Sobol sequence

256 points from a pseudorandom number source (top); compared with the first 256 points from the 2,3 Sobol sequence (below). The Sobol sequence covers the space more evenly. (red=1,..,10, blue=11,..,100, green=101,..,256)

Sobol sequences (also called LP_τ sequences or (t, s) sequences in base 2) are an example of quasi-random low-discrepancy sequences. They were first introduced by the Russian mathematician I. M. Sobol (Илья Меерович Соболь) in 1967.^[1]

These sequences use a base of two to form successively finer uniform partitions of the unit interval, and then reorder the coordinates in each dimension.

Good distributions in the s-dimensional unit hypercube

Let I^s = [0,1]^s be the s-dimensional unit hypercube and f a real integrable function over I^s. The original motivation of Sobol was to construct a sequence x_n in I^s so that

\lim_{n\to\infty}\; \frac{1}{n}\sum_{i=1}^n f(x_i) \; =\; \int_{I^s} f

and the convergence be as fast as possible.

It is more or less clear that for the sum to converge towards the integral, the points x_n should fill I^s minimizing the holes. Another good property would be that the projections of x_n on a lower-dimensional face of I^s leave very few holes as well. Hence the homogeneous filling of I^s does not qualify ; because in lower-dimensions many points will be at the same place, therefore useless for the integral estimation.

These good distributions are called (t,m,s)-nets and (t,s)-sequences in base b. To introduce them, define first an elementary s-interval in base b a subset of I^s of the form

\prod_{j=1}^s \left[ \frac{a_j}{b^{d_j}}, \frac{a_j+1}{b^{d_j}} \right]

, where a_j, d_j are non-negative integers and

a_j < b^{d_j}

for all j in {1, ...,s}

Given 2 integers $0\leq t\leq m$ , a (t,m,s)-net in base b is a sequence x_n of b^m points of I^s such that $\text{Card} \, P \cap \{x_1, ..., x_{b^m}\} = b^t$ for all elementary interval P in base b of hypervolume λ(P) = b^t-m.

Given a non-negative integer t, a (t,s)-sequence in base b is an infinite sequence of points x_n such that for all integers $k\geq0, \; m\geq t$ , the sequence $\{x_{kb^m}, ..., x_{(k+1)b^m-1}\}$ is a (t,m,s)-net in base b.

In his article, Sobol described Π_τ-meshes and LP_τ sequences, which are (t,m,s)-nets and (t,s)-sequences in base 2 respectively. The terms (t,m,s)-nets and (t,s)-sequences in base b (also called Niederreiter sequences) were coined in 1988 by Harald Niederreiter.^[2] The term Sobol sequences was introduced in late English-speaking papers in comparison with Halton, Faure and other low-discrepancy sequences.

A fast algorithm for the construction of Sobol sequences

A more efficient Gray code implementation was proposed by Antonov and Saleev.^[3]

As for the generation of Sobol numbers, they are clearly aided by the use of Gray code $G(n)=n \oplus \lfloor n/2 \rfloor$ instead of n for constructing the n-th point draw.

Suppose we have already generated all the Sobol sequence draws up to n − 1, and kept in memory the values x_n−1,j for all the required dimensions. Since the Gray code G(n) differs from that of the preceding one G(n − 1) by just a single, say the k-th, bit (which is a rightmost bit of n − 1), all that needs to be done is a single XOR operation for each dimension in order to propagate all of the x_n−1 to x_n, i.e.

x_{n,i}=x_{n-1,i} \oplus v_{k,i}. \,

Additional uniformity properties

Sobol introduced additional uniformity conditions known as property A and A’.^[4]

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 2^d there is exactly one draw in each 2^d 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 4^d there is exactly one draw in each 4^d 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 V^d 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 U^d 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.

The initialization of Sobol numbers

To construct a Sobol sequence a set of direction numbers v_i,j needs to be selected. There is some freedom in the selection of initial direction numbers.^{[note 1]} Therefore, it is possible to receive different realisations of the Sobol sequence for selected dimensions. A bad selection of initial numbers can considerably reduce the efficiency of Sobol sequences when used for computation.

