importance_sampling

Importance sampling

Topics

uncertainty_quantification, aleatory_uncertainty_quantification_methods, sampling

Specification

• Alias: nond_importance_sampling

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional		samples	Number of samples for sampling-based methods
Optional		seed	Seed of the random number generator
Required (Choose One)	Importance Sampling Approach	import	Importance sampling option for probability refinement
		adapt_import	Importance sampling option for probability refinement
		mm_adapt_import	Importance sampling option for probability refinement
Optional		refinement_samples	Number of samples used to refine a probabilty estimate or sampling design.
Optional		max_iterations	Number of iterations allowed for optimizers and adaptive UQ methods
Optional		convergence_tolerance	Stopping criterion based on objective function or statistics convergence
Optional		response_levels	Values at which to estimate desired statistics for each response
Optional		probability_levels	Specify probability levels at which to estimate the corresponding response value
Optional		gen_reliability_levels	Specify generalized relability levels at which to estimate the corresponding response value
Optional		distribution	Selection of cumulative or complementary cumulative functions
Optional		rng	Selection of a random number generator
Optional		model_pointer	Identifier for model block to be used by a method

Description

The importance_sampling method is based on ideas in reliability modeling.

An initial Latin Hypercube sampling is performed to generate an initial set of samples. These initial samples are augmented with samples from an importance density as follows:

• The variables are transformed to standard normal space.

• In the transformed space, the importance density is a set of normal densities centered around points which are in the failure region.

• Note that this is similar in spirit to the reliability methods, in which importance sampling is centered around a Most Probable Point (MPP).

• In the case of the LHS samples, the importance sampling density will simply by a mixture of normal distributions centered around points in the failure region.

Options

Choose one of the importance sampling options:

• import

• adapt_import

• mm_adapt_import

The options for importance sampling are as follows: import centers a sampling density at one of the initial LHS samples identified in the failure region. It then generates the importance samples, weights them by their probability of occurence given the original density, and calculates the required probability (CDF or CCDF level). adapt_import is the same as import but is performed iteratively until the failure probability estimate converges. mm_adapt_import starts with all of the samples located in the failure region to build a multimodal sampling density. First, it uses a small number of samples around each of the initial samples in the failure region. Note that these samples are allocated to the different points based on their relative probabilities of occurrence: more probable points get more samples. This early part of the approach is done to search for “representative” points. Once these are located, the multimodal sampling density is set and then mm_adapt_import proceeds similarly to adapt_import (sample until convergence).

Theory

Importance sampling is a method that allows one to estimate statistical quantities such as failure probabilities (e.g. the probability that a response quantity will exceed a threshold or fall below a threshold value) in a way that is more efficient than Monte Carlo sampling. The core idea in importance sampling is that one generates samples that preferentially samples important regions in the space (e.g. in or near the failure region or user-defined region of interest), and then appropriately weights the samples to obtain an unbiased estimate of the failure probability [Sri02]. In importance sampling, the samples are generated from a density which is called the importance density: it is not the original probability density of the input distributions. The importance density should be centered near the failure region of interest. For black-box simulations such as those commonly interfaced with Dakota, it is difficult to specify the importance density a priori: the user often does not know where the failure region lies, especially in a high-dimensional space. [SW10]. We have developed two importance sampling approaches which do not rely on the user explicitly specifying an importance density.