Illustration of a low-discrepancy sequence
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Here is an illustration of a low-discrepancy sequence.
The points plotted below are the first 100, 1000, and 10000 elements in a sequence of the Sobol type. For comparison, 10000 elements of a sequence of pseudorandom points are also shown.
The low-discrepancy sequence was generated by TOMS algorithm 659, described by P. Bratley and B.L. Fox in ACM Transactions on Mathematical Software, vol. 14, no. 1, pp 88--100. An implementation of the algorithm in Fortran may be downloaded from Netlib, URL: http://www.netlib.org/toms/659
The first 100 points in a low-discrepancy sequence of the Sobol type.
The first 1000 points in the same sequence. These 1000 comprise the first 100, with 900 more points.
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Low_discrepancy_10000.png
Low_discrepancy_10000.png
The first 10000 points in the same sequence. These 10000 comprise the first 1000, with 9000 more points.
For comparison, here are the first 10000 points in a sequence of uniformly distributed pseudorandom numbers. Regions of higher and lower density are evident.