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Topic: Extrapolation methods

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	Extrapolation - Wikipedia, the free encyclopedia
		Extrapolation often looks sensible at first glance, but its results may sometimes be invalid or subject to substantial uncertainty.
		This divergence is a specific property of extrapolation methods and is only circumvented when the functional forms assumed by the extrapolation method (inadvertently or intentionally due to additional information) accurately represent the nature of the function being extrapolated.
		Another problem of extrapolation is loosely related to the problem of analytic continuation, where (typically) a power series representation of a function is expanded at one of its points of convergence to produce a power series with a larger radius of convergence.
	en.wikipedia.org /wiki/Extrapolation (823 words)

	Extrapolation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-29)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, extrapolation is a type of ((mathematics) calculation of the value of a function between the values already known) interpolation.
		Extrapolation often looks sensible at first glance, but its results may sometimes be invalid or subject to substantial (Being unsettled or in doubt) uncertainty.
		A problem of extrapolation may be converted into an ((mathematics) calculation of the value of a function between the values already known) interpolation problem by the change of variable z → 1/z.
	www.absoluteastronomy.com /encyclopedia/e/ex/extrapolation.htm (691 words)

	Numerical integration - Wikipedia, the free encyclopedia
		Methods developed for ordinary differential equations, such as the Runge-Kutta method, can be applied to the restated problem.
		An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function of the number of integrand evaluations.
		Monte Carlo methods and quasi-Monte Carlo methods are easy to apply to multi-dimensional integrals, and may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods.
	en.wikipedia.org /wiki/Numerical_integration (1466 words)

	Extrapolation Methods (Site not responding. Last check: 2007-10-29)
		An extrapolation method for solving an ODE applies a single-step method with various step sizes and then computes a more accurate estimate for the solution by extrapolating to a step size of zero based on the known dependence of the error on the step size.
		The error is known to be approximately linear for Euler's method and quadratic for Heun's method, so the extrapolation is calculated by fitting the appropriate polynomial to the two data points in the right panel and evaluating it at zero.
		The fitted function and extrapolated solution value are displayed in fl in the right-hand panel, and in the left panel the approximate solution generated by the extrapolation method, shown in fl, replaces the dark green approximate solution computed using two steps of the basic method.
	www.cse.uiuc.edu /eot/modules/ode/extrap (710 words)

	Alroy in review, Paleobiology
		The major criterion for evaluating the methods is the degree to which their correction of the diversity curve corresponds with known variation in sampling intensity, as measured by the number of faunal lists or taxonomic records in each interval.
		Unlike other extrapolation methods, this one has the disadvantage of never predicting the existence of species that are as yet completely unsampled: instead, it focuses only on extending age-ranges of known species.
		The acceptable performance of the Lazarus method is due to its simple and persuasive logic: unless there is some systematic preservational difference between taxa that range through intervals and taxa whose ranges truncate in intervals, the proportion of range-throughs that are preserved truly must be an accurate estimate of the overall preservation probability (Paul 1982).
	www.nceas.ucsb.edu /~alroy/Alroy.sampling.html (12572 words)

	Abstract (Site not responding. Last check: 2007-10-29)
		This limitation is lifted substantially for MTS methods based on extrapolation in combination with stochastic dynamics, as demonstrated for the LN method in the companion paper for protein dynamics.
		We show that two-class impulse methods are generally stable except at integer multiples of half the period of the fastest motion, with the severity of the instability worse at larger timesteps.
		Extrapolation methods are generally unstable for the Newtonian model problem, but the instability is bounded for increasing timesteps.
	monod.biomath.nyu.edu /index/papdir/pap_1_58.html (273 words)

	Solving linear equations on parallel distributed memory architectures by extrapolation - Abstract (Site not responding. Last check: 2007-10-29)
		Extrapolation methods can be used to accelerate the convergence of vector sequences.
		Numerically the extrapolation methods are not as stable as the Krylov subspace methods since they require the solution of ill-conditioned overdetermined systems.
		The use of regularization methods and a slightly modified stationary method have proved to be especially useful.
	www.student.nada.kth.se /~christe/abstract_master.html (243 words)

