IDEAS home Printed from
MyIDEAS: Log in (now much improved!) to save this paper

An alternative method to the scrambled Halton sequence for removing correlation between standard Halton sequences in high dimensions

Listed author(s):
  • Stephane Hess


  • John Polak


Registered author(s):

    Halton sequences were first introduced in the 1960s as an alternative to pseudo-random number sequences, with the aim of providing better coverage of the area of integration and negative correlation in the simulated probabilities between observations. This is needed in order to achieve variance reduction when using simulation to approximate an integral that does not have a closed-form expression. Such integrals arise in many areas of regional science, for example in the evaluation and estimation of certain types of discrete choice models. While the performance of standard Halton sequences is very good in low dimensions, problems with correlation have been observed between sequences generated from higher primes. This can cause serious problems in the estimation of models with high-dimensional integrals (e.g., models of aspects of spatial choice, such as route or location). Various methods have been proposed to deal with this; one of the most prominent solutions is the scrambled Halton sequence, which uses special predetermined permutations of the coefficients used in the construction of the standard sequence. In this paper, we conduct a detailed analysis of the ability of scrambled Halton sequences to remove the problematic correlation that exists between standard Halton sequences for high primes in the two-dimensional space. The analysis shows that although the scrambled sequences exhibit a lower degree of overall correlation than the standard sequences, for some choices of primes, correlation remains at an unacceptably high level. This paper then proposes an alternative method, based on the idea of using randomly shuffled versions of the one-dimensional standard Halton sequences in the construction of multi-dimensional sequences. We show that the new shuffled sequences produce a significantly higher reduction in correlation than the scrambled sequences, without loss of quality of coverage. Another substantial advantage of this new method is that it can, without any modifications, be used for any number of dimensions, while the use of the scrambled sequences requires the a-priori computation of a matrix of permutations, which for high dimensional problems could lead to significant runtime disadvantages. Repeated runs of the shuffling algorithm will also produce different sequences in different runs, which nevertheless maintain the same quality of one-dimensional coverage. This is not at all the case for the scrambled sequences. In view of the clear advantages in its ability to remove correlation, combined with its runtime and generalization advantages, this paper recommends that this new algorithm should be preferred to the scrambled Halton sequences when dealing with high correlation between standard Halton sequences.

    If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

    File URL:
    Download Restriction: no

    Paper provided by European Regional Science Association in its series ERSA conference papers with number ersa03p406.

    in new window

    Date of creation: Aug 2003
    Handle: RePEc:wiw:wiwrsa:ersa03p406
    Contact details of provider: Postal:
    Welthandelsplatz 1, 1020 Vienna, Austria

    Web page:

    References listed on IDEAS
    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

    in new window

    1. Train,Kenneth E., 2009. "Discrete Choice Methods with Simulation," Cambridge Books, Cambridge University Press, number 9780521766555, March.
    Full references (including those not matched with items on IDEAS)

    This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

    When requesting a correction, please mention this item's handle: RePEc:wiw:wiwrsa:ersa03p406. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Gunther Maier)

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If references are entirely missing, you can add them using this form.

    If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.