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Journal articles on the topic 'Data approximation'

Author: Grafiati

Published: 7 December 2024

Last updated: 16 June 2025

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1

FROYLAND, GARY, KEVIN JUDD, ALISTAIR I. MEES, DAVID WATSON, and KENJI MURAO. "CONSTRUCTING INVARIANT MEASURES FROM DATA." International Journal of Bifurcation and Chaos 05, no. 04 (1995): 1181–92. http://dx.doi.org/10.1142/s0218127495000843.

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We present a method of approximating an invariant measure of a dynamical system from a finite set of experimental data. Our reconstruction technique automatically provides us with a partition of phase space, and we assign each set in the partition a certain weight. By refining the partition, we may make our approximation to an invariant measure of the reconstructed system as accurate as we wish. Our method provides us with both a singular and an absolutely continuous approximation, so that the most suitable representation may be chosen for a particular problem.

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Grubas, Serafim I., Georgy N. Loginov, and Anton A. Duchkov. "Traveltime-table compression using artificial neural networks for Kirchhoff-migration processing of microseismic data." GEOPHYSICS 85, no. 5 (2020): U121—U128. http://dx.doi.org/10.1190/geo2019-0427.1.

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Massive computation of seismic traveltimes is widely used in seismic processing, for example, for the Kirchhoff migration of seismic and microseismic data. Implementation of the Kirchhoff migration operators uses large precomputed traveltime tables (for all sources, receivers, and densely sampled imaging points). We have tested the idea of using artificial neural networks for approximating these traveltime tables. The neural network has to be trained for each velocity model, but then the whole traveltime table can be compressed by several orders of magnitude (up to six orders) to the size of less than 1 MB. This makes it convenient to store, share, and use such approximations for processing large data volumes. We evaluate some aspects of choosing neural-network architecture, training procedure, and optimal hyperparameters. On synthetic tests, we find a reasonably accurate approximation of traveltimes by neural networks for various velocity models. A final synthetic test shows that using the neural-network traveltime approximation results in good accuracy of microseismic event localization (within the grid step) in the 3D case.

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STOJANOVIĆ, MIRJANA. "PERTURBED SCHRÖDINGER EQUATION WITH SINGULAR POTENTIAL AND INITIAL DATA." Communications in Contemporary Mathematics 08, no. 04 (2006): 433–52. http://dx.doi.org/10.1142/s0219199706002180.

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We consider linear Schrödinger equation perturbed by delta distribution with singular potential and the initial data. Due to the singularities appearing in the equation, we introduce two kinds of approximations: the parameter's approximation for potential and the initial data given by mollifiers of different growth and the approximation for the Green function for Schrödinger equation with regularized derivatives. These approximations reduce the perturbed Schrödinger equation to the family of singular integral equations. We prove the existence-uniqueness theorems in Colombeau space [Formula: see text], 1 ≤ p,q ≤ ∞, employing novel stability estimates (w.r.) to singular perturbations for ε → 0, which imply the statements in the framework of Colombeau generalized functions. In particular, we prove the existence-uniqueness result in [Formula: see text] and [Formula: see text] algebra of Colombeau.

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FRAHLING, GEREON, PIOTR INDYK, and CHRISTIAN SOHLER. "SAMPLING IN DYNAMIC DATA STREAMS AND APPLICATIONS." International Journal of Computational Geometry & Applications 18, no. 01n02 (2008): 3–28. http://dx.doi.org/10.1142/s0218195908002520.

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A dynamic geometric data stream is a sequence of m ADD/REMOVE operations of points from a discrete geometric space {1,…, Δ} d ?. ADD (p) inserts a point p from {1,…, Δ} d into the current point set P , REMOVE(p) deletes p from P . We develop low-storage data structures to (i) maintain ε-nets and ε-approximations of range spaces of P with small VC-dimension and (ii) maintain a (1 + ε)-approximation of the weight of the Euclidean minimum spanning tree of P . Our data structure for ε-nets uses [Formula: see text] bits of memory and returns with probability 1 – δ a set of [Formula: see text] points that is an e-net for an arbitrary fixed finite range space with VC-dimension [Formula: see text]. Our data structure for ε-approximations uses [Formula: see text] bits of memory and returns with probability 1 – δ a set of [Formula: see text] points that is an ε-approximation for an arbitrary fixed finite range space with VC-dimension [Formula: see text]. The data structure for the approximation of the weight of a Euclidean minimum spanning tree uses O ( log (1/δ)( log Δ/ε) O ( d )) space and is correct with probability at least 1 – δ. Our results are based on a new data structure that maintains a set of elements chosen (almost) uniformly at random from P .

