Sensitivity Analysis function

## Proposal
Add a Sensitivity Analysis (SA) function.

The function would compute _Sobol'_ indices [1,2]. Consider a function `f` with parameters `x1`, `x2` and `x3`. Hence `y=f(x1,x2,x3)`. We are interested to know which parameter has the most impact, in terms of variance, on the value `y`.

I believe scikit-learn, and the wider scientific community, would greatly benefit to have such tool. I believe scikit-learn has something related with `feature_importances_` in some regressors.

As an expert in the field, I would propose to develop the initial functionalities and provide long term support.

## Background
_Sobol’_ indices are the cornerstone of SA and UQ and its application is not restricted to any field. It’s a non intrusive method which makes the only assumption that the variables are independent (this constraint can be alleviated).

Being able to compute sensitivity indices allows to reduce the dimensionality of a problem, better understand the importance of each factors and also see how parameters are interacting with each other. As such, it’s an important engineering tool. If you have 2 or more variables and only have the budget to improve your knowledge on one of them, this can help to make an informed choice. There are a lot of successful usage of SA in the literature and in real world applications. The EU (through the JRC), is now requiring to conduce uncertainty analysis when evaluating a system. They are recommending the use of _Sobol’_ indices.

## Example
Let’s take an example with the _Ishigami_ function:

`y = sin(x1) + 7*sin(x2)^2 + 0.1*x3^4*sin(x1)`

It’s not obvious to tell which variable would impact more `y`.

The Sobol’ indices are bounded from 0 to 1, with 1 meaning more important. Here they would be:

| Variable | First order Sobol’ | Total Sobol’ |
| -------- | ------------------ | ------------ |
|   x1     | 0.31               | 0.56         |
|   x2     | 0.41               | 0.44         |
|   x3     | 0.00               | 0.24         |


The difference between the first and total indices indicate an interaction between variables. The total indices allow to rank the variable by importance. `x1` is the most important. Looking at the first orders, `x3` by itself does not have an impact on the variance of the output. It’s its combination with another variable which makes it have a total impact of 0.24. `x1` in this case also have a difference in first and total indices while `x2` is the same. We can say that `x1` and `x2` have a second order interaction.

Computing the indices requires a large sample size, to alleviate this constraint, a common approach is to construct a surrogate model with Gaussian Process or Polynomial Chaos (to name the most used strategies). This allows to conduct a SA with a low computational budget (we see lots of engineering applications with expensive HPC codes taking advantage of this strategy).

## Implementation

Here is an example implementation I did a while back:

https://gist.github.com/tupui/09f065d6afc923d4c2f5d6d430e11696

Note that this implementation lack a few things such as higher order indices, other methods, input validation, doc, tests, other wrapping, etc. This is just to show how it works.

## Alternatives
Practitioners might be more familiarized with gradient based technics. Keywords include: gradient, adjoint. These methods are _local_ sensitivity analysis methods. And as such they are only capturing the sensitivity of the model in specific areas of the parameter space.

_Sobol’_ indices are variance based indices. There are other indices using higher moments, namely: _moment independant_ based sensitivity analysis. These methods are very attractive and provide lot of information while being simple to compute/analyse. They have been less studied but there is an increasing interest in the community. I believe this would make a good second step.

## References
Upon request, I can provide more information. Following are some fundational references–first 2 cited a few thousand times.

* [1] Sobol',I.M. (2001), _Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates_, MATH COMPUT SIMULAT,55(1–3),271-280, [doi:10.1016/S0378-4754(00)00270-6](http://dx.doi.org/10.1016/S0378-4754(00)00270-6)

* [2] Saltelli, A. et al., (2008), _Global Sensitivity Analysis. The Primer_, John Wiley & Sons, [doi:10.1002/9780470725184](http://dx.doi.org/10.1002/9780470725184)

* [3] Saltelli, A. et al., (2020), _The Future of Sensitivity Analysis: An essential discipline for systems modeling and policy support_, Environmental Modelling & Software, [doi:10.1016/j.envsoft.2020.104954](http://dx.doi.org/10.1016/j.envsoft.2020.104954)

For some examples of industrial applications (Part. 3), a visual explanation of some methods (Chap. 2.2) or more bibliography, you can also see my thesis:

https://github.com/tupui/PHD-Thesis/blob/master/phd_thesis_pamphile_roy.pdf

COMMENT:
> IIUC, sensitivity analysis can be viewed as a global method measuring drop in R2.

Considering `y = f(x1, x2)`, sensitivity analysis is looking at whether `x1` or `x2` matters by looking at `var(y|x_i)/var(y)`.

Which is different than measuring a variation on R2. Depending on your model, one parameter could matter more for R2 than it actually matter for `var(f)`.

> The only thing I'm wary of, is that it assumes features are independent, and they pretty much never are. So in reality, this is only useful if we do alleviate those issues.

It depends what you are talking about. Sobol' indices do capture parameters interactions. But it assumes that parameters are independently draws/distributed. So x1 varies in function of x2.

There are ways to deal with parameter dependence with Sobol'. But other methods might be more appropriate/easy. E.g. Shapley values or moment independent methods.

In my experience though, parameter independence does not represent the majority of cases.

> This would make sense, if we could also give users a nice easy guide on how to choose the way they should be inspecting their model.

100% on more tutorials/doc on feature importance/sensitivity analysis.

Labels: module:inspection, Needs Decision - Include Feature