	SSR135 Critical review of methods to derive water quality guidelines for toxicants and a proposal for a new framework
		The original method called the assessment factor method divided the lowest toxicity value by an assessment factor, the magnitude of which was based on the number, character and quality of the available toxicity data.
		Of the various versions of the AF method and the statistical extrapolation method the current ANZECC AF method and the Aldenberg and Slob method used by The Netherlands were chosen for detailed analysis.
		In the section comparing the precautionary natures of the AF and Aldenberg and Slob methods it was highlighted that a major limitation was that the toxicity of mixtures, accumulation of toxicant in the animal tissue, and transfer of chemicals between the various compartments of the environment were not considered.
	www.deh.gov.au /ssd/publications/ssr/135.html (995 words)

	Implicit extrapolation methods for multilevel finite element computations
		Extrapolation methods for the solution of partial differential equations are commonly based on the existence of error expansions for the approximate solution.
		Implicit extrapolation, in the contrast, is based on applying extrapolation indirectly, by using it on quantities like the residual.
		In the context of multigrid methods, a special technique of this type is known as \034 -extrapolation.
	archiv.tu-chemnitz.de /pub/1998/0051/index_en.html (156 words)

	EVALUATION OF SEVERAL EXTRAPOLATION METHODS FOR ESTIMATION OF MACROINVERTEBRATE SPECIES RICHNESS IN NEOTROPICAL STREAMS ...
		Until the moment there are many methods in the literature and to choose one them is difficult because a good performance of a method for a specific data set may be not so good for others.
		We have examined the performance of several methods for the richness estimation of stream macroinvertebrates inhabiting riffles both at local (stream) and regional (catchment) scales by subsampling a large data set assumed be the total richness.
		Of the 14 tested methods, Jackknife2 and Stout and Vandermeer's method estimated the total richness with the smaller sampling size, while Chao1, Chao2 and Jackknife1 with a sample size in which the discovery rate of new species begins to levels off.
	www.benthos.org /Database/nabstracts99.cfm/ID/393 (260 words)

	methods * Manual of Chemical Methods for...
		Methods of Direct Solving the Boltzmann Equation and Study of Nonequilibrium Flows Fluid Mechanics and Its Applications, 60.
		Methods used in longterm forecasting of domestic telecommunications demand and required resources overall and by main sector.
		Methods of Nonconvex Analysis Lectures Given at the 1st Session of the Centro Internazionale Matematico Estivo C.I.M.E. Held at.
	www.ciyoda.de /ciyouuumethods.html (1085 words)

	Kamvar, Sepandar; Haveliwala, Taher; Manning, Chris; Golub, Gene: Extrapolation Methods for Accelerating PageRank ...
		The algorithm presented here, called Quadratic Extrapolation, accelerates the convergence of the Power Method by periodically subtracting off estimates of the nonprincipal eigenvectors from the current iterate of the Power Method.
		In Quadratic Extrapolation, we take advantage of the fact that the first eigenvalue of a Markov matrix is known to be 1 to compute the nonprincipal eigenvectors using successive iterates of the Power Method.
		Our contribution is useful to the PageRank community and the numerical linear algebra community in general, as it is a fast method for determining the dominant eigenvector of a matrix that is too large for standard fast methods to be practical.
	dbpubs.stanford.edu:8090 /pub/2003-16 (474 words)

	Edge-preserving filtering
		I experienced that extrapolation by edge-preserving filtering is able to reconstruct the original image perfectly in the case of simple phantoms.
		Next, The Gaussian filter method is applied to extrapolate the truncated 2D real-world data set of Fig.
		The final image shows that the extrapolated data have reduced the ringing artefacts significantly but high frequency information on fine image details could not be fully retrieved.
	dutnsic.tn.tudelft.nl:8080 /main/node12.html (718 words)