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Chen, Jing-Bo, Hong Liu, and Zhi-Fu Zhang. "A separable-kernel decomposition method for approximating the DSR continuation operator." GEOPHYSICS 72, no. 1 (2007): S25—S31. http://dx.doi.org/10.1190/1.2399368.

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We develop a separable-kernel decomposition method for approximating the double-square-root (DSR) continuation operator in one-way migrations in this paper. This new approach is a further development of separable approximations of the single-square-root (SSR) operator. The separable approximation of the DSR operator generally involves solving a complicated nonlinear system of integral equations. Instead of solving this nonlinear system directly, our new method consists of repeatedly applying the separable-kernel technique developed for the two-variable SSR operator to the multivariable DSR operator. Numerical experiments demonstrate the efficiency of the proposed method. We illustrate the fast convergence of the obtained separable approximation. We also demonstrate the capability of this novel approximation for imaging an area with geologic complexities through synthetic data.

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Mardia, K. V., and I. L. Dryden. "Shape distributions for landmark data." Advances in Applied Probability 21, no. 4 (1989): 742–55. http://dx.doi.org/10.2307/1427764.

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The paper obtains the exact distribution of Bookstein's shape variables under his plausible model for landmark data. We consider its properties including invariances, marginal distributions and the relationship with Kendall's uniform measure. Particular cases for triangles and quadrilaterals are highlighted. A normal approximation to the distribution is obtained, extending Bookstein's result for three landmarks. The adequacy of these approximations is also studied.

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Mardia, K. V., and I. L. Dryden. "Shape distributions for landmark data." Advances in Applied Probability 21, no. 04 (1989): 742–55. http://dx.doi.org/10.1017/s0001867800019029.

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The paper obtains the exact distribution of Bookstein's shape variables under his plausible model for landmark data. We consider its properties including invariances, marginal distributions and the relationship with Kendall's uniform measure. Particular cases for triangles and quadrilaterals are highlighted. A normal approximation to the distribution is obtained, extending Bookstein's result for three landmarks. The adequacy of these approximations is also studied.

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8

Birch, A. C., and A. G. Kosovichev. "Towards a Wave Theory Interpretation of Time-Distance Helioseismology Data." Symposium - International Astronomical Union 203 (2001): 180–82. http://dx.doi.org/10.1017/s0074180900219025.

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Time-distance helioseismology, which measures the time for acoustic waves to travel between points on the solar surface, has been used to study small-scale three-dimensional features in the sun, for example active regions, as well as large-scale features, such as meridional flow, that are not accessible by standard global helioseismology. Traditionally, travel times have been interpreted using geometrical ray theory, which is not always a good approximation. In order to develop a wave interpretation of time-distance data we employ the first Born approximation, which takes into account finite-wavelength effects and is expected to provide more accurate inversion results. In the Born approximation, in contrast with ray theory, travel times are sensitive to perturbations to sound speed which are located off the ray path. In an example calculation of travel time perturbations due to sound speed perturbations that are functions only of depth, we see that that the Born and ray approximations agree when applied to perturbations with large spatial scale and that the ray approximation fails when applied to perturbations with small spatial scale.

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Dong, Bin, Zuowei Shen, and Jianbin Yang. "Approximation from Noisy Data." SIAM Journal on Numerical Analysis 59, no. 5 (2021): 2722–45. http://dx.doi.org/10.1137/20m1389091.

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Piegl, L. A., and W. Tiller. "Data Approximation Using Biarcs." Engineering with Computers 18, no. 1 (2002): 59–65. http://dx.doi.org/10.1007/s003660200005.

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Nawar, A. S., R. Abu-Gdairi, M. K. El-Bably, and H. M. Atallah. "Enhancing Rheumatic Fever Analysis via Tritopological Approximation Spaces for Data Reduction." Malaysian Journal of Mathematical Sciences 18, no. 2 (2024): 321–41. http://dx.doi.org/10.47836/mjms.18.2.07.

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This paper introduces the concept of tritopological approximation space, extending conventional approximation space by drawing upon topological spaces and precisely defined binary relations within a universe of discourse. Through meticulous construction of subbases, this progressive paradigm shift facilitates a comprehensive analysis of rough sets within the domain of tritopological approximation spaces. Additionally, the study pioneer's multiple membership functions and inclusion functions, enhancing the analytical framework and enabling more effective redefinition of rough approximations. To illustrate the practical advantages, real-life application examples are presented, focusing on the implementation of data reduction methods within the context of rheumatic fever---a prevalent disease characterized by diverse symptoms among patients, despite a consistent diagnosis. This research contributes to the advancement of rough set theory and its applications in addressing complex, real-world problems.

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Ullah, Insha, Sudhir Paul, Zhenjie Hong, and You-Gan Wang. "Significance tests for analyzing gene expression data with small sample sizes." Bioinformatics 35, no. 20 (2019): 3996–4003. http://dx.doi.org/10.1093/bioinformatics/btz189.

