Low discrepancy sequence

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	Low-discrepancy sequence - Wikipedia, the free encyclopedia
		Low-discrepancy sequences are also called quasi-random or sub-random sequences, due to their use in situations similar to those when pseudorandom or random numbers are used instead.
		In mathematics, a low-discrepancy sequence is a sequence with the property that for all N, the subsequence x
		Roughly speaking, the discrepancy of a sequency is low if the number of points falling into a set B is close to the number one would expect from the measure of B.
	en.wikipedia.org /wiki/Low-discrepancy_sequence (680 words)

	Illustration of a low-discrepancy sequence - Wikipedia, the free encyclopedia
		The low-discrepancy sequence was generated by TOMS algorithm 659, described by P. Bratley and B.L. Fox in ACM Transactions on Mathematical Software, vol.
		The points plotted below are the first 100, 1000, and 10000 elements in a sequence of the Sobol type.
		For comparison, 10000 elements of a sequence of pseudorandom points are also shown.
	en.wikipedia.org /wiki/Illustration_of_a_low-discrepancy_sequence (185 words)

	Quasi-Monte Carlo Simulation
		Let X be a vector of low discrepancy sequence (for example the van der Corput base 2) with NI elements, and we need a random swap between the elements X(i) and X(p) for all NI elements.
		Dupire and Savine (1998, p.54) on this problem wrote: "Low-discrepancy sequences fill the space in an additive way, which means that the (N+1)th point does not modify the position of the first N points (sequences tend to place it in the centre of the biggest gap left by the previous points).
		Discrepancy is related the worse deviation of the percentage of points in any sub-volume (or sub-rectangle, if you imagine a thin slice in the space) in relation to the exact percentage of points for a perfect evenly distributed points.
	www.puc-rio.br /marco.ind/quasi_mc2.html (3270 words)

	GNU Scientific Library -- Reference Manual - Quasi-Random Sequences
		The quasi-random sequence generators use an interface that is similar to the interface for random number generators, except that seeding is not required--each generator produces a single sequence.
		A quasi-random sequence progressively covers a d-dimensional space with a set of points that are uniformly distributed.
		This chapter describes functions for generating quasi-random sequences in arbitrary dimensions.
	www.gnu.org /software/gsl/manual/gsl-ref_18.html (463 words)

	Project-Team-armor
		is a low discrepancy sequence) is bounded by the product of a quantity depending on the discrepancy of the sequence and the variation of the integrand.
		As another remark, due to the correlation structure of the sequence, necessary to ``minimize'' the error, the direct application of QMC methods to the analysis of Markov chains was inefficient.
		In [76], it is also shown how by using QMC for a small amount of coordinates, and MC for the remaining ones, the accuracy of the simulation can be improved.
	www.inria.fr /rapportsactivite/RA2004/armor2004/uid32.html (485 words)

	Diophantine_approximation
		That is, more abstractly, look at the sequence in R/Z, which is a circle.
		Hermann Weyl proved a basic result showing that this was equivalent to bounds for exponential sums formed from the sequence.
		For any interval I on the circle we look at the proportion of the sequence's elements that lie in it, up to some integer N, and compare it to the proportion of the circumference occupied by I.
	en-cyclopedia.com /wiki/Diophantine_approximation (354 words)

	QR Streams
		The construction and application of low-discrepancy sequences recieved a lot of attention over the last decades in various fields of numerical computation, as well as in theoretical mathematics.
		The general study of uniformly distributed sequences was initiated by Herman Weyl in 1916 with a famous paper "Über die Gleichverteilung von Zahlen mod.
		He defined the notion of discrepancy to quantify the quality of uniformity of a finite point set.
	www.mathdirect.com /products/qrn (520 words)

