How to fit a distribution whose parameters depend on other parameters?

Hi!

How to fit a distribution is clear, in dfittool for example. An example is a gas at a fixed temperature T. A priori, the distribution of Vx velocities of its molecules is proportional to exp(-mVx^2/2kT). If I know the velocity Vx of every molecule, I will fit normal distribution to data containind Vx and will obtain normal fit with zero mean and standard deviation proportional to T (variance kT/m).

But how to fit a distribution whose parameters depend on other parameters??? An example - a gas in a volume in a gravity field with gas temperature depending on height T(z). Now every molecule has its own height z, and my data look like (Vx,z). Do I have a function to feed such data to it? I mean, to fit a normal distribution exp(-mVx^2/2kT(z)) to data, getting in the result T(z)?

The question is related to another distribution and another system, I used normal and gas as an example. The question is more general whether distribution fitting and curve fitting can be done at one step.

Answer 1

[Torsten]

In this case, you must assume a parametric expression for T(z) as well (e.g. T(z) = a + b*z) and add a and b to the parameters to be fitted.

And be careful to put the denominator into brackets - so use

exp(-m*Vx^2/(2*k*T))

instead of

exp(-m*Vx^2/2*k*T)

in MATLAB.

[Sergey Abaimov]

Thanks for the answer, but I do not get it. For curve fitting, yes, clear. But how do I feed additional parameters to distribution fitting?

pd = fitdist(x,'Normal',Name,Value)

How do I tell that x above are pairs (Vx,z). How do I put T(z) = a + b*z as Name and Value? Can you refer to an example?

Best regards

[Torsten]

Sorry, I was in the realm of curve fitting, not distribution fitting.

If your data look like (Vx,z), I think a multidimensional Gaussian would be the way to go:

https://uk.mathworks.com/help/stats/fitgmdist.html

This model can even handle the case that Vx and z are correlated.

Or is only Vx a random variable and z is deterministic ?