Arguably the easiest choice for the initialisation numbers is just to have the l-th leftmost bit set, and all other bits to be zero, i.e. m_k,j = 1 for all k and j. This initialisation is usually called unit initialisation. However, such a sequence fails the test for Property A and A’ even for low dimensions and hence this initialisation is bad.

Implementation and availability of Sobol sequences

Good initialisation numbers for different numbers of dimensions are provided by several authors. For example, Sobol provides initialisation numbers for dimensions up to 51.^[5] The same set of initialisation numbers is used by Bratley and Fox.^[6]

Initialisation numbers for high dimensions are available on Joe and Kuo.^[7] Peter Jäckel provides initialisation numbers up to dimension 32 in his book "Monte Carlo methods in finance".^[8]

Other implementations are available as C, Fortran 77, or Fortran 90 routines in the Numerical Recipes collection of software.^[9] A free/open-source implementation in up to 1111 dimensions, based on the Joe and Kuo initialisation numbers, is available in C^[10] and Julia.^[11] A different free/open-source implementation is available for C++, Fortran 90, Matlab, and Python.^[12]

Finally, commercial Sobol sequence generators are available within, for example, the NAG Library,^[13] or from the British-Russian Offshore Development Agency (BRODA).^[14]^[15]

Notes

↑ These numbers are usually called initialisation numbers

References

↑ Sobol,I.M. (1967), "Distribution of points in a cube and approximate evaluation of integrals". Zh. Vych. Mat. Mat. Fiz. 7: 784–802 (in Russian); U.S.S.R Comput. Maths. Math. Phys. 7: 86–112 (in English).
↑ Niederreiter, H. (1988). "Low-Discrepancy and Low-Dispersion Sequences", Journal of Number Theory 30: 51–70.
↑ Antonov, I.A. and Saleev, V.M. (1979) "An economic method of computing LP_τ-sequences". Zh. Vych. Mat. Mat. Fiz. 19: 243–245 (in Russian); U.S.S.R Comput. Maths. Math. Phys. 19: 252–256 (in English).
↑ Sobol, I.M. (1976) "Uniformly distributed sequences with an additional uniform property". Zh. Vych. Mat. Mat. Fiz. 16: 1332–1337 (in Russian); U.S.S.R Comput. Maths. Math. Phys. 16: 236–242 (in English).
↑ Sobol, I.M. and Levitan, Y.L. (1976). "The production of points uniformly distributed in a multidimensional cube" Tech. Rep. 40, Institute of Applied Mathematics, USSR Academy of Sciences (in Russian).
↑ Bratley, P. and Fox, B. L. (1988), "Algorithm 659: Implementing Sobol’s quasirandom sequence generator". ACM Trans. Math. Software 14: 88–100.
↑ "Sobol sequence generator". University of New South Wales. 2010-09-16. Retrieved 2013-12-20.
↑ Jäckel, P. (2002) "Monte Carlo methods in finance". New York: John Wiley and Sons. (ISBN 0-471-49741-X.)
↑ Press, W.H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992) "Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed." Cambridge University Press, Cambridge, U.K.
↑ C implementation of the Sobol sequence in the NLopt library (2007).
↑ Sobol.jl package: Julia implementation of the Sobol sequence.
↑ The Sobol Quasirandom Sequence, code for C++/Fortran 90/Matlab/Python by J. Burkardt
↑ "Numerical Algorithms Group". Nag.co.uk. 2013-11-28. Retrieved 2013-12-20.
↑ I. Sobol’, D. Asotsky, A. Kreinin, S. Kucherenko (2011). "Construction and Comparison of High-Dimensional Sobol’ Generators" (PDF). Wilmott Journal Nov: 64–79.
↑ "Broda". Broda. 2004-04-16. Retrieved 2013-12-20.

External links

Collected Algorithms of the ACM (See algorithms 647, 659, and 738.)
Collection of Sobol sequences generator programming codes
Freeware C++ generator of Sobol sequence

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