Hammersley

Hammersley sampling belongs to the category of quasi-Monte Carlo methods. This technique uses a quasi-random number generator, based on the Hammersley points, to uniformly sample a unit hypercube.

hammersley

Latin Hypercube (left) and Hammersley (right) for 100 runs.

Hammersley sampling generates the input variable values for designs as:

where is the number of designs, is the number of variables, is the design index (in our case starting from 0), are the first n-1 prime numbers (2,3,5,7…) and are calculated as:

where are the coefficients of integer, in radix-R notation which is represented as:

where

Usability Characteristics

•	Hammersley sampling is an efficient sampling technique that provides reliable estimates of output descriptive statistics using fewer samples than random sampling. For example, for the same number of runs, a Hammersley sample will be closer to the theoretical mean than a truly random sample.

•	Hammersley provides good, uniform properties on a k-dimensional hypercube. This is an advantage over Latin Hypercube sampling, which provides good uniform properties of each dimension individually.

•

To get a good quality fitting function, a minimum number of runs should be evaluated. (N+1)(N+2)/2 runs are needed to fit a second order polynomial, assuming that most output responses are close to a second order polynomial within the commonly used input variable ranges of -+10%. An additional number of runs equal to 10% is recommended to provide redundancy, which results in more reliable post-processing. As a result, this equation is recommend to calculate the number of runs needed or a minimum of 1.1*(N+1)(N+2)/2 runs.

•	Any data in the inclusion matrix is combined with the run data for post-processing. Any run matrix point which is already part of the inclusion data will not be rerun.

Settings

In the Specifications step, you can change the following setting of Hammersley from the Settings tab.

Parameter

Default

Range

Description

Number of runs

(npt)

> 0 integer

Number of new designs to be evaluated.

Use Inclusion Matrix

false

true or false

Concatenation without duplication between the inclusion and the generated run matrix.