I'm searching for a method that transforms a normal distribution into a normal distribution. I've looked everywhere, but I'm not sure if I just missed something completely obvious, if this actually is a really hard question or if I searched in the wrong places.
Say, I have some data which is normally distributed (by reading from a device which creates such data from sensor inputs for example, which are distributed this way). How can I then get a uniform distribution of the data? I have seen plenty of algorithms for the other way around.
Thank you very much for your time.
Let $N$ be a normally distributed variable, with mean 0 and variance 1 (substract the mean and divide by the standard deviation in order be in this case).
You look for a certain (smooth, increasing) $f:\Bbb R\to[0,1]$ such as $ f(N) $ is uniform, that is: $$ P(f(N) \le q) = q $$for every $q\in(0,1)$.
Under regularity assumptions, this is $$ q = P(N \le f^{-1}(q)) = \mathcal N(f^{-1}(q)) \\ \iff f^{-1}(q) = \mathcal N^{-1}(q) \iff f(q) = \mathcal N(q) $$ with $\mathcal N(x) = P(N \le x)$.
Now check the smoothness assumptions (in particular, $\mathcal N$ is increasing).