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Abstract Motivation Under two biologically different conditions, we are often interested in identifying differentially expressed genes. It is usually the case that the assumption of equal variances on the two groups is violated for many genes where a large number of them are required to be filtered or ranked. In these cases, exact tests are unavailable and the Welch’s approximate test is most reliable one. The Welch’s test involves two layers of approximations: approximating the distribution of the statistic by a t-distribution, which in turn depends on approximate degrees of freedom. This study attempts to improve upon Welch’s approximate test by avoiding one layer of approximation. Results We introduce a new distribution that generalizes the t-distribution and propose a Monte Carlo based test that uses only one layer of approximation for statistical inferences. Experimental results based on extensive simulation studies show that the Monte Carol based tests enhance the statistical power and performs better than Welch’s t-approximation, especially when the equal variance assumption is not met and the sample size of the sample with a larger variance is smaller. We analyzed two gene-expression datasets, namely the childhood acute lymphoblastic leukemia gene-expression dataset with 22 283 genes and Golden Spike dataset produced by a controlled experiment with 13 966 genes. The new test identified additional genes of interest in both datasets. Some of these genes have been proven to play important roles in medical literature. Availability and implementation R scripts and the R package mcBFtest is available in CRAN and to reproduce all reported results are available at the GitHub repository, https://github.com/iullah1980/MCTcodes. Supplementary information Supplementary data is available at Bioinformatics online.

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Chervov, Alexander, Jonathan Bac, and Andrei Zinovyev. "Minimum Spanning vs. Principal Trees for Structured Approximations of Multi-Dimensional Datasets." Entropy 22, no. 11 (2020): 1274. http://dx.doi.org/10.3390/e22111274.

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Construction of graph-based approximations for multi-dimensional data point clouds is widely used in a variety of areas. Notable examples of applications of such approximators are cellular trajectory inference in single-cell data analysis, analysis of clinical trajectories from synchronic datasets, and skeletonization of images. Several methods have been proposed to construct such approximating graphs, with some based on computation of minimum spanning trees and some based on principal graphs generalizing principal curves. In this article we propose a methodology to compare and benchmark these two graph-based data approximation approaches, as well as to define their hyperparameters. The main idea is to avoid comparing graphs directly, but at first to induce clustering of the data point cloud from the graph approximation and, secondly, to use well-established methods to compare and score the data cloud partitioning induced by the graphs. In particular, mutual information-based approaches prove to be useful in this context. The induced clustering is based on decomposing a graph into non-branching segments, and then clustering the data point cloud by the nearest segment. Such a method allows efficient comparison of graph-based data approximations of arbitrary topology and complexity. The method is implemented in Python using the standard scikit-learn library which provides high speed and efficiency. As a demonstration of the methodology we analyse and compare graph-based data approximation methods using synthetic as well as real-life single cell datasets.

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Abd El-Raheem, Abd El-Raheem M., and Mona Hosny. "Saddlepoint p-values for a class of nonparametric tests for the current status and panel count data under generalized permuted block design." AIMS Mathematics 8, no. 8 (2023): 18866–80. http://dx.doi.org/10.3934/math.2023960.

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<abstract><p>Current status and panel count data appear in many applied fields, including medicine, clinical trials, epidemiology, econometrics, demography, engineering and public health. Therefore, in this article, we use the saddlepoint approximation method to approximate the exact p-value of a number of nonparametric tests for the current status and panel count data under a generalized permuted block design. The saddlepoint approximation is referred to as higher-order approximation and it is more accurate than the methods that lead to approximations that are accurate to the first order, such as the asymptotic normal approximation method. To verify the accuracy and efficiency of the saddlepoint approximation method, a simulation study is conducted. The simulation study results confirm that the saddlepoint approximation method is more powerful than the existing approximation method. Furthermore, number of real current status and panel count data sets are analyzed and displayed as illustrative examples.</p></abstract>

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15

Siswantining, Titin, Muhammad Ihsan, Saskya Mary Soemartojo, Devvi Sarwinda, Herley Shaori Al-Ash, and Ika Marta Sari. "MULTIPLE IMPUTATION FOR ORDINARY COUNT DATA BY NORMAL DISTRIBUTION APPROXIMATION." MEDIA STATISTIKA 14, no. 1 (2021): 68–78. http://dx.doi.org/10.14710/medstat.14.1.68-78.

