On the Scrambled Halton Sequence

Mascagni Michael and Chi Hongmei
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Mascagni Michael: Department of Computer Science and School of Computational Science and Information Technology, Florida State University, Tallahassee, FL 32306-4530, USA, E-mail: E-mail: , URL: http://www.cs.fsu.edu/∼mascagni Michael.Mascagni@fsu.edu
Chi Hongmei: Department of Computer Science, Florida State University, Tallahassee, FL 32306-4120, USA, Email: chi@cs.fsu.edu

Monte Carlo Methods and Applications, 2004, vol. 10, issue 3-4, 435-442

Abstract: 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.

Date: 2004
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DOI: 10.1515/mcma.2004.10.3-4.435

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