Refined Stratified Sampling

Refined Stratified Sampling (RSS) is a sequential sampling procedure that adaptively refines the stratification of the parameter space to add samples. There are four variations of RSS currently available in UQpy. First, the procedure works with either rectangular stratification (i.e. using Rectangular) or Voronoi stratification (i.e. using Voronoi). For each of these, two refinement procedures are available. The first is a randomized algorithm where strata are selected at random according to their probability weight. This algorithm is described in [19]. The second is a gradient-enhanced version (so-called GE-RSS) that draws samples in strata that possess both large probability weight and have high variance. This algorithm is described in [20].

Refined Stratified Sampling Class

All variations of Refined Stratifed Sampling are implemented in the RefinedStratifiedSampling class. This class provides the framework for refined stratified sampling. with the aid of an underlying stratification generated in a previous sampling of type TrueStratifiedSampling, as well as a Refinement algorithm an adaptive sampling and refinement of the stratification can be performed.

The RefinedStratifiedSampling class is imported using the following command:

>>> from UQpy.sampling.stratified_sampling.RefinedStratifiedSampling import RefinedStratifiedSampling

Methods

class RefinedStratifiedSampling(stratified_sampling, refinement_algorithm, nsamples=None, samples_per_iteration=1, random_state=None)[source]
Parameters:
run(nsamples)[source]

Executes refined stratified sampling.

Parameters:

nsamples (int) – Total number of samples to be drawn (including the initial samples). If nsamples is provided when instantiating the class, the run() method will automatically be called. If nsamples is not provided, an RefinedStratifiedSampling subclass can be executed by invoking the run() method and passing nsamples

Attributes

RefinedStratifiedSampling.samples: ndarray

The generated stratified samples following the prescribed distribution.

RefinedStratifiedSampling.samplesU01: ndarray

The generated samples on the unit hypercube.

Examples

Stratification Refinement Algorithms

The RefinedStratifiedSampling allows for an adaptive refinement of existing strata. This adaptive refinement procedure can be performed based on different algorithms. Each algorithm provides a different approach on selecting and refining the available Strata, which can be either randomly, or based on advanced selection techniques. In order to accommodate all possible refinement procedures, the Refinement baseclass is created. The user only needs to implement the update_samples() method, thus allowing the implementation of different adaptive strata refinement techniques.

class Refinement[source]

Baseclass of all available strata refinement methods. Provides the methods that each existing and new refinement algorithm must implement in order to be used in the RefinedStratifiedSampling class.

abstract update_samples(nsamples, samples_per_iteration, random_state, index, dimension, samples_u01, training_points)[source]

Method that need to be overridden in case of new Refinement techniques.

Parameters:

  • nsamples (int) – Total number of samples to be drawn

  • samples_per_iteration (int) – New samples to be drawn at each refinement iteration.

  • random_state (Union[None, int, RandomState]) – Random seed used to initialize the pseudo-random number generator. Default is None. If an int is provided, this sets the seed for an object of numpy.random.RandomState. Otherwise, the object itself can be passed directly.

  • index (int) – Iteration index

  • dimension (int) – Number of dimension of the sampling problem.

  • samples_u01 (ndarray) – Existing samples drawn at the unit hypercube space.

  • training_points (ndarray) – Training points required in case of advanced refinement techniques.

The RandomRefinement class is imported using the following command:

>>> from UQpy.sampling.stratified_sampling.refinement.RandomRefinement import RandomRefinement
class RandomRefinement(strata)[source]

Randomized refinement algorithm. Strata to be refined are selected at random according to their probability weights.

Parameters:

strata – Strata object containing already stratified domain to be adaptively sampled using RefinedStratifiedSampling

The GradientEnhancedRefinement class is imported using the following command:

>>> from UQpy.sampling.stratified_sampling.refinement.GradientEnhancedRefinement import GradientEnhancedRefinement
class GradientEnhancedRefinement(strata, runmodel_object, surrogate, nearest_points_number=None, qoi_name=None, step_size=0.005)[source]

Gradient-enhanced version (so-called GE-RSS) refinement algorithm. Draws samples in strata that possess both large probability weight and have high variance.

Parameters:

  • strata (Strata) – Strata object containing already stratified domain to be adaptively sampled using RefinedStratifiedSampling

  • runmodel_object (RunModel) – A RunModel object, which is used to evaluate the model. It is used to compute the gradient of the model in each stratum.

  • surrogate (Union[Surrogate, GaussianProcessRegressor, object]) – An object defining a surrogate model.This object must have the fit() and predict() methods. This parameter aids in computing the gradient.

  • nearest_points_number (Optional[int]) – Specifies the number of nearest points at which to update the gradient.

  • qoi_name (Optional[str]) – Name of the quantity of interest from the runmodel_object. If the quantity of interest is a dictionary, this used to convert it to a list.

  • step_size (float) – Defines the size of the step to use for gradient estimation using the central difference method.