fsu_quasi_mc

Design of Computer Experiments - Quasi-Monte Carlo sampling

Topics

package_fsudace, design_and_analysis_of_computer_experiments

Specification

  • Alias: None

  • Arguments: None

Child Keywords:

Required/Optional

Description of Group

Dakota Keyword

Dakota Keyword Description

Required (Choose One)

Sequence Type

halton

Generate samples from a Halton sequence

hammersley

Use Hammersley sequences

Optional

latinize

Adjust samples to improve the discrepancy of the marginal distributions

Optional

quality_metrics

Calculate metrics to assess the quality of quasi-Monte Carlo samples

Optional

variance_based_decomp

Activates global sensitivity analysis based on decomposition of response variance into contributions from variables

Optional

samples

Number of samples for sampling-based methods

Optional

fixed_sequence

Reuse the same sequence and samples for multiple sampling sets

Optional

sequence_start

Choose where to start sampling the sequence

Optional

sequence_leap

Specify how often the sequence is sampled

Optional

prime_base

The prime numbers used to generate the sequence

Optional

max_iterations

Number of iterations allowed for optimizers and adaptive UQ methods

Optional

model_pointer

Identifier for model block to be used by a method

Description

Quasi-Monte Carlo methods produce low discrepancy sequences, especially if one is interested in the uniformity of projections of the point sets onto lower dimensional faces of the hypercube (usually 1-D: how well do the marginal distributions approximate a uniform?)

This method generates sets of uniform random variables on the interval [0,1]. If the user specifies lower and upper bounds for a variable, the [0,1] samples are mapped to the [lower, upper] interval.

The user must first choose the sequence type:

  • halton or

  • hammersley

Then three keywords are used to define the sequence and how it is sampled:

  • prime_base

  • sequence_start

  • sequence_leap

Each of these has defaults, so specification is optional.

Theory

The quasi-Monte Carlo sequences of Halton and Hammersley are deterministic sequences determined by a set of prime bases. Generally, we recommend that the user leave the default setting for the bases, which are the lowest primes. Thus, if one wants to generate a sample set for 3 random variables, the default bases used are 2, 3, and 5 in the Halton sequence. To give an example of how these sequences look, the Halton sequence in base 2 starts with points 0.5, 0.25, 0.75, 0.125, 0.625, etc. The first few points in a Halton base 3 sequence are 0.33333, 0.66667, 0.11111, 0.44444, 0.77777, etc. Notice that the Halton sequence tends to alternate back and forth, generating a point closer to zero then a point closer to one. An individual sequence is based on a radix inverse function defined on a prime base. The prime base determines how quickly the [0,1] interval is filled in. Generally, the lowest primes are recommended.

The Hammersley sequence is the same as the Halton sequence, except the values for the first random variable are equal to 1/N, where N is the number of samples. Thus, if one wants to generate a sample set of 100 samples for 3 random variables, the first random variable has values 1/100, 2/100, 3/100, etc. and the second and third variables are generated according to a Halton sequence with bases 2 and 3, respectively.

For more information about these sequences, see [Hal60], [HS64], and [KW97].