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Missing values are a problem that is often encountered in various fields and must be addressed to obtain good statistical inference such as parameter estimation. Missing values can be found in any type of data, included count data that has Poisson distributed. One solution to overcome that problem is applying multiple imputation techniques. The multiple imputation technique for the case of count data consists of three main stages, namely the imputation, the analysis, and pooling parameter. The use of the normal distribution refers to the sampling distribution using the central limit theorem for discrete distributions. This study is also equipped with numerical simulations which aim to compare accuracy based on the resulting bias value. Based on the study, the solutions proposed to overcome the missing values in the count data yield satisfactory results. This is indicated by the size of the bias parameter estimate is small. But the bias value tends to increase with increasing percentage of observation of missing values and when the parameter values are small.

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Bochkov, A. P., V. A. Prourzin, and O. V. Prourzin. "BIG DATA ANALYSIS OF RELIABILITY OF NON-RESTORABLE MULTICHANNEL SYSTEMS." H&ES Research 13, no. 4 (2021): 49–55. http://dx.doi.org/10.36724/2409-5419-2021-13-4-49-55.

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Introduction: methods for analyzing big data of reliability of multichannel systems with a loaded reserve with nonrecoverable elements are considered. Big data contains information on the operating time to failure of elements, obtained by monitoring the operation of similar systems. The main problem that arises when analyzing big data is related to its variety and veracity. Reliability data of system elements correspond to different operating conditions and different laws of failure distribution. The exponential approximation of failure distributions greatly simplifies reliability analysis. However, it can lead to significant errors and requires a separate justification. Purpose: development of approaches to the analysis of heterogeneous big data of reliability of system elements characterized by different distributions of failures. Derivation of estimates of the accuracy of approximation of the distribution of failures by exponential laws and criteria for the possibility of such an approximation. Results: methods for calculating the mean time to failure of systems with a monotonic structure are described. An estimate of the error of exponential approximation of the distributions of failures of elements of a multichannel system is obtained. The relationship between the error of exponential approximation and the coefficient of variation of non-exponential distribution of failures is shown. The case of two channel systems is investigated in detail. For uniform, lognormal, gamma and Weibull distributions, the dependences of the average operating time to rejection of the coefficient of variation are plotted. Areas of variation of the coefficient of variation of these distributions are constructed, for which exponential approximations are justified. An algorithm for constructing the mean time to failure in the analysis of large reliability data of a non-recoverable two-channel system is presented. Practical relevance: analysis of reliability data obtained from monitoring the operation of similar systems eliminates costly reliability tests. The relationship between the sample coefficient of variation and the error of exponential approximation of failure distributions in the analysis of big data of system reliability is shown. This connection forms the basis of the criterion for the possibility of such an approximation.

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Sil, Samik. "Fracture parameter estimation from well-log data." GEOPHYSICS 78, no. 3 (2013): D129—D134. http://dx.doi.org/10.1190/geo2012-0407.1.

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We evaluated a method of deriving seismic fracture parameters from vertical-well-log data with the assumption that the fractured medium is transversely isotropic with a horizontal axis of symmetry (HTI). One approximation we used is that the observed vertical P-wave velocity is the same as the background isotropic P-wave velocity of the HTI medium. Another assumption was that the fractures and cracks are noninteractive and penny shaped. Using these approximations, we generated the fracture compliance matrix for each layer. Fracture parameters were then derived by constructing the HTI stiffness matrix for those layers. We tested our method using vertical-well-log data from a tight sand reservoir in Colorado, USA. “Thomsen-style” parameters were derived, and gas-filled fractures were identified on this log. The identified gas-filled fractures were compared to the production log data. The fracture density was also obtained at the well location within the depth of interest. We also found some problems and limitations caused by approximating vertical P-wave velocity the same as the background isotropic P-wave velocity.

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Pratiwi, Indah Nur, Mohammad Syamsu Rosid, and Humbang Purba. "Reducing Residual Moveout for Long Offset Data in VTI Media Using Padé Approximation." E3S Web of Conferences 125 (2019): 15005. http://dx.doi.org/10.1051/e3sconf/201912515005.

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Modification of the hyperbolic travel time equation into non-hyperbolic travel time equation is important to increase the reduction residual moveout for long offset data. Some researchers have modified hyperbolic travel time equation into a non-hyperbolic travel time equation to obtain a more accurate value NMO velocity and parameter an-ellipticity or etha on the large offset to depth ratio (ODR) so that the residual moveout value is smaller mainly in large offset to depth ratio. The aims of research is to increase the reduction value of error residue at long offset data using Padé approximation then compare with several approximations. The method used in this study is to conduct forward modeling of the subsurface coating structure. The results of the three-dimensional analysis show that the Padé approximation has the best accuracy compared to the other travel time equations for ODR value up to 4 with an-ellipticity parameter is varying from 0 to 0.5. Testing of synthetic data for single layer on vertical transverse isotropy (VTI) medium obtained the maximum residual error value produced by the Padé approximation is 0.25% in ODR=4. Therefore, Padé approximation is better than other methods for reducing residual moveout.

