Q:

Quantile Transformation with Gaussian Distribution - Sklearn Implementation

This might be a vague question, but I'm wondering how is Scikit-Learn's quantile transformation implemented?
I'm wondering how can a skewed dataset be transformed into a normal distribution like this?
Usually scikit-learn provides a link to the wiki, but not this transformation.
Can someone point me to the right direction?
Thanks

A:

Yes, it appears to be described in a few different places, with no link to any papers.
The class documentation summarises the algorithm as follows:

The transformation is applied on each feature independently. The
  cumulative density function of a feature is used to project the
  original values. Features values of new/unseen data that fall below or
  above the fitted range will be mapped to the bounds of the output
  distribution. Note that this transform is non-linear. It may distort
  linear correlations between variables measured at the same scale but
  renders variables measured at different scales more directly
  comparable.

And the user guide adds some new information:

However, by performing a rank transformation, it smooths out unusual
  distributions and is less influenced by outliers than scaling methods.
  It does, however, distort correlations and distances within and across
  features.

Specifically, for a Normal transformation:

Thus the median of the input becomes the mean of the output, centered
  at 0. The normal output is clipped so that the input’s minimum and
  maximum — corresponding to the 1e-7 and 1 - 1e-7 quantiles
  respectively — do not become infinite under the transformation.

The GitHub pull request for this estimator references an older one that shows it was originally going to be named a "rank scaler". 
In broader terms, this paper provides a good summary of the various ways that such "inverse normal transformations (INTs)" may be implemented:

INTs are ways of transforming the sample distribution of a continuous
  variable to make it appear more normally distributed. There are
  several types of INTs. The first distinction we make is between
  rank-based and non-rank-based INTs. Non-rank-based INTs entail
  assuming a particular cumulative distribution function (CDF) for the
  observed data, estimating the parameters of that distribution,
  converting observed scores to estimated quantiles from the CDF, and
  then converting these quantiles to standard normal deviates using the
  inverse normal (or probit function). Such non-rank-based INTs are
  usually referred to as copulas (Basrak et al. 2004; Li et al. 2006)
  and will not be considered further. It is worth noting, however, that
  the rank-based INTs can be expressed as a special case of the copula
  method in which the empirical CDF is used instead of restricting the
  CDF to some family of distributions. That is, every moment is in
  effect estimated from the data and the quantiles become simple
  functions of the ranks.
Rank-based INTs involve a preliminary step of converting a variable to
  ranks and can be further subdivided into two classes: those that
  involve a stochastic element and those that are deterministic. We are
  aware of only one INT that involves a stochastic element and this
  approach has been referred to as the use of “random normal deviates”
  (Conover 1980). One deterrent to this approach is that each
  investigator applying the same method to the same dataset will obtain
  a slightly different answer, which might be unsatisfying to some. This
  approach has the theoretical advantage of avoiding the granularity in
  the distribution of P values, an issue which often plagues many
  nonparametric tests. Nevertheless, the stochastic nature of these INTs
  seems to discourage researchers and they are rarely, if ever, used.
Deterministic rank-based INTs can be classified into those that use
  expected normal scores (Fisher and Yates 1938) versus those that use
  back transformation of sample quantile (or fractional rank) to
  approximate the expected normal scores. Using numerical integration,
  Harter (1961) has provided the most complete table of expected normal
  scores. INTs that involve back transformation of fractional ranks to
  approximate the expected normal scores of Fisher and Yates (Maritz
  1982) appear to be the most commonly used in genetic research and will
  be the primary focus of attention. In back-transforming ranks, a
  fractional offset is needed to avoid having the minimum and maximum
  observations transformed to negative and positive infinity,
  respectively. Perhaps the most commonly used rank-based INT
  transformation entails creating a modified rank variable and then
  computing a new transformed value of the phenotype for the ith
  subject.

Looking at the QuantileTransformer code, it looks like the very last item in the list: a deterministic rank-based INT that calculates a modified rank variable. 
However, it's a relatively simple implementation: 

calculate empirical ranks, using numpy.percentile
modify the ranking through interpolation, using numpy.interp
map to a Normal distribution by inverting the CDF, using scipy.stats.norm.ppf

taking care to deal with bounds at the extremities.
Representing as a very simplified mapping, i.e. ignoring the interpolation and bounding logic, it would just be $y_i = \Phi^{-1}(F(x_i))$, where $F$ and $\Phi$ represent the CDFs of an empirical and standard Normal distribution, respectively.