	ACM TOMS: Bibliographic record for `Kocis:1997:CIL'
		An empirical formula for the error of the quasi Monte Carlo integration is suggested.", keywords = "discrepancy, error of numerical integration, Faure sequence, generalized Halton sequence, Halton sequence, low-discrepancy sequences, Monte Carlo and quasi Monte Carlo integration, Sobol sequence", subject = "{\bf G.1.4} Mathematics of Computing, NUMERICAL ANALYSIS, Quadrature and Numerical Differentiation.
		A modification of the Halton sequence (the Halton sequence leaped) and a new construction of the generalized Halton sequence are suggested for unrestricted number of dimensions and are show to improve considerably on the original Halton sequence.
		Then an estimate of the maximum error of the quasi Monte Carlo integration of nine test functions is computed for up to 400 dimensions and is used to evaluate the known generators mentioned above and the two new generators.
	math.nist.gov /toms/cgi-bin/TOMSbibget?Kocis:1997:CIL (166 words)

	Caltech Multi-Res Modeling Group - low discrepancy sequence library
		This C++ library is an implementation of several low discrepancy sequences accompanying the new techniques presented in the article Fast Generation of Randomized Low Discrepancy Point Sets by Ilja Friedel and Alexander Keller.
		Fast Generation of Randomized Low Discrepancy Point Sets [ps][pdf]
		In its curent version the library implements Latin Hypercube sampling, the Halton sequence, digital sequences, randomized (t,s)-sequences, and Latin supercube sampling.
	www.multires.caltech.edu /software/libseq (259 words)

	chung.kyusik.kc74
		The classical application of low-discrepancy sequences is in the choosing of points over which to compute a Riemann Sum for a multidimensional integral.
		Low-discrepancy sequences are deterministic sequences with high uniformity of distribution.
		Author: Kyusik Chung Title: Efficient Approximation of Asian Options Using Low-Discrepancy Sequences Advisor: Ming-Yang Kao Major: Computer Science and Mathematics Asian options are path dependent options whose strike price is determined as the arithmetic average of the stock price over the life of the option.
	zoo.cs.yale.edu /classes/cs490/99-00b/Abstracts/chung.kyusik.kc74 (301 words)

	HAMMERSLEY - The Hammersley Quasirandom Sequence
		In a generalized Hammersley sequence, each coordinate is allowed to be a fractional or van der Corput sequence.
		The first dimension of entries in the sequence will have the form J/N for J from 1 to N. The remaining dimensions are computed using the 1-dimensional van der Corput sequence, using successive primes as bases.
		By contrast, you can compute 100 points of a Halton sequence, and then 100 more, and this will be the same as computing the first 200 points of the Halton sequence in one calculation.
	www.csit.fsu.edu /~burkardt/f_src/hammersley/hammersley.html (934 words)

	An introduction to quasi-random numbers by George Levy
		The principal aim in the construction of low-discrepancy sequences is thus to find sequences in which the constant is as small as possible.
		The discrepancy of a sequence is a measure of its uniformity and is defined as follows:
		The discrepancy is therefore computed by comparing the actual number of sample points in a given volume of multidimensional space with the number of sample points that should be there assuming a uniform distribution.
	www.fenews.com /fen24/levy.html (1260 words)

	Simple Halton sequence generator
		The availability of open-source low discrepancy sequence generators over the Internet, however, is very scarce, especially when compared with the rather large number of applications which claim to make use of such techniques.
		Among quasi-random numbers, the Halton sequence is perhaps the easiest to translate into computer language: basically, a number gets converted to binary form, "reflected", and brought back to base 10; dimensionality can be achieved by using a radix other than 2, chosen among prime numbers.
		The following Java code was written with the objective of clarity in mind, rather than speed, and it can be easily converted into a variety of other languages, C++ in primis.
	www.atomproject.org /other_halton.shtml (302 words)

	Quasi-Monte Carlo Simulation
		The main challenge for the low discrepancy sequences is to avoid the multi-dimensional clustering caused by the correlations between the dimensions.
		However, for low discrepancy sequences has been reporting that this is not the better way because it damage the low discrepancy sequence properties (alters the order of the sequence or scramble the sequence uniformity), see Moro (1995), Galanti and Jung (1997), and Jackson and Staunton (2001).
		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.
	www.puc-rio.br /marco.ind/quasi_mc.html (7263 words)

