Theta Criterion

The technique enables one-by-one extension of an experimental design while trying to obtain an optimal sample at each stage of the adaptive sequential surrogate model construction process. The sequential sampling strategy based on \(\Theta\) criterion selects from a pool of candidate points by trying to cover the design domain proportionally to their local variance contribution. The proposed criterion for the sample selection balances both exploitation of the surrogate model using variance density derived analytically from Polynomial Chaos Expansion and exploration of the design domain. The active learning technique based on \(\Theta\) criterion can be combined with arbitrary sampling technique employed for construction of a pool of candidate points. More details can be found in:

L. Novák, M. Vořechovský, V. Sadílek, M. D. Shields, Variance-based adaptive sequential sampling for polynomial chaos expansion, 637 Computer Methods in Applied Mechanics and Engineering 386 (2021) 114105. doi:10.1016/j.cma.2021.114105

ThetaCriterionPCE Class

The ThetaCriterionPCE class is imported using the following command:

>>> from UQpy.sampling.ThetaCriterionPCE import ThetaCriterionPCE

Methods

class ThetaCriterionPCE(surrogates)[source]

Active learning for polynomial chaos expansion using Theta criterion balancing between exploration and exploitation.

Parameters:

surrogates (list[PolynomialChaosExpansion]) – list of objects of the UQpy() PolynomialChaosExpansion class

run(existing_samples, candidate_samples, nsamples=1, samples_weights=None, candidate_weights=None, pce_weights=None, enable_criterium=False)[source]

Execute the ThetaCriterionPCE active learning.

Parameters:

  • existing_samples (ndarray) – Samples in existing ED used for construction of PCEs.

  • candidate_samples (ndarray) – Candidate samples for selecting by Theta criterion.

  • samples_weights – Weights associated to X samples (e.g. from Coherence Sampling).

  • candidate_weights – Weights associated to candidate samples (e.g. from Coherence Sampling).

  • nsamples – Number of samples selected from candidate set in a single run of this algorithm

  • pce_weights – Weights associated to each PCE (e.g. Eigen values from dimension-reduction techniques)

  • enable_criterium (bool) – If True, values of Theta criterion (variance density, average variance density, geometrical part, total Theta criterion) for all candidates are returned instead of a positions of best candidates The run() method is the function that performs iterations in the ThetaCriterionPCE class. The run() method of the ThetaCriterionPCE class can be invoked many times for sequential sampling.

Returns:

Position of the best candidate in candidate set. If enable_criterium = True, values of Theta criterion (variance density, average variance density, geometrical part, total Theta criterion) for all candidates are returned instead of a position.

Examples