consapp_.md

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references

author id issued publisher title type url literal Wallis, J. wallis1656arithmetica 1656 Oxonii, Typis LEON: LICHFIELD Academiæ Typographi, Impensis THO. ROBINSON [Arithmetica infinitorum sive nova methodus inquirendi in curvilineorum quadraturam aliaque difficiliora matheseos problemata]{.nocase} book https://books.google.com/books?id=Z5w_AAAAcAAJ author edition id issued publisher publisher-place title type url literal Newton, I. 1 Newton1687 1687 Jussu Societatis Regiae ac typis Josephi Streater, prostant venales apud Sam. Smith Londini [Philosophiae naturalis principia mathematica]{.nocase} book https://books.google.at/books?id=XJwx0lnKvOgC&pg=PP2&redir_esc=y#v=onepage&q&f=false author id issued publisher publisher-place title type url literal Taylor, B. taylor1715methodus 1715 Typis Pearsonianis: Prostant apud Gul. Innys ad Insignia Principis in Coemeterio Paulino MDCCXV Londini [Methodus incrementorum directa & inversa. Auctore Brook Taylor, LL. D. & Regiae Societatis Secretario]{.nocase} book https://books.google.com/books?id=iXN1xgEACAAJ abstract author container-title doi id issue issued page title type url volume Si p sit numerus casuum quibus eventus aliquis contingere possit, &amp; q numerus casuum quibus possit non-contingere; literal de Moivre, A. Philosophical Transactions of the Royal Society of London 10.1098/rstl.1710.0018 DeMoivre1711 329 1711 213-264 [De mensura sortis, seu, de probabilitate eventuum in ludis a casu fortuito pendentibus]{.nocase} article-journal https://royalsocietypublishing.org/doi/abs/10.1098/rstl.1710.0018 27 author id issued publisher publisher-place title type url literal Bernoulli, J. Bernoulli1713 1713 Impensis Thurnisiorum, Fratrum Basileae [Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis]{.nocase} book https://www.e-rara.ch/zut/doi/10.3931/e-rara-9001 author edition id issued publisher title type url literal de Montmort, P. R. 2 de1713essay 1713 A Paris, Chez Laurent Le Conte; Quay des Augustins, à la Ville de Montpiller [Essay D'analyse Sur Les Jeux De Hazard]{.nocase} book https://books.google.fr/books?id=DIM_AAAAcAAJ author id issued publisher-place title url literal Bernoulli, D. Bernoulli1729 1729-10 St.-Petersbourg ce 6. octobre 1729 [Lettre XLVII. D. Bernoulli a Goldbach]{.nocase} https://commons.m.wikimedia.org/wiki/File:DanielBernoulliLetterToGoldbach-1729-10-06.jpg author edition id issued publisher publisher-place title type url literal de Moivre, A. 2 DeMoivre1738 1738 H. Woodfall London [The Doctrine of Chances: Or, A Method of Calculating the Probability of Events in Play]{.nocase} book https://books.google.com/books?id=PII\\\_AAAAcAAJ author id issued publisher publisher-place title type url literal Fourier, J. B. J. Fourier1822 1822 Chez Firmin Didot, pere et fils A Paris [Théorie analytique de la chaleur]{.nocase} book https://archive.org/details/bub_gb_TDQJAAAAIAAJ/mode/1up author id issued page publisher publisher-place title type url literal Poisson, S.-D. Poisson1837 1837 205-207 Bachelier Paris [Recherches sur la probabilité des jugements en matière criminelle et en matière civile]{.nocase} book https://gallica.bnf.fr/ark:/12148/bpt6k110193z/f218.image author container-title id issued page title type url volume literal Riemann, B. Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen Riemann1868 1868 87-131 [Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. (Mitgetheilt durch R. Dedekind)]{.nocase} article-journal http://eudml.org/doc/135759 13 author container-title id issued page title type url volume literal Helmert, F. R. Zeitschrift für Mathematik und Physik Helmert1876 1876 192-219 [Ueber die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und über einige damit im Zusammenhange stehende Fragen]{.nocase} article-journal https://gdz.sub.uni-goettingen.de/id/PPN599415665_0021 21 author container-title doi id issue issued page title type url volume literal Lüroth, J. Astronomische Nachrichten 10.1002/asna.18760871402 Lüroth1876 14 1876 209-220 [Vergleichung von zwei Werthen des wahrscheinlichen Fehlers]{.nocase} article-journal https://doi.org/10.1002/asna.18760871402 87 author container-title doi id issue issued page title type volume literal Gosset, W. S. Biometrika 10.2307/2331554 Gosset1908 1 1908 1-25 The probable error of a mean article-journal 6 abstract author container-title doi id issn issue issued page title type url volume 1. Introduction.--- 1·0. The object of this paper is to develop methods where by the differential equations of physics may be applied more freely than hitherto in the approximate form of difference equations to problems concerning irregular bodies. Though very different in method, it is in purpose a continuation of a former paper by the author, on a "Freehand Graphic Way of Determining Stream Lines and Equipotentials" ('Phil. Mag.,'February, 1908; also 'Proc. Physical Soc.,' London, vol. xxi.). And all that was there said, as to the need for new methods, may be taken to apply here also. In brief, analytical methods are the foundation of the whole subject, and in practice they are the most accurate when they will work, but in the integration of partial equations, with reference to irregular-shaped boundaries, their field of application is very limited. literal Richardson, L. F. Philosophical Transactions of the Royal Society of London, Series A: Containing Papers of a Mathematical or Physical Character 10.1098/rsta.1911.0009 10.1098/rsta.1911.0009 0264-3952 459-470 1911-01 307-357 IX. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam article-journal https://doi.org/10.1098/rsta.1911.0009 210 author container-title doi id issn issue issued page publisher title type url volume literal Fisher, R. A. Biometrika 10.2307/2331838 Fisher1915 26404 4 1915 507-521 \[Oxford University Press, Biometrika Trust\] Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population article-journal http://www.jstor.org/stable/2331838 10 author container-title doi id issue issued page title type volume literal Fisher, R. A. Journal of the Royal Statistical Society 10.2307/2340521 Fisher1922 1 1922 87-94 On the interpretation of [χ2]{.nocase} from contingency tables, and the calculation of p article-journal 85 author container-title id issued page title type url volume literal Fisher, R. A. Proceedings International Mathematical Congress, Toronto Fisher1924 1924 805-813 On a distribution yielding the error functions of several well-known statistics article-journal https://repository.rothamsted.ac.uk/item/8w2q9/on-a-distribution-yielding-the-error-functions-of-several-well-known-statistics 2 author id issued publisher publisher-place title type url family given Cardano G. Cardano1545 1545 ANDREAE OSIANDRO viro eruditiff S. P. D ARTIS MAGNAE, SIVE DE REGVLIS ALGEBRAICIS, LIBER VNVS book https://web.archive.org/web/20220201093634/http://www.filosofia.unimi.it/cardano/testi/operaomnia/vol_4\_s_4.pdf abstract author container-title doi id issued page title type url volume In der Arbeit soll versucht werden, Grundlagen zu gewinnen für eine quantentheoretische Mechanik, die ausschließlich auf Beziehungen zwischen prinzipiell beobachtbaren Größen basiert ist. literal Heisenberg, W. Zeitschrift für Physik 10.1007/BF01328377 Heisenberg1925 1925 879-893 [Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen]{.nocase} article-journal https://doi.org/10.1007/BF01328377 33 abstract author container-title doi id issn issue issued page title type url volume For a given function B and a non-zero real number h, Schoenberg's approximation defines from some data (jh, yj)jϵZd the function ΣjϵZd yj B(•h − j). For people not used to this kind of approximation, this paper intends to do a summary of the main definitions, properties and utilizations of Schoenberg's approximation: we show that the main tool to handle Schoenberg's approximation is the Fourier transform of B and even more its modified version, the transfer function of B; we give conditions for convergence of ΣjϵZd f(jh) B(•h − j) when h tends to zero, and we give various ways to define various B as combinations of translates of some function ϕ (usually ϕ is either some radial function, or obtained by a tensor product of some radial function), depending on the properties we want for the associated Schoenberg's approximation. Last, we show how multi-resolution analysis, subdivision techniques, and wavelets techniques, are nicely connected to Schoenberg's approximation. literal Rabut, C. Computers & Mathematics with Applications 10.1016/0898-1221(92)90177-J RABUT1992149 0898-1221 12 1992 149-175 An introduction to schoenberg's approximation article-journal https://www.sciencedirect.com/science/article/pii/089812219290177J 24 author container-title doi id issued page title type volume literal Schoenberg, I. J. Quarterly of Applied Mathematics 10.1090/qam/15914 Schoenberg1946 1946 45-99 Contributions to the problem of approximation of equidistant data by analytic functions. Part a. On the problem of smoothing or graduation. A first class of analytic approximation formulae article-journal 4 abstract author container-title doi id issue issued page title type volume The present study aimed at developing a rather easily applicable method of testing physiological reactions to images of threats and misery. To this end, rapid-changing, transient heart rate orienting responses were used for gaining physiologically based, objective responses to the images. Additionally, subjective ratings were obtained. A significant insensitivity to other's welfare and well-being was already demonstrated as a core feature of callous-unemotional personalities. Thus, physiologically based methods may supplement and possibly improve existing assessments and, in particular, may contribute to a multimodal assessment of psychopathic traits. Out of a non-forensic community sample of 122 men, we selected two extreme groups of 30 participants with the lowest and highest callous-unemotional traits respectively, ascertained by questionnaires. As expected, participants with higher scores of callous-unemotional traits showed smaller responses to distress cues in both heart rate responses and subjective ratings. Moreover, within the group with high callous-unemotional traits heart rate responses to threatening as well as distress cues did not significantly differ from responses to neutral pictures. The study provides further evidence for the idea that a lack of responsiveness to distress cues may be seen as a central feature of callous-unemotional personalities. literal Schulter, G., Milek, B., Lackner, H. K., Weber, B., Fink, A., Rominger, C., Perchtold-Stefan, C., & Papousek, I. Scientific Reports 10.1038/s41598-023-49307-7 Schulter2023 1 2023 22063 [Diagnosing callous-unemotional personality traits by heart rate orienting responses to images inducing threat and distress]{.nocase} article-journal 13 author container-title id issue issued note page publisher publisher-place title type url volume literal Romberg, W. Forhandlinger / Det Kongelige Norske Videnskabers Selskab Romberg1955 7 1955 https://katalog.ub.uni-heidelberg.de/cgi-bin/titel.cgi?katkey=68187317 30-36 F. Bruns Bokhandel Trondheim Vereinfachte numerische integration article-journal https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://web.eng.fiu.edu/LEVY/images/EGM5346/romberg%2520textbook%2520example.pdf&ved=2ahUKEwiSxr-u846SAxVFQvEDHd83Bj8QFnoECHMQAQ&usg=AOvVaw3vtOkKCF8WeDkOB49i_03i 28 author doi id issued publisher title type url version literal Schrausser, D. G. 10.5281/zenodo.19422127 Schrausser19422127 2006 Zenodo Schrausser/ATINA: 1.5 book https://doi.org/10.5281/zenodo.19422127 v1.5.2 author container-title doi id issued publisher title url literal Schrausser, D. G. Code 10.13140/RG.2.2.14377.26727 SchrausserRG.2.2.14377.26727 2022-06 ResearchGate SCHRAUSSER-MAT http://doi.org/10.13140/RG.2.2.14377.26727 author container-title id issued publisher title url literal Schrausser, D. G. Software Schrausser81800920 2009 Academia ThetaWin Overview https://www.academia.edu/81800920 author container-title doi id issued publisher title url literal Schrausser, D. G. Method 10.13140/RG.2.2.32114.17601 SchrausserRG.2.2.32114.17601 2000-05 ResearchGate Development of a parameter to indicate the focussation-level of cortical activation https://rgdoi.net/10.13140/RG.2.2.32114.17601 author container-title doi id issued publisher title literal Schrausser, D. G. Method 10.13140/RG.2.2.28637.90083 SchrausserEEGalgorithms 2000 ResearchGate [Spectral and Coherence Analysis: Algorithms]{.nocase} abstract author container-title doi id issue issued keyword page publisher title type url volume Graphical user interface for Theta applications within ConsoleApp_Distribution_Functions (Schrausser, 2024), generating distributions and estimators for several parameters via bootstrap method. literal Schrausser, D. G. Zenodo Software documentation 10.5281/zenodo.17691241 schrausser_2025_17691241 11/23 2025-11 bootstrapping myown probability resampling software statistics 1 Zenodo ThetaWin article-journal https://doi.org/10.5281/zenodo.17691241 2025 abstract author container-title doi id issue issued keyword page publisher title type url volume Tools for calculations, bootstrapping and permutation tests, also for practice purposes when learning programming languages (C, Basic, Fortran etc.), see e.g. Halvorson and Rygmyr (1991), Chivers and Sleightholme (2018), Joyce (2019), Streib (2020) or Gonzalez-Morris and Horton (2024). literal Schrausser, D. G. Zenodo Software documentation 10.5281/zenodo.17743756 schrausser_2025_17743756 11/28 2025-11 bootstrapping myown permutation_test programs simulation software statistics 1 Zenodo Various programs article-journal https://doi.org/10.5281/zenodo.17743756 2025 abstract author container-title doi id issue issued keyword page publisher title title-short type url volume Graphical MS Windows user interface for ConsoleApp_DistributionFunctions. literal Schrausser, D. G. Zenodo Software documentation 10.5281/zenodo.17880113 schrausser_2025_17880113 12/10 2025-12 myown probability software statistics 1 Zenodo FunktionWin: Windows interface for distribution functions FunktionWin article-journal https://doi.org/10.5281/zenodo.17880113 2025 abstract author doi id issued keyword publisher title title-short type Summarizing presentation of the fundamentally most important distribution functions with distribution graphics and formulation. Examples for the calculation in SCHRAUSSER-MAT syntax, as well as an extensive collection of formulas with transformations according to parameters of interest in the appendix. literal Schrausser, D. G. 10.31234/osf.io/rvzxa Schrausser10.31234/osf.io/rvzxa 2024 myown probability statistics PsyArXiv Handbook: Distribution functions (verteilungs funktionen) Handbook book author doi id issued publisher title type literal Schrausser, D. G. 10.5281/zenodo.7651660 Schrausser7651660 2023 Zenodo [Schrausser/FunktionWin: Windows Interface for distribution functions]{.nocase} book author doi id issued publisher title type literal Schrausser, D. G. 10.5281/zenodo.7659263 Schrausser7659263 2023 Zenodo [Schrausser/ThetaWin: Distribution simulator]{.nocase} book author doi id issued publisher title type literal Schrausser, D. G. 10.5281/zenodo.7664141 Schrausser7664141 2023 Zenodo [Schrausser/ConsoleApp_DistributionFunctions: Console applications for distribution functions]{.nocase} book author doi id issued publisher title type literal Schrausser, D. G. 10.5281/zenodo.7655046 Schrausser7655046 2023 Zenodo [Schrausser/ConsoleApp_Matrix: Console applications for matrix calculation and tools]{.nocase} book author doi id issued publisher title type literal Schrausser, D. G. 10.5281/zenodo.7655239 Schrausser7655239 2023 Zenodo [Schrausser/ConsoleApp_Tools: Console tool applications]{.nocase} book author doi id issued publisher title type literal Schrausser, D. G. 10.5281/zenodo.7653790 Schrausser7653790 2023 Zenodo [Schrausser/ConsoleApp_String: Console applications for string and transformation]{.nocase} book author doi id issued publisher title type literal Schrausser, D. G. 10.5281/zenodo.7655056 Schrausser7655056 2023 Zenodo [Schrausser/ConsoleApp_Integral: Console applications for integral and interpolation]{.nocase} book author doi id issued publisher title title-short type literal Schrausser, D. G. 10.5281/zenodo.10701350 schrausser_2024_10701350 2023 Zenodo Schrausser/ConsoleApp_EEG: 2.0 Schrausser/ConsoleApp_EEG book abstract author doi edition id issued keyword title title-short type url Mathematical and statistical applications for HP Prime. Algorithms are presented in context with the corresponding scope of application. CAS programs (1), HP Prime User functions (2) and functions for HP Prime Applications (3) are listed in alphabetical order, a comparison to corresponding SCHRAUSSER-MAT functions is given. In addition to the source codes of the functions, raw data sets are provided for correlation- as well as resampling-methods. literal Schrausser, D. G. 10.5281/zenodo.15713317 1 schrausser_2025_15713317 2025-06 bootstrapping calculus complex_plane hp_prime myown permutation_test probability randomization_test resampling software statistics HP_prime_MATH: manual HP_prime_MATH book https://doi.org/10.5281/zenodo.15713317 abstract author doi edition id isbn issued publisher publisher-place title type The C programming language was created in the 1970s, yet it is still in extensive use today and is the basis of many other languages. For this reason I have used C as the language for the solution of the numerical problems demonstrated in this book. literal Joyce, P. 10.1007/978-1-4842-5064-8 1 Joyce2019 978-1-4842-5064-8 2019 Apress Berkeley, CA [Numerical C: Applied Computational Programming with Case Studies]{.nocase} book author doi edition id isbn issued publisher publisher-place title type literal Gonzalez-Morris, G., & Horton, I. 10.1007/979-8-8688-0149-5 7 Gonzalez2024 979-8-8688-0148-8 2024 Apress Berkeley, CA [Beginning C: From Beginner to Pro]{.nocase} book abstract author container-title doi id isbn issued page publisher publisher-place title type url DOSSHELL als Befehl: Mit DOSSHELL wird die Menü-Oberfläche von MS-DOS von der Befehlszeilen-Oberfläche aus gestartet. Beispiel. literal Kaier, E. [Informatik: Referenzbuch. Mit den vollständigen Befehlslisten zu MS-DOS, Turbo Pascal, dBase und Multiplan]{.nocase} 10.1007/978-3-322-89035-1_1 Kaier1990 978-3-322-89035-1 1990 1-12 Vieweg+Teubner Verlag Wiesbaden Referenz zu MS-DOS chapter https://doi.org/10.1007/978-3-322-89035-1_1 abstract author container-title doi id isbn issued page publisher publisher-place title type url Obwohl die in diesem Abschnitt verwendeten Schlüsselwörter MS-DOS-spezifisch sind und daher nicht der ANSI C-Norm entsprechen, soll an einigen Beispielen gezeigt werden, wie man in C Zugang zum MS-DOS-Betriebssystem erhält. Eine vollständige Darstellung ist im Rahmen des Buchs natürlich nicht möglich. Es wird hier auf die Literatur \[131 bzw. \[22\] verwiesen. literal Herrmann, D. [Effektiv Programmieren in C und C++: Eine aktuelle Einführung mit Beispielen aus Mathematik, Naturwissenschaft und Technik]{.nocase} 10.1007/978-3-322-94365-1_16 Herrmann2001 978-3-322-94365-1 2001 295-318 Vieweg+Teubner Verlag Wiesbaden System-programmierung (MS-DOS) chapter https://doi.org/10.1007/978-3-322-94365-1_16 abstract author container-title doi id isbn issued page publisher publisher-place title type url Ein wichtiger Teil der Computerphysik ist die Implementierung eines Programms, d. h. das Programmieren selbst. Es gibt zwar Simulationspakete und Umgebungen, die spezielle Probleme ohne Programmierung lösen können, aber erst mit der Programmierung besteht die Möglichkeit, eigene Probleme zu implementieren und zu lösen. In diesem Kapitel werden zuerst die Grundlagen der Programmierung im Detail erläutert. Danach folgt eine Einführung in die Programmiersprache C, die eine hohe Verbreitung genießt und auch zu großen Teilen in diesem Buch verwendet wird. Am Ende des Kapitels werden Hilfsmittel zur