

Claudia
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  • Welcome to Cross Validated! It seems like you have measured values and models making predictions about those values that will have some degree of inaccuracy. Why do you not see this as a job for $R^2?$ What about the nonlinearity invalidates $R^2$ in your mind? Sure, plenty of people have dismissed $R^2$ for nonlinear regression, sure, but that is problematic. // Do you have some sense about what you value in the predictions, absolute or squared difference between predictions and observations, perhaps? – Dave Jun 23 '23 at 17:38
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    Please include more information about the data. Is the response variable plotted on the y-axis and the predictor plotted on the x-axis? Can the responses reasonably be treated as independent and normally distributed? – Rachel Altman Jun 23 '23 at 17:28
  • Yes, the y-values depend on the x-axis, but not in a way that is easily calculated i.e. not a simple mathematical function. Because of how complex the calculations are, I'm not sure if they can be considered normally distributed. I just need some way to compare each curve to each other. I'm hoping for a kind of statistics quantity that will say "this curve is inside the green envelope 80% of the time, and the other 20% is off by some amount, which is better than the other curves" or something like that. I don't understand the nuances of R^2 enough to know if it is applicable for my purpose. – Claudia Jun 24 '23 at 23:32
  • Can you calculate the predicted values and compare them with the real observations, something like predicting $7$ when the observation was $5$ and knowing you overshot by $2?$ // What is the shades green region, and why is it important? – Dave Jun 25 '23 at 00:56
  • Hello, no, I don't think I can do that. The green is experimental data and the curves are possible models. I'm trying to narrow down what shape particle can make data like the green, so I need some way to tell if I'm closer with one shape than another. – Claudia Jun 25 '23 at 17:11
  • What do you mean that the green region is experimantal data? – Dave Jun 25 '23 at 18:53
  • Is the green area calculated from similar curves as the simulated ones? And if you simulate again for a particular aspect ratio, say 1:1, would you get a different curve, or is it deterministic? – Ute Jun 25 '23 at 22:26
  • The green is data from a satellite and the curves are simulated data that I'm trying to match to the satellite. But yes, each curve will yield the same result always for that certain shape. The curves are not random in any way. – Claudia Jun 26 '23 at 15:12
  • But what is the green region, not just the green points? – Dave Jun 26 '23 at 15:14
  • Oh, sorry, those are error bars. That is why I am unsure if I can use the points, or do two comparisons (one for the upper bounds, one for the lower bounds) – Claudia Jun 26 '23 at 15:48
  • I’m still trying to figure out what you want to do. Is it that you want to see which of the orange, brown, and black (those look like the colors) curves have the best fit to the green points? – Dave Jun 26 '23 at 16:25

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