Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
 All Pages
importance_sampling


Importance sampling

Topics

This keyword is related to the topics:

Specification

Alias: nond_importance_sampling

Argument(s): none

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)
Group 1 import Sampling option
adapt_import Importance sampling option
mm_adapt_import Sampling option
Optional refinement_samples

Number of samples used to refine a probabilty estimate or sampling design.

Optional response_levels

Values at which to estimate desired statistics for each response

Optional max_iterations

Stopping criterion based on number of iterations

Optional convergence_tolerance

Stopping criterion based on convergence of the objective function or statistics

Optional distribution

Selection of cumulative or complementary cumulative functions

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 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 [76]. 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.[78]. We have developed two importance sampling approaches which do not rely on the user explicitly specifying an importance density.

See Also

These keywords may also be of interest: