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# Topic: Halton sequences

###### In the News (Thu 18 Apr 19)

 Random sequence - Wikipedia, the free encyclopedia In short, a random sequence is a sequence of random variables. Also, the statement of the central limit theorem (essentially that the average of a number of observations converges to the mean value) involves an infinite sequence of independent identically-distributed random variables. The term "random sequence" can also describe a sequence of random numbers : Algorithmic information theory defines a random sequence as one that is shorter than any computer program that can produce that sequence (Chaitin-Kolmogorov randomness). www.wikipedia.org /wiki/Random_sequence   (183 words)

 HALTON - The Halton Quasirandom Sequence The DIM_NUM-dimensional Halton sequence is derived from the 1-dimensional van der Corput sequence. The DIM_NUM-dimensional Halton sequence is related to the DIM_NUM+1 dimensional Hammersley sequence of length NMAX. While the Hammersley sequence has better dispersion properties in technical measures such as the discrepancy, it suffers from the problem that you must know, beforehand, the number of points you are going to generate. www.csit.fsu.edu /~burkardt/m_src/halton/halton.html   (844 words)

 [No title]   (Site not responding. Last check: 2007-10-21) The NDIM-dimensional Halton sequence is derived from the 1-dimensionsal van der Corput sequence. The NDIM-dimensional Halton sequence is related to the NDIM+1 dimensional Hammersley sequence of length NMAX. As an example of the use of Halton sequences, we also use them to compute "random" points on or in the the unit circle in 2D, and on the unit sphere in 3D. www.mlahanas.de /Software/halton.htm   (498 words)

 HALTON - The Halton Quasirandom Sequence HALTON is a C++ library, using double precision arithmetic, that computes elements of a Halton quasirandom sequence. HALTON includes routines to make it easy to manipulate this computation, to compute the next N entries, to compute a particular entry, to restart the sequence at a particular point, or to compute NDIM-dimensional versions of the sequence. The NDIM-dimensional Halton sequence is derived from the 1-dimensional van der Corput sequence. www.csit.fsu.edu /~burkardt/cpp_src/halton/halton.html   (941 words)

 ERSA 2003 Congress - University of Jyväskylä - Finland 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. 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. www.jyu.fi /ersa2003/cdrom/abstracts/a406.html   (471 words)

 Search Encyclopedia.com   (Site not responding. Last check: 2007-10-21) Stratigraphy is based on the law of superposition, which states that in a normal sequence of rock layers the youngest is on top and the oldest on the bottom. The complete amino acid sequence of the insulin molecule was described in the early 1950s; insulin was the... series series, in mathematics, indicated sum of a sequence of terms. www.encyclopedia.com /search.asp?target=Halton+sequences&rc=10&fh=15&fr=11   (536 words)

 Halton - Wikipedia, the free encyclopedia Halton is the name of several places in the United Kingdom: Halton is the name of several places in Canada: In the British military, Halton is an RAF base: see RAF Halton. www.encyclopedia-online.info /Halton   (101 words)

 Randomized Halton Sequences - Wang, Hickernell (ResearchIndex)   (Site not responding. Last check: 2007-10-21) Abstract: The Halton sequence is a well-known multi-dimensional low discrepancy sequence. In this paper, we propose a new method for randomizing the Halton sequence: we randomize the start point of each component of the sequence. This method combines the potential accuracy advantage of Halton sequence in multi-dimensional integration with the practical error estimation advantage of Monte Carlo methods. citeseer.ist.psu.edu /wang00randomized.html   (493 words)

 System and method for rendering images using a strictly-deterministic methodology for generating a coarse sequence of ... Both coarse and scrambled coarse Halton sequences are low-discrepancy sequences with a discrepancy that is similar to that exhibited by a Halton sequence as described above. The emit_photons_all sequence, comprising lines (1) through (8) of Code Segment 1, enables the computer graphics system 10 to store all of the photons that are to be stored during a rendering operation in connection with at least a portion of an image to be rendered. Generally, in lines (2) through (6b) of the emit_photons_all sequence, the number of jobs is determined by determining the lowest number that is not divisible by the first "m" prime numbers, where "m" is a selected number. www.freepatentsonline.com /6911976.html   (6438 words)

 Quasi-Monte Carlo Simulation The base of a Faure sequence is the smallest primer number that is larger than or equal to the number of dimensions in the problem, or equal 2 for one dimensional problem. As occurred with high-dimensional Halton sequence, there is the problem of low speed at which the Faure sequence generates increasing finer grid points to cover the unit hypercube. For example, if the dimension of the problem is 50, the last Halton sequence (in dimension 50) uses the 50th prime number that is 229, whereas the Faure sequence uses the first prime number after 50, that is a base 53, which is much smaller than 229. www.puc-rio.br /marco.ind/quasi_mc.html   (7263 words)

 Re: st: halton sequence The full source for our generator is in Appendix B of the paper (http://www.nber.org/papers/w4147), and looking at it I don't see why it couldn't be done in Stata. Halton's paper describes the sequences in a number-theoretic way which looks like something only a LISP interpreter could do, but there is an alternative floating point generater that Halton also describes (and which is referenced in our paper). However, it has been suggested in the literature that using Halton draws (which are systematic and cover better the 0-1 interval) is better. www.stata.com /statalist/archive/2004-08/msg00221.html   (434 words)

 19th GAMM-Seminar Leipzig - Abstracts   (Site not responding. Last check: 2007-10-21) The first one is that the quality of distribution of the quasi-random sequence usually depends on the choice of a certain set of generation parameters, and finding the optimal choice is not easy. The second one is that the generation of these sequences requires thorough understanding of their theory, and thus the low-level computer hardware issues are usually neglected. The idea of scrambling the low-discrepancy sequences was introduced with the purpose to add some automatic statistical error estimation, and even to improve the equidistribution of the sequence, in the case of Owen type scrambling. www.mis.mpg.de /conferences/gamm/2003/GAMM_2003_Abstracts_4.html   (206 words)

 Quasi-Monte Carlo Rendering with Adaptive Sampling The generator for the Sobol' sequence is based on the program in [15], and the algorithm for Halton sequence is described in [14] and others. For example, if a 2D point sequence is generated, the first dimension (horizontal axis of the plots in Figure 2) of the Sobol's sequence generated by the code in [15] alternates between the upper and the lower half of the interval [0,1]. In this experiment, the convergence of QMC with the Halton sequence appears to be the fastest, followed by QMC with the Sobol' sequence, while MC with PRS was the slowest. www.kki.yamanashi.ac.jp /~ohbuchi/online_pubs/eg96_html/eg96.htm   (4154 words)

 TRB Annual Meeting Program   (Site not responding. Last check: 2007-10-21) 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 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. These sequences, like scrambled and shuffled Halton sequences, avoid the undesirable correlation patterns that arise in standard Halton sequences. gulliver.trb.org /am/ip/paper_detail.asp?paperid=1955&p=yes&h=   (235 words)

 QuasiMonteCarlo.nb Extensions of the van-der-Corput sequences are the Halton sequences (For further details on these sequences see Ötken (1999)). Therefore a sequence of 2000 numbers is needed to generate 1000 normal distributed random numbers. Following the described approach for low discrepancy sequences the implementation of a simple option is the same as with the normal Monte-Carlo method. www.weber-und-partner.com /resources/QuasiMonteCarlo   (426 words)

 bunched_beam is an array of three flags that permit turning on Halton sequence generation for the horizontal, vertical, or longitudinal planes. One should never use the same prime for two sequences, unless one randomizes the order of the sequences relative to each other (see the next item). This is used with Halton sequences to remove banding. www.aps.anl.gov /asd/oag/manuals/elegant_ver14.7.1/node15.html   (827 words)

 [No title]   (Site not responding. Last check: 2007-10-21) Bhat (2003) has suggested that scrambled Halton sequences avoid the problem of poor coverage of the integration domain in high dimensions, and has used random shifts to evaluate the quality of the sequences in the context of MMNL estimation. Since the sequences used in QMC approaches are deterministic, it is not possible to use the classical analytical tools for error estimation as we have done in the MC variable sample size strategy by using the delta method. In particular, we are interested in seeing how randomized QMC sequences can reduce the variance in comparison with MC methods and how they can improve the performance of the original deterministic sequences, in combination with the variable sample size strategy. www.fundp.ac.be /~phtoint/pubs/TR04-11.abstract   (531 words)

 Sven's Homepages The NPR rendering is constructed as follows: I start by using Halton pseudo-random sequences to scatter a number of particles over the image. Here is a link to the page where I found the implementation of the Halton sequence generator used to initialize the brush stroke positions. And just for the heck of it, here is the cool siggraph paper by Alexander Keller which was responsible for getting me interested in Halton sequences in the first place. www.sccs.swarthmore.edu /users/03/sven/cs140/assignment2.htm   (1429 words)

 QuasiRandomDemonstration.nb   (Site not responding. Last check: 2007-10-21) The first 10 sequence elements of the Van der Corput sequence in base 7, as rational numbers. and the permuted Halton sequence are the multi-dimensional analogons. Low discrepancy sequences are designed to be as uniform as possible. www.mathdirect.com /products/qrn/resources/Links/QRDemonstration_lnk_3.html   (220 words)

 Halton sequences - Encyclopedia, History, Geography and Biography Halton sequences - Encyclopedia, History, Geography and Biography This page was last modified 02:41, 17 May 2004. This encyclopedia, history, geography and biography article about Halton sequences contains research on www.arikah.net /encyclopedia/Halton_sequences   (301 words)

 eo Halton Halton is the name of several places in... Halton West Halton West, Yorkshire Yorkshire "Halton" is the name of several places in Canada Canada: Halton (electoral district) Halton (electoral district) In the British British military military, "Halton" is an RAF base: see RAF Halton RAF Halton. In statistics statistics, "Halton sequences Halton sequences" are quasi-random sequence quasi-random sequences. www.biodatabase.de /Halton   (140 words)

 Wilmott Forums - sub-random sequences in high dimension MC sims? For high dimensionality Halton sequences, there seems to be a lot of clumping and recognisable patterns. Right, it's non-trivial to extend quasi-random sequance generation to be valid in extra dimensions, especially up in the 100 range. Morevoer, I agree that spectral decomposition can enhance the convergence when combined with quasi random sequences, but the advantage of cholesky is that you get a lower triangular structure that cuts the number of elementary operations to half when correlating your iid random numbers. www.wilmott.com /messageview.cfm?catid=4&threadid=20934   (1425 words)

 Low-discrepancy sampling methods Their potential use in motion planning is no less reasonable than using pseudo-random sequences, which were also designed with a different intention in mind (satisfying statistical tests of randomness). 427] is an adaptation of the Halton sequence. Beware that although these sequences produce asymptotically optimal discrepancy, their performance degrades substantially in higher dimensions (e.g., beyond 10). msl.cs.uiuc.edu /planning/node222.html   (495 words)

 Centre for Transport Studies One-dimensional Halton sequences are generated through a sequence of individual cycles that each progressively fill the 0-1 space. For a Halton sequence based on prime r, the first such cycle will be of length r-1, with subsequent cycles being of length r. It is clear that for most choices of N, the length of the sequence, and r, the prime number used, the final cycle in the sequence will not be a complete cycle. www.cts.cv.ic.ac.uk /html/ResearchActivities/publicationDetails.asp?PublicationID=298   (336 words)

 Re: QMC - sequences whose ``periodic'' differences are low discrepancy s Choose a[1] to be the zero >> sequence and a[2] to any low-discrepancy sequence and we have satisfied >> my condition. The >resulting sequences are uniformly distributed but I do not know if they have >better than an a Sqrt(N) dependence. The first two Halton sequences give: > >N phi_2(N) phi_3(N) phi_2-phi_3 >1 1/2 1/3 1/6 >2 1/4 2/3 5/12 >3 3/4 1/9 23/36 >4 1/8 4/9 23/72 >5 5/8 7/9 13/72 >etc. Unfortunately, low discrepancy of the differences of the sequences is my *only* requirement. www.mcqmc.org /listarchive/mcqmc/discussion/98/Nov/msg00003.html   (662 words)

 John H. Halton's Home Page Halton, J. "On the Efficiency of Generalized Antithetic Transformations for Monte Carlo Integration," Nuclear Science and Engineering, 98, 1988, 299-316. Halton, J. "A Retrospective and Prospective Survey of the Monte Carlo Method," Society for Industrial and Applied Mathematics Review, 12, 1970, 1-63. Halton, J. "On the Efficiency of Certain Quasi-random Sequences of Points in Evaluating Multi-dimensional Integrals," Numerische Mathematik, 2, 1960, 84-90. www.cs.unc.edu /~halton   (363 words)

 Halton   (Site not responding. Last check: 2007-10-21) Halton is the name of several places in the UnitedKingdom : Halton is the name of several places in Canada : In the British military,Halton is an RAF base: see RAF Halton. www.therfcc.org /halton-367341.html   (90 words)

 umontreal.iro.lecuyer.hups (Java Libraries for Stochastic Simulation)   (Site not responding. Last check: 2007-10-21) The terminology low-discrepancy sequence (LDS) is often used for infinite sequences of points such that the discrepancy between the distribution of the first n points of the sequence and the uniform distribution converges to zero at a certain rate when Historically, Halton sequences were defined as extensions of Hammersley point sets. For digital sequences, the matrices [tex2html_wrap_inline1256] actually have an infinite number of columns, although only the first k columns are needed to generate the first b www.iro.umontreal.ca /~lecuyer/ssj/dist/doc/html/umontreal/iro/lecuyer/hups/package-summary.html   (3317 words)

 [No title]   (Site not responding. Last check: 2007-10-21) Identical algorithms are run almost independently on each processor, except that the random number sequences used are different on each processor. Since the results from each processor differ only due to the different random number sequences used, it is important to ensure that the sequences on different processors are uncorrelated. It is clear that the scrambled Halton sequence implementation is superior to the Halton sequence implementation. www.cs.fsu.edu /~sudhir/courses/CIS5935/Latex_1/hw1.tex   (366 words)

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