How to evaluate the "importance" of a variable in a function

Question

Let's say that we have
$$f(x,y,z) = x/k - (y/k) ((z - x/k)/(z - y/k))$$
$$k = constant in ]0,1[$$
And  I need to show in some way that the variable $x$ is more important in  some metric that I don't know which one could be good. I thought about  analyze the partial derivatives of that functions, but I don't think  that is a good way, because one will only see some restricted path  through the surface.
Another  approach would be do monte carlo simulation in some domain of interest  (it exists, actually is a real world problem) and see that the variance  of the function value increases more varying x fixing the another  variables rather than the same simulation with $x$ fixed.
In some extent one can thinking in this function as a model, and I need to calculate the variable importance, it could be a way as well.
But I'm really  puzzled how to approach this problem.
Any metric or some approach is welcomed!
Thank you very much!

Nikos M. · Answer

The way to start would be to understand intuitively what makes some variable important.
For example, the output value of the function might depend to a large extend on the value on that variable (so correlation would be quite high between output and that variable).
Another example would be to make that variable zero and measure the correlation between the original function and the new function (eg it would exhibit low correlation).
Still another way would be to compute the rate of change (derivative) of the function with respect to the rates of change (derivatives) of each variable and see how much a small perturbation of that variable affects the output of the function.
Similar other approaches are possible as well. Since you have the formula of the function wrt the variables one of the above approaches is fine.
There is no agreed upon universal metric for variable/feature importance but there are various approaches one can take depending on problem at hand.
Further reading:

Variable importance analysis: A comprehensive review

Assessing Variable Importance for Predictive Models of Arbitrary Type

Variable-importance Measures

Methods to quantify variable importance: implications for the analysis of noisy ecological data

How to evaluate the "importance" of a variable in a function

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