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Jauch, Jens, Felix Bleimund, Michael Frey, and Frank Gauterin. "An Iterative Method Based on the Marginalized Particle Filter for Nonlinear B-Spline Data Approximation and Trajectory Optimization." Mathematics 7, no. 4 (2019): 355. http://dx.doi.org/10.3390/math7040355.

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The B-spline function representation is commonly used for data approximation and trajectory definition, but filter-based methods for NWLS approximation are restricted to a bounded definition range. We present an algorithm termed NRBA for an iterative NWLS approximation of an unbounded set of data points by a B-spline function. NRBA is based on a MPF, in which a KF solves the linear subproblem optimally while a PF deals with nonlinear approximation goals. NRBA can adjust the bounded definition range of the approximating B-spline function during run-time such that, regardless of the initially chosen definition range, all data points can be processed. In numerical experiments, NRBA achieves approximation results close to those of the Levenberg–Marquardt algorithm. An NWLS approximation problem is a nonlinear optimization problem. The direct trajectory optimization approach also leads to a nonlinear problem. The computational effort of most solution methods grows exponentially with the trajectory length. We demonstrate how NRBA can be applied for a multiobjective trajectory optimization for a BEV in order to determine an energy-efficient velocity trajectory. With NRBA, the effort increases only linearly with the processed data points and the trajectory length.

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Li, Renpu. "Set Approximation in Incomplete Data." Journal of Applied Sciences 13, no. 9 (2013): 1621–28. http://dx.doi.org/10.3923/jas.2013.1621.1628.

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Piegl, L. A., and W. Tiller. "Surface approximation to scanned data." Visual Computer 16, no. 7 (2000): 386–95. http://dx.doi.org/10.1007/pl00013393.

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Can, Emine. "Piecewise Cubic Approximation for Data." American Journal of Applied Mathematics 1, no. 2 (2013): 24. http://dx.doi.org/10.11648/j.ajam.20130102.11.

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Grohs, Philipp, Markus Sprecher, and Thomas Yu. "Scattered manifold-valued data approximation." Numerische Mathematik 135, no. 4 (2016): 987–1010. http://dx.doi.org/10.1007/s00211-016-0823-0.

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Gorban, A. N., N. R. Sumner, and A. Y. Zinovyev. "Topological grammars for data approximation." Applied Mathematics Letters 20, no. 4 (2007): 382–86. http://dx.doi.org/10.1016/j.aml.2006.04.022.

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Hřebíček, Jiří, Jan Kučera, Petr Švenda, and Vladimír A. Vasilenko. "IDA — interactive data approximation package." Computer Physics Communications 61, no. 1-2 (1990): 231–33. http://dx.doi.org/10.1016/0010-4655(90)90121-g.

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Wille, Emilio. "APPROXIMATING PROBABILITY DISTRIBUTION FUNCTIONS WITH FEW MOMENTS." Latin American Applied Research - An international journal 50, no. 1 (2019): 21–25. http://dx.doi.org/10.52292/j.laar.2020.132.

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A procedure is presented for approximating a given probability distribution function or statistical data considering a subset of their moments.This is done by a method of fitting moments of a piecewise linear functionto the moments of the known data. The approach has many advantages over popular approximation approaches. The procedure is demonstrated with commonly used cdfs (Exponential, Gamma, Log-Normal, Normal) andmore difficult problems involving sum and product of random variables,obtaining good agreement between the theoretical/simulation curves and the piecewise linear approximations.

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Ma, Shuai, and Jinpeng Huai. "Approximate computation for big data analytics." ACM SIGWEB Newsletter, Winter (January 2021): 1–8. http://dx.doi.org/10.1145/3447879.3447883.

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Over the past a few years, research and development has made significant progresses on big data analytics. A fundamental issue for big data analytics is the efficiency. If the optimal solution is unable to attain or unnecessary or has a price to high to pay, it is reasonable to sacrifice optimality with a "good" feasible solution that can be computed efficiently. Existing approximation techniques can be in general classified into approximation algorithms, approximate query processing for aggregate SQL queries and approximation computing for multiple layers of the system stack. In this article, we systematically introduce approximate computation, i.e. , query approximation and data approximation, for efficient and effective big data analytics. We explain the ideas and rationales behind query and data approximation, and show efficiency can be obtained with high effectiveness, and even without sacrificing for effectiveness, for certain data analytic tasks.

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Narayanan, Swathi J., Ilango Paramasivam, Rajen B. Bhatt, and M. Khalid. "A Study on the Approximation of Clustered Data to Parameterized Family of Fuzzy Membership Functions for the Induction of Fuzzy Decision Trees." Cybernetics and Information Technologies 15, no. 2 (2015): 75–96. http://dx.doi.org/10.1515/cait-2015-0030.

