Linearly scaling a distribution (X) to have the same 5th and 95th percentile of distribution (Y)

Question

As done in this paper here:

I would like to linearly scale a distribution such that it has the same 5th and 95th percentile (0.05 and 0.95 quantile) as another distribution.

That is, given two genes on two different platforms, how do I scale them to have the same 5th and 95th % percentile.

Answer 1

Let's say the 5th percentile of $X$ is $l_x$ and the 95th percentile is $u_x$, and similarly for $Y$.

You now want to construct a linear transformation $Z=aX+b$ such that $l_z = l_y$ and $u_z=u_y$.

Note that $\frac{X-l_x}{u_x-l_x}$ has the required percentiles at 0 and 1.

Hence $Z=l_y+(u_y-l_y)\frac{X-l_x}{u_x-l_x}$ should match those percentiles with $Y$.

Here's an example in R:

 x=rnorm(100)
 y=rgamma(100,3,.2)

 quantile(x,c(.05,.95))
       5%       95% 
-1.685679  1.517378 

 quantile(y,c(.05,.95))
       5%       95% 
 4.724867 29.434846 

 lx=quantile(x,.05)
 ux=quantile(x,.95)
 ly=quantile(y,.05)
 uy=quantile(y,.95)
 z=ly+(uy-ly)*(x-lx)/(ux-lx)

 quantile(z,c(.05,.95))
       5%       95% 
 4.724867 29.434846