83 research outputs found
Goodness-of-fit tests in many dimensions
A method is presented to construct goodness-of-fit statistics in many
dimensions for which the distribution of all possible test results in the limit
of an infinite number of data becomes Gaussian if also the number of dimensions
becomes infinite. Furthermore, an explicit example is presented, for which this
distribution as good as only depends on the expectation value and the variance
of the statistic for any dimension larger than one.Comment: 14 page
The dynamical point of view of low-discrepancy sequences
International audienceIn this overview we show by examples, how to associate certain sequences in the higher-dimensional unit cube to suitable dynamical systems. We present methods and notions from ergodic theory that serve as tools for the study of low-discrepancy sequences and discuss an important technique, cutting- and-stacking of intervals
Quantum field theory for discrepancies
The concept of discrepancy plays an important role in the study of uniformity
properties of point sets. For sets of random points, the discrepancy is a
random variable. We apply techniques from quantum field theory to translate the
problem of calculating the probability density of (quadratic) discrepancies
into that of evaluating certain path integrals. Both their perturbative and
non-perturbative properties are discussed.Comment: 26 page
Gaussian limits for discrepancies. I: Asymptotic results
We consider the problem of finding, for a given quadratic measure of
non-uniformity of a set of points (such as star-discrepancy or
diaphony), the asymptotic distribution of this discrepancy for truly random
points in the limit . We then examine the circumstances under which
this distribution approaches a normal distribution. For large classes of
non-uniformity measures, a Law of Many Modes in the spirit of the Central Limit
Theorem can be derived.Comment: 25 pages, Latex, uses fleqn.sty, a4wide.sty, amsmath.st
On the use of a Modified Latin Hypercube Sampling (MLHS) approach in the estimation of a Mixed Logit model for vehicle choice
Quasi-random number sequences have been used extensively for many years in the simulation of integrals that do not have a closed-form expression, such as Mixed Logit and Multinomial Probit choice probabilities. Halton sequences are one example of such quasi-random number sequences, and various types of Halton sequences, including standard, scrambled, and shuffled versions, have been proposed and tested in the context of travel demand modeling. In this paper, we propose an alternative to Halton sequences, based on an adapted version of Latin Hypercube Sampling. These alternative sequences, like scrambled and shuffled Halton sequences, avoid the undesirable correlation patterns that arise in standard Halton sequences. However, they are easier to create than scrambled or shuffled Halton sequences. They also provide more uniform coverage in each dimension than any of the Halton sequences. A detailed analysis, using a 16-dimensional Mixed Logit model for choice between alternative-fuelled vehicles in California, was conducted to compare the performance of the different types of draws. The analysis shows that, in this application, the Modified Latin Hypercube Sampling (MLHS) outperforms each type of Halton sequence. This greater accuracy combined with the greater simplicity make the MLHS method an appealing approach for simulation of travel demand models and simulation-based models in general
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