How can I find out the dependency of two variables from one or multiple equations.
Example:
Set 1 -
x1,a
f(x1,a)=z1
Set 2 -
x2,a
f(x2,a)=y2
f(y2)=z2
How highly ( z1 and x1 ) and ( z2 and x2 ) are related among themselves and can we quantify it?
Which Set's relation/dependency of the variables are stronger in comparison with the other Set ?
How easy will it be to get back x from z from both the Sets ? Which one will be harder and how to express that ?
You need to work under a set of hypotheses allowing you to apply the
Inverse function theorem for f ( ⋅ , ⋅ ) in the first case, and for the composition f∘f ( ⋅ , ⋅ ) in the latter case.
Hi, z1 depends on x1 and a. On the other side, z2 depends on y2 thus it depends on x2 and a. If x1 and x2 are independent variables, the only link between z1 and z2 is through a.
To obtain x1 from z1 you have to set the value of a and solve numerically the equation z1=f(x1,a) unless f is sufficiently simple (linear, quadratic, ...) to be inverted analytically.
I think Statistics has a topic called correlation and covariance. Perhaps that is what you would like to be revising here.http://www.math.uah.edu/stat/expect/Covariance.html
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