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Let $X$ be a normally-distributed random variable. How can I transform this $X \in \mathbb{R}$ into another r.v. $Y \in [0; 1]$ whilst maintaining its normal-like shape?

One common way of performing a one-to-one transformation from $\mathbb{R}$ into $[0; 1]$ is to calculate the CDF of the original variable. The "problem" with that approach is that the result will be uniformly-distributed. This is a well-known property, but here is an example to illustrate:

set.seed(2345)
x <- rnorm(1000)
hist(x)
y <- pnorm(x)
hist(y)

The histograms of $x$ and $y$ will look like this: Histograms of x and y

However, as I mentioned, I would like to transform $X \in \mathbb{R}$ into $Y \in [0; 1]$ and have $Y$ bell-shaped?

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    $\begingroup$ I'm probably misunderstanding something, but what are you trying to get? If $x$ is already normally distributed, what do you mean by "one-to-one transform into a normally-distributed $Y$"? You could put a qnorm on the runif, but that will only get you the original variable... $\endgroup$ – juod Feb 27 '17 at 10:08
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    $\begingroup$ Your question (see title and first paragraph) still calls Y normal. It can't be normal if it's bounded. "Bell shaped" could be almost anything unimodal and more or less symmetric. What properties should this thing on (0,1) have? Is Beta(3,3) close enough, for example? If you don't actually need normality, what is it supposed to achieve? $\endgroup$ – Glen_b -Reinstate Monica Feb 27 '17 at 10:26
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    $\begingroup$ It's not clear what are you trying to achieve, if you just want bell-shape data in $(0,1)$ why not simply do (x - min(x))/(max(x) - min(x))? $\endgroup$ – Francis Feb 27 '17 at 10:41
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    $\begingroup$ @WaldirLeoncioL it's in fact simple re-scaling. $\endgroup$ – Francis Feb 27 '17 at 10:55
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    $\begingroup$ @Waldir If you're transforming a normal variable, $X$ (as you clearly state in your question), then there is no finite maximum value of $X$ (it is on the whole real line). If you're transforming a sample based on its sample characteristics - like the sample maximum and minimum) then the shape of the resulting distribution is not what you expect. For example in R try res=replicate(10000,{x=rnorm(6);(x-min(x))/(max(x)-min(x))});hist(res,n=50) to see what it does to samples of size 6. $\endgroup$ – Glen_b -Reinstate Monica Feb 28 '17 at 11:14
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As you correctly point out, $U=F_X(X)$ will be uniform. Consider now that we have (somehow) identified some sufficiently "normal-like" continuous strictly monotonic distribution function on (0,1) (which we'll call $G$).

Then $Y=G^{-1}(U)$ has distribution $G$. So $Y=G^{-1}(F_X(X))\sim G$

So if we can find some suitable $G$, we're done.


Possible distributions with the kind of properties you want:

  1. The beta distribution. This is very often used as a distribution for the binomial parameter in Bayesian statistics. If you choose the two parameters to be approximately equal and larger than 2, then you have something that looks roughly normal. The inverse cdf is widely available.

  2. A suitably truncated normal. Rescale a normal so that it has mean near the middle of (0,1) and most of its probability in (0,1). For example, $N(\frac12,\frac16^2)$ truncated to the unit interval. The inverse cdf is fairly convenient and (if you don't need the tails of the original distribution) this can avoid the need to go via the uniform.

  3. Logit-normal, for some parameter values; to get those, rescale your normal to have a suitable parameter combination such that the logit-normal looks like you want. Typically you'll want $\mu$ not too far from $0$ and $\sigma<1$ (a value like $\frac12$ or $\frac13$ -- or less -- will often be reasonable). Incidentally I think your own answer is a particular case of this with $\sigma=1$. The inverse cdf is quite convenient and this also avoids the need to convert to uniform first.

  4. A standard Bates distribution. Take the mean of $k$ independent standard uniforms. The inverse cdf isn't particularly convenient if you're doing this from scratch, but if you can find an existing function to do this (or the related Irwin-Hall, followed by a rescaling), this may be convenient.

  5. Raised cosine. Specifically, the one with $f_Y(y) = 1+\cos(\pi(2x-1))\mathbb{I}_{(0,1)}$.

There are many other fairly obvious choices; for example a suitably truncated $t$ would allow you to push up the kurtosis a little (I think all of the previous examples have lower kurtosis than the normal; you may in some circumstances want to get it a bit closer to that of the normal); you could also consider scale mixtures for example.

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Thanks to Francis, I ended up doing a simple rescaling of the data in order to achieve what I wanted.

Scaling is achieved by performing the following transformation:

$$ y = \frac{x - \min(x)}{\max(x) - \min(x)} $$

This is a great solution for me because it's fairly shape-constant. No matter how my original variable $X$ looks, $Y$ will be a horizontally-distorted version of it, squished from its original range to $(0, 1)$. This works for me because I had a bell-shaped $X$ which could take any real value and wanted a bell-shaped $Y$ which was $\in (0, 1)$.


One drawback of the transformation above for my case was that $\min(x)$ becomes 0 and $\max(x)$ becomes 1 by definition. I wanted $Y$ to be a little more concentrated around 0.5, so the transformation I've actually used was this:

$$ y = \frac{x - [\min(x) - \text{IQR}(x)]}{[\max(x) + \text{IQR}(x)] - [\min(x) - \text{IQR}(x)]} = \frac{x - \min(x) + \text{IQR}(x)}{\max(x) - \min(x) + 2\text{IQR}(x)}, $$ where $\text{IQR}(x) = Q_3(x) - Q_1(x)$ is the interquartile range of $X$.

What happens here is I'm giving the minimum and maximum values extra padding, thus increasing the "range of the observed values of $X$" and, consequently, having the transformed values lay farther from the extremities 0 and 1.

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One solution would be to use the CDF of a logistic distribution. In the example:

set.seed(2345)
x <- rnorm(1000)
y <- 1 / (1 + exp(-x))  # assumes mu = 0 and sd = 1 for Y.

This should give the following histogram for $y$: histogram of y

At first this seems to answer my question, but I welcome other solutions very much, since this transformation may be too similar to another method I am using and comparing.

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  • $\begingroup$ Why not just sample from the logistic distribution directly? $\endgroup$ – Neil G Feb 28 '17 at 11:04
  • $\begingroup$ @NeilG, because I need the original data for something else. $X$ in my case is a matrix of test answers (correct/incorrect per individual and question); I am exploring several pre-smoothing techniques for this matrix. $\endgroup$ – Waldir Leoncio Feb 28 '17 at 11:30
  • $\begingroup$ Then $X$ isn't normally distributed since it's bounded? $\endgroup$ – Neil G Feb 28 '17 at 12:02
  • $\begingroup$ @NeilG, sorry for misleading you, $X$ is actually the skills of the test takers, which is modeled as a normally-distributed latent trait. From that I am trying to derive the probability of acing a test ($Y$). $\endgroup$ – Waldir Leoncio Feb 28 '17 at 12:20

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