1 to 1 Transformation of Random Vector
 

Motivation: Generate complex random variable from simpler one.

Most software language support functions that are able to create uniform distributed pseudorandom number. This number can be use to generate other random variable with different distribution or density, by evaluating the inverse function of the distribution function of the random variable we wish to generate. However for some density functions, like Gaussian, this approach is limited, since the distribution function can’t be easily evaluate, or would require high computational effort. For these cases the 1 to 1 transformation provides an alternative that is simpler and requires less computational stress.

To illustrate the value of the 1 to 1 transformation, we are providing an experimental and a mathematical prove, which will show how two Gaussian R.V. can be generated from one uniform R.V. and one Rayleigh R.V. using a some variable transformation.

Let θ be a uniform Random (U[0,2π]) Variable and R be a Rayleigh Random Variable

And consider the following Variable Transformation

h1=X=Rcos(θ), h2= Y=Rsin(θ)

Where X and Y are going to be to independent Gaussian R.V. generated from the Uniform and Rayleigh R.V.s

Experimental Prove:
Samples Range
              
 
For our experiment we have written in html and java script a code that generate our random variable and plot the histogram to verify they have the correct density. JavaScript provides a random function under the library Math (Math.ramdom()) that generates a pseudorandom random number with a uniform distribution from 0 to 1 with double precision. On the above interface, we can use Uniform, Rayleigh and Gaussian buttons, to generate a plot for the histogram of the respective random variable. The histogram uses a sample of 10,000 to 500,000 numbers, and a integer range of 20 to 200 points representing numbers from 0 to 2 π.

To generate the density U[0,2π]  we can use the code 2*π*Math.random(). For the Rayleigh density we evaluate a uniform density random variable in the inverse of the distribution function of the Rayleigh.

So we can use the code Math.squrt(-2*Math.log(Math.random())) to get a Rayleigh random value
Finaly, we generate the two Gaussian random variables by valuating two previous in X=Rcos(θ), Y=Rsin(θ) transform functions. Where R is the U[0,2π], θ is the Rayleigh, and X and Y are the Gaussian random variable.
Mathematical Prove:

Let θ be a uniform Random (U[0,2π]) Variable and R be a Rayleigh Random Variable
with join density:



And consider the following Variable Transformation h1=X=Rcos(θ), h2= Y=Rsin(θ)

To know the density function for the new variables, we can use the following join density variable transform:



Where                   is the Jacobian matrix of the variable exchange:



Inverse functions for h1 and h2:

As we can see the final result are two independent Gaussian random variables.