	An Optimal Statistical Model for Extrapolation of the Maryland Child Support Guidelines by Hans Dutt
		The objective of this analysis is not theoretical academic work, but to provide extrapolation of the child support tables which would likely reflect the intent of the legislators when developing the guidelines.
		The most important contribution of this work is not the fact that extrapolation tables are provided, but extrapolation tables are provided that are objective and whose accuracy is quantifiable.
		Statistical methods make assumptions about the randomness of the data and can therefore make statements regarding how reliable a given estimate is. Given the assumptions, non linear statistical methods generally approximate reliability while linear statistical methods can frequently employ exact tests of reliability.
	www.deltabravo.net /custody/dutt-study2.php (1569 words)

	[No title]
		The basic algorithmic variants are Richardson extrapolation, truncation error extrapolation and extrapolation of the functional.
		We are particularly interested in the special case of (variational) multigrid methods, based on a nested system of subspaces and the multiplicative combination of the subspace corrections, or the corresponding additive methods that lead to efficient preconditioners.
		We describe higher order techniques, like extrapolation and sparse grid combination that are particularly interesting in the context of multilevel algorithms, because they are based on discretizing the problems on grids with different mesh sizes.
	www.netlib.org /bibnet/authors/r/ruede-ulrich.bib (1066 words)

	Monthly Labor Review: How price indexes affect BLS productivity measures - Bureau of Labor Statistics
		BEA uses extrapolation methods to calculate real trends for approximately 6 percent of output for the business sector, and for about 14 percent of GDP.
		For the remaining components of output that are extrapolated, physical quantity data are used as the quantity indicators.
		Finally, the direct-valuation method is used to calculate quantity indexes by multiplying the base-year price by the actual quantity data for the index period; the resulting series is then expressed as an index with the base year equal to 100.
	www.findarticles.com /p/articles/mi_m1153/is_2_122/ai_54552262/pg_2 (1085 words)

	[No title]
		Extrapolation is used to determine the behavior for a channel before the first or after the last keyframe.
		Extrapolation is useful when you want to create a cyclical effect, such as a shutter flicker (as you would find in an old-time movie clip).
		Cycles the animation as it appears between the first and last keyframe, similarly to Loop extrapolation, but with a relative offset based on the channel value at the last keyframe.
	web.tiscali.it /olgab/09_Keyanim31.html (309 words)

	Extrapolation Methods for Accelerating PageRank Computations
		In Quadratic Extrapolation, we take advantage of the fact that the first eigenvalue of a Markov matrix is known to be 1 to compute estimates of the nonprincipal eigenvectors using successive iterates of the Power Method.
		The Power Method is the oldest method for computing the principal eigenvector of a matrix, and is at the heart of both the motivation and implementation of the original PageRank algorithm (in conjunction with Algorithm 1).
		The Aitken and Epsilon Extrapolation methods take advantage of the fact that the first eigenvalue of the Markov hyperlink matrix is 1 to find an approximation to the principal eigenvector.
	www2003.org /cdrom/papers/refereed/p270/kamvar-270-xhtml (5040 words)

	Precision in the extrapolation of creep rupture data
		To be able to determine the accuracy in the extrapolation of creep rupture data, expressions for estimating the error are presented.
		Comparison to previously performed extrapolation analyses shows that the actual variation between extrapolation methods lies in general between the least square error and the upper limit estimate.
		It is proposed that design values derived from extrapolated creep rupture data should be reduced, with an additional data factor if the estimated error in the extrapolation is larger than what is typical in the materials codes and standards.
	www.astm.org /JOURNALS/TESTEVAL/PAGES/436.htm (230 words)

	Tech Report: HPL-2000-37: Extensions: Extrapolation Methods for
		This means that one has to extrapolate the underlying curves or surfaces.
		This operation causes instability since extrapolation is inherently an unstable process.
		I present the methods currently used, and illustrate their advantages and disadvantages.
	www.hpl.hp.com /techreports/2000/HPL-2000-37.html (296 words)

	[No title]
		Computational aspects of Richardson's extrapolation methods implementation and nonlinear iterations are investigated.
		Section 5 describes the multigrid algorithms with Richardson's extrapolation that allow us to obtain the solution of higher-order accuracy on the basis of simple piecewise linear approximations.
		Method 2 is an improved variant of upper relaxation algorithm where the initial approximation is taken from the coarser grid and the relaxations on the finer grid perform until the level of truncation error is reached.
	www.mgnet.org /mgnet/Conferences/ParMGM98/Papers/trofimov.doc (1685 words)