Programmierung, d.h. sog. Makefiles, Debugger und Versionsverwaltungen und deren Verwendung angesprochen. literal Gerlach, S. [Computerphysik: Einführung, Beispiele und Anwendungen]{.nocase} 10.1007/978-3-662-59246-5_4 Gerlach2019 978-3-662-59246-5 2019 31-54 Springer Berlin, Heidelberg [Programmieren in C]{.nocase} chapter https://doi.org/10.1007/978-3-662-59246-5_4 author container-title doi editor id isbn issued page publisher publisher-place title type url literal Philippou, A. N., & Antzoulakos, D. L. International encyclopedia of statistical science 10.1007/978-3-662-69359-9_71 literal Lovric, M. Philippou2025 978-3-662-69359-9 2025 306-309 Springer Berlin, Heidelberg Binomial distribution chapter https://doi.org/10.1007/978-3-662-69359-9_71 author collection-title doi edition id issued publisher publisher-place title type literal Collani, E., & Dräger, K. Graduate texts in mathematics 10.1007/978-1-4612-0215-8 1 Collani2001 2001 Birkhäuser Boston, MA Binomial distribution handbook for scientists and engineers book abstract author container-title doi id issn issue issued title title-short type url volume The increase in available probabilistic information and its usefulness for understanding the world has made it necessary to promote probabilistic literate citizens. For this, the binomial distribution is fundamental as one of the most important distributions for understanding random phenomena and effective decision making, and as a facilitator for the understanding of mathematical and probabilistic notions such as the normal distribution. However, to understand it effectively, it is necessary to consider how it has developed throughout history, that is, the components that gave it the form and meaning that we know today. To address this perspective, we identify the problem situations that gave origin to the binomial distribution, the operational and discursive practices developed to find solutions, and the conflicts that caused a leap in mathematical and probability heuristics, culminating in what is now known as the binomial distribution formula. As a result, we present five historical links to the binomial phenomenon where problem situations of increasing complexity were addressed: a case study using informal means (such as direct counting), the formalization of numerical patterns and constructs related to counting cases, specific probability calculus, the study and modeling of probability in variable or complex phenomena, and the use of the distribution formula as a tool to approaching notions such as the normal distribution. The periods and situations identified correspond to a required step in the design of binomial distribution learning from a historical epistemological perspective and when solving conflicts. literal García-García, J. I., Fernández Coronado, N. A., Arredondo, E. H., & Imilpán Rivera, I. A. Mathematics 10.3390/math10152680 math10152680 2227-7390 15 2022 The binomial distribution: Historical origin and evolution of its problem situations The binomial distribution article-journal https://www.mdpi.com/2227-7390/10/15/2680 10 abstract accessed author container-title id issn issue issued page publisher title type url volume The sum of k independent and identically distributed (0, 1) variables has a binomial distribution. If the variables are identically distributed but not independent, this may be generalized to a two-parameter distribution where the k variables are assumed to have a symmetric joint distribution with no second- or higher-order \"interactions\". Two distinct generalizations are obtained, depending on whether the \"multiplicative\" or \"additive\" definition of \"interaction\" for discrete variables is used. The multiplicative generalization gives rise to a two-parameter exponential family, which naturally includes the binomial as a special case. Whereas with a beta-binomially distributed variable the variance always exceeds the corresponding binomial variance, the \"additive\" or \"multiplicative\" generalizations allow the variance to be greater or less than the corresponding binomial quantity. The properties of these two distributions are discussed, and both distributions are fitted, successfully, to data given by Skellam (1948) on the secondary association of chromosomes in Brassica. 2026-01-12 literal Altham, P. M. E. Journal of the Royal Statistical Society. Series C (Applied Statistics) f22e64ae-c859-380d-809e-72c74d13957d 00359254, 14679876 2 1978 162-167 \[Royal Statistical Society, Oxford University Press\] Two generalizations of the binomial distribution article-journal http://www.jstor.org/stable/2346943 27 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-12 literal Goel, S. K., & Rodriguez, D. M. Mathematics Magazine fbd8e02f-8560-36a2-a00c-e74f040810fc 0025570X, 19300980 4 1987 225-228 \[Mathematical Association of America, Taylor & Francis, Ltd.\] A note on evaluating limits using riemann sums article-journal http://www.jstor.org/stable/2689344 60 abstract accessed author container-title id issn issue issued page publisher title type url volume We examine how the binomial distribution B(n, p) arises as the distribution Sn = ∑i=1n Xi of an arbitrary sequence of Bernoulli variables. It is shown that B (n, p) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases B(2, p) and B(3, p) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which Sn follows B(n, p), given that Sn-1 is not necessarily a binomial variable, are investigated. Several natural characterizations of B(n , p), including one which relates the binomial distributions and the Poisson process, are also given.These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility. 2026-01-12 literal Vellaisamy, P., & Punnen, A. P. Journal of Applied Probability fa0257d0-e7e0-3c60-a790-1bc8b1adf0c3 00219002 1 2001 36-44 Applied Probability Trust On the nature of the binomial distribution article-journal http://www.jstor.org/stable/3215739 38 abstract author container-title doi id isbn issued page publisher publisher-place title type url The geometric distribution applies when an experiment is run repeatedly until a successful outcome occurs, and the probability of a success is the same for all trials. The random variable could be defined as the number of fails till the first success, and has a range of integers zero and above. The random variable could also be labeled as the number of trials till the first success and the range is integers one and above. Both scenarios are described in the chapter. This situation occurs, for example, when a process produces units that need to meet acceptable engineering standards, and the process is repeated until an acceptable unit is produced. When the probability of a successful outcome is not known, sample data can be used to estimate the probability. Sometimes, no sample data is available, and a person of experience offers an approximation on the distribution and this data is used to estimate the probability of a successful outcome. The chapter also describes how the geometric distribution is the only discrete distribution that has a memory less property. literal Thomopoulos, N. T. Statistical distributions: Applications and parameter estimates 10.1007/978-3-319-65112-5_15 Thomopoulos2017 978-3-319-65112-5 2017 127-133 Springer International Publishing Cham Geometric chapter https://doi.org/10.1007/978-3-319-65112-5_15 abstract accessed author container-title id issn issue issued page publisher title type url volume Let X1, X2, ⋯, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, ⋯, Xn:n. We prove that if the random variable X2:n - X1:n is independent of the events \[X1:n = m\] and \[X1:n = k\], for fixed k \> m \> 1, then the Xi's are geometric. This is related to a characterization problem raised by Arnold (1980). 2026-01-13 literal Sreehari M. Journal of Applied Probability 90db79eb-3312-380e-b218-3292416a6de2 00219002 1 1983 209-212 Applied Probability Trust A characterization of the geometric distribution article-journal http://www.jstor.org/stable/3213738 20 author container-title doi id issue issued page title type volume literal Zijlstra, M. Journal of Applied Probability 10.2307/3213595 Zijlstra_1983 4 1983 843-850 Characterizations of the geometric distribution by distributional properties article-journal 20 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-16 literal Dutka, J. Archive for History of Exact Sciences 30097c0e-8012-3a25-bb7a-d6c982c320c6 00039519, 14320657 1 1984 15-34 Springer The early history of the hypergeometric function article-journal http://www.jstor.org/stable/41133728 31 abstract author container-title doi id isbn issued page publisher publisher-place title type url H Inc sequitur, quod. Si ex Tabellæ prop. 184. locis vacuis unus quilibet numero noto suppleatur, enurt & reliqui omnes cogniti. literal Berggren, L., Borwein, J., & Borwein, P. Pi: A source book 10.1007/978-1-4757-3240-5_11 Berggren2000 978-1-4757-3240-5 2000 78-80 Springer New York, NY Wallis. Arithmetica infinitorum (1655) chapter https://doi.org/10.1007/978-1-4757-3240-5_11 abstract author container-title doi editor id isbn issued page publisher publisher-place title type url Publisher Summar The Arithmetica infinitorum was a key text in the 17th-century transition from geometry to algebra and in the development of infinite series and the integral calculus. The text contains the infinite fraction for 4/π that is now his chief claim to fame; but for his contemporaries, the most significant feature was the introduction of new methods, new concepts, and new vocabulary. Wallis came to Oxford with no more than a little self-taught mathematics, but within a few years, produced his most important work, the Arithmetica infinitorum. The Arithmetica infinitorum stands both chronologically and mathematically at the mid point of the 17th century, drawing together the best ideas from the first half of the century, the algebraic geometry of René Descartes, and the theory of indivisibles of Bonaventura Cavalieri (1598--1647), and preparing the ground for some of the astonishing advances of the second half: the discovery of the general binomial theorem, applications of infinite series, and the integral calculus. literal Stedall, J. Landmark writings in western mathematics 1640-1940 10.1016/B978-044450871-3/50083-8 literal Grattan-Guinness, I., Cooke, R., Corry, L., Crépel, P., & Guicciardini, N. STEDALL200523 978-0-444-50871-3 2005 23-32 Elsevier Science Amsterdam Chapter 2 - john wallis, arithmetica infinitorum (1965) chapter https://www.sciencedirect.com/science/article/pii/B9780444508713500838 abstract author container-title doi id isbn issued page publisher publisher-place title type url The function represented by the infinite series \$\$\\sum\\limits\_{n = 0}\^\\infty {\\frac{{{{(a)}\_n}{{(b)}\_n}}}{{{{(c)}\_n}}}\\frac{{{z\^n}}}{{n!}}} \$\$within its circle of convergence and all the analytic continuations is called the hypergeometric function 2F1(a, b; c;z).\* literal Magnus, W., Oberhettinger, F., & Soni, R. P. Formulas and theorems for the special functions of mathematical physics 10.1007/978-3-662-11761-3_2 Magnus1966b 978-3-662-11761-3 1966 37-65 Springer Berlin, Heidelberg The hypergeometric function chapter https://doi.org/10.1007/978-3-662-11761-3_2 abstract author container-title doi editor id isbn issued page publisher publisher-place title type url The Fisher Exact test was proposed by Fisher (1934) in the fifth edition of Statistical Methods for Research Workers. It is a test for independence as opposed to association in 2 `\texttimes`{=latex} 2 contingency tables. literal Sprent, P. International encyclopedia of statistical science 10.1007/978-3-662-69359-9_693 literal Lovric, M. Sprent2025 978-3-662-69359-9 2025 961-962 Springer Berlin, Heidelberg The fisher exact test chapter https://doi.org/10.1007/978-3-662-69359-9_693 abstract author container-title doi id isbn issued keyword publisher title title-short type url Abstract For a finite population of subjects of two types, suppose we select a random sample without replacement. The probability distribution of the number in the sample of one of the two types is the hypergeometric distribution. This article presents the hypergeometric distribution, summarizes its properties, discusses binomial and normal approximations, and presents a multivariate generalization. literal Shuster, J. J. Wiley StatsRef: Statistics reference online 10.1002/9781118445112.stat04869 doi:https://doi.org/10.1002/9781118445112.stat04869 9781118445112 2014 binomial distribution, contingency tables, exact tests, independence, normal distribution, sampling without replacement, Yates's continuity correction John Wiley & Sons, Ltd Hypergeometric distribution: introduction Hypergeometric distribution chapter https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat04869 abstract author container-title doi id isbn issued keyword publisher title type url Abstract Exact tests evaluate the significance of a test statistic by using the test statistic's empirically derived sampling distribution instead of a theoretical sampling distribution. Preference for exact tests may occur in situations in which the sample size is small, or when the data have distributional deficiencies. Two well-known exact tests, Pitman's test and Fisher's test, are described. literal Hershberger, S. L. Wiley StatsRef: Statistics reference online 10.1002/9781118445112.stat06165 doi:https://doi.org/10.1002/9781118445112.stat06165 9781118445112 2014 asymptotic theory, nonparametric statistics, sampling distribution, resampling procedures, the bootstrap, Pitman test, Fisher's exact test, contingency table, hypergeometric distribution John Wiley & Sons, Ltd Exact methods for categorical data chapter https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat06165 abstract accessed author container-title id issn issue issued page publisher title type url volume This paper reviews the problems that bedevil the selection of an appropriate test for the analysis of a 2 × 2 table. In contradiction to an earlier paper, the author now argues the case for the use of Fisher's exact test. It is noted that all test statistics for the 2 × 2 table have discrete distributions and it is suggested that it is irrational to prescribe an unattainable fixed significance level. The use of mid-P is suggested, if a formula is required for prescribing a variable tail probability. The problems of two-tail tests are discussed. 2026-01-12 literal Upton, G. J. G. Journal of the Royal Statistical Society. Series A (Statistics in Society) 5d8fa06c-0d2a-391a-b97f-85223f1f3b0f 09641998, 1467985X 3 1992 395-402 \[Wiley, Royal Statistical Society\] Fisher's exact test article-journal http://www.jstor.org/stable/2982890 155 abstract accessed author container-title id issn issue issued page publisher title type url volume A relationship is derived between the posterior probability of negative association of rows and columns of a 2 × 2 contingency table and Fisher's \"exact\" probability, as given in existing tables for testing the hypothesis of no association of rows and columns. The result for the 2 × 2 table is generalized to provide the posterior probability that one discrete-valued random variable is stochastically larger than another. 2026-01-12 literal Altham, P. M. E. Journal of the Royal Statistical Society. Series B (Methodological) 202b9adf-f7a7-32c5-94dd-9f269c7f5a86 00359246 2 1969 261-269 \[Royal Statistical Society, Oxford University Press\] Exact bayesian analysis of a 2 × 2 contingency table, and fisher's \"exact\" significance test article-journal http://www.jstor.org/stable/2984209 31 abstract author container-title doi id issue issued title title-short type volume It is demonstrated in this paper that two major tests for 2 × 2 talbes are highly related from a Bayesian perspective. Although it is well-known that Fisher's exact and Pearson's chi-square tests are asymptotically equivalent, the present analysis shows that a formal similarity also exists in small samples. The key assumption that leads to the resemblance is the presence of a continuous parameter measuring association. In particular, it is shown that Pearson's probability can be obtained by integrating a two-moment approximation to the posterior distribution of the log-odds ratio. Furthermore, Pearson's chi-square test gave an excellent approximation to the actual Bayes probability in all 2×2 tables examined, except for those with extremely disproportionate marginal frequencies. literal Camilli, G. Psychometrika 10.1007/BF02301418 Camilli1995 2 1995 The relationship between fisher's exact test and pearson's chi-square test: A bayesian perspective The relationship between fisher's exact test and pearson's chi-square test article-journal 60 abstract author container-title doi id isbn issued page publisher publisher-place title type url Apart from the elementary transcendental functions such as the exponential and trigonometric functions and their inverses, the Gamma function is probably the most important transcendental function. It was defined by Euler to interpolate the factorials at noninteger arguments. literal Koepf, W. Hypergeometric summation: An algorithmic approach to summation and special function identities 10.1007/978-1-4471-6464-7_1 Koepf2014 978-1-4471-6464-7 2014 1-10 Springer London The gamma function chapter https://doi.org/10.1007/978-1-4471-6464-7_1 abstract author container-title doi id isbn issued page publisher publisher-place title type url The function $\Gamma$(z) is a meromorphic function of z with simple poles at z = −n, (n = 0, 1, 2,\...) with the respective residue \$\$\\frac{{{{( - 1)}\^n}}}{{n!}}.\$\$. literal Magnus, W., Oberhettinger, F., & Soni, R. P. Formulas and theorems for the special functions of mathematical physics 10.1007/978-3-662-11761-3_1 Magnus1966a 978-3-662-11761-3 1966 1-37 Springer Berlin, Heidelberg The gamma function and related functions chapter https://doi.org/10.1007/978-3-662-11761-3_1 abstract author container-title doi id issn issue issued keyword page title type url volume In this paper we prove a complete monotonicity theorem and establish some upper and lower bounds for the gamma function in terms of digamma and polygamma functions. literal Batir, N. Expositiones Mathematicae 10.1016/j.exmath.2007.10.001 BATIR2008187 0723-0869 2 2008 Gamma function, Digamma function, Polygamma functions, Complete monotonicity, Riemann zeta-function 187-196 On some properties of the gamma function article-journal https://www.sciencedirect.com/science/article/pii/S0723086907000497 26 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-12 literal Rasch, G. Annals of Mathematics 5bcebaa1-7494-336c-85e9-de3c540e9f64 0003486X, 19398980 3 1931 591-599 \[Annals of Mathematics, Trustees of Princeton University on Behalf of the Annals of Mathematics, Mathematics Department, Princeton University\] Notes on the gamma-function article-journal http://www.jstor.org/stable/1968254 32 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-12 literal Gronwall, T. H. Annals of Mathematics 6b95b756-79df-32e7-8452-988d85f07718 0003486X, 19398980 2 1918 35-124 \[Annals of Mathematics, Trustees of Princeton University on Behalf of the Annals of Mathematics, Mathematics Department, Princeton University\] The gamma function in the integral calculus article-journal http://www.jstor.org/stable/1967180 20 abstract accessed author container-title id issn issue issued page publisher title type url volume Since its inception in 1894, the Monthly has printed 50 articles on the Γ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the Γ function in future Monthly issues. We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of Γ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of n! and the first few terms of a new asymptotic formula for invΓ. 2026-01-12 literal Borwein, J. M., & Corless, R. M. The American Mathematical Monthly 7c7965cf-2aff-3860-a01c-d57527fbdc42 00029890, 19300972 5 2018 400-424 \[Taylor & Francis, Ltd., Mathematical Association of America\] Gamma and factorial in the monthly article-journal https://www.jstor.org/stable/48663320 125 accessed author container-title id issn issue issued page publisher title title-short type url volume 2026-01-12 literal Davis, P. J. The American Mathematical Monthly f70ecf4e-6c25-3732-9fda-1e8986c07b3c 00029890, 19300972 10 1959 849-869 \[Taylor & Francis, Ltd., Mathematical Association of America\] Leonhard euler's integral: A historical profile of the gamma function: In memoriam: Milton abramowitz Leonhard euler's integral article-journal http://www.jstor.org/stable/2309786 66 abstract author container-title doi id isbn issued page publisher publisher-place title type url This chapter first describes in Sect. 15.1 the t-distribution introduced by Gosset (1908) and its multivariate extensions, in the form of the multivariate t-distribution, Hotelling's T2-statistic and the F-distribution, all derived from sampling theory of a normal (or multivariate normal) distribution with unknown variance (or covariance matrix). It then develops the asymptotic distributions of these self-normalized sample means even when the population has infinite second moment. Related results such as the law of the iterated logarithm (LIL) for these self-normalized statistics are also described. literal Peña, V. H., Lai, T. L., & Shao, Q.