Sobol Sequence - Additional Uniformity Properties

Additional Uniformity Properties

Sobol introduced additional uniformity conditions known as property A and A’.

Definition. A low-discrepancy sequence is said to satisfy Property A if for any binary segment (not an arbitrary subset) of the d-dimensional sequence of length 2d there is exactly one draw in each 2d hypercubes that result from subdividing the unit hypercube along each of its length extensions into half.

Definition. A low-discrepancy sequence is said to satisfy Property A’ if for any binary segment (not an arbitrary subset) of the d-dimensional sequence of length 4d there is exactly one draw in each 4d hypercubes that result from subdividing the unit hypercube along each of its length extensions into four equal parts.

There are mathematical conditions that guarantee properties A and A'.

Theorem. The d-dimensional Sobol sequence possesses Property A iff

$\det(\bold{V}_d) \equiv 1 (\mod 2),$

where Vd is the d × d binary matrix defined by

$\bold{V}_d := \begin{pmatrix} {v_{1,1,1}}&{v_{2,1,1}}&{\dots}&{v_{d,1,1}}\\ {v_{1,2,1}}&{v_{2,2,1}}&{\dots}&{v_{d,2,1}}\\ {\vdots}&{\vdots}&{\ddots}&{\vdots}\\ {v_{1,d,1}}&{v_{2,d,1}}&{\dots}&{v_{d,d,1}} \end{pmatrix}$ ,

with v_k,j,m denoting the m-th digit after the binary point of the direction number v_k,j = (0.v_k,j,1v_k,j,2 . . .)₂.

Theorem. The d-dimensional Sobol sequence possesses Property A' iff

$\det(\bold{U}_d) \equiv 1 \mod 2,$

where Ud is the 2d × 2d binary matrix defined by

$\bold{U}_d := \begin{pmatrix} {v_{1,1,1}}&{v_{1,1,2}}&{v_{2,1,1}}&{v_{2,1,2}}&{\dots}&{v_{d,1,1}}&{v_{d,1,2}}\\ {v_{1,2,1}}&{v_{1,2,2}}&{v_{2,2,1}}&{v_{2,2,2}}&{\dots}&{v_{d,2,1}}&{v_{d,2,2}}\\ {\vdots}&{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}\\ {v_{1,2d,1}}&{v_{1,2d,2}}&{v_{2,2d,1}}&{v_{2,2d,2}}&{\dots}&{v_{d,2d,1}}&{v_{d,2d,2}} \end{pmatrix}$ ,

with v_k,j,m denoting the m-th digit after the binary point of the direction number v_k,j = (0.v_k,j,1v_k,j,2 . . .)₂.

Tests for properties A and A’ are independent. Thus it is possible to construct the Sobol sequence which satisfies both properties A and A’ or only one of them.

Read more about this topic: Sobol Sequence

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