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About 10,000 papers for “Approximation error”

Approximation error.

Wen Zou (389375), Qingbin Zhang (557437), Qingyu Gao (1446913) et al. · Figshare · 2018

Cited by 0Open access

Approximation error introduces estimation error.

Fabian Fröhlich (848026), Philipp Thomas (319681), Atefeh Kazeroonian (848025) et al. · Figshare · 2016

<p>(a) Mean monomer concentration in the trimerization process for Ω = 6<i>μ</i>m<sup>3</sup> computed from 10<sup>5</sup> SSA trajectory realizations (black line). Approximate mean monomer concentrations obtained using RRE, EMRE and 2MA (colored lines). (b) Mean monomer concentration for RRE, EMRE and 2MA obtained after parameter estimation using the SSA mean as artificial dataset. (c) True (black ×) and optimized parameter values (colored ×) for RRE, EMRE and 2MA. Contour lines of objective function are colored. The opacity increases with increasing likelihood values.</p>

Cited by 0Open access

Approximation Error Maps

Anamaria Gomide, Anamaria Gomide, Jorge Stolfi · Defense Technical Information Center (DTIC) · 2001

In order to analyze the accuracy of a fixed, finite-dimensional approximation space which is not uniform over its domain Omega, we define approximation error map, a description of how the error is distributed over Omega-not for a single test function but for a general class of such functions. We show how to compute such a map from the best approximations to an orthonormal basis of the target function space.

Cited by 0

Type I error rate approximation.

Moritz Fabian Danzer (12618814), Jannik Feld (11563841), Andreas Faldum (5482229) et al. · Figshare · 2022

<p>Type I error rate approximation given by as a function of the allocation ratio <i>π</i> for different durations <i>f</i><sub><i>B</i></sub> ∈ {1, 2, 3} of the follow-up period in the new trial. Calculations were done for exponentially distributed survival times with a 1 year survival rate of 50%. Accrual <i>a</i> for the historic control and new treatment groups was set to 5 years, follow-up <i>f</i><sub><i>A</i></sub> of the historic trial was set to 3 years. To satisfy the conditions of Theorem 1 (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0271094#pone.0271094.s003" target="_blank">S1 Appendix</a>), we choose <i>s</i><sub>max</sub> = <i>a</i> + <i>f</i><sub><i>B</i></sub>−10<sup>−8</sup>.</p>

Cited by 0Open access

Mean Square Error Approximation for

Marc Antonini, Hal Id Hal, Wavelet-based Semiregular Mesh et al. · 2008

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Cited by 0

Approximation error and network size.

Christian Donner (3663007), Klaus Obermayer (53675), Hideaki Shimazaki (330023) · Figshare · 2017

<p>Error analysis on networks consisting of subpopulations with 10 neurons, constructed by the same procedure as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005309#pcbi.1005309.g001" target="_blank">Fig 1</a>. <b>A</b>: The average value of the macroscopic properties over time as a function of network size. Black line is the true value, while colored lines show the estimated ones (solid line fit with 500 trials and dashed with 1000 trials) <b>B</b>: The corresponding errors (only for <b><i>θ</i></b><sub><i>t</i></sub> the RMSE is shown) for 500 trials (solid) and 1000 trials (dashed).</p>

Cited by 0Open access

Sample Complexity versus Approximation Error

Peter L. Bartlett, Robert C. Williamson · 2007

We consider the problem of learning an unknown real-valued function from a sequence of values of the function at randomly chosen points when the estimates are constrained to some class of functions called the hypothesis class. Ideally, thehypothesis class should be able to approximate a wide variety of target functions, yet not need an excessive number of examples to ensure that a learning algorithm can choose a near-optimal approximation to the target function. We show that these two objectives are incompatible, in the sense that as the approximation error of the hypothesis class decreases, a lower bound on the sample complexity must increase. The approximation error is the largest distance between an element of the class of candidate target functions and its best approximation in the hypothesis class. The sample complexity isthenumber of examples necessary for a nearoptimal estimate of any target function. The relationship between approximation error and sample complexity depends on a combinatorial dimension of the class of candidate target functions which are to be approximated. We give an example of this relationship for the