-M. [Self-Normalized Processes: Limit Theory and Statistical Applications]{.nocase} 10.1007/978-3-540-85636-8_15 delaPeña2009 978-3-540-85636-8 2009 207-221 Springer Berlin, Heidelberg [The t-Statistic and Studentized Statistics]{.nocase} chapter https://doi.org/10.1007/978-3-540-85636-8_15 author collection-title doi edition id issued publisher publisher-place title type literal Grigelionis, B. Springer briefs in statistics 10.1007/978-3-642-31146-8 1 Grigelionis2012 2012 Springer Berlin, Heidelberg [Student's t-Distribution and Related Stochastic Processes]{.nocase} book abstract author container-title doi id issn issue issued title type url volume In this work, we present a new generalization of the student's t distribution. The new distribution is obtained by the quotient of two independent random variables. This quotient consists of a standard Normal distribution divided by the power of a chi square distribution divided by its degrees of freedom. Thus, the new symmetric distribution has heavier tails than the student's t distribution and extensions of the slash distribution. We develop a procedure to use quantile regression where the response variable or the residuals have high kurtosis. We give the density function expressed by an integral, we obtain some important properties and some useful procedures for making inference, such as moment and maximum likelihood estimators. By way of illustration, we carry out two applications using real data, in the first we provide maximum likelihood estimates for the parameters of the generalized student's t distribution, student's t, the extended slash distribution, the modified slash distribution, the slash distribution generalized student's t test, and the double slash distribution, in the second we perform quantile regression to fit a model where the response variable presents a high kurtosis. literal Reyes, J., Rojas, M. A., & Arrué, J. Symmetry 10.3390/sym13122444 sym13122444 2073-8994 12 2021 [A New Generalization of the Student's t Distribution with an Application in Quantile Regression]{.nocase} article-journal https://www.mdpi.com/2073-8994/13/12/2444 13 abstract author container-title doi id issn issue issued page title type url volume In this paper we provide rather weak conditions on a distribution which would guarantee that the t-statistic of a random vector of order n follows the t-distribution with n-1 degrees of freedom. The results sharpen the earlier conclusions of Mauldon \[Characterizing properties of statistical distributions, Quart. J. Math. 2(7) (1956) 155--160\] and the more recent advances due to Bondesson \[When is the t-statistic t-distributed, Sankhyā, Ser. A 45 (1983) 338--345\]. The basic tool involved in the derivations is the vertical density representation originally suggested by Troutt \[A theorem on the density of the density ordinate and an alternative interpretation of the Box--Muller method, Statistics 22(3) (1991) 463--466; Vertical density representation and a further remark on the Box--Muller method, Statistics 24 (1993) 81--83\]. Several illustrative examples are presented. literal Yang, Z., Fang, K.-T., & Kotz, S Journal of Multivariate Analysis 10.1016/j.jmva.2006.11.003 YANG20071293 0047-259X 6 2007 1293-1304 [On the Student's t-distribution and the t-statistic]{.nocase} article-journal https://www.sciencedirect.com/science/article/pii/S0047259X06002004 98 abstract accessed author container-title id issn issue issued page publisher title type url volume We develop a Bayesian analysis based on two different Jeffreys priors for the Student-t regression model with unknown degrees of freedom. It is typically difficult to estimate the number of degrees of freedom: improper prior distributions may lead to improper posterior distributions, whereas proper prior distributions may dominate the analysis. We show that Bayesian analysis with either of the two considered Jeffreys priors provides a proper posterior distribution. Finally, we show that Bayesian estimators based on Jeffreys analysis compare favourably to other Bayesian estimators based on priors previously proposed in the literature. 2026-01-17 literal Fonseca, T. C. O., Ferreira, M. A. R., & Migon, H. S. Biometrika 8bbd2dac-610f-3a49-b88b-7ff72e396d2e 00063444, 14643510 2 2008 325-333 \[Oxford University Press, Biometrika Trust\] [Objective Bayesian Analysis for the Student-t Regression Model]{.nocase} article-journal http://www.jstor.org/stable/20441467 95 abstract author container-title doi id issue issued page title type volume The Student's t distribution is the most popular model for economic and financial data. In recent years, many generalizations of the Student's t distribution have been proposed. This paper provides a review of generalizations, including software available for them. A real data application is presented to compare some of the reviewed distributions. literal Li, R., & Nadarajah, S. Empirical Economics 10.1007/s00181-018-1570-0 Li2020 3 2020 1461-1490 [A review of Student's t distribution and its generalizations]{.nocase} article-journal 58 author container-title doi id issued title type literal Edelman, D. arXiv 10.48550/arXiv.2508.13226 edelman2025worstcasenonparametricboundsstudent 2025-09 [Worst-case Nonparametric Bounds for the Student t-statistic]{.nocase} article-journal author container-title doi id issued title title-short type literal Kirkby, J. L., Nguyen, D., & Nguyen, D. arXiv 10.48550/arXiv.1912.01607 kirkby2024momentsstudentstdistributionunified 2024-12-17 Moments of student's t-distribution: A unified approach Moments of student's t-distribution article-journal abstract author container-title doi id isbn issued page publisher publisher-place title type url The distribution of $\chi$`<!-- -->`{=html}2(chi squared) is a continuous and asymmetrical distribution which ranges from 0 to + ∞ and is followed by a sum of squares of independent standardized normal variates. The $\chi$`<!-- -->`{=html}2distribution and its application to frequency tables (chapter 15) were discovered by the British biometrician Karl Pearson (1857--1936), who was considered as the "father of biometry" and was one of the founders of the periodical Biometrika. The $\chi$`<!-- -->`{=html}2distribution is exact when hypotheses must be tested or confidence intervals must be determined (chapter 10) concerning the parametric variance $\sigma$`<!-- -->`{=html}2of a continuous variate X which follows a normal distribution (chapter 5). Moreover, the $\chi$`<!-- -->`{=html}2distribution may be used as an approximation in many cases, including Bartlett's test of the homogeneity of variance (section 12.7), tests of hypotheses concerning frequency tables (chapter 15), tests of goodness of fit (chapters 16, 17 and 18), the binomiality (chapter 17) and Poissonianity (chapter 18) tests, etc.. Paradoxically, the approximate utilizations of the $\chi$`<!-- -->`{=html}2distribution are perhaps better known than the exact ones. literal Jolicoeur, P. Introduction to biometry 10.1007/978-1-4615-4777-8_8 Jolicoeur1999 978-1-4615-4777-8 1999 38-39 Springer US Boston, MA The distribution of $\chi$`<!-- -->`{=html}2 (chi squared) chapter https://doi.org/10.1007/978-1-4615-4777-8_8 author id issued title url literal Das, A. das2025newmethodscomputegeneralized 2025 New methods to compute the generalized chi-square distribution https://arxiv.org/abs/2404.05062 author container-title doi id isbn issued page publisher publisher-place title type url literal Chattamvelli, R., & Shanmugam, R. Continuous distributions in engineering and the applied sciences -- part II 10.1007/978-3-031-02435-1_7 Chattamvelli2021 978-3-031-02435-1 2021 227-234 Springer International Publishing Cham F Distribution chapter https://doi.org/10.1007/978-3-031-02435-1_7 abstract author container-title doi id isbn issued page publisher publisher-place title type url Die letzte der drei wichtigsten Test-Verteilungen ist die F-Verteilung mit der Verteilungsfunktion Die Bedeutung der F-Verteilung liegt darin, daß mit zwei $\chi$`<!-- -->`{=html}2 -verteilten Größen X2 und Y2 mit den Freiheitsgraden f1 bzw. f2F-verteilt ist. Die F-Verteilung hat somit zwei Freiheitsgrade. Sie ist eine sehr allgemeine Funktion, denn sie geht für f1 = 1, t = `\textsurd`{=latex}F in die t-Verteilung f1 = 1, f2 = ∞, z = `\textsurd`{=latex}F in die Normalverteilung über. literal Herrmann, D. Wahrscheinlichkeitsrechnung und statistik --- 30 BASIC-programme 10.1007/978-3-322-96320-8_21 Herrmann1984 978-3-322-96320-8 1984 40-41 Vieweg+Teubner Verlag Wiesbaden F-Verteilung chapter https://doi.org/10.1007/978-3-322-96320-8_21 abstract author container-title doi id isbn issued page publisher publisher-place title type url Die Standardnormalverteilung ist ohne Zweifel die wichtigste und grundlegendste Verteilungsform in der Statistik. Sie ist von fundamentaler Bedeutung für die Schätzung von Parametern, aber auch zur Konstruktion von Teststatistiken. Nicht alle Teststatistiken aber können so konstruiert werden, dass sie normalverteilt sind. Die Teststatistiken von wichtigen statistischen Tests haben oft auch andere Verteilungsformen, die allerdings aus der Standardnormalverteilung abgeleitet werden können. Die wichtigsten dieser weiteren Verteilungen sind die so genannte Chi -Verteilung und die F-Verteilung. Die Chi2 -Verteilung wird unter anderem bei der Untersuchung von Zusammenhängen zwischen nominalskalierten Variablen eingesetzt, aber sie kann in einem wesentlich weiteren Sinn in vielen Fällen für den Vergleich einer theoretischen Verteilung mit einer empirischen Verteilung herangezogen werden. Die F-Verteilung spielt vor allem für den Vergleich von Varianzen eine herausragende Rolle. Schon bekannt ist uns außerdem die T-Verteilung, die im Prinzip überall dort Verwendung findet, wo auch die Standard-normalverteilung eingesetzt werden kann, wo aber die Standardabweichung der Grundgesamtheit erst mit Hilfe der Standardabweichung der Stichprobe geschätzt werden muss. Wir greifen die T-Verteilung in diesem Kapitel noch einmal kurz auf, um ihre Beziehung zur Standardnormalverteilung noch etwas genauer zu klären. literal Behnke, J., & Behnke, N. Grundlagen der statistischen datenanalyse: Eine einführung für politikwissenschaftler 10.1007/978-3-531-90003-2_26 Behnke2006 978-3-531-90003-2 2006 344-355 VS Verlag für Sozialwissenschaften Wiesbaden Verteilungen, die von der standardnormalverteilung abgeleitet werden können chapter https://doi.org/10.1007/978-3-531-90003-2_26 abstract author container-title doi id isbn issued page publisher publisher-place title type url Vorgelegt seien zwei Normalverteilungen 1 und 2, die beispielsweise zu zwei Erzeugungsvorgängen gehören, mit unbekannten Varianzen $\sigma$`<!-- -->`{=html}12und $\sigma$`<!-- -->`{=html}22. Man will entscheiden, ob beide Vorgänge mit der gleichen Varianz für die Merkmalwerte ablaufen. Mit anderen Worten: Es soll die Hypothese $\sigma$`<!-- -->`{=html}12= $\sigma$`<!-- -->`{=html}22getestet werden. Verfügbar ist aus jeder Verteilung eine Zufallsprobe der Größe n1 bzw. n2 mit der Varianz s12bzw. s22. Als Prüfgröße, mit der man die Entscheidung fällt, wählt man das Verhältnis s12/s22der Stichprobenvarianzen. literal Stange, K. Angewandte statistik: Erster teil eindimensionale probleme 10.1007/978-3-642-85602-0_11 Stange1970 978-3-642-85602-0 1970 357-372 Springer Berlin, Heidelberg Die F-Verteilung chapter https://doi.org/10.1007/978-3-642-85602-0_11 abstract author container-title doi id isbn issued page publisher publisher-place title type url Die in den Kapiteln 11, 12 und 13 behandelten statistischen Hypothesen waren Parameterhypothesen in dem Sinne, daß Aussagen über einzelne Verteilungsparameter (z.B.: geprüft werden sollten. Häufig hat man aber allgemeinere Aussagen über die den Daten zugrundeliegende Verteilung zu testen. Man möchte zum Beispiel wissen, ob die Annahme, daß die Daten normal-verteilt sind, haltbar ist, oder ob zwei Verteilungen deutlich oder kaum verschieden sind. literal Hafner, R. Statistik: Für sozial- und wirtschaftswissenschaftler 10.1007/978-3-7091-3420-7_14 Hafner1992 978-3-7091-3420-7 1992 164-170 Springer Vienna Der chi-quadrat-test chapter https://doi.org/10.1007/978-3-7091-3420-7_14 author container-title doi id issue issued page title type url volume literal Brereton, R. G. Journal of Chemometrics 10.1002/cem.2734 https://doi.org/10.1002/cem.2734 11 2015 582-586 [The F distribution and its relationship to the chi squared and t distributions]{.nocase} article-journal https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/cem.2734 29 abstract author container-title doi id isbn issued keyword publisher title type url Abstract The F-distribution is fundamental to much of statistical analysis, particularly the analysis of variance. The article entitled "F-Distribution" is a brief but complete description of this probability distribution. Its properties are first discussed, followed by a description of the relationship between the F-distribution and other common statistical distributions (e.g. beta, t-, chi-square, and binomial distributions). The article concludes with a typical application of this distribution to a small set of sampled observations. literal Selvin, S. Wiley StatsRef: Statistics reference online 10.1002/9781118445112.stat05856 doi:https://doi.org/10.1002/9781118445112.stat05856 9781118445112 2014 -distribution, variance, probability distribution John Wiley & Sons, Ltd F Distributions chapter https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112.stat05856 abstract author container-title doi id issn issue issued keyword page title type url volume An accurate normal approximation for the cumulative distribution function of the chi-square distribution with n degrees of freedom is proposed. This considers a linear combination of appropriate fractional powers of chi-square. Numerical results show that the maximum absolute error associated with the new transformation is substantially lower than that found for other power transformations of a chi-square random variable for all the degrees of freedom considered (1⩽n⩽1000). literal Canal, L. Computational Statistics & Data Analysis 10.1016/j.csda.2004.04.001 CANAL2005803 0167-9473 4 2005 Chi-square distribution, Maximum absolute error, Normal approximation, Power transformation 803-808 A normal approximation for the chi-square distribution article-journal https://www.sciencedirect.com/science/article/pii/S0167947304001069 48 abstract accessed author container-title id issn issue issued page publisher title type url volume The distribution of the χ2 test under the null hypothesis is studied, when the parameters are estimated by the method of moments. A general formula, applicable also to other situations, is given. Three examples are studied in more detail and numerical results are given, indicating how unsafe it can be to use a χ2 distribution with a number of degrees of freedom smaller than or equal to the number of cells. 2026-01-13 literal Molinari, L. Biometrika a71e7dcf-67b3-3b81-a0c2-17153341fab1 00063444, 14643510 1 1977 115-121 \[Oxford University Press, Biometrika Trust\] Distribution of the chi-squared test in nonstandard situations article-journal http://www.jstor.org/stable/2335780 64 abstract accessed author container-title id issn issue issued page publisher title type url volume A procedure for interpolating in tables of the F-distribution is proposed. This procedure is already known by many statisticians, although its original source has been forgotten. A table gives an upper bound for the relative error in the interpolation between some tabulated values. An example is given. 2026-01-13 literal Zinger, A. Journal of the Royal Statistical Society. Series C (Applied Statistics) b3bc6554-760a-35aa-92dd-1ef8068d343e 00359254, 14679876 1 1964 51-53 \[Royal Statistical Society, Oxford University Press\] [On Interpolation in Tables of the F-Distribution]{.nocase} article-journal http://www.jstor.org/stable/2985223 13 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-13 literal Wilson, E. B., & Hilferty, M. M. Proceedings of the National Academy of Sciences of the United States of America b71582f9-54b0-3efd-95a7-a60d1005dd3b 00278424, 10916490 12 1931 684-688 National Academy of Sciences The distribution of chi-square article-journal http://www.jstor.org/stable/86022 17 abstract accessed author container-title id issn issue issued page publisher title type url volume A relation is derived between the percentile points of a t-distribution with n degrees of freedom and those of an F-distribution with n and n degrees of freedom. In effect, the t-percentiles can be obtained by a simple transformation from the \"diagonal\" entries of an F-table. 2026-01-13 literal Cacoullos, T. Journal of the American Statistical Association c9bb1879-be2e-3251-9489-81b3cdab8ab2 01621459, 1537274X 310 1965 528-531 \[American Statistical Association, Taylor & Francis, Ltd.