Ensemble sampling: PCA + Unscented Transform?

[mdietze]

Two observations suggesting that we need an uncertainty propagation option that's much faster than Monte Carlo:

• in the current PDA tutorial, the running of the ensemble analysis before and after PDA takes longer than the PDA
• ESMs currently run with no uncertainty in the land and, even if they could convince the atmospheric scientists to run land ensembles, they'd be severely limited computationally in terms of ensemble number.

Therefore, we want an option for ensemble error propagation that uses the minimum number of possible ensemble members but gives a better approximation than a small random sample.

Two interrelated ideas for how to achieve this:

1. Reduce the dimensionality of parameter space by performing a PCA on the joint posterior probability distribution. Do this having transformed parameter space through the sensitivity functions (so that output variability reflects parameter importance). Truncate to the top N eigenvectors which explain some threshold value of variability.
2. Rather than sampling the PCA'ed posterior randomly, choose 2*N + 1 ensemble members systematically (e.g. using the Unscented Transform), run the model for these points, and then back-transform to the output mean and variance analytically rather than estimating summary statistics (mean, median, SD, quantiles) from the sample

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