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Abstract This paper investigates the Triangular, Trapezoidal and Gaussian approximation methods for the purpose of induction of Fuzzy Decision Trees (FDT). The generation of FDT is done using a Fuzzy ID3 induction algorithm. In this work three fuzzy partitioning techniques which form the basis for our investigation are given attention, namely Fuzzy C Means clustering (FCM), Grid partitioning and Subtractive clustering (Subclust). Our contribution lies in studying the effect of various approximations on the generation of FDT giving specific attention to the classification accuracy of FDT. In this study we show that the accuracy levels of FDT generated using FCM clustered raw data, bypassing the approximation step, is acceptable and this method has several advantages too. Several computational experiments are conducted and non parametric statistical tests are performed to find if any significant differences exist between the method of bypassing the approximation step and the other methods which include approximation. Ten FDTs are developed and used in this study. These FDT’s differ in their fuzzy partitioning techniques and the approximation methods used.

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Borovsky, Andrei Viktorovich, Andrey Leonidovich Galkin, and Svetlana Sergeevna Kozlova. "Mathematical modeling of statistical data on the incidence of new coronavirus infection, taking into account the stratification by concomitant diagnoses." Vestnik of Astrakhan State Technical University. Series: Management, computer science and informatics 2024, no. 3 (2024): 95–106. http://dx.doi.org/10.24143/2072-9502-2024-3-95-106.

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The article considers the stratification of concomitant diagnoses of Covid-19 recovery statistics for the city of Irkutsk for 2020-2021. The previous study was conducted without taking into account such stratification. Various options for approximating real statistics by Gaussian and Lorentz functions, gamma distribution, and Johnson curves are considered. It is shown that the stratification of recovery statistics improves the approximation of Gaussian and Lorentz functions in comparison with integral statistics, and the construction of an approximation based on the Lorentz function always describes the real statistics better. Estimates of mathematical expectation and variance based on statistical data are consistent with estimates of these values based on the Gaussian approximation of statistics by the least squares method, i.e. the approaches are equivalent. At the same time, calculations of the Pearson Chi-squared criterion reject the hypothesis that empirical data correspond to the assumed theoretical distribution. Therefore, we cannot talk about finding the distribution function, but only about approximating statistics by certain types of curves. The fitting of empirical data by Gaussian and Lorentz functions was carried out using the least squares method. In general, the approximation error due to the stratification of statistics on concomitant diagnoses decreases from 6% to 3%.

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Wang, Ziming, Chang Liu, Junjie Zhao, and Lijing Shao. "Extending the Fisher Information Matrix in Gravitational-wave Data Analysis." Astrophysical Journal 932, no. 2 (2022): 102. http://dx.doi.org/10.3847/1538-4357/ac6b99.

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Abstract The Fisher information matrix (FM) plays an important role in forecasts and inferences in many areas of physics. While giving fast parameter estimation with Gaussian likelihood approximation in the parameter space, the FM can only give the ellipsoidal posterior contours of the parameters and it loses the higher-order information beyond Gaussianity. We extend the FM in gravitational-wave (GW) data analysis by using the Derivative Approximation for LIkelihoods (DALI), a method to expand the likelihood, while keeping it positive definite and normalizable at every order, for more accurate forecasts and inferences. When applied to two real GW events, GW150914 and GW170817, DALI can reduce the difference between the FM approximation and the real posterior by 5 times in the best case. The calculation times of DALI and the FM are at the same order of magnitude, while obtaining the real full posterior will take several orders of magnitude longer. Besides more accurate approximations, higher-order correction from DALI provides a fast assessment of the FM analysis and gives suggestions for complex sampling techniques that are computationally intensive. We recommend using the DALI method as an extension to the FM method in GW data analysis to pursue better accuracy while still keeping the speed.

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DeVore, Ronald, Boris Hanin, and Guergana Petrova. "Neural network approximation." Acta Numerica 30 (May 2021): 327–444. http://dx.doi.org/10.1017/s0962492921000052.

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Neural networks (NNs) are the method of choice for building learning algorithms. They are now being investigated for other numerical tasks such as solving high-dimensional partial differential equations. Their popularity stems from their empirical success on several challenging learning problems (computer chess/Go, autonomous navigation, face recognition). However, most scholars agree that a convincing theoretical explanation for this success is still lacking. Since these applications revolve around approximating an unknown function from data observations, part of the answer must involve the ability of NNs to produce accurate approximations.This article surveys the known approximation properties of the outputs of NNs with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis, such as approximations using polynomials, wavelets, rational functions and splines. Comparisons are made with traditional approximation methods from the viewpoint of rate distortion, i.e. error versus the number of parameters used to create the approximant. Another major component in the analysis of numerical approximation is the computational time needed to construct the approximation, and this in turn is intimately connected with the stability of the approximation algorithm. So the stability of numerical approximation using NNs is a large part of the analysis put forward.The survey, for the most part, is concerned with NNs using the popular ReLU activation function. In this case the outputs of the NNs are piecewise linear functions on rather complicated partitions of the domain of f into cells that are convex polytopes. When the architecture of the NN is fixed and the parameters are allowed to vary, the set of output functions of the NN is a parametrized nonlinear manifold. It is shown that this manifold has certain space-filling properties leading to an increased ability to approximate (better rate distortion) but at the expense of numerical stability. The space filling creates the challenge to the numerical method of finding best or good parameter choices when trying to approximate.