	Patent information
		We have made improvements in the Sobol’ sequence and, since the filing date of the patent, improvements in the generalized Faure low discrepancy sequence in FinDer.
		The study of low discrepancy sequences is part of number theory.
		Columbia has patented a method for valuing a complex security using samples derived from points belonging to a low discrepancy sequence, whenever the security value is represented by an integral in at least fifty dimensions.
	www1.cs.columbia.edu /~traub/html/body_patent_information.html (563 words)

	Wilmott Forums - High Dimension MC
		any low discrepancy sequence tends to work very badly in high dimensions as the problem of covering the space evenly becomes impossible without enormous numbers of simulations.
		If you do not use the Sobol sequence, do you use any other low discrepancy numbers?
		so the low discrepancy points are used to generate points on the path widely spaced apart (=high variance) and brownian bridges are used to calculate the intermediate points with the remaining pseudo-random numbers.
	www.wilmott.com /messageview.cfm?catid=4&threadid=1862 (806 words)

	FaureSequence (JMSL Numerical Library)
		Discrepancy measures the deviation from uniformity of a point set.
		It is faster to compute a shuffled Faure sequence than to compute the Faure sequence itself.
		Returns the number of points skipped at the beginning of the sequence.
	vni.com /products/imsl/jmsl/v25/api/com/imsl/stat/FaureSequence.html (330 words)

	On the Scrambled Halton Sequence
		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.
		This derandomized Halton sequence is then numerically tested and shown empirically to be far superior to the original sequence.
		One of its important advantages is that the Halton sequence is easy to implement due to its definition via the radical inverse function.
	www.cs.fsu.edu /~mascagni/abstracts/scrambled_halton.html (210 words)

	Wilmott Forums - Matlab code for Quasi Monte Carlo
		This is simply the way the Halton sequence works and is the main reason why this sequence is not a very good sequence to use for high dimensional problems.
		This is the reason for using the n:th prime as the base for the n:th dimension, and the 3rd dimension is therefor a base-5 sequence.
		Also, if you use the Halton sequence, try to use b^x-1 numbers from the sequence (b = base, x = any integer), as this will make the drawings "complete" (using a couple of drawings more or less than b^x-1 will only make the distribution less uniform, however, the distribution is simply an equally spaced abscissa...).
	www.wilmott.com /messageview.cfm?catid=4&threadid=4144 (838 words)

	MONTE CARLO METHODS IN FINANCE FACTS AND INFORMATION
		The selection of points is a low-discrepancy_sequence such as a Sobol sequence.
		Taking averages of derivative payoffs at points in a low-discrepancy sequence is often more efficient than taking averages of payoffs at random points.
		Instead of generating sample paths randomly, it is possible to systematically (and in fact completely deterministically, despite the "quasi-random" in the name) select points in a probability spaces so as to optimally "fill up" the space.
	www.livingflowers.com /Monte_Carlo_methods_in_finance (955 words)

	GrayFaure package
		This page is related to Eric Thiémard's 1998 paper Economic generation of low-discrepancy sequences with a b-ary Gray code (133 Ko).
		As an application, we present an implementation of a Faure sequence generator and compare its properties with other implementations of low-discrepancy sequence generators.
		GrayFaure is a fast ANSI-C implementation of a Faure low-discrepancy sequence generator that can be freely downloaded from this page.
	rosowww.epfl.ch /papers/grayfaure (211 words)

	Software.htm
		A quasirandom or low discrepancy sequence, such as the Faure, Halton, Hammersley, Niederreiter or Sobol sequences, is less random than a pseudorandom number sequence, but more useful for such tasks as approximation of integrals in higher dimensions, and in global optimization.
		This is because low discrepancy sequences tend to sample space more uniformly than random numbers.
		Computation of bounds on the star discrepancy of an M-dimensional pointset
	www.mlahanas.de /Software/Software.htm (99 words)

