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
where Vd is the d × d binary matrix defined by
- ,
with vk,j,m denoting the m-th digit after the binary point of the direction number vk,j = (0.vk,j,1vk,j,2 . . .)2.
Theorem. The d-dimensional Sobol sequence possesses Property A' iff
where Ud is the 2d × 2d binary matrix defined by
- ,
with vk,j,m denoting the m-th digit after the binary point of the direction number vk,j = (0.vk,j,1vk,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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