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Skala, Vaclav, and Eliska Mourycova. "Meshfree Interpolation of Multidimensional Time-Varying Scattered Data." Computers 12, no. 12 (2023): 243. http://dx.doi.org/10.3390/computers12120243.

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Interpolating and approximating scattered scalar and vector data is fundamental in resolving numerous engineering challenges. These methodologies predominantly rely on establishing a triangulated structure within the data domain, typically constrained to the dimensions of 2D or 3D. Subsequently, an interpolation or approximation technique is employed to yield a smooth and coherent outcome. This contribution introduces a meshless methodology founded upon radial basis functions (RBFs). This approach exhibits a nearly dimensionless character, facilitating the interpolation of data evolving over time. Specifically, it enables the interpolation of dispersed spatio-temporally varying data, allowing for interpolation within the space-time domain devoid of the conventional “time-frames”. Meshless methodologies tailored for scattered spatio-temporal data hold applicability across a spectrum of domains, encompassing the interpolation, approximation, and assessment of data originating from various sources, such as buoys, sensor networks, tsunami monitoring instruments, chemical and radiation detectors, vessel and submarine detection systems, weather forecasting models, as well as the compression and visualization of 3D vector fields, among others.

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Zhang, Jiani, Jennifer Erway, Xiaofei Hu, Qiang Zhang, and Robert Plemmons. "Randomized SVD Methods in Hyperspectral Imaging." Journal of Electrical and Computer Engineering 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/409357.

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We present a randomized singular value decomposition (rSVD) method for the purposes of lossless compression, reconstruction, classification, and target detection with hyperspectral (HSI) data. Recent work in low-rank matrix approximations obtained from random projections suggests that these approximations are well suited for randomized dimensionality reduction. Approximation errors for the rSVD are evaluated on HSI, and comparisons are made to deterministic techniques and as well as to other randomized low-rank matrix approximation methods involving compressive principal component analysis. Numerical tests on real HSI data suggest that the method is promising and is particularly effective for HSI data interrogation.

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Lin, Zi Zhi, and Si Hui Shu. "B-Spline Surface Approximation to Scanned Data Using Least Square Approximation." Applied Mechanics and Materials 571-572 (June 2014): 711–16. http://dx.doi.org/10.4028/www.scientific.net/amm.571-572.711.

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Lofting is widely used to approximate the scanned data in row-wise fashion, but this method is prone to result an astonishing number of control points in the process of making the rows curve compatible. A novel algorithm of B-spline surface approximation to the scanned data is presented in this paper to solve this problem. Firstly, the scanned data are interpolated by rows of curves; then these curves are approximated by other curves using least square approximation. In this process, all curves are approximated by a common knot vector, and it is different form the traditional method that each curve is approximated by a different knot vector, so we needn’t insert many knots in each curve to make curves compatible. We also can meet high accuracy without losing the shape of lofting surface because we firstly interpolate the data, the best least square approximation substitute insertion of knots in lofting. Numerical example shows that the proposed method is efficient in reducing control points of the lofting surface.

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Marshalko, Grigory, and Julia Trufanova. "Polynomial Approximations for Several Neural Network Activation Functions." Informatics and Automation 21, no. 1 (2021): 161–80. http://dx.doi.org/10.15622/ia.2022.21.6.

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Active deployment of machine learning systems sets a task of their protection against different types of attacks that threaten confidentiality, integrity and accessibility of both processed data and trained models. One of the promising ways for such protection is the development of privacy-preserving machine learning systems, that use homomorphic encryption schemes to protect data and models. However, such schemes can only process polynomial functions, which means that we need to construct polynomial approximations for nonlinear functions used in neural models. The goal of this paper is the construction of precise approximations of several widely used neural network activation functions while limiting the degree of approximation polynomials as well as the evaluation of the impact of the approximation precision on the resulting value of the whole neural network. In contrast to the previous publications, in the current paper we study and compare different ways for polynomial approximation construction, introduce precision metrics, present exact formulas for approximation polynomials as well as exact values of corresponding precisions. We compare our results with the previously published ones. Finally, for a simple convolutional network we experimentally evaluate the impact of the approximation precision on the bias of the output neuron values of the network from the original ones. Our results show that the best approximation for ReLU could be obtained with the numeric method, and for the sigmoid and hyperbolic tangent – with Chebyshev polynomials. At the same time, the best approximation among the three functions could be obtained for ReLU. The results could be used for the construction of polynomial approximations of activation functions in privacy-preserving machine learning systems.

