Add Stratified Sampling to ERT

[sondreso]

Feature Request:

Implementing stratified sampling in ERT to improve the sampling process. Stratified sampling could potentially provide a better coverage of the sample space and reduce the risk of clustering. The suggestion includes the possibility of adding a configuration option for the parameter group to define the sampling strategy, such as RANDOM or STRATIFIED, with the potential to add LATIN_HYPERCUBE in the future.

Suggested Feature

1. Stratified Sampling Implementation: Introduce stratified sampling as an option for parameter updates within ERT. This would involve setting d=1 and converting to normal distribution, following the method described in this Stack Overflow post.

2. Configuration Option: Add a new configuration option SAMPLING_STRATEGY on the parameter group level in the ERT configuration files. The user could specify SAMPLING_STRATEGY:STRATIFIED to enable stratified sampling for that parameter group.

3. Naming Consideration: Instead of naming the random sampling strategy as RANDOM, consider using MONTE_CARLO, if this is appropriate.

Benefits

• Better sample space coverage.
• Reduced risk of clustering in parameter sampling.
• More intuitive user experience with control over parameter sampling.
• Alignment with requests for support of Latin Hypercube sampling.

Considerations

• Determine if the stratified sampling should be applied per parameter or for the entire parameter group.
• Assess the impact of the proposed stratified sampling on the update process and correlated parameters.
• Discuss the naming and approach with statisticians for validation.
  • In ERT we understand the update through the statistical properties of the ensemble X (independently sampled). We estimate Cov(X), have monte carlo samples of Y=g(X), estimate Cov(Y) and regress Y on X. The properties are understood. Changing the sampling, one risks loosing the understanding. The statistical properties, at least of these specific estimators, as a function of an LHC sample, must be understood before changing ERT.
• Consider making STRATIFIED the default sampling method. This is a breaking change and would require communication to users.

Additional Context

• Support for Latin Hypercube has been requested multiple times, with an issue dating back to 2018.
• Stratified sampling is seen as a more straightforward approach and could be the first step before considering more complex sampling strategies like Latin Hypercube.
• See this PR for context on how the UPDATE:FALSE option was added

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This feature request has been compiled from an internal discussion (link).

[Blunde1]

This should be under Considerations:

In ERT we understand the update through the statistical properties of the ensemble X (independently sampled). We estimate Cov(X), have monte carlo samples of Y=g(X), estimate Cov(Y) and regress Y on X. The properties are understood. Changing the sampling, one risks loosing the understanding. The statistical properties, at least of these specific estimators, as a function of an LHC sample, must be understood before changing ERT.

[sondreso]

If we use the d=1 option, the variables would still be independently sampled, or no? 🤔

[Blunde1]

If we use the d=1 option, the variables would still be independently sampled, or no? 🤔

I think across dimensions, yes, but not across samples.