Cross cuts

[mbaudin47]

This creates cross-cuts for functions, distributions and events.

• The CrossCutFunction is presented in https://github.com/mbaudin47/otbenchmark/blob/CrossCuts/examples/CrossCutFunction_Demo.ipynb
• The theory of ConditionalDistribution is presented in https://github.com/mbaudin47/otbenchmark/blob/CrossCuts/examples/ConditionalDistribution_Theory.ipynb and a demo in https://github.com/mbaudin47/otbenchmark/blob/CrossCuts/examples/ConditionalDistribution_Demo.ipynb.
• The CrossCutDistribution is in https://github.com/mbaudin47/otbenchmark/blob/CrossCuts/examples/CrossCutDistribution-3D_Demo.ipynb
• The DrawEvent demo is in https://github.com/mbaudin47/otbenchmark/blob/CrossCuts/examples/DrawEvent_demo.ipynb
• A full demo of these cross-cuts is in https://github.com/mbaudin47/otbenchmark/blob/CrossCuts/examples/draw-cross-cuts.ipynb
• The application to RP8 is in https://github.com/mbaudin47/otbenchmark/blob/CrossCuts/examples/RP8_Demo.ipynb

[mbaudin47]

@vchabri, @efekhari27 : Please review this PR.

[vchabri]

Great work! Thanks Michael.
Here are a few comments about each document:

1. [CrossCutFunction_Demo.ipynb]:

• For the last cell with the various figures (A, b, C, D, E, F), I don't get the reason why Figs. D and E show contours which are orthogonal to the ones presented here (https://rprepo.readthedocs.io/en/latest/reliability_problems.html#rp33). If the gradient follows a given stepwise direction (e.g., from left to right), why in these plots it is from bottom to top?
• Nice explanation at the end of the Notebook!!! Just a question: would it be possible to use/create a method like ".getDesignPoint()" in the X-space or in the U-space as a reference point?

2. [ConditionalDistribution_Theory.ipynb]:

• The theory reminders are clear and well-explained. However, there are a few typos in the text ("and", "integeger", typo in the equation f_{X|Xc}).

3. [ConditionalDistribution_Demo.ipynb]:

• Nice example. Maybe I could help to provide a few more examples (issued from my PhD manuscript or other resources).

4. [DrawEvent_demo.ipynb]:

• Really nice. I got the same question as previously. What is missing in the code to draw similar graphs from a reference point taken as the design point P* ?

5. [RP8_Demo.ipynb]:

• Amazing!! Question: Is it the white square in the plots (e.g., top left, or top right) the "removed legend" ?
• For the other sets of plots, I would suggest to add a marker to the reference point as it is not that clear on the plots.

Hope these remarks will help. The work is great!

[mbaudin47]

• [CrossCutFunction_Demo.ipynb] There is a bug in the (x1, x3) projection. Either the RP or myself is wrong.

Below is my version:

Here is reference RP33:

The first difference is the bounds: [-4, 4] in my graph and [-2, 2] in the reference.

If I use the same bounds, here is the result:

If we set x2=0, we get g=min(-x1-x3+3*sqrt(3), -x3 + 3). Here is the result from Mathematica:

https://www.wolframalpha.com/input/?i=min%28-x1-x3%2B3*sqrt%283%29%2C-x3%2B3%29

Mathematica produces the same output as mine: the reference output is wrong. The detailed mathematical analysis shows that the reference picture is wrong and this is why I created the associated ticket: https://gitlab.com/rozsasarpi/rprepo/-/issues/20

• Reference point from design point. With this development, it is possible to draw the function in the X-space. In the U-space, it will be possible by composition of the probabilistic transformation and the function itself. Creating the CrossCut object with this composed function will do the job: this would be very nice example indeed. See Fig3.5 page 40 in Chabridon's thesis for an example. I create the issue in Draw the limit state function in the U-space. #44, as this does not prevent the current PR from being accepted.

• [ConditionalDistribution_Demo.ipynb] Replace X1=mu1 with X1=2.0. FIxed. Replace conditionalRefencePoint with conditionalReferencePoint: fixed. BayesianDistribution is for Theta, not X. TruncatedDistribution would fit. computeConditionalPDF() is limited with respect to one marginal of X, not all. Useful for Shapley. Dedicated implementation for Normal.

• CrossCutDistribution / drawConditionalPDF has a bug in the XTitle. Fixed.

• DrawEvent : specialized in 1D e.g. Y=sin(X) where X is gaussian. I created the following ticket for this: The DrawEvent class should draw 1D events #45.

• DrawEvent / buildCrossCutFunction currently uses the mean of the distribution as the reference point. It may use another reference point: The buildCrossCutFunction of the DrawEvent class may manage an arbitrary reference point #46.

• DrawEvent / drawSampleCrossCut currently uses the marginal distribution. It may use the conditional distribution instead: The drawSampleCrossCut method of DrawEvent should use the conditional distribution. #47. Indeed, if we plot the CrossCutFunction by setting conditioned variables and a sample with unconditioned distribution, we compare very different objects. This is why the DrawEvent class must take into account conditional distributions.

• DrawEvent : add an example with a reference point equal to the design point: There is no example of DrawEvent with design point as reference #48.

• It would be interesting to draw the marginal distribution and ConditionalDistribution. Add the orange hole example. This will be done within the tech. report on kernel smoothing.

• draw-cross-cuts.ipynb : insist on the fact that there are 3 different samples. Add a new method with a given sample, drawing projections. This would be an extension of drawInputOutputSample: DrawEvent should be able to manage a given sample #49.

[mbaudin47]

This development is not completely finished, but the current version runs fine and provides interesting features. This is why I merge it.