\] [A Relation Between t and F-Distributions]{.nocase} article-journal http://www.jstor.org/stable/2282687 60 author container-title doi id issue issued page title type volume literal Saunders, I. W., & Moran, P. A. P. Journal of Applied Probability 10.2307/3213414 Saunders_Moran_1978 2 1978 426-432 [On the quantiles of the gamma and F distributions]{.nocase} article-journal 15 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-13 literal Patnaik, P. B. Biometrika e800b171-dd42-3678-b651-8375c1fde110 00063444, 14643510 1/2 1949 202-232 \[Oxford University Press, Biometrika Trust\] [The Non-Central χ2- and F-Distribution and their Applications]{.nocase} article-journal http://www.jstor.org/stable/2332542 36 abstract author container-title doi id issn issue issued page title type url volume One of the most interesting problems studied in Random Field Theory (RFT) is to approximate the distribution of the maximum of a random field. This problem usually appears in a general hypothesis testing framework, where the statistics of interest are the maximum of a random field of a known distribution. In this paper, we use the RFT approach to compare two independent correlation random fields, R1 and R2. Our statistics of interest are the maximum of a random field G, resulting from the difference between the Fisher's Z transformation of R1 and R2, respectively. The Fisher's Z transformation guarantees a Gaussian distribution at each point of G but, unfortunately, G is not transformed into a Gaussian random field. Hence, standard results of RFT for Gaussian random fields are not longer available for G. We show here that the distribution of the maximum of G can still be approximated by the distribution of the maximum of a Gaussian random field, provided there is some correction by its spatial smoothness. Indeed, we present a general setting to obtain this correction. This is done by allowing different smoothness parameters for the components of G. Finally, the performance of our method is illustrated by means of both numerical simulations and real Electroencephalography data, recorded during a face recognition experimental paradigm. literal Carbonell, F., Worsley, K. J., & Trujillo-Barreto, N. J. Statistics & Probability Letters 10.1016/j.spl.2008.11.007 CARBONELL2009780 0167-7152 6 2009 780-788 [On the Fisher's Z transformation of correlation random fields]{.nocase} article-journal https://www.sciencedirect.com/science/article/pii/S0167715208005154 79 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-13 literal Hjelm, H. F., & Norris, R. C. The Journal of Experimental Education 67d2aa04-e169-3e9d-9c2d-31d37c595ef4 00220973, 19400683 3 1962 269-277 Taylor & Francis, Ltd. [Empirical Study of the Efficacy of Fisher's Z-Transformation]{.nocase} article-journal http://www.jstor.org/stable/20156574 30 author container-title doi id issue issued page title type volume literal Bond, C. F., & Richardson, K. Psychometrika 10.1007/BF02295945 Bond_Richardson_2004 2 2004 291-303 [Seeing the Fisher Z-transformation]{.nocase} article-journal 69 author container-title doi id issue issued page title type volume literal Mendoza, J. L. Psychometrika 10.1007/BF02294830 Mendoza_1993 4 1993 601-615 Fisher transformations for correlations corrected for selection and missing data article-journal 58 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-13 literal David, F. N. Biometrika fcdfa5ac-5e28-3f56-8f5f-82b584b870a3 00063444, 14643510 3/4 1949 394-403 \[Oxford University Press, Biometrika Trust\] [The Moments of the z and F Distributions]{.nocase} article-journal http://www.jstor.org/stable/2332676 36 abstract author container-title doi id issued page title type volume A new explicit quadratic radical function is found by numerical experiments, which is simpler and has only 70.778 literal Yang, Z., Duan, Z., Wang, J., Wang, T., Song, Y., & Zhang, J. Transactions of Tianjin University 10.1007/s12209-013-1978-8 Yang2013 2013 381-384 [Quadratic radical function better than fisher Z transformation]{.nocase} article-journal 5 abstract author container-title doi id isbn issued page publisher publisher-place title type url There are many different frequency distributions in statistics, some of which we shall meet later in the book, but one in particular applies to the theory of errors. This is the normal or Gaussian distribution. The word 'normal' is here used in its original sense of 'ideal', for it was believed for a long time that this distribution was a reflection of a universal and fundamental law of nature. If certain assumptions are made it can be 'proved' that the accumulation of a large number of small independent random deviations follows a normal law, but belief in the rigour of this proof is no longer widespread. There is a well-known quotation on the subject: 'Everybody believes in the Gaussian law of errors; the experimenters because they think it will be proved by mathematics; and the mathematicians because they believe it has been established by observation.' However, for our present purposes we can assume that the Gaussian distribution is the best possible representation of the variation of small random errors. literal Leaver, R. H., & Thomas, T. R. Analysis and presentation of experimental results 10.1007/978-1-349-01942-7_4 Leaver1974 978-1-349-01942-7 1974 30-37 Macmillan Education UK London Normal distribution chapter https://doi.org/10.1007/978-1-349-01942-7_4 abstract author container-title doi id isbn issued page publisher publisher-place title type url The phrases 'it is probable that', 'it is very likely that', 'there is virtually no chance' are often used in conversation. They are used to describe events which could possibly happen. They represent our estimates of the probability of the events happening. They are sometimes based on personal feelings, but more often they are based on relative frequency, i.e. the proportion of times such an event has happened previously. These subjective or intuitive ideas of probability are basic and fundamental in life and they give rise to two definitions of probability which are now discussed. literal Mulholland, H., & Jones, C. R. Fundamentals of statistics 10.1007/978-1-4899-6507-3_3 Mulholland1968a 978-1-4899-6507-3 1968 32-58 Springer US Boston, MA Elementary probability chapter https://doi.org/10.1007/978-1-4899-6507-3_3 abstract author container-title doi id isbn issued page publisher publisher-place title type url The normal distribution is probably the most widely studied distribution in statistics because :(a)Distributions which are approximately normal are frequently encountered, e.g. most sets of random errors follow the normal distribution.(b)The normal distribution is important as a 'limiting distribution', i.e. it can be used as an approximation to other distributions (see sections 8.7 and 8.8).(c)The normal distribution is easy to use.(d)It has been shown that the results obtained by assuming a non-normal population to be normally distributed are reasonably accurate when the departure from normality is not too severe.(e)The central limit theorem shows that the means of samples of size n from any population are approximately normally distributed. The approximation improves as n gets bigger. literal Mulholland, H., & Jones, C. R. Fundamentals of statistics 10.1007/978-1-4899-6507-3_8 Mulholland1968b 978-1-4899-6507-3 1968 122-138 Springer US Boston, MA The normal distribution chapter https://doi.org/10.1007/978-1-4899-6507-3_8 abstract author container-title doi id isbn issued page publisher publisher-place title type url Bedeutung der Normalverteilung: Die Normalverteilung ist ein wichtiges Modell: es beschreibt die Streuung von Meßwerten um ihren Mittelwert. Meßfehler sind als zufällige Fehler \[unvermeidbar, symmetrisch, meist klein\] oft angenähert normalverteilt. Der Praktiker muß sich damit abfinden, daß es, streng genommen, in der Empirie keine Normalverteilung gibt. Indessen lassen sich viele mehr oder weniger symmetrisch-eingipflig verteilte Beobachtungen zumindest in ihrem mittleren Bereich als angenähert normalverteilt auffassen. literal Sachs, L. Statistische methoden: Planung und auswertung 10.1007/978-3-642-77717-2_4 Sachs1993 978-3-642-77717-2 1993 41-54 Springer Berlin, Heidelberg Normalverteilung chapter https://doi.org/10.1007/978-3-642-77717-2_4 author container-title doi id isbn issued page publisher publisher-place title type url literal Saneii, S. H., & Doosti, H. Practical biostatistics for medical and health sciences 10.1007/978-981-97-3083-4_6 Saneii2024 978-981-97-3083-4 2024 115-137 Springer Singapore Normal distribution chapter https://doi.org/10.1007/978-981-97-3083-4_6 author container-title doi editor id issued publisher publisher-place title type literal Wesolowski, B., & Musselwhite Thompson, D. The SAGE encyclopedia of educational research, measurement, and evaluation 10.4135/9781506326139.n476 literal Frey, B. B. Wesolowski2018 2018 University of Kansas USA Normal distribution chapter author container-title doi id issued page title type volume literal Benzon, B. St open 10.48188/so.2.6 Benzon2021 2021-07 1-16 The normal distribution, an epistemological view article-journal 2 abstract author container-title doi id issue issued keyword page title type url volume Abstract It is less emphasized in scientific literature that centering and scaling of data may drastically change the original distribution of the data for small samples. The destruction of the original distribution depends on the source of the estimation of the mean (centering) and the divisor (scaling) where the latter is connected to the spread of data. In our comparative study we focus on cases, where the sample is taken from normally distributed data; the means and the standard deviations are population or sample based. We discuss six cases in transforming the data or the sample means. Most of them are studied previously, but some of them have not been theoretically investigated. The transformed data follow normal distribution in three cases, if the scaling is performed with population standard deviation. In one case, the final distribution is related to β-distribution with astonishing density functions for N = 2 Dirac-delta, N = 3 Viking helmet like and N = 4 uniform distributions. Another case is the well-known t-distribution. For one of the transformed data, we were not able to identify the general form. Here we obtained only numerical results for 3 ≤ N. The effect of the transformations was tested on experimental data representing more or less normally distributed variables. We found that transformations using the sample standard deviation were significantly less normally distributed-like than the original data for small samples, but the other transformations enhanced the normal distribution-like feature. The results point out that centering and especially scaling require consideration for small samples up to questioning the reality of subsequent data evaluation processes where normal distribution is assumed. Copyright © 2011 John Wiley & Sons, Ltd. literal Tóth, G. Journal of Chemometrics 10.1002/cem.1382 https://doi.org/10.1002/cem.1382 5 2011 centering, scaling, studentization, standardization, normal distribution 247-253 Destruction of normal distribution in small samples by centering and scaling article-journal https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/abs/10.1002/cem.1382 25 author container-title doi id issue issued page title type volume literal Brereton, R. G. Journal of Chemometrics 10.1002/cem.2655 https://doi.org/10.1002/cem.2655 11 2014 789-792 The normal distribution article-journal 28 abstract author container-title doi id issn issue issued keyword page title type url volume Let X1:3≤X2:3≤X3:3 be the order statistics of the independent and identically distributed random variables X1, X2, and X3. Using the regressional property E\[(aX2:3−X1:3)2\|X2:3=x\]=E\[(X3:3−aX2:3)2\|X2:3=x\], we characterize some distributions, including Student's tν distributions (ν\>2) and normal distributions. literal Wang, J. Statistics & Probability Letters 10.1016/j.spl.2011.03.037 WANG2011903 0167-7152 8 2011 Characterization, Order statistics, Regressional property 903-906 A simple characterization of student's distributions and normal distributions article-journal https://www.sciencedirect.com/science/article/pii/S0167715211001283 81 abstract collection-title container-title doi editor id issn issued page publisher title type url volume Publisher Summary This chapter focuses on the concept of normal distribution. It also describes population parameters and their estimators. To summarize the characteristics of a distribution, its moments can be used. The best known probability distribution is the normal distribution. Distributions can be non-normal and procedures or tests are needed to detect this departure from normality. A graphical procedure is described in the chapter that permits to indicate whether a distribution is normal or not. The graphical procedure applied is called the "rankit procedure." A bacteriological example provides a clue to the way to make non-normal distributions normal--- namely, by transformation. The transformation carried out in the chapter is called the "log-transformation." Log-normal distributions are frequently found in nature, particularly when the variable studied has a natural zero (such as weight, length, etc.). In this case, simple normality around the mean could include negative values. Data handling in science and technology Handbook of chemometrics and qualimetrics: Part a 10.1016/S0922-3487(97)80033-0 literal Massart, D. L., Vandeginste, B. G. M., Buydens, L. M. C., De Jong, S., Lewi, P. J., & Smeyers-Verbeke, J. 199847 0922-3487 1998 47-72 Elsevier Chapter 3 the normal distribution chapter https://www.sciencedirect.com/science/article/pii/S0922348797800330 20 abstract author container-title doi id issue issued page title type volume It is usually supposed that the central limit theorem explains why various quantities we find in nature are approximately normally distributed---people's heights, examination grades, snowflake sizes, and so on. This sort of explanation is found in many textbooks across the sciences, particularly in biology, economics, and sociology. Contrary to this received wisdom, I argue that in many cases we are not justified in claiming that the central limit theorem explains why a particular quantity is normally distributed, and that in some cases, we are actually wrong. 1 Introduction2 Normal Distributions and the Central Limit Theorem  2.1 Normal distributions  2.2 The central limit theorem  2.3 Terminology3 Explaining Normality  3.1 Loaves of bread  3.2 Varying variances and probability densities  3.3 Tensile strengths and problems with summation  3.4 Products of factors and log-normal distributions  3.5 Transforming factors and sub-factors  3.6 Transformations of quantities  3.7 Quantitative genetics  3.8 Inference to the best explanation4 Maximum Entropy Explanations5 Conclusion literal Lyon, A. The British Journal for the Philosophy of Science 10.1093/bjps/axs046 doi:10.1093/bjps/axs046 3 2014 621-649 Why are normal distributions normal? article-journal 65 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-14 literal Stahl, S. Mathematics Magazine 2e210a69-a2e7-3fa7-96e2-b30efac9ed56 0025570X, 19300980 2 2006 96-113 Mathematical Association of America The evolution of the normal distribution article-journal http://www.jstor.org/stable/27642916 79 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-14 literal Breitenberger, E. Biometrika 8ccd5f0b-f3d0-3894-a051-95d6e3d4ed98 26404 1/2 1963 81-88 \[Oxford University Press, Biometrika Trust\] Analogues of the normal distribution on the circle and the sphere article-journal http://www.jstor.org/stable/2333749 50 accessed author container-title id issn issue issued page publisher title title-short type url 2026-01-14 literal Thome, H. Historical Social Research / Historische Sozialforschung. Supplement 57fba293-9d42-3fd4-9780-f77567f507d7 09366784 3 1990 1-275 GESIS - Leibniz-Institute for the Social Sciences, Center for Historical Social Research GRUNDKURS STATISTIK FÜR HISTORIKER TEIL II: INDUKTIVE STATISTIK UND REGRESSIONSANALYSE GRUNDKURS STATISTIK FÜR HISTORIKER TEIL II article-journal http://www.jstor.org/stable/40986007 abstract accessed author container-title id issn issue issued page publisher title type url volume We study the asymptotic covariance function of the sample mean and quantile, and derive a new and surprising characterization of the normal distribution: the asymptotic covariance between the sample mean and quantile is constant across all quantiles, if and only if the underlying distribution is normal. This is a powerful result and facilitates statistical inference. Utilizing this result, we develop a new omnibus test for normality based on the quantile-mean covariance process. Compared to existing normality tests, the proposed testing procedure has several important attractive features. Monte Carlo evidence shows that the proposed test possesses good finite sample properties. In addition to the formal test, we suggest a graphical procedure that is easy to implement and visualize in practice. Finally, we illustrate the use of the suggested techniques with an application to stock return datasets. 2026-01-14 literal Bera, A. K., Galvao, A. F., Wang, L., & Xiao, Z. Econometric Theory adfd1cdd-be2e-33b1-bf94-693f4754181e 02664666, 14694360 5 2016 1216-1252 Cambridge University Press A New Characterisazion Of The Normal Distribution And Test For Normality article-journal http://www.jstor.org/stable/43948011 32 author container-title doi id issued page publisher title type literal von Storch, H., & Zwiers, F. W. Statistical analysis in climate research 10.1017/CBO9780511612336 Storch_Zwiers_1999 1999 419-420 Cambridge University Press Normal density and cumulative distribution function chapter abstract author container-title doi id isbn issued page publisher publisher-place title type url The Poisson distribution is a discontinuous distribution which was described by the French mathematician Siméon Denis Poisson (1781-2013;1840) in a book published in 1837. The Poisson distribution is a particular case of the binomial distribution (chapter 17) where the probability p becomes smaller and smaller (p `\textrightarrow`{=latex} 0) while the exponent k becomes larger and larger (k`\textrightarrow`{=latex}∞) in such a way that their product, the mean, $\mu$= kp, keeps a finite value (0 \< $\mu$ = kp \< ∞). The Poisson distribution might thus be defined by the following equation literal Jolicoeur, P. Introduction to biometry 10.1007/978-1-4615-4777-8_19 Jolicoeur1999 978-1-4615-4777-8 1999 124-133 Springer Boston, MA The poisson distribution chapter https://doi.org/10.1007/978-1-4615-4777-8_19 author container-title doi id issued title type volume literal Zhang, Y. Advances in Economics, Management and Political Sciences 10.54254/2754-1169/2025.BJ24761 Zhang2025 2025 Poisson distribution and its applications article-journal 196 abstract author container-title doi id isbn issued page publisher publisher-place title type url The Poisson distribution is of great importance in theory and in practice. It has the added virtue of being a simple mathematical object. We could have introduced it at an earlier stage in the book, and the reader was alerted to this in `\textsection`{=latex}4.4. However, the belated entrance will give it more prominence, as well as a more thorough discussion than would be possible without the benefit of the last two chapters. literal Chung, K. L. Elementary probability theory with stochastic processes 10.1007/978-1-4757-3973-2_7 Chung1974 978-1-4757-3973-2 1974 192-239 Springer New York, NY Poisson and normal distributions chapter https://doi.org/10.1007/978-1-4757-3973-2_7 abstract accessed author container-title id issn issue issued page publisher title title-short type url volume Social determinists have argued that the occurrence of independent discoveries and inventions demonstrates the inevitability of techno-scientific progress. Yet the frequency of such multiples may be adequately predicted by a probabilistic model, especially the Poisson model suggested by Price. A detailed inquiry reveals that the Poisson distribution can predict almost all the observed variation in the frequency distribution of multiples collected by Merton, and by Ogburn and Thomas. This study further indicates that: (a) the number of observed multiples may be greatly underestimated, particularly those involving few independent contributors; (b) discoveries and inventions are not sufficiently probable to avoid a large proportion of total failures, and hence techno-scientific advance is to a large measure indeterminate; (c) chance or 'luck' seems to play such a major part that the 'great genius' theory is no more tenable than the social deterministic theory. 