# LowDiscrepancySequence¶

class LowDiscrepancySequence(*args)

Base class to generate low discrepancy sequences.

Parameters:
dimensionint

Dimension of the points of the low discrepancy sequence.

Notes

The low discrepancy sequences, also called ‘quasi-random’ sequences, are a deterministic alternative to random sequences for use in Monte Carlo methods. These sequences are sets of equidistributed points which the error in uniformity is measured by its discrepancy.

The discrepancy of a set is defined, using Niederreiter’s notation, as:

where is the s-dimensional Lebesgue measure, is the number of points in that fall into , and is the set of s-dimensional intervals or boxes of the form:

where .

The star-discrepancy is defined similarly, except that the supremum is taken over the set of intervals of the form:

where is in the half-open interval .

A low-discrepancy sequence can be generated only through the derived classes of LowDiscrepancySequence. The sequences implemented are Faure, Halton, Reverse Halton, Haselgrove and Sobol sequences.

Examples

>>> import openturns as ot
>>> # Create a sequence of 3 points of 2 dimensions
>>> sequence = ot.LowDiscrepancySequence(ot.SobolSequence(2))
>>> print(sequence.generate(3))
0 : [ 0.5  0.5  ]
1 : [ 0.75 0.25 ]
2 : [ 0.25 0.75 ]


Methods

 computeStarDiscrepancy(sample) Compute the star discrepancy of a sample uniformly distributed over [0, 1). generate(*args) Generate a sample of pseudo-random vectors of numbers uniformly distributed over [0, 1). Accessor to the object's name. Accessor to the dimension of the points of the low discrepancy sequence. Accessor to the object's id. Accessor to the underlying implementation. Accessor to the object's name. initialize(dimension) Initialize the sequence. setName(name) Accessor to the object's name.
__init__(*args)
computeStarDiscrepancy(sample)

Compute the star discrepancy of a sample uniformly distributed over [0, 1).

Parameters:
sample2-d sequence of float
Returns:
starDiscrepancyfloat

Star discrepancy of a sample uniformly distributed over [0, 1).

Examples

>>> import openturns as ot
>>> # Create a sequence of 3 points of 2 dimensions
>>> sequence = ot.LowDiscrepancySequence(ot.SobolSequence(2))
>>> sample = sequence.generate(16)
>>> print(sequence.computeStarDiscrepancy(sample))
0.12890625
>>> sample = sequence.generate(64)
>>> print(sequence.computeStarDiscrepancy(sample))
0.0537109375

generate(*args)

Generate a sample of pseudo-random vectors of numbers uniformly distributed over [0, 1).

Parameters:
sizeint

Number of points to be generated. Default is 1.

Returns:
sampleSample

Sample of pseudo-random vectors of numbers uniformly distributed over [0, 1).

Examples

>>> import openturns as ot
>>> # Create a sequence of 3 points of 2 dimensions
>>> sequence = ot.LowDiscrepancySequence(ot.SobolSequence(2))
>>> print(sequence.generate(3))
0 : [ 0.5  0.5  ]
1 : [ 0.75 0.25 ]
2 : [ 0.25 0.75 ]

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

The object class name (object.__class__.__name__).

getDimension()

Accessor to the dimension of the points of the low discrepancy sequence.

Returns:
dimensionint

Dimension of the points of the low discrepancy sequence.

getId()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

getImplementation()

Accessor to the underlying implementation.

Returns:
implImplementation

A copy of the underlying implementation object.

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

initialize(dimension)

Initialize the sequence.

Parameters:
dimensionint

Dimension of the points of the low discrepancy sequence.

Examples

>>> import openturns as ot
>>> # Create a sequence of 3 points of 2 dimensions
>>> sequence = ot.LowDiscrepancySequence(ot.SobolSequence(2))
>>> print(sequence.generate(3))
0 : [ 0.5  0.5  ]
1 : [ 0.75 0.25 ]
2 : [ 0.25 0.75 ]
>>> print(sequence.generate(3))
0 : [ 0.375 0.375 ]
1 : [ 0.875 0.875 ]
2 : [ 0.625 0.125 ]
>>> sequence.initialize(2)
>>> print(sequence.generate(3))
0 : [ 0.5  0.5  ]
1 : [ 0.75 0.25 ]
2 : [ 0.25 0.75 ]

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.