The image shows the original marginal data $u$ and $v$ on the left, which has a bounded support, and their probit transformations $r$ and $s$ on the right, which has an unbounded support.

enter image description here

The function $\Phi^{-1}(\cdot)$ looks alot like a probability integral transform function, yet the transformation from left to right is called "probit transformation". Looking at it, I would probably say right to left is the probability integral transform, not left to right as shown, since the right looks more like real data and the left looks like transformed marginals, not the other way around. Anyway,

Are probit transformation and probability integral transform the same thing? If not, what is the function being applied by this probit transformation (e.g. the logit function for logit transformation is $\ln \frac{p}{(1-p)}$.)

Also, if the grid lines in the right graph are called "equally-spaced", what name can be given to the grid lines in the left graph?

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    $\begingroup$ PIT will depend on the assumed distribution for the variable you are transforming. $\endgroup$ – Richard Hardy Sep 15 at 18:33
  • $\begingroup$ What explains the different grid spacings in the first graph? The datapoints are just datapoints, and if the grid were'nt shown, how would an observer know there is a contour $\endgroup$ – develarist Sep 15 at 19:02
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    $\begingroup$ The grid lines in the left graph are the images, under the transformation $\Phi,$ of the grid lines in the right graph. $\endgroup$ – whuber Sep 16 at 13:19

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