	Extrapolation methods
		As a consequence there is not one standard reconstruction method but a large collection of proposed methods that make use of some kind of general image and/or data properties.
		A major problem of the methods in the second category is the determination and next the quantification of the available prior knowledge.
		The maximum entropy method [15, 16] for instance quantifies the entropy of an image and maximizes it.
	dutnsic.tn.tudelft.nl:8080 /main/node11.html (755 words)

	SEP-114 -- TABLE OF CONTENTS
		We obtain one-way wavefield extrapolation methods which are not dip-limited, and which can even be used to image overturning waves.
		Examples of wavefields successfully extrapolated in phase-ray coordinates are presented, and the merits and drawbacks of this approach, relative to conventionally traced ray coordinates, are discussed.
		However, the conventional residual moveout method assumes that the reflectors in the subsurface are all flat, and thus estimates curvature parameters for velocity analysis that are inaccurate for steeply dipping reflectors.
	sepwww.stanford.edu /public/docs/sep114 (2937 words)

	Publisher description for Library of Congress control number 2002024669 (Site not responding. Last check: 2007-10-29)
		This book is concerned with the coherent treatment, including the derivation, analysis, and applications, of the most useful scalar extrapolation methods.
		Its importance is rooted in the fact that the methods it discusses are geared towards problems that arise commonly in scientific and engineering disciplines.
		It differs from existing books on the subject in that it concentrates on the most powerful nonlinear methods, presents in-depth treatments of them, and shows which methods are most effective for different classes of practical nontrivial problems, and also shows how to apply these methods to obtain best results.
	www.loc.gov /catdir/description/cam022/2002024669.html (240 words)

	[No title]
		This kind of extrapolation are scientifically well accepted and established methods in risk assessment procedures in the existing chemicals program as well as for biocides and pesticides.
		These extrapolation methods are use for all types of chemicals; a recent example is glufosinate-ammonium where the a classification proposal with R48/22 is based on a one-year dog study.
		An extrapolation technique should also be used for maneb and mancozeb.
	ecb.jrc.it /classlab/14004a3-I_S_maneb-mancozeb.doc (169 words)

	Papers by U. Ruede
		In Preliminary Proceedings of the Colorado Conference on Iterative Methods, Breckenridge, Colorado, April 4-10, 1994, 1994.
		In Proceedings of the GAMM-Seminar on Multigrid Methods, Sept. 21 - 25, 1992 in Gosen, Germany, Berlin, 1993.
		Extrapolation techniques for constructing higher order finite element methods.
	www10.informatik.uni-erlangen.de /~ruede/info/refs.html (511 words)

	Practical Extrapolation Methods - Cambridge University Press
		These limits can be approximated economically and with high accuracy by applying suitable extrapolation (or convergence acceleration) methods to a small number of terms.
		This state-of-the art reference for mathematicians, scientists and engineers is concerned with the coherent treatment, including derivation, analysis, and applications, of the most useful scalar extrapolation methods.
		It differs from existing books by concentrateing on the most powerful nonlinear methods, presenting in-depth treatments of them, and showing which methods are most effective for different classes of practical nontrivial problems.
	www.cup.cam.ac.uk /catalogue/catalogue.asp?isbn=0521661595 (408 words)

	Squared Extrapolation Methods (SQUAREM): A New Class of Simple and Efficient Numerical Schemes for Accelerating the ...
		We derive a new class of iterative schemes for accelerating the convergence of the EM algorithm, by exploiting the connection between fixed point iterations and extrapolation methods.
		First, we present a general formulation of one-step iterative schemes, which are obtained by cycling with the extrapolation methods.
		Roland, "Squared Extrapolation Methods (SQUAREM): A New Class of Simple and Efficient Numerical Schemes for Accelerating the Convergence of the EM Algorithm" (November 2004).
	www.bepress.com /jhubiostat/paper63 (486 words)

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