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 
Number of samples for samplingbased methods 

Optional 
Seed of the random number generator 

Required (Choose One) 
Importance Sampling Approach 
Importance sampling option for probability refinement 

Importance sampling option for probability refinement 

Importance sampling option for probability refinement 

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

Optional 
Number of iterations allowed for optimizers and adaptive UQ methods 

Optional 
Stopping criterion based on objective function or statistics convergence 

Optional 
Values at which to estimate desired statistics for each response 

Optional 
Specify probability levels at which to estimate the corresponding response value 

Optional 
Specify generalized relability levels at which to estimate the corresponding response value 

Optional 
Selection of cumulative or complementary cumulative functions 

Optional 
Selection of a random number generator 

Optional 
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 userdefined 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 blackbox 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 highdimensional space. [SW10]. We have developed two importance sampling approaches which do not rely on the user explicitly specifying an importance density.