2026-01-12 literal Simonton, D. K. Social Studies of Science a2ab11c2-825b-3d03-aa88-542d1bf1629c 812631 4 1978 521-532 Sage Publications, Ltd. Independent discovery in science and technology: A closer look at the poisson distribution Independent discovery in science and technology article-journal http://www.jstor.org/stable/284821 8 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-12 literal Crow, E. L. Biometrika beaf2bb2-3f3e-3d37-837e-422ad120324f 00063444, 14643510 3/4 1958 556-559 \[Oxford University Press, Biometrika Trust\] The mean deviation of the poisson distribution article-journal http://www.jstor.org/stable/2333201 45 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-12 literal Rao, C. R., & Chakravarti, I.M. Biometrics c5daa63c-82d0-3414-9df1-e64522b8f21e 0006341X, 15410420 3 1956 264-282 \[Wiley, International Biometric Society\] Some small sample tests of significance for a poisson distribution article-journal http://www.jstor.org/stable/3001466 12 author collection-title id isbn issued publisher title type url literal Haight, F. A. Operations research society of america. Publications in operations research haight1967handbook 9788391014424 1967 Wiley Handbook of the poisson distribution book https://books.google.com/books?id=l8Y-AAAAIAAJ author container-title doi id issue issued page title title-short type volume literal Finkelshtein, D., Malyarenko, A., Mishura, Y., & Ralchenko, K. Methodology and Computing in Applied Probability 10.1007/s11009-025-10171-9 Finkelshtein2025 2 2025 45 Entropies of the poisson distribution as functions of intensity: "Normal" and "anomalous" behavior Entropies of the poisson distribution as functions of intensity article-journal 27 author id isbn issued publisher publisher-place title type url literal Cohen, J. Cohen1977 978-0-12-179060-8 1977 Elsevier Academic Press Amsterdam Statistical power analysis for the behavioral science book https://doi.org/10.1016/C2013-0-10517-X author doi edition id issued publisher publisher-place title type literal Cohen, J. 10.4324/9780203771587 2 Cohen1988 1988 Lawrence Erlbaum Associates Hillsdale, NJ Statistical power analysis for the behavioral science book author container-title id issue issued page title type url volume literal Cohen, J. Psychological Bulletin Cohen1992 1 1992 155-159 A power primer article-journal https://doi.org/10.1037/0033-2909.112.1.155 112 abstract author container-title doi id isbn issued page publisher publisher-place title type url Obwohl signifikante Ergebnisse oft gewünscht werden, sagt -- wie in diesem Kapitel aufgezeigt wird -- die bloße Signifikanz eines Tests nichts über die Stärke eines Effekts aus, und man kann sich berechtigt fragen: Bedeutet statistische Signifikanz auch immer ˝inhhaltliche Relevanz˝ bzw. ˝praktische Bedeutsamkeit˝? In diesem Kapitel werden diejenigen Konzepte eingeführt, die zur Beantwortung dieser Frage benötigt werden, und begonnen wird dabei mit einer systematischen Betrachtung statistischer Entscheidungen. Im Anschluss wird Cohen's d als Maß der Effektstärke für die bisher behandelten t-Tests betrachtet. Das Konzept der Effektstärke führt schließlich zur Power (Teststärke) eines Signifikanztests und der Frage nach dem optimalen Stichprobenumfang. literal Janczyk, M., & Pfister, R. Inferenzstatistik verstehen: Von a wie signifikanztest bis z wie konfidenzintervall 10.1007/978-3-642-34825-9_7 Janczyk2013 978-3-642-34825-9 2013 77-90 Springer Berlin, Heidelberg Fehlertypen, effektstärken und power chapter https://doi.org/10.1007/978-3-642-34825-9_7 author container-title doi id issued title type literal Brandmaier, A. M. PsyArXiv 10.31234/osf.io/n9ta7_v1 Brandmaier2025 2025 Stop calling cohen's d an effect size article-journal author container-title doi id issue issued page title title-short type volume literal Fritz, C. O., Morris, P. E., & Richler, J. J. Journal of Experimental Psychology: General 10.1037/a0024338 Fritz2012 1 2012 2-18 Effect size estimates: Current use, calculations, and interpretation Effect size estimates article-journal 141 abstract author container-title doi id issn issued title title-short type url volume Effect sizes are the currency of psychological research. They quantify the results of a study to answer the research question and are used to calculate statistical power. The interpretation of effect sizes---when is an effect small, medium, or large?---has been guided by the recommendations Jacob Cohen gave in his pioneering writings starting in 1962: Either compare an effect with the effects found in past research or use certain conventional benchmarks. The present analysis shows that neither of these recommendations is currently applicable. From past publications without pre-registration, 900 effects were randomly drawn and compared with 93 effects from publications with pre-registration, revealing a large difference: Effects from the former (median r = .36) were much larger than effects from the latter (median r = .16). That is, certain biases, such as publication bias or questionable research practices, have caused a dramatic inflation in published effects, making it difficult to compare an actual effect with the real population effects (as these are unknown). In addition, there were very large differences in the mean effects between psychological sub-disciplines and between different study designs, making it impossible to apply any global benchmarks. Many more pre-registered studies are needed in the future to derive a reliable picture of real population effects. literal Schäfer, T., & Schwarz, M. A. Frontiers in Psychology 10.3389/fpsyg.2019.00813 10.3389/fpsyg.2019.00813 1664-1078 2019 The meaningfulness of effect sizes in psychological research: Differences between sub-disciplines and the impact of potential biases The meaningfulness of effect sizes in psychological research article-journal https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2019.00813 Volume 10 - 2019 abstract author container-title doi id issn issued keyword page title type url volume Current statistical inference methods for task-fMRI suffer from two fundamental limitations. First, the focus is solely on detection of non-zero signal or signal change, a problem that is exacerbated for large scale studies (e.g. UK Biobank, N=40,000+) where the 'null hypothesis fallacy' causes even trivial effects to be determined as significant. Second, for any sample size, widely used cluster inference methods only indicate regions where a null hypothesis can be rejected, without providing any notion of spatial uncertainty about the activation. In this work, we address these issues by developing spatial Confidence Sets (CSs) on clusters found in thresholded Cohen's d effect size images. We produce an upper and lower CS to make confidence statements about brain regions where Cohen's d effect sizes have exceeded and fallen short of a non-zero threshold, respectively. The CSs convey information about the magnitude and reliability of effect sizes that is usually given separately in a t-statistic and effect estimate map. We expand the theory developed in our previous work on CSs for literal Bowring, A., Telschow, F. J. E., Schwartzman, A., & Nichols, T. E. NeuroImage 10.1016/j.neuroimage.2020.117477 BOWRING2021117477 1053-8119 2021 Confidence sets, fMRI, Task fmri, Cohen's , Effect sizes 117477 Confidence sets for cohen's d effect size images article-journal https://www.sciencedirect.com/science/article/pii/S1053811920309629 226 abstract author container-title doi id issn issue issued keyword page title title-short type url volume Objectives Meta-analyses typically compute a treatment effect size (Cohen's d), which is readily converted to another common measure, the binomial effect size display (BESD). BESD is the correlation coefficient and represents a percentage difference in outcome attributable to an intervention. Both d and BESD are in arbitrary units; neither measures the absolute change resulting from intervention. The method used to estimate absolute change from BESD assumes both a 50-50 split of the outcome and a balanced design. Consequently, inaccurate assumptions underpin most meta-analytic estimates of the gain resulting from an intervention (and of its cost effectiveness). This article develops an exact formula without these assumptions. Methods The formula is developed algebraically from 1) the formula for the correlation coefficient represented as a 2-by-2 contingency table constructed from the relative size of the treatment and control groups and the percentage of people who would have the condition absent intervention, and 2) the BESD correlation coefficient formula showing change in success probability with treatment. Results Simulation reveals that BESD only approximates the reduction in the outcome an intervention might well achieve when the problem outcome occurs in 35 Conclusion It is time to retire BESD. Our equations can also guide effect size estimation from difficult articles. literal Miller, T. R., Hendrie, D., & Derzon, J. Value in Health 10.1016/j.jval.2010.10.013 MILLER2011144 1098-3015 1 2011 Absolute change, BESD, Effect size, Meta-analysis 144-151 Exact method for computing absolute percent change in a dichotomous outcome from meta-analytic effect size: Improving impact and cost-outcome estimates Exact method for computing absolute percent change in a dichotomous outcome from meta-analytic effect size article-journal https://www.sciencedirect.com/science/article/pii/S1098301510000148 14 abstract accessed author container-title id issn issue issued page publisher title title-short type url volume Given the long history of discussion of issues surrounding statistical testing and effect size indices and various attempts by the American Psychological Association and by the American Educational Research Association to encourage the reporting of effect size, most journals in education and psychology have witnessed an increase in effect size reporting since 1999. Yet, effect size was often reported in three indices, namely, the unadjusted R², Cohen's d, and η² with a simple labeling of small, medium, or large, according to Cohen's (1969) criteria. In this article, the authors present several alternatives to Cohen's d to help researchers conceptualize effect size beyond standardized mean differences for between-subject designs with two groups. The alternative effect size estimators are organized into a typology and are empirically contrasted with Cohen's d in terms of purposes, usages, statistical properties, interpretability, and the potential for meta-analysis. Several sound alternatives are identified to supplement the reporting of Cohen's d. The article concludes with a discussion of the choice of standardizers, the importance of assumptions, and the possibility of extending sound alternative effect size indices to other research contexts. 2026-01-14 literal Peng, C.-Y. J., & Chen, L.-T. The Journal of Experimental Education abb9e70f-419e-3a38-8754-462a7041475a 00220973, 19400683 1 2014 22-50 Taylor & Francis, Ltd. Beyond cohen's d: Alternative effect size measures for between-subject designs Beyond cohen's d article-journal https://www.jstor.org/stable/26594399 82 abstract accessed author container-title id issn issue issued page publisher title title-short type url volume Reporting effect size plays an integral role in educational and psychological research and is required by many journals. Certainly, the best-known measure of effect size is Cohen's d, which represents a substantial improvement over using p values. But Cohen's d is known to suffer from some fundamental concerns. The author's goal was to illustrate some graphical methods aimed at addressing those concerns. These methods can be applied in a wide range of situations, including situations in which the Wilcoxon-Mann-Whitney test is used. 2026-01-14 literal Wilcox, R. R. The Journal of Experimental Education d04b907b-1c2b-329b-902f-ade87a9fd109 00220973, 19400683 4 2006 353-367 Taylor & Francis, Ltd. Graphical methods for assessing effect size: Some alternatives to cohen's d Graphical methods for assessing effect size article-journal http://www.jstor.org/stable/20157436 74 abstract author container-title doi id issue issued page title type volume The size of the effect of the difference in two groups with respect to a variable of interest may be estimated by the classical Cohen's d. A recently proposed generalized estimator allows conditioning on further independent variables within the framework of a linear regression model. In this note, it is demonstrated how unbiased estimation of the effect size parameter together with a corresponding standard error may be obtained based on the non-central t distribution. The portrayed estimator may be considered as a natural generalization of the unbiased Hedges' g. In addition, confidence interval estimation for the unknown parameter is demonstrated by applying the so-called inversion confidence interval principle. The regarded properties collapse to already known ones in case of absence of any additional independent variables. The stated remarks are illustrated with a publicly available data set. literal Groß, J., & Möller, A. Statistical Papers 10.1007/s00362-023-01527-9 Groß2024 6 2024 3971-3979 Some additional remarks on statistical properties of cohen's d in the presence of covariates article-journal 65 abstract author container-title doi id issue issued page title type volume In this note, we introduce a generalized formula for Cohen's d under the presence of additional independent variables, providing a measure for the size of a possible effect concerning the size of a difference location effect of a variable in two groups. This is done by employing the so-called Frisch--Waugh--Lovell theorem in a partitioned linear regression model. The generalization is motivated by demonstrating the relationship to appropriate t and F statistics. Our discussion is further illustrated by inference about a publicly available data set. literal Groß, J., & Möller, A. Journal of Statistical Theory and Practice 10.1007/s42519-023-00323-w Groß2023 2 2023 22 A note on cohen's d from a partitioned linear regression model article-journal 17 abstract author container-title doi id issue issued page title type volume In psychological science, the "new statistics" refer to the new statistical practices that focus on effect size (ES) evaluation instead of conventional null-hypothesis significance testing (Cumming, Psychological Science, 25, 7--29, 2014). In a two-independent-samples scenario, Cohen's (1988) standardized mean difference (d) is the most popular ES, but its accuracy relies on two assumptions: normality and homogeneity of variances. Five other ESs---the unscaled robust d (dr\*; Hogarty & Kromrey, 2001), scaled robust d (dr; Algina, Keselman, & Penfield, Psychological Methods, 10, 317--328, 2005), point-biserial correlation (rpb; McGrath & Meyer, Psychological Methods, 11, 386--401, 2006), common-language ES (CL; Cliff, Psychological Bulletin, 114, 494--509, 1993), and nonparametric estimator for CL (Aw; Ruscio, Psychological Methods, 13, 19--30, 2008)---may be robust to violations of these assumptions, but no study has systematically evaluated their performance. Thus, in this simulation study the performance of these six ESs was examined across five factors: data distribution, sample, base rate, variance ratio, and sample size. The results showed that Awand drwere generally robust to these violations, and Awslightly outperformed dr. Implications for the use of Awand drin real-world research are discussed. literal Li, J. C.-H. Behavior Research Methods 10.3758/s13428-015-0667-z Li2016 4 2016 1560-1574 Effect size measures in a two-independent-samples case with nonnormal and nonhomogeneous data article-journal 48 author container-title doi id issue issued page publisher title type url volume literal Efron, B. The Annals of Statistics 10.1214/aos/1176344552 Efron1979 1 1979 1-26 Institute of Mathematical Statistics [Bootstrap Methods: Another Look at the Jackknife]{.nocase} article-journal https://doi.org/10.1214/aos/117634455 7 abstract author container-title doi id issn issue issued page title title-short type url volume We discuss several nonparametric methods for attaching a standard error to a point estimate: the jackknife, the bootstrap, half-sampling, subsampling, balanced repeated replications, the infinitesimal jackknife, influence function techniques and the delta method. The last three methods are shown to be identical. All the methods derive from the same basic idea, which is also the idea underlying the common parametric methods. Extended numerical comparisons are made for the special case of the correlation coefficient. literal Efron, B. Biometrika 10.1093/biomet/68.3.589 Efron1981 0006-3444 3 1981-12 589-599 Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods Nonparametric estimates of standard error article-journal https://doi.org/10.1093/biomet/68.3.589 68 author collection-title doi id issued publisher publisher-place title type url literal Efron, B. CBMS-NSF regional conference series in applied mathematics, monograph 38 10.1137/1.9781611970319 Efron1982 1982 SIAM, Society for Industrial and Applied Mathematics Philadelphia The jackknife, the bootstrap and other resampling plans book https://epubs.siam.org/doi/abs/10.1137/1.9781611970319 abstract author container-title doi id isbn issued page publisher publisher-place title type url When testing hypotheses (or computing confidence intervals) with the one-sample Student's T method described in Chapter 5, the central limit theorem tells us that Student's T performs better as the sample size increases. That is, under random sampling the discrepancy between the nominal and actual Type I error probability will go to zero as the sample size goes to infinity. But unfortunately, for reasons outlined in Section 5.3 of Chapter 5, there are realistic situations where about two hundred observations are needed to get satisfactory control over the probability of a Type I error or accurate probability coverage when computing confidence intervals. When comparing the population means of two groups of individuals, using Student's T is known to be unsatisfactory when sample sizes are small or even moderately large. In fact, it might be unsatisfactory no matter how large the sample sizes are because under general conditions it does not converge to the correct answer (Cressie and Whitford, 1986). Switching to the test statistic W given by Equation 5.3, the central limit theorem now applies under general conditions, so using W means we will converge to the correct answer as the sample sizes increase, but in some cases we again need very large sample sizes to get accurate results. (There are simple methods for improving the performance of W using what are called estimated degrees of freedom, but the improvement remains highly unsatisfactory for a wide range of situations.) Consequently, there is interest in finding methods that beat our reliance on the central limit theorem as it applies to these techniques. That is, we would like to find a method that converges to the correct answer more quickly as the sample sizes get large, and such a method is described here. literal Wilcox, R. R. Fundamentals of modern statistical methods: Substantially improving power and accuracy 10.1007/978-1-4757-3522-2_6 Wilcox2001 978-1-4757-3522-2 2001 93-115 Springer New York, NY The bootstrap chapter https://doi.org/10.1007/978-1-4757-3522-2_6 author container-title doi id issue issued page title type volume literal Politis, D. N. Statistical Science 10.1214/ss/1063994977 politis2003 2 2003 219-230 The impact of bootstrap methods on time series analysis article-journal 18 abstract author container-title doi id isbn issued page publisher publisher-place title type url In this chapter we introduce basic concepts and methods of statistical inference. The focus is on estimating the parameters of statistical distributions and testing hypotheses about them. Problems of testing if certain distributions fit observed data are also considered. literal Kenett, R., Zacks, S., & Gedeck, P. Modern statistics: A computer-based approach with python 10.1007/978-3-031-07566-7_3 Kenett2022 978-3-031-07566-7 2022 139-223 Springer International Publishing Cham Statistical inference and bootstrapping chapter https://doi.org/10.1007/978-3-031-07566-7_3 abstract author container-title doi id isbn issued page publisher publisher-place title type url In this chapter, we will learn how to estimate the standard error of an estimate and how to construct confidence intervals for a parameter of interest. Both problems can be solved by the bootstrap method, which was inspired by the jackknife. The bootstrap, introduced in 1979 by Bradley Efron, is a general simulation-based method for measuring the uncertainty of estimates, in particular, for estimating their standard errors and constructing confidence intervals. Its beauty lies in its simplicity and universality: the bootstrap is fully automatic, requires no theoretical calculations, and is always available. literal Zuev, K. M. Fundamentals of statistical inference: Foundations of data analysis 10.1007/978-3-032-03848-7_6 Zuev2026 978-3-032-03848-7 2026 115-129 