Expressions used in change of variables

Click For Summary
SUMMARY

The discussion focuses on the challenge of performing a change of variables from a function f(x,y) to a predefined function g(u,v), specifically seeking expressions for u(x,y) and v(x,y) along with the Jacobian. It is established that there is no universal method applicable to all forms of f and g, but rather techniques that may apply to specific cases. The Box-Muller method is referenced as an example of a transformation that generates Gaussian random deviates from uniform deviates, prompting the inquiry into alternative, more efficient transformations.

PREREQUISITES
  • Understanding of multivariable calculus, particularly change of variables
  • Familiarity with Jacobian determinants in transformation of variables
  • Knowledge of probability distributions, specifically Gaussian and uniform distributions
  • Experience with mathematical transformations and their applications in statistics
NEXT STEPS
  • Research the Box-Muller transform and its applications in generating random variables
  • Explore alternative methods for variable transformations, such as the Inverse Transform Sampling
  • Study Jacobian matrices and their role in multivariable calculus
  • Investigate specific cases of f(x,y) and g(u,v) that allow for successful transformations
USEFUL FOR

Mathematicians, statisticians, and data scientists interested in variable transformations, particularly those working with probability distributions and computational efficiency in random variable generation.

eccefeles
Messages
11
Reaction score
0
Hi. I know the title is not very informative. Here's what I'm trying to do:
I have f(x,y). I want to perform a change of variables to obtain a pre-defined g(u,v). How can I work out the actual expressions u(x,y) and v(x,y) so that it works out (including the Jacobian as well)?

I have a vague feeling that there may not be a general approach but only techniques for specific forms of f and g. I have Googled and searched for "change of variables" on this forum. 9 pages later, I think no one has asked this before. Some directions (even Google keywords) would be much appreciated. :smile:
 
Physics news on Phys.org
Actually, you can't! There may not be any u(x,y), v(x,y) that converts a given f(x,y) into a given g(u,v)!
 
OK, I accept there is no general approach. However, what about some "special" forms of f(x, y) and g(u, v)?

I started pondering this question after I read about the Box-Muller method of generating random deviates with a Gaussian distribution from uniform deviates. (http://en.wikipedia.org/wiki/Box-Muller_transform" ) It made me wonder if there are systematic methods of working out alternative transforms that would achieve the same purpose and be superior in terms of computing efficiency.
 
Last edited by a moderator:

Similar threads

Undergrad Change of variables clarification
  • · Replies 5 ·
Replies
5
Views
2K
MHB How to use change of variables technique here?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Change of variables/ Transformations
  • · Replies 3 ·
Replies
3
Views
2K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Integral change of variables formula confusion
  • · Replies 4 ·
Replies
4
Views
2K
High School Question about change of variables
  • · Replies 29 ·
Replies
29
Views
5K
Undergrad Changing variables in multiple integrals
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Change of variable and Jacobians
  • · Replies 15 ·
Replies
15
Views
5K
Undergrad Multi-dimensional Integral by Change of Variables
  • · Replies 1 ·
Replies
1
Views
2K
MHB Joint probability distribution of functions of random variables
  • · Replies 3 ·
Replies
3
Views
2K