	The Fast Calculation Of Form Factors Using Low Discrepancy Sequences - Keller (ResearchIndex)
		The low discrepancy sequence has been generated by the incremental Halton generator proposed in
		The fast Calculation of Form Factors using Low Discrepancy Sequences.
		In this paper we show, that using deterministic low discrepancy sample points is superior to random sampling, resulting in an acceleration of more...
	citeseer.ist.psu.edu /78549.html (640 words)

	HaltonSequence Class
		Quasi-random sequences, also called low discrepancy sequences, are sequences of vectors that progressively cover a multi-dimensional space with points that are uniformly distributed.
		A Halton sequence is one kind of quasi-random sequence.
		Every sequence is generated from a different prime number.
	www.extremeoptimization.com /Statistics/Reference/Extreme.Statistics.Random.HaltonSequence.html (196 words)

	Global Derivatives Cliquet / Ratchet Options
		Using a low-discrepancy sequence and the inversion method given by Moro or Acklam, convergence of the option value is quicker and faster than other methods.
		Winiarski (1998) show that the use of low-discrepancy sequences, coupled with a Brownian bridge or an incremental approach method produces cliquet values with reduced error levels (relative to standard MCS or QMCS without the mentioned methods).
		There has been extensive research into the use of Quasi-MCS employing low discrepancy sequences in order to price path-dependent options.
	www.global-derivatives.com /options/cliquet-options.php (950 words)

	GloriaMundi Resource Detail page
		We explore the performance of a new low discrepancy sequence when applied to the computation of the standard deviation of two very simple, low dimensional value at risk problems.
		The new sequence is constructed by a hybrid, known as scrambled nets, of Monte Carlo and quasi-Monte Carlo methods.
		In the examples we consider, scrambled nets appear to be at least as accurate as Sobol sequences and require on the order of one fiftieth as many function evaluations as Monte Carlo sequences.
	www.gloriamundi.org /detailpopup.asp?keywords=Monte&ID=453055045 (169 words)

	Caltech Multi-Res Modeling Group - Software
		The low-discrepancy sequence library written by Ilja Friedel and Alexander Keller provides fast sample-point generators for the quasi-Monte Carlo and randomized quasi-Monte Carlo methods of integration.
		Dual Quadrilateral Subdivision is demonstrated in a code primarily written by Jianhui Zhang as a teaching tool for the Siggraph Course on Subdivision for Modeling and Animation.
		Jeff Bolz wrote a fast subdivision demo and library using the techniques described in Rapid Evaluation of Catmull-Clark Subdivision Surfaces.
	www.multires.caltech.edu /software/software.htm (225 words)

	Self-similar Image Sampling Schemes: Holographic and Low Discrepancy Properties (ResearchIndex)
		We show that such sampling schemes generate 2-D low discrepancy sequences, and as such are also useful for the Monte Carlo evaluation of area integrals.
		6 An improved low-discrepancy sequence for multidimensional qu..
		26 Quasi-random sequences and their discrepancies - Morokoff, Caflisch - 1994 ACM
	citeseer.ist.psu.edu /bruckstein99selfsimilar.html (374 words)

	Low-discrepancy Sequences
		My guess is that it is due to either that the problem is too simple for QMC to show its advantage or that FINDER's generalized Faure sequence is superior to my simple Faure sequence algorithm (described in "Quasi-Monte Carlo methods in Numerical Finance", by C. Joy, P. Boyle and K.S. Tan).
		I have written a C++ program using Faure sequence and tested it on Asian option pricing with dimension ranged 10 - 40.
		The website address is www.cs.columbia.edu/~traub The software is available to academic researchers (assumed free of charge), but is not free for financial people.
	www.csc.fi /math_topics/Mail/NANET98-4/msg00259.html (322 words)

	Low-discrepancy sequences
		This picture is an illustration that low-discrepancy sequences have better uniformity properties than pseudo-random sequences.
		On the right: 2'000 points from a low-discrepancy sequence
		On the left: 2'000 points from a pseudo-random sequence
	rosowww.epfl.ch /et/lds.html (33 words)

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