The Halton sequence is one of the standard (along with (t, s)-sequences and lattice points) low-discrepancy sequences, and thus is widely used in quasi-Monte Carlo applications. One of its important advantages is that the Halton sequence is easy to implement due to its definition via the radical inverse function. However, the original Halton sequence suffers from correlations between radical inverse functions with different bases used for different dimensions. These correlations result in poorly distributed two-dimensional projections. A standard solution to this is to use a randomized (scrambled) version of the Halton sequence. Here, we analyze the correlations in the standard Halton sequence, and based on this analysis propose a new and simpler modified scrambling algorithm. We also provide a number theoretic criterion to choose the optimal scrambling from among a large family of random scramblings. Based on this criterion, we have found the optimal scrambling for up to 60 dimensions for the Halton sequence. This derandomized Halton sequence is then numerically tested and shown empirically to be far superior to the original sequence.
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On the Scrambled Halton Sequence
* Department of Computer Science and School of Computational Science and Information Technology, Florida State University, Tallahassee, FL 32306-4530, USA, E-mail: E-mail: Michael.Mascagni@fsu.edu, URL: http://www.cs.fsu.edu/∼mascagni
† Department of Computer Science, Florida State University, Tallahassee, FL 32306-4120, USA, Email: chi@cs.fsu.edu
Citation Information: Monte Carlo Methods and Applications mcma. Volume 10, Issue 3-4, Pages 435–442, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: 10.1515/mcma.2004.10.3-4.435, May 2008
- Published Online:
- 2008-05-09
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