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Hunt, P. J. "Asymptotic Behaviour of an Integrated Video-Data Network." Probability in the Engineering and Informational Sciences 5, no. 4 (1991): 429–47. http://dx.doi.org/10.1017/s0269964800002217.

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We consider a communication network that can support both wideband video calls and narrowband data traffic. First we consider a single link and prove a weak convergence result to justify a piecewise-deterministic Markov process approximation to the system. We then generalize this approximation to allow priorities and more than one link. This second approximation is a generalization of the Erlang fixed-point approximation for loss networks and is justified via a diverse routing limit theorem.

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Zhang, Xiao Lei, and Jin Ming Wu. "Positive Approximation for Positive Scattered Data." Applied Mechanics and Materials 50-51 (February 2011): 683–87. http://dx.doi.org/10.4028/www.scientific.net/amm.50-51.683.

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The curve and surface fitting problem is very important in CAD and CAGD. However, it is important to construct a suitable function to interpolate or approximate which satisfies the underlying constraints since we have some additional information that is confined to interpolation or approximation. In this paper, we discuss the positive approximation for positive scattered data of any dimensionality by using radial basis functions. The approach is presented to compute positive approximation by solving a quadratic optimization problem. Numerical experiments are provided to illustrate the proposed algorithm is flexible.

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Potts, Daniel, and Michael Schmischke. "Interpretable Approximation of High-Dimensional Data." SIAM Journal on Mathematics of Data Science 3, no. 4 (2021): 1301–23. http://dx.doi.org/10.1137/21m1407707.

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Markovsky, Ivan, and Florian Dörfler. "Data-driven dynamic interpolation and approximation." Automatica 135 (January 2022): 110008. http://dx.doi.org/10.1016/j.automatica.2021.110008.

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Das, A., J. Gehrke, and M. Riedewald. "Semantic approximation of data stream joins." IEEE Transactions on Knowledge and Data Engineering 17, no. 1 (2005): 44–59. http://dx.doi.org/10.1109/tkde.2005.17.

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Lazzeretti, Riccardo, Tommaso Pignata, and Mauro Barni. "Piecewise Function Approximation With Private Data." IEEE Transactions on Information Forensics and Security 11, no. 3 (2016): 642–57. http://dx.doi.org/10.1109/tifs.2015.2503268.

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Heinis, Thomas. "Approximation aids handling of big data." Nature 515, no. 7526 (2014): 198. http://dx.doi.org/10.1038/515198d.

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FLOATER, MICHAEL S. "PARAMETRIC TILINGS AND SCATTERED DATA APPROXIMATION." International Journal of Shape Modeling 04, no. 03n04 (1998): 165–82. http://dx.doi.org/10.1142/s021865439800012x.

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Cao, Yang, and Wenfei Fan. "Data driven approximation with bounded resources." Proceedings of the VLDB Endowment 10, no. 9 (2017): 973–84. http://dx.doi.org/10.14778/3099622.3099628.

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Ghashami, Mina, Jeff M. Phillips, and Feifei Li. "Continuous matrix approximation on distributed data." Proceedings of the VLDB Endowment 7, no. 10 (2014): 809–20. http://dx.doi.org/10.14778/2732951.2732954.

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Karczewicz, Marta, and Moncef Gabbouj. "ECG data compression by spline approximation." Signal Processing 59, no. 1 (1997): 43–59. http://dx.doi.org/10.1016/s0165-1684(97)00037-6.

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von Freyberg, Axel, and Andreas Fischer. "Holistic approximation of combined surface data." Precision Engineering 54 (October 2018): 396–402. http://dx.doi.org/10.1016/j.precisioneng.2018.07.009.

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Zhu, Xibin, Andrej Gisbrecht, Frank-Michael Schleif, and Barbara Hammer. "Approximation techniques for clustering dissimilarity data." Neurocomputing 90 (August 2012): 72–84. http://dx.doi.org/10.1016/j.neucom.2012.01.033.

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Dong, Bo, Matthew M. Lin, and Haesun Park. "Integer Matrix Approximation and Data Mining." Journal of Scientific Computing 75, no. 1 (2017): 198–224. http://dx.doi.org/10.1007/s10915-017-0531-7.

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Hońko, Piotr. "Compound approximation spaces for relational data." International Journal of Approximate Reasoning 71 (April 2016): 89–111. http://dx.doi.org/10.1016/j.ijar.2016.02.002.

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