Springer Nature Switzerland Cham The bootstrap method chapter https://doi.org/10.1007/978-3-032-03848-7_6 abstract author container-title doi id issn issue issued title type url volume Typically, the traditional bootstrap methods for parameter inference of spatial error models suffer from high computational costs, so this study proposes fast cluster bootstrap methods for spatial error models to deal with the dilemma. The key idea is to calculate the sufficient statistics for each cluster before performing the bootstrap loop of the spatial error model, and based on these sufficient statistics, all quantities needed for bootstrap inference can be computed. Furthermore, this study performed Monte Carlo simulations, and the result reveals that compared with traditional bootstrap methods, our proposed methods can reduce the computational cost substantially and improve the reliability for obtaining the bootstrap test statistics and confidence intervals of the parameters for spatial error models. literal Zheng, Y., & Fan, H. Mathematics 10.3390/math13182913 math13182913 2227-7390 18 2025 Fast cluster bootstrap methods for spatial error models article-journal https://www.mdpi.com/2227-7390/13/18/2913 13 author container-title doi id issued title type literal Eidous, O. ResearchGate 10.13140/RG.2.2.12238.11846 eidous2025 2025-09 Does the bootstrap method effectively evaluate estimators? article-journal author container-title doi id issue issued keyword page title type volume literal Politis, D. N. IEEE Signal Processing Magazine 10.1109/79.647042 647042 1 1998 Statistical analysis;Sampling methods;Statistical distributions;Gaussian distribution;Shape;Density functional theory;Statistics 39-55 Computer-intensive methods in statistical analysis article-journal 15 abstract author container-title doi id issn issued keyword note page title type url volume Sustainability is of essential interest for many organizations and is defined as the ability to maintain existing resources at a certain rate or level when encountered with barriers. Factors affecting sustainability are categorized as enablers (capacities) and barriers (challenges) that have positive and negative effects on sustainability, respectively. To evaluate the status of sustainability, organizations need a measurement method to account for all the aspects of the sustainability classified into social, economic, and environmental tiers. Previously many researchers have provided indexes and measure specific to the studied field, which is not applicable to other areas. Furthermore, the proposed methods fail to cover all the aspects of sustainability. This study investigates a statistical method to measure the sustainability and the application of the bootstrap re-sampling method in order to overcome the problem with normality assumption when the sample size is not large enough and thus develop a more realistic stochastic model. The Bootstrap re-sampling method enables the unbiased estimation of population parameters such as mean and standard deviation. The proposed method is evaluated by comparing its results with those found in the literature. literal Khosravi, F., Izbirak, G., & Shavarani, S. M. Socio-Economic Planning Sciences 10.1016/j.seps.2020.100781 KHOSRAVI2021100781 0038-0121 2021 Sustainability, Bootstrap Re-Sampling, Statistical performance measurement, Enablers, Barriers Decision-Making for Environmental Sustainability 100781 Application of bootstrap re-sampling method in statistical measurement of sustainability article-journal https://www.sciencedirect.com/science/article/pii/S0038012119303416 75 abstract author collection-title doi editor id issn issued keyword page publisher title type url volume The bootstrap is a method for estimating the distribution of an estimator or test statistic by resampling one's data or a model estimated from the data. Under conditions that hold in a wide variety of econometric applications, the bootstrap provides approximations to distributions of statistics, coverage probabilities of confidence intervals, and rejection probabilities of hypothesis tests that are more accurate than the approximations of first-order asymptotic distribution theory. The reductions in the differences between true and nominal coverage or rejection probabilities can be very large. The bootstrap is a practical technique that is ready for use in applications. This chapter explains and illustrates the usefulness and limitations of the bootstrap in contexts of interest in econometrics. The chapter outlines the theory of the bootstrap, provides numerical illustrations of its performance, and gives simple instructions on how to implement the bootstrap in applications. The presentation is informal and expository. Its aim is to provide an intuitive understanding of how the bootstrap works and a feeling for its practical value in econometrics. family given Horowitz Joel L. Handbook of econometrics 10.1016/S1573-4412(01)05005-X literal Heckman, J. J., & Leamer, E. HOROWITZ20013159 1573-4412 2001 JEL classification:, C12, C13, C15 3159-3228 Elsevier Chapter 52 - the bootstrap chapter https://www.sciencedirect.com/science/article/pii/S157344120105005X 5 abstract accessed author container-title id issn issue issued page publisher title type url volume The contemporary development of bootstrap methods, from the time of Efron's early articles to the present day, is well documented and widely appreciated. Likewise, the relationship of bootstrap techniques to certain early work on permutation testing, the jackknife and cross-validation is well understood. Less known, however, are the connections of the bootstrap to research on survey sampling for spatial data in the first half of the last century or to work from the 1940s to the 1970s on subsampling and resampling. In a selective way, some of these early linkages will be explored, giving emphasis to developments with which the statistics community tends to be less familiar. Particular attention will be paid to the work of P. C. Mahalanobis, whose development in the 1930s and 1940s of moving-block sampling methods for spatial data has a range of interesting features, and to contributions of other scientists who, during the next 40 years, developed half-sampling, subsampling and resampling methods. 2026-01-14 literal Hall, P. Statistical Science 5d7fe90b-50e5-3e2c-9890-dc7b7f71d323 08834237 2 2003 158-167 Institute of Mathematical Statistics A short prehistory of the bootstrap article-journal http://www.jstor.org/stable/3182845 18 abstract accessed author container-title id issn issue issued page publisher title type url volume The bootstrap provides a simple and powerful means of assessing the quality of estimators. However, in settings involving large data sets---which are increasingly prevalent---the calculation of bootstrap-based quantities can be prohibitively demanding computationally. Although variants such as subsampling and the m out of n bootstrap can be used in principle to reduce the cost of bootstrap computations, these methods are generally not robust to specification of tuning parameters (such as the number of subsampled data points), and they often require knowledge of the estimator's convergence rate, in contrast with the bootstrap. As an alternative, we introduce the 'bag of little bootstraps' (BLB), which is a new procedure which incorporates features of both the bootstrap and subsampling to yield a robust, computationally efficient means of assessing the quality of estimators. The BLB is well suited to modern parallel and distributed computing architectures and furthermore retains the generic applicability and statistical efficiency of the bootstrap. We demonstrate the BLB's favourable statistical performance via a theoretical analysis elucidating the procedure's properties, as well as a simulation study comparing the BLB with the bootstrap, the m out of n bootstrap and subsampling. In addition, we present results from a large-scale distributed implementation of the BLB demonstrating its computational superiority on massive data, a method for adaptively selecting the BLB's tuning parameters, an empirical study applying the BLB to several real data sets and an extension of the BLB to time series data. 2026-01-14 literal Kleiner, A., Talwalkar, A., Sarkar, P., & Jordan, M. I. Journal of the Royal Statistical Society. Series B (Statistical Methodology) 71a9795f-441b-3377-aa30-ddc5824ffbf2 13697412, 14679868 4 2014 795-816 \[Royal Statistical Society, Wiley\] A scalable bootstrap for massive data article-journal http://www.jstor.org/stable/24774569 76 author container-title doi id issn issue issued page title type url volume literal Pewsey, A. Journal of the Royal Statistical Society Series D: The Statistician 10.2307/2348962 10.2307/2348962 2515-7884 1 2018-12 215-216 Exploring the limits of bootstrap article-journal https://doi.org/10.2307/2348962 43 abstract accessed author container-title id issn issue issued page publisher title type url volume Consistency of the bootstrap second moments does not usually follow from the proofs of consistency of the distribution of the bootstrap. Here it is shown that the convergence of the bootstrap distribution to a normal variate implicitly defines a consistent estimator for the asymptotic second moments. The estimator is based on the L-estimation of the scale parameter of arbitrary linear combinations of the bootstrap sequence and uses Classical Minimum Distance techniques to impose the positive semi-definiteness restrictions. 2026-01-14 literal Machado, J. A. F., & Parente, P. The Econometrics Journal e841cec6-d2b9-36dd-bee3-92abf3c9e224 13684221, 1368423X 1 2005 70-78 \[Royal Economic Society, Wiley\] Bootstrap estimation of covariance matrices via the percentile method article-journal http://www.jstor.org/stable/23114968 8 abstract accessed author container-title id issn issued page publisher title type url volume We propose a unifying principle which identifies a very broad class of hypotheses and statistics for which a suitable application of the n out of n bootstrap yields asymptotically correct critical values and power for contiguous alternatives. We also show that this attractive principle can fail in situations which the m out of n bootstrap can deal with (Bickel, Götze and van Zwet, 1997) (BGvZ). We formalize the m out of n bootstrap theory for testing and show that under mild conditions, it provides correct significance level, asymptotic power under contiguous alternatives, and consistency. We conclude with simulation results supporting the asymptotics. 2026-01-14 literal Bickel, P. J., & Ren, J.-J. Lecture Notes-Monograph Series ee45c30f-1813-36a2-b2a9-57ffa08fcc80 07492170 2001 91-112 Institute of Mathematical Statistics The bootstrap in hypothesis testing article-journal http://www.jstor.org/stable/4356107 36 abstract author container-title doi id issue issued page title type volume We introduce the almost goodness-of-fit test, a procedure to assess whether a (parametric) model provides a good representation of the probability distribution generating the observed sample. Specifically, given a distribution function F and a parametric family $$\mathcal {G}=\{ G(\varvec{\theta }): \varvec{\theta } \in \Theta \}$$, we consider the testing problem $$H_0: \Vert F - G(\varvec{\theta }_F) \Vert _p \ge \epsilon \quad \text {vs} \quad H_1: \Vert F - G(\varvec{\theta }_F) \Vert _p < \epsilon ,$$H0:‖F-G(θF)‖p≥ϵvsH1:‖F-G(θF)‖p\<ϵ,where $$\epsilon >0$$is a margin of error and $$G(\varvec{\theta }_F)$$denotes a representative of F within the parametric class. The approximate model is determined via an M-estimator of the parameters. The methodology also quantifies the percentage improvement of the proposed model relative to a non-informative (constant) benchmark. The test statistic is the $$\textrm{L}^p$$-distance between the empirical distribution function and that of the estimated model. We present two consistent, easy-to-implement, and flexible bootstrap schemes to carry out the test. The performance of the proposal is illustrated through simulation studies and analysis and real-data applications. literal Baíllo, A., & Cárcamo, J. Statistics and Computing 10.1007/s11222-025-10762-z Baíllo2025 1 2025 10 Bootstrap tests for almost goodness-of-fit article-journal 36 abstract author container-title doi id issued title title-short type Bootstrapping offers a flexible approach to estimating statistical power, when planning a new study based on a previous one. In this article, we explore both parametric and non-parametric bootstrap power calculations and demonstrate how to incorporate uncertainty about effect size in bootstrap power analysis. We use real examples to illustrate bootstrap power calculations in independent t-test and ANCOVA. Finally, we discuss the broader implications of bootstrap power calculation for a variety of research designs. literal Liu, X. S. Behaviormetrika 10.1007/s41237-025-00283-4 Liu2025 2025 Bootstrap power calculation: A flexible alternative to conventional power analysis for prospective and replication studies Bootstrap power calculation article-journal author doi id issued publisher publisher-place title type literal Galvani, A. 10.5479/sil.324681.39088000932442 galvani1791 1791 Ex Typographia Instituti Scientiarium Bononiae De viribus electricitatis in motu musculari commentarius book author container-title id issue issued title type url volume literal Caton, R. The Journal of Nervous and Mental Disease caton1875 4 1875 Electrical currents of the brain article-journal https://journals.lww.com/jonmd/fulltext/1875/10000/electrical_currents_of_the_brain.13.aspx 2 author container-title id issued page title type url volume literal Beck, A. A. Zentralblatt für Physiologie beck1890 1890 572-573 Die ströme der nervenzentren article-journal https://scholar.google.com/scholar_lookup?&title=Die%20Str%C3%B6me%20der%20Nervenzentren&journal=Zbl%20Physiol&volume=4&pages=572-573&publication_year=1890&author=Beck%2CA 4 author container-title doi id issue issued page title type volume literal Berger, H. Archiv für Psychiatrie und Nervenkrankheiten 10.1007/BF01797193 Berger1929 1 1929 527-570 Über das elektrenkephalogramm des menschen article-journal 87 author doi id isbn issued publisher publisher-place title title-short type literal Adrian, E. D. 10.9783/9781512809794 Adrian2016 9781512809794 2016 University of Pennsylvania Press Philadelphia The mechanism of nervous action: Electrical studies of the neurone The mechanism of nervous action book author container-title doi id issn issue issued page title type volume literal Tassinari C. A. Clinical Neurophysiology 10.1016/j.clinph.2019.10.010 TASSINARI20192258 1388-2457 12 2019 2258-2263 An electroencephalographer recalls the history of the federation on the 70th anniversary of its journal, clinical neurophysiology article-journal 130 abstract author container-title doi id issn issued page title title-short type volume On July 6th of 1924 Hans Berger --a German psychiatrist- first recorded electric signals from the human brainvia scalp electrodes. This date marks the beginning of Electroencephalography. In this review a representative panel of past and present Officers of the International Federation of Clinical Neurophysiology (IFCN) and of its Official Journal briefly summarizes the past, present and future of Electroencephalographic and related neurophysiological techniques' impact and the role of the IFCN in global collaboration, education, standardization, research innovation, and clinical practice. literal Rossini, P. M., Cole, J., Paulus, W., Ziemann, U., & Chen, R. Clinical Neurophysiology 10.1016/j.clinph.2024.11.021 ROSSINI2025132 1388-2457 2025 132-135 1924--2024: First centennial of EEG 1924--2024 article-journal 170 author collection-title doi edition id issued publisher publisher-place title type literal Grafakos, L. Graduate texts in mathematics 10.1007/978-3-031-56500-7 1 Grafakos2023 2024 Springer Cham Fundamentals of fourier analysis book author collection-title doi edition id issued publisher publisher-place title type literal Brigola, R. Texts in applied mathematics 10.1007/978-3-031-81311-5 1 Brigola2025 2025 Springer Cham Fourier analysis and distributions book author collection-title doi edition id issued publisher publisher-place title type literal Siuly, S., Li, Y., & Zhang, Y. Health information science 10.1007/978-3-319-47653-7 1 Siuly2017 2017 Springer Cham EEG signal analysis and classification book abstract author container-title doi id issue issued title type volume Electroencephalography (EEG) is a non-invasive measurement method for brain activity. Due to its safety, high resolution, and hypersensitivity to dynamic changes in brain neural signals, EEG has aroused much interest in scientific research and medical fields. This article reviews the types of EEG signals, multiple EEG signal analysis methods, and the application of relevant methods in the neuroscience field and for diagnosing neurological diseases. First, three types of EEG signals, including time-invariant EEG, accurate event-related EEG, and random event-related EEG, are introduced. Second, five main directions for the methods of EEG analysis, including power spectrum analysis, time--frequency analysis, connectivity analysis, source localization methods, and machine learning methods, are described in the main section, along with different sub-methods and effect evaluations for solving the same problem. Finally, the application scenarios of different EEG analysis methods are emphasized, and the advantages and disadvantages of similar methods are distinguished. This article is expected to assist researchers in selecting suitable EEG analysis methods based on their research objectives, provide references for subsequent research, and summarize current issues and prospects for the future. literal Zhang, H., Zhou, Q.-Q., Chen, H., Hu, X.-Q., Li, W.-G., Bai, Y., Han, J.-X., Wang, Y., Liang, Z.-H., Chen, D., Cong, F.-Y., Yan, J.-Q., & Li, X.