Cited by 0

Error Bounds in Constrained Best Approximation

Morgan A Hanson · Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis · 1965

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 04 January 1965Published online: 14 July 2006Publication DataISSN (print): 0887-459XISSN (online): 2168-3581Publisher: Society for Industrial and Applied MathematicsCODEN: sjnaam.1

Cited by 1

Approximation with Error Bounds in Spark

Guangyan Hu, Desheng Zhang, Sandro José Rigo et al. · arXiv (Cornell University) · 2018

We introduce a sampling framework to support approximate computing with estimated error bounds in Spark. Our framework allows sampling to be performed at the beginning of a sequence of multiple transformations ending in an aggregation operation. The framework constructs a data provenance tree as the computation proceeds, then combines the tree with multi-stage sampling and population estimation theories to compute error bounds for the aggregation. When information about output keys are available early, the framework can also use adaptive stratified reservoir sampling to avoid (or reduce) key losses in the final output and to achieve more consistent error bounds across popular and rare keys. Finally, the framework includes an algorithm to dynamically choose sampling rates to meet user specified constraints on the CDF of error bounds in the outputs. We have implemented a prototype of our framework called ApproxSpark, and used it to implement five approximate applications from different domains. Evaluation results show that ApproxSpark can (a) significantly reduce execution time if users can tolerate small amounts of uncertainties and, in many cases, loss of rare keys, and (b) automatically find sampling rates to meet user specified constraints on error bounds. We also explore and discuss extensively trade-offs between sampling rates, execution time, accuracy and key loss.

Cited by 2Open access

Laplace approximation in measurement error models

Michela Battauz · Biometrical Journal · 2011

Likelihood analysis for regression models with measurement errors in explanatory variables typically involves integrals that do not have a closed-form solution. In this case, numerical methods such as Gaussian quadrature are generally employed. However, when the dimension of the integral is large, these methods become computationally demanding or even unfeasible. This paper proposes the use of the Laplace approximation to deal with measurement error problems when the likelihood function involves high-dimensional integrals. The cases considered are generalized linear models with multiple covariates measured with error and generalized linear mixed models with measurement error in the covariates. The asymptotic order of the approximation and the asymptotic properties of the Laplace-based estimator for these models are derived. The method is illustrated using simulations and real-data analysis.

Cited by 3Open access

Relative Error Tensor Low Rank Approximation

Zhao Song, David P. Woodruff, Peilin Zhong · arXiv (Cornell University) · 2017

We consider relative error low rank approximation of $tensors$ with respect to the Frobenius norm: given an order-$q$ tensor $A \in \mathbb{R}^{\prod_{i=1}^q n_i}$, output a rank-$k$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+ε)$OPT, where OPT $= \inf_{\textrm{rank-}k~A'} \|A-A'\|_F^2$. Despite the success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that there may be no rank-$k$ tensor $A_k$ achieving the above infinum. Another, computational issue, is that an efficient relative error low rank approximation algorithm for tensors would allow one to compute the rank of a tensor, which is NP-hard. We bypass these issues via (1) bicriteria and (2) parameterized complexity solutions: (1) We give an algorithm which outputs a rank $k' = O((k/ε)^{q-1})$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+ε)$OPT in $nnz(A) + n \cdot \textrm{poly}(k/ε)$ time in the real RAM model. Here $nnz(A)$ is the number of non-zero entries in $A$. (2) We give an algorithm for any $δ>0$ which outputs a rank $k$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+ε)$OPT and runs in $ ( nnz(A) + n \cdot \textrm{poly}(k/ε) + \exp(k^2/ε) ) \cdot n^δ$ time in the unit cost RAM model. For outputting a rank-$k$ tensor, or even a bicriteria solution with rank-$Ck$ for a certain constant $C > 1$, we show a $2^{Ω(k^{1-o(1)})}$ time lower bound under the Exponential Time Hypothesis. Our results give the first relative error low rank approximations for tensors for a large number of robust error measures for which nothing was known, as well as column row and tube subset selection. We also obtain new results for matrices, such as $nnz(A)$-time CUR decompositions, improving previous $nnz(A)\log n$-time algorithms, which may be of independent interest.

Cited by 0Open access