-L. Military Medical Research 10.1186/s40779-023-00502-7 Zhang2023 1 2023 The applied principles of EEG analysis methods in neuroscience and clinical neurology article-journal 10 abstract author container-title doi editor id isbn issued page publisher publisher-place title title-short type url The use of mathematical models for electroencephalography (EEG) analysis has been going on for many years, and currently they are starting to find a place both in clinical practice and in parallel with other methods for evaluation of brain activity. The use and interpretation of these methods are not possible without knowledge of the basic mechanisms and processes that underlie the results obtained from their application. The advantages of the quantitative methods, the processes underlying their presentation, and the optimal parameters used, such as the number of electrodes and time intervals, have been evaluated. Attention has been paid to the advantages of the individual possibilities for EEG signal analysis---absolute and relative power as well as coherence. An analysis of these defined quantities provides an opportunity to enter into the intimate mechanisms underlying specific psychopathological phenomena. literal Panov, G. Computational neuroscience 10.1007/978-1-0716-3230-7_5 literal Stoyanov, D., Draganski, B., Brambilla, P., & Lamm, C. Panov2023 978-1-0716-3230-7 2023 85-91 Springer US New York, NY Quantitative EEG analysis: Introduction and basic principles Quantitative EEG analysis chapter https://doi.org/10.1007/978-1-0716-3230-7_5 abstract author container-title doi id issn issue issued keyword page title type url volume Studies of hemisphere function involving the use of coherence analysis are reviewed. Despite the fact that many effects related to subject group and type of task have been reported, few seem to be reliable across studies. The assumptions which seem to underlie the use of coherence analysis are made explicit and discussed. Most investigators consider coherence levels to reflect certain 'baseline' patterns, possibly related to structural connectivity, upon which task and/or strategy effects may be superimposed. Use of a cephalic reference renders coherence effects virtually uninterpretable for a variety of reasons which are fully discussed. literal French, C. C, & Graham, J. B. International Journal of Psychophysiology 10.1016/0167-8760(84)90044-8 FRENCH1984241 0167-8760 3 1984 coherence---hemisphere function---psychopathology---task effects---sex---handedness---methodology 241-254 A critical review of EEG coherence studies of hemisphere function article-journal https://www.sciencedirect.com/science/article/pii/0167876084900448 1 abstract author container-title doi id issn issued keyword note page title type url volume Human brain consists of millions of neurons which are playing an important role for controlling behavior of human body with respect to internal/external motor/sensory stimuli. These neurons will act as information carriers between human body and brain. Understanding cognitive behaviour of brain can be done by analyzing either signals or images from the brain. Human behaviour can be visualized in terms of motor and sensory states such as, eye movement, lip movement, remembrance, attention, hand clenching etc. These states are related with specific signal frequency which helps to understand functional behavior of complex brain structure. Electroencephalography (EEG) is an efficient modality which helps to acquire brain signals corresponds to various states from the scalp surface area. These signals are generally categorized as delta, theta, alpha, beta and gamma based on signal frequencies ranges from 0.1Hz to more than 100Hz. This paper primarily focuses on EEG signals and its characterization with respect to various states of human body. It also deals with experimental setup used in EEG analysis. literal Kumar, J. S., & Bhuvaneswari, P. Procedia Engineering 10.1016/j.proeng.2012.06.298 KUMAR20122525 1877-7058 2012 Electroencephalography, brain signal, modality, brain states INTERNATIONAL CONFERENCE ON MODELLING OPTIMIZATION AND COMPUTING 2525-2536 Analysis of electroencephalography (EEG) signals and its categorization--a study article-journal https://www.sciencedirect.com/science/article/pii/S1877705812022114 38 author collection-title container-title doi editor id issued page publisher title type literal Panitz, C., Ward, R. T., Pouliot, J., & Keil, A. Cambridge handbooks in psychology The cambridge handbook of research methods and statistics for the social and behavioral sciences: Volume 2: Performing research 10.1017/9781009000796.024 literal Edlund, J. E., & Nichols, A. L. Panitz_Ward_Pouliot_Keil_2024 2024 519-544 Cambridge University Press EEG and ERP chapter author container-title doi id issued title title-short type volume literal Guevara, M., Hernández-González, M., Sanz-Martin, A., & Amezcua, C. Journal of Biomedical Science and Engineering 10.4236/jbise.2011.412096 Guevara2011 2011 EEGcorco: A computer program to simultaneously calculate and statistically analyze EEG coherence and correlation EEGcorco article-journal 4 author container-title doi id issue issued title title-short type volume literal Chiarion, G., Sparacino, L., Antonacci, Y., Faes, L., & Mesin, L. Bioengineering (Basel, Switzerland) 10.3390/bioengineering10030372 Chiarion2023 3 2023 Connectivity analysis in EEG data: A tutorial review of the state of the art and emerging trends Connectivity analysis in EEG data article-journal 10 author doi edition editor id issued publisher publisher-place title title-short type literal Gerthsen, C. 10.1007/978-3-662-30201-9 9 literal Kneser, H. O. Gerthsen1966 1966 Springer Berlin, Heidelberg Physik: Ein lehrbuch zum gebrauch neben vorlesungen Physik book author doi edition id issued publisher publisher-place title type literal Meschede, D., ed., Feld, L., Gross, R., Müller, R., Niedner-Schatteburg, G., Schäfer, G., Sokolowski, M., Vewinger, F., Reinhard F., Werner, R. F., & Zohm, H. 10.1007/978-3-662-30201-9 26 Gerthsen2026 2026 Springer Spektrum Berlin, Heidelberg Gerthsen physik book abstract author container-title doi id issn issue issued keyword page title type url volume We review research on the neural efficiency hypothesis of intelligence, stating that brighter individuals display lower (more efficient) brain activation while performing cognitive tasks \[Haier, R.J., Siegel, B.V., Nuechterlein, K.H., Hazlett, E., Wu, J.C., Paek, J., Browning, H.L., Buchsbaum, M.S., 1988. Cortical glucose metabolic rate correlates of abstract reasoning and attention studied with positron emission tomography. Intelligence 12, 199--217\]. While most early studies confirmed this hypothesis later research has revealed contradictory evidence or has identified some moderating variables like sex, task type, task complexity or brain area. Neuroscientific training studies suggest that neural efficiency also seems to be a function of the amount and quality of learning. From integrating this evidence we conclude that neural efficiency might arise when individuals are confronted with tasks of (subjectively) low to moderate task difficulty and it is mainly observable for frontal brain areas. This is true for easier novel cognitive tasks or after sufficient practice allowing participants to develop appropriate (efficient) strategies to deal with the task. In very complex tasks more able individuals seem to invest more cortical resources resulting in positive correlations between brain usage and cognitive ability. Based on the reviewed evidence we propose future empirical approaches in this field. literal Neubauer, A. C., & Fink, A Neuroscience & Biobehavioral Reviews 10.1016/j.neubiorev.2009.04.001 NEUBAUER20091004 0149-7634 7 2009 Intelligence, Neural efficiency, Training, Task complexity, Neuroscience 1004-1023 Intelligence and neural efficiency article-journal https://www.sciencedirect.com/science/article/pii/S0149763409000591 33 abstract author container-title doi id issn issue issued keyword page title title-short type url volume In studying physiological correlates of human intelligence, new brain imaging techniques like positron emission tomography (PET) and electroencephalography (EEG) mapping methods focus on the level and topographical distribution of cortical activation. Actually, there is strong empirical evidence that more intelligent individuals display a more focused cortical activation during cognitive performance resulting in lower total brain activation than in less intelligent individuals (i.e., neural efficiency hypothesis). Former studies have used only single, homogeneous tasks and most of the studies have been performed using males. Therefore, here the influence of different task content and of sex on the relationship between intelligence and cortical activation has been tested. In a sample of 26 males and 25 females, we administered verbal, numerical, and figural versions of a well-known elementary cognitive task, the so-called Posner task. Our results suggest a comparatively low cortical activation in brighter as compared to less intelligent individuals but this expected neural efficiency pattern interacted with sex and task content: In the verbal Posner task, the females were more likely to produce cortical activation patterns in line with the neural efficiency hypothesis, whereas in the figural task, primarily the males displayed the expected inverse relationship between IQ and cortical activation. literal Neubauer, A. C., Fink, A., & Schrausser, D. G. Intelligence 10.1016/S0160-2896(02)00091-0 NEUBAUER2002515 0160-2896 6 2002 Intelligence, EEG sex, Elementary cognitive task, Task specificity 515-536 Intelligence and neural efficiency: The influence of task content and sex on the brain--IQ relationship Intelligence and neural efficiency article-journal https://www.sciencedirect.com/science/article/pii/S0160289602000910 30 abstract author container-title doi id issn issue issued keyword page title type url volume Previous studies demonstrated that intelligence is significantly related to an impressive array of psychological, social, biological and genetic factors and that working memory (WM) can be considered as a general cognitive resource strongly related with a wide variety of higher order cognitive competencies and intelligence. Also, evaluating the WM of subjects might allow one to test the neural efficiency hypothesis (NEH). WM typically involves functional interactions between frontal and parietal cortices. We recorded EEG signals to study neuronal interactions during one WM test in individuals who had few years of formal education (LE) as compared to individuals with university degrees (UE). The two groups of individuals differed in the scores they obtained in psychological tests. To quantify the synchronization between EEG channels in several frequency bands, we evaluated the "synchronization likelihood" (SL), which takes into consideration nonlinear processes as well as linear ones. SL was then converted into graphs to estimate the distance from "small-world network" (SWN) organization, i.e., an optimally organized network that would give rise to the data. In comparison to LE subjects, those with university degrees exhibited less prominent SWN properties in most frequency bands during the WM task. This finding supports the NEH and suggests that the connections between brain areas of well-educated subjects engaged in WM tasks are not as well-organized in the sense of SWN. literal Micheloyannis, S., Pachou, E., Stam, C. J., Vourkas, M., Erimaki, S., & Tsirka, V. Neuroscience Letters 10.1016/j.neulet.2006.04.006 MICHELOYANNIS2006273 0304-3940 3 2006 Neural efficiency, Graph theory, Small-world network, Synchronization likelihood, Working memory 273-277 Using graph theoretical analysis of multi channel EEG to evaluate the neural efficiency hypothesis article-journal https://www.sciencedirect.com/science/article/pii/S030439400600382X 402 abstract author container-title doi id issn issue issued keyword page title title-short type url volume In the field of physiological study of human intelligence, strong evidence of a more efficient operation (i.e., less activation) of the brain in brighter individuals (the neural efficiency hypothesis) can be found. Most studies in this field have used single, homogeneous tasks and have not examined sex differences. In analyzing the extent of Event-related Desynchronization (ERD) in the EEG during the performance of a verbal and a visuo-spatial task, we recently found that males and females display neural efficiency primarily in the domain where they usually perform better (i.e., verbal in females and spatial in males; cf. A.C. Neubauer, A. Fink, D.G. Schrausser, Intelligence and neural efficiency: the influence of task content and sex on brain--IQ relationship. Intelligence, 30 (2002) 515--536). However, this interpretation was complicated by differences in the complexity of the two tasks. By using a verbal (semantic) and a spatial (rotation) task of comparable complexity in this research, we sought to replicate and extend our earlier findings by additionally considering the individual differences in intelligence structure and the topographical distribution over the cortex. Findings were similar to the previous study: Females (n = 35) display neural efficiency (i.e., less brain activation in brighter individuals) primarily during the verbal task, males (n = 31) in the spatial task. However, the strength of this brain activation--IQ relationship varies with the intelligence factor: In males, the highest correlations were observed for spatial IQ, in females for verbal IQ. Furthermore, the sexes displayed topographical differences of neural efficiency patterns. literal Neubauer, A. C., Grabner, R. H., Fink, A., & Neuper, C. Cognitive Brain Research 10.1016/j.cogbrainres.2005.05.011 NEUBAUER2005217 0926-6410 1 2005 EEG, ERD, Neural efficiency, Sex difference, Task content, Verbal IQ, Visuo-spatial IQ, Topographical difference 217-225 Intelligence and neural efficiency: Further evidence of the influence of task content and sex on the brain--IQ relationship Intelligence and neural efficiency article-journal https://www.sciencedirect.com/science/article/pii/S092664100500162X 25 author container-title doi id issued publisher publisher-place title type literal Schrausser, D. G., Fink, A., & Neubauer, A. C. 10th biennial meeting of the international society for the study of individual differences (ISSID) 10.5281/zenodo.13738772 schrausser_2001_13738772 2001 The University of Edinburgh Edinburgh, UK Intelligence and neural efficiency as determined by EEG-coherence paper-conference abstract author container-title doi id issue issued page title title-short type url volume Background: Reliable EEG biomarkers of brain-network engagement could personalize action-observation (AO) therapy after stroke. Objective: Identify functional-connectivity (FC) metrics that most robustly decode AO stimuli. Methods: Five right-handed adults (21-29 y) viewed 120 video trials (robot/human limb actions + controls) while 32-channel EEG was recorded. Ten central-region channels were filtered (alpha and beta bands) and FC matrices (10×10) computed using coherence (COH), imaginary coherence (iCOH), phase-locking value, partial directed coherence (PDC) and spectral Granger causality (SpcG). A graph neural network (GNN) was trained with stratified 15-fold cross validation for two tasks: (1) AO-condition decoding: six classes -- human-left, human-right, robot-left, robot-right, baseline, and landscape; (2) Action-type decoding: five upper-limb actions -- air punch, back-and-forth arm swing, lateral arm swing, overhead arm raise, and wave. Results: iCOH achieved the highest performance across both tasks (macro-AUC 0.997&amp;1.000; balanced accuracy 0.96 - 1.00). Directed metrics PDC and SpcG also performed strongly (macro-AUC ≥ 0.99). Findings persisted despite class imbalance and small sample size. Conclusions: Volume-conduction-invariant (iCOH) and directed FC measures provide robust signatures of motor-and-cognitive-network engagement during AO. These EEG markers may inform adaptive AO therapy or BCI-guided rehabilitation post-stroke. Larger cohorts will validate clinical utility. literal Nguyen, T. A., Rentala, Z., & Johnson, M. Circulation 10.1161/circ.152.suppl\\\_3.4373687 Nguyen2025 Suppl_3 2025 A4373687-A4373687 Abstract 4373687: Robust EEG functional connectivity metrics for decoding action observation conditions and observed actions Abstract 4373687 article-journal https://www.ahajournals.org/doi/abs/10.1161/circ.152.suppl_3.4373687 152 abstract author container-title doi id issue issued page title type volume Analysis of functional connectivity can provide insights into how the brain performs various cognitive and behavioral tasks as well as the neural mechanisms underlying several pathologies. In this work, we describe a novel approach to estimate functional connectivity from electroencephalography (EEG) data using the concept of coherence potentials (CPs), which are defined as clusters of high-amplitude deflections with similar waveform shapes. We define connectivity measures based on features of CPs, including the time intervals between CP peaks and their co-occurrence on different electrodes or channels. We used EEG data from 25 healthy subjects performing three tasks - resting state (eyes closed and eyes open), working memory and pattern completion tasks to investigate the ability of CP based connectivity measures to distinguish between these tasks. When compared with traditional connectivity measures including several spectral-based measures and mutual information, our results showed that CP based connectivity measures more robustly and significantly distinguished between all the tasks both at group-level and subject-level. In conclusion, CP based EEG connectivity measures provide a reliable way to distinguish between different cognitive task conditions and could pave way in the early detection of neurological disorders such as Alzheimer's disease that affect various cognitive tasks. literal Puthanmadam Subramaniyam, N., & Thiagarajan, T. C. Scientific Reports 10.1038/s41598-025-94076-0 PuthanmadamSubramaniyam2025 1 2025 10723 A novel method for estimating functional connectivity from EEG coherence potentials article-journal 15 author doi id isbn issued publisher publisher-place title title-short type url literal Meyberg, K., & Vachenauer, P. 10.1007/978-3-642-56654-7_4 Meyberg2001 978-3-642-56654-7 2001 Springer Berlin, Heidelberg Höhere mathematik 1: Differential- und integralrechnung vektor- und matrizenrechnung Höhere mathematik 1 book https://doi.org/10.1007/978-3-642-56654-7 abstract author container-title doi id isbn issued page publisher publisher-place title type url Mit der Integration löst man das Umkehrproblem, aus der Ableitung f' die ursprüngliche Funktion f zu rekonstruieren. Die wesentliche, zur Lösung führende Idee stammt aus dem Mittelwertsatz der Differentialrechnung (`\textrightarrow`{=latex} Kap. 3, `\textsection`{=latex}2): Zu jeder Zerlegung a = x0 \< x1 \< \... \< xn−1 \< xn= x des Intervalls \[a, x\], auf dem f differenzierbar ist, gibt es Zwischenpunkte $\xi$ ∈ \[xi−1, xi\], so daß gilt\$\$f\\left( {{x\_i}} \\right) - f\\left( {{x\_{i - 1}}} \\right) = f'\\left( {{\\xi \_i}} \\right)\\left( {{x\_i} - {x\_{i - 1}}} \\right)\\left( {i = 1,2, \\ldots ,n} \\right).\$\$ literal Meyberg, K., & Vachenauer, P. Höhere mathematik 1: Differential- und integralrechnung vektor- und matrizenrechnung 10.1007/978-3-642-56654-7_4 Meyberg2001int 978-3-642-56654-7 2001 161-211 Springer Berlin, Heidelberg Integration chapter https://doi.org/10.1007/978-3-642-56654-7_4 abstract author container-title doi editor id isbn issued page publisher publisher-place title type url This note contains some results obtained while ruminating about Riemann sums. We know that nothing here is truly new, but we know of no other place in which these ideas are presented in the way that they occurred to us. In particular, some of them are so elementary that we hope they will find their way into calculus texts. literal Guillemin, V. W., & Stroock, D. W. Representations, wavelets, and frames: A celebration of the mathematical work of lawrence w. baggett 10.1007/978-0-8176-4683-7_1 literal Jorgensen, P. E. T., Merrill, K. D., & Packer, J. A. Guillemin2008 978-0-8176-4683-7 2008 3-12 Birkhäuser Boston, MA Some riemann sums are better than others chapter https://doi.org/10.1007/978-0-8176-4683-7_1 abstract author container-title doi editor id isbn issued page publisher publisher-place title type url Simple integrability of a function f (defined by Haber and Shisha in \[2\]) is shown to be equivalent to the convergence of the infinite Riemann sum \<m:math display='block'\>\<m:mrow\>\<m:mstyle displaystyle='true'\>\<m:munderover\>\<m:mo\>&\#x2211;\</m:mo\>\<m:mrow\>\<m:mi\>k\</m:mi\>\<m:mo\>=\</m:mo\>\<m:mn\>1\</m:mn\>\</m:mrow\>\<m:mi\>&\#x221E;\</m:mi\>\</m:munderover\>\<m:mrow\>\<m:mi\>f\</m:mi\>\<m:mrow\>\<m:mo\>(\</m:mo\>\<m:mrow\>\<m:msub\>\<m:mi\>&\#x03BE;\</m:mi\>\<m:mi\>k\</m:mi\>\</m:msub\>\</m:mrow\>\<m:mo\>)\</m:mo\>\</m:mrow\>\<m:mrow\>\<m:mo\>(\</m:mo\>\<m:mrow\>\<m:msub\>\<m:mi\>x\</m:mi\>\<m:mi\>k\</m:mi\>\</m:msub\>\<m:mo\>&\#x2212;\</m:mo\>\<m:msub\>\<m:mi\>x\</m:mi\>\<m:mrow\>\<m:mi\>k\</m:mi\>\<m:mo\>&\#x2212;\</m:mo\>\<m:mn\>1\</m:mn\>\</m:mrow\>\</m:msub\>\</m:mrow\>\<m:mo\>)\</m:mo\>\</m:mrow\>\</m:mrow\>\</m:mstyle\>\</m:mrow\>\</m:math\>\$\$\\sum\\limits\_{k = 1}\^\\infty {f\\left( {{\\xi \_k}} \\right)\\left( {{x\_k} - {x\_{k - 1}}} \\right)} \$\$to the improper Riemann integral \<m:math display='block'\>\<m:mrow\>\<m:mstyle displaystyle='true'\>\<m:mrow\>\<m:msubsup\>\<m:mo\>&\#x222B;\</m:mo\>\<m:mn\>0\</m:mn\>\<m:mi\>&\#x221E;\</m:mi\>\</m:msubsup\>\<m:mi\>f\</m:mi\>\</m:mrow\>\</m:mstyle\>\</m:mrow\>\</m:math\>\$\$\\int\_0\^\\infty f \$\$f as the gauge of the partition \<m:math display='block'\>\<m:mrow\>\<m:msubsup\>\<m:mrow\>\<m:mrow\>\<m:mo\>(\</m:mo\>\<m:mrow\>\<m:msub\>\<m:mi\>x\</m:mi\>\<m:mi\>k\</m:mi\>\</m:msub\>\</m:mrow\>\<m:mo\>)\</m:mo\>\</m:mrow\>\</m:mrow\>\<m:mrow\>\<m:mi\>k\</m:mi\>\<m:mo\>=\</m:mo\>\<m:mn\>0\</m:mn\>\</m:mrow\>\<m:mi\>&\#x221E;\</m:mi\>\</m:msubsup\>\</m:mrow\>\</m:math\>\$\$\\left( {{x\_k}} \\right)\_{k = 0}\^\\infty \$\$of \[0,∞)converges to O. An analogous result is obtained for dominant integrability (defined by Osgood and Shisha in \[5\]). Also certain results of Bromwich and Hardy \[1\] are recovered. literal Lewis, J. T., Osgood, C. F., & Shisha, O. General inequalities 1 / allgemeine ungleichungen 1: Proceedings of the first international conference on general inequalities held in the mathematical research institute at oberwolfach, black forest, may 10--14, 1976 / abhandlung zur ersten internationalen tagung [ü]{.nocase}ber allgemeine ungleichungen im mathematischen forschungsinstitut oberwolfach, schwarzwald vom 10. Bis 14. Mai 1976 10.1007/978-3-0348-5563-1_25 literal Beckenbach, E. F. Lewis1978 978-3-0348-5563-1 1978 233-242 Birkhäuser Basel Infinite riemann sums, the simple integral, and the dominated integral chapter https://doi.org/10.1007/978-3-0348-5563-1_25 abstract author container-title doi id isbn issued page publisher publisher-place title type url In diesem Kapitel beschäftigen wir uns mit stochastischen Riemann-Integralen, d. h. mit gewöhnlichen Riemann-Integralen mit einem stochastischen Prozess als Integrand1. Mathematisch sind diese Gebilde relativ wenig anspruchsvoll, sie ließen sich pfadweise wie in der konventionellen (deterministischen) Analysis über stetige Funktionen definieren. Diese pfadweise Definition wird aber z. B. bei Ito-Integralen im übernächsten Kapitel nicht mehr möglich sein. Daher schlagen wir hier den auch später nützlichen Weg ein, Integrale als Grenzwert (im quadratischen Mittel) zu definieren. Ist der stochastische Integrand speziell ein Wiener-Prozess, so folgt das Riemann-Integral einer Normalverteilung mit Erwartungswert Null und bekannter Formel für die Varianz. Eine Reihe von Beispielen soll die Einführung erleichtern. literal Hassler, U. [Stochastische Integration und Zeitreihenmodellierung: Eine Einführung mit Anwendungen aus Finanzierung und Ökonometrie]{.nocase} 10.1007/978-3-540-73568-7_7 Hassler2007 978-3-540-73568-7 2007 137-153 Springer Berlin, Heidelberg Riemann-integrale chapter https://doi.org/10.1007/978-3-540-73568-7_7 author collection-title doi edition id issued publisher publisher-place title type literal Torchinsky, A. Lecture notes in mathematics 10.1007/978-3-031-11799-2 1 Torchinsky2022 2022 Springer Cham A modern view of the riemann integral book abstract author container-title doi id isbn issued page publisher publisher-place title type url Die Integration ist neben der Differentiation die wichtigste Anwendung des Grenzwertbegriffs in der Analysis. Wir definieren das Integral zunächst für Treppenfunktionen, wobei noch keine Grenzwertbetrachtungen nötig sind und der elementargeometrische Flächeninhalt von Rechtecken zugrundeliegt. Das Integral allgemeinerer Funktionen wird dann durch Approximation mittels Treppenfunktionen definiert. Mithilfe sog. Riemannscher Summen berechnen wir explizit die Integrale einiger elementarer Funktionen. literal Forster, O., & Lindemann, F. [Analysis 1: Differential- und Integralrechnung einer Veränderlichen]{.nocase} 10.1007/978-3-658-40130-6_18 Forster2023 978-3-658-40130-6 2023 283-302 Springer Fachmedien Wiesbaden Das riemannsche integral chapter https://doi.org/10.1007/978-3-658-40130-6_18 abstract author container-title doi id isbn issued page publisher publisher-place title title-short type url In diesem Kapitel wollen wir nach der Differenzial- auch die Integralrechnung auf eine mathematisch präzisere Grundlage stellen, sodass wir im nächsten Kapitel -- gewissermaßen als Höhepunkt unserer Theorieentwicklung -- den Hauptsatz, der Differenzial- und Integralrechnung miteinander verbindet (vgl. 3.3), ebenso präzise formulieren und beweisen können. literal Büchter, A., & Henn, H.-W. Elementare analysis: Von der anschauung zur theorie 10.1007/978-3-8274-2680-2_6 Büchter2010 978-3-8274-2680-2 2010 221-235 Spektrum Akademischer Verlag Heidelberg Grenzwerte von riemann'schen summen: Das integral Grenzwerte von riemann'schen summen chapter https://doi.org/10.1007/978-3-8274-2680-2_6 abstract accessed author container-title id issn issue issued page publisher title type url volume It is well known that Riemann sums converge toward the integral of any Riemannintegrable function f on the segment \[a, b\]. In some cases this property is still valid for generalized integrals and can be useful to solve problems involving series and sums. In this note we prove (Theorem 3) that the result still holds for a generalized integral under the assumption that the function is decreasing on the interval. We give two counterexamples: the first one with a linear piecewise function, and a second one based on an integral suggested by Hardy. The last section shows recent applications of the Riemann sums in computer science and applied mathematics. 2026-01-12 literal Truc, J.-P. The College Mathematics Journal e941eb5c-b072-3fd7-a80c-cfa5c7173324 07468342, 19311346 2 2019 pp. 123-132 \[Mathematical Association of America, Taylor & Francis, Ltd.\] Riemann sums for generalized integrals article-journal https://www.jstor.org/stable/48662056 50 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-12 literal Goel, S. K., & Rodriguez, D. M. Mathematics Magazine fbd8e02f-8560-36a2-a00c-e74f040810fc 0025570X, 19300980 4 1987 225-228 \[Mathematical Association of America, Taylor & Francis, Ltd.\] A note on evaluating limits using riemann sums article-journal http://www.jstor.org/stable/2689344 60 abstract author container-title doi id isbn issued page publisher publisher-place title type url Die bei Programm 22 verwendete Romberg-Extrapolation läßt sich auch auf die numerische Integration anwenden. literal Herrmann, D. Numerische mathematik --- 40 BASIC-programme 10.1007/978-3-322-96321-5_27 Herrmann1983 978-3-322-96321-5 1983 84-86 Vieweg+Teubner Verlag Wiesbaden Romberg-integration chapter https://doi.org/10.1007/978-3-322-96321-5_27 abstract author container-title doi id issn issue issued keyword page title type url volume Let (Xt) be the solution of a stochastic differential system. We consider the following situations: computation of Eƒ(Xt) by a Monte-Carlo method (for example, computation of moments of the solution); integration of ƒ(·) with respect to the invariant probability law of (Xt) (when this process is ergodic); or of the upper Lyapunov exponent, by simulating a single trajectory. We propose to perform an extrapolation between approximate values due to first-order schemes; we show that this algorithm (simpler to implement than second-order schemes) provides a second-order accuracy, and we give results of numerical tests. literal Talay, D., & Tubaro, L. Structural Safety 10.1016/0167-4730(90)90036-O TALAY1990143 0167-4730 1 1990 Monte Carlo method, numerical analysis, probability distribution, random process, simulation 143-150 Romberg extrapolations for numerical schemes solving stochastic differential equations article-journal https://www.sciencedirect.com/science/article/pii/016747309090036O 8 abstract author chapter-number container-title doi id isbn issued keyword page publisher title type url Summary Elementary techniques for computing a definite integral use the fundamental theorem of calculus. This chapter discusses more sophisticated techniques that yield accurate approximations with less computational effort. It surveys two methods for enhancing the accuracy of composite quadrature formulas. The first method, Romberg quadrature, uses approximations to the integral that have low-order accuracy to compute high-order approximations. The second approach, adaptive quadrature, encompasses a class of strategies for tailoring composite formulas to local, idiosyncratic behavior in the integrand. The chapter also explores the theory of Gauss quadrature. In doing so, it generalizes the framework based on the Legendre polynomials, to include other Gauss quadrature methods based on different orthogonal systems of polynomials. In each case, there is a specific interval of integration associated with the basic quadrature method. However, each method readily extends to more general intervals via the change-of-variables tactic. literal Allen, M. B., & Isaacson, E. L 7 Numerical analysis for applied science 10.1002/9781119245476.ch7 doi:https://doi.org/10.1002/9781119245476.ch7 9781119245476 2019 adaptive quadrature, Gauss quadrature, Legendre polynomials, numerical integration, Romberg quadrature 363-401 John Wiley & Sons, Ltd Numerical integration chapter https://onlinelibrary.wiley.com/doi/abs/10.1002/9781119245476.ch7 accessed author container-title id issn issue issued page publisher title type url volume 2026-01-11 literal von Petersdorff, T. The American Mathematical Monthly 30c28040-c164-336f-b877-2f793977628d 00029890, 19300972 8 1993 783-785 \[Taylor & Francis, Ltd., Mathematical Association of America\] A short proof for romberg integration article-journal http://www.jstor.org/stable/2324787 100 abstract author container-title doi id issue issued title type volume Subject to rather mild assumptions on the integrand, the Romberg table (theT-table) and the modified Romberg table (theU-table) yield asymptotically upper and lower bounds for the value of the integral, and the convergence of the columns of the two tables is asymptotically monotone. This is verified for arbitrary sequences of step sizes satisfying the usual condition of convergence for Romberg integration. literal Jetter, K. Numerische Mathematik 10.1007/BF01389471 Jetter1984 2 1984 Ein kurze anmerkung zur romberg-integration article-journal 45 abstract accessed author container-title id issn issue issued page publisher title type url volume Simulation methods \[particularly Efron's (1979) bootstrap\] are being applied more and more frequently in statistical inference. Given data (X1, \..., Xn) distributed according to P, which belongs to a hypothesized model P, the basic goal is to estimate the distribution LP of a function Tn(X1, \..., Xn, P). The bootstrap presupposes the existence of an estimate P̂(X1, \..., Xn) and involves estimating Lp by the distribution L\*n of Tn(X\*1, \..., X\*n, P̂), where (X\*1, \..., X\*n) is distributed according to P̂. The method is of particular interest when L\*n, though known in principle, can realistically only be computed by simulation. Such computation can be expensive if n is large and Tn is complex (e.g., see the multivariate goodness-of-fit tests of Beran and Millar 1986). Even when bootstrap application to a single data set is not excessively expensive, Monte Carlo studies of the bootstrap are another matter. We propose a method based on the classical ideas of Richardson extrapolation for reducing the computational cost inherent in bootstrap simulations and Monte Carlo studies of the bootstrap, by performing simulations for statistics based on two smaller sample sizes. We study theoretically which ratio of the two small sample sizes is apt to give best results. We show how our method works for approximating the χ2, t, and smoothed binomial distributions, and for setting bootstrap percentile confidence intervals for the variance of a normal distribution with a mean of 0. 2026-01-11 literal Bickel, P. J., & Yahav, J. A. Journal of the American Statistical Association cc7b552a-2f22-3972-bf87-40a114018835 01621459, 1537274X 402 1988 387-393 \[American Statistical Association, Taylor & Francis, Ltd.\] Richardson extrapolation and the bootstrap article-journal http://www.jstor.org/stable/2288854 83 author collection-title container-title doi id issued page publisher title type literal Sidi, A. Cambridge monographs on applied and computational mathematics Practical extrapolation methods: Theory and applications 10.1017/CBO9780511546815 Sidi_2003 2003 19-20 Cambridge University Press The richardson extrapolation process and its generalizations chapter abstract author container-title doi id issue issued page title type volume Richardson extrapolation is a classical technique from numerical analysis that can improve the approximation error of an estimation method by combining linearly several estimates obtained from different values of one of its hyperparameters without the need to know in details the inner structure of the original estimation method. The main goal of this paper is to study when Richardson extrapolation can be used within data science beyond the existing applications to step-size adaptations in stochastic gradient descent. We identify two situations where Richardson interpolation can be useful: (1) when the hyperparameter is the number of iterations of an existing iterative optimization algorithm with applications to averaged gradient descent and Frank--Wolfe algorithms (where we obtain asymptotically rates of \$O(1/k\^2)\$ on polytopes, where \$k\$ is the number of iterations) and (2) when it is a regularization parameter with applications to Nesterov smoothing techniques for minimizing nonsmooth functions (where we obtain asymptotically rates close to \$O(1/k\^2)\$ for nonsmooth functions) and kernel ridge regression. In all these cases, we show that extrapolation techniques come with no significant loss in performance but with sometimes strong gains, and we provide theoretical justifications based on asymptotic developments for such gains, as well as empirical illustrations on classical problems from machine learning. literal Bach, F. SIAM Journal on Mathematics of Data Science 10.1137/21M1397349 doi:10.1137/21M1397349 4 2021 1251-1277 On the effectiveness of richardson extrapolation in data science article-journal 3 abstract author container-title doi id issn issued keyword page title type url volume Richardson's extrapolation process is a well known method to improve the order of several approximation processes. Here we observe that for numerical differentiation, Richardson's process can be applied not only to improve the order of a numerical differentiation formula but also to find in fact the original formula. literal Dubeau, F. Journal of Computational Physics: X 10.1016/j.jcpx.2019.100017 DUBEAU2019100017 2590-0552 2019 Numerical differentiation, Richardson's extrapolation process 100017 A remark on richardson's extrapolation process and numerical differentiation formulae article-journal https://www.sciencedirect.com/science/article/pii/S2590055219300332 2 abstract author container-title doi editor id isbn issued page publisher publisher-place title type url The application of extrapolation (or convergence acceleration) methods to the solution of problems in numerical analysis has become very widespread in recent years. One important area in which extrapolation methods have proved to be remarkably efficient is that of numerical integration. Of the methods that have proved to be useful in this area, the Richardson extrapolation process and some of its generalizations have been the subject of intense research. It is the purpose of this paper to survey briefly the recent developments relating to the generalizations of the Richardson extrapolation process with special emphasis on their application to the numerical evaluation of finite and infinite range integrals. literal Sidi, A. Numerical integration III: Proceedings of the conference held at the mathematisches forschungsinstitut, oberwolfach, nov. 8 -- 14, 1987 10.1007/978-3-0348-6398-8_22 literal Brass, H., & Hämmerlin, G. Sidi1988 978-3-0348-6398-8 1988 237-250 Birkhäuser Basel Generalizations of richardson extrapolation with applications to numerical integration chapter https://doi.org/10.1007/978-3-0348-6398-8_22 abstract author container-title doi editor id isbn issued page publisher publisher-place title type url In a recent paper by H. Blum, Lin Q., and the author,\[l\], it has been shown that the Ritz projection method with linear finite elements admits an asymptotic error expansion for certain classes of "uniform" meshes. This provides the theoretical justification for the use of Richardson extrapolation or related correction processes for increasing the accuracy of the scheme. The proofs are entirely based on finite element techniques and extend to situations where no "discrete" maximum principle is available. This is examplarily demonstrated here for the second-order Lamé-Navier system in plain linear elasticity and for a mixed formulation of the fourth-order Kirchhoff plate bending model. literal Rannacher, R. Numerical techniques in continuum mechanics: Proceedings of the second GAMM-seminar, kiel, january 17 to 19, 1986 10.1007/978-3-322-85997-6_9 literal Hackbusch, W., & Witsch, K. Rannacher1987 978-3-322-85997-6 1987 90-101 Vieweg+Teubner Verlag Wiesbaden Richardson extrapolation with finite elements chapter https://doi.org/10.1007/978-3-322-85997-6_9 abstract author container-title doi id isbn issued page publisher publisher-place title type url Manche Anwendungsprobleme lassen sich auf die Berechnung eines bestimmten Integrals zurückführen. Ist keine Stammfunktion des Integranden bekannt, muss die Integrationsaufgabe näherungsweise gelöst werden. Verschiedene geeignete Verfahren werden vorgestellt und analysiert. literal Neher, M. [Numerische Mathematik: Eine anschauliche modulare Einführung]{.nocase} 10.1007/978-3-662-68815-1_7 Neher2024 978-3-662-68815-1 2024 195-222 Springer Berlin, Heidelberg Numerische integration chapter https://doi.org/10.1007/978-3-662-68815-1_7 abstract author container-title doi id isbn issued page publisher publisher-place title type url Manche Probleme der angewandten Mathematik führen auf die Berechnung von Integralen, die meistens nicht in expliziter Form dargestellt werden können. Die numerische Integralberechnung, die man kurz als Quadratur bezeichnet, spielt deshalb eine wichtige Rolle. Wir befassen uns im folgenden mit der genäherten Berechnung von bestimmten Integralen. Unbestimmte Integrale (Stammfunktionen) werden zweckmäßig als Anfangswertprobleme gewöhnlicher Differentialgleichungen behandelt (vgl. Kapitel 9). Von den zahlreichen Anwendungen der numerischen Quadratur nennen wir die Berechnung von Oberflächen, Volumina, Wahrscheinlichkeiten und Wirkungsquerschnitten, die Auswertung von Integraltransformationen und Integralen im Komplexen, die Konstruktion von konformen Abbildungen für Polygonbereiche nach der Formel von Schwarz-Christoffel \[Hen 85\], die Behandlung von Integralgleichungen etwa im Zusammenhang mit der Randelementmethode und schließlich die Methode der finiten Elemente \[Scw91\]. literal Schwarz, H. R. Numerische mathematik 10.1007/978-3-663-01227-6_8 Schwarz1997 978-3-663-01227-6 1997 375-411 Vieweg+Teubner Verlag Wiesbaden Integralberechnung chapter https://doi.org/10.1007/978-3-663-01227-6_8 abstract author container-title doi id issue issued page publisher title type url volume Numerical Methods have attracted of research community for solving engineering problems. This interest is due to its practicality and the improvement of highspeed calculations done on current century processors. The increase in numerical method tools in engineering software, such as Matlab, is an example of the increased interest. In this paper, we are present a new improved numerical integration method, that is based on the well-known trapezoidal rule. The proposed method gives a great enhancement to the trapezoidal rule and overcomes the issue of the error value when dealing with some higher order functions even when solving for a single interval. After literature review, the proposed system is mathematically explained along with error analysis. Few examples are illustrated to prove improved accuracy of the proposed method over traditional trapezoidal method. literal Abdulhameed, A. F., & Memon, Q. A. Journal of Physics: Conference Series 10.1088/1742-6596/2090/1/012104 Abdulhameed_2021 1 2021-11 5188 IOP Publishing An improved trapezoidal rule for numerical integration article-journal https://doi.org/10.1088/1742-6596/2090/1/012104 2090 abstract author container-title doi id issn issued keyword page title type url volume We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in Kublik et al. (2013) \[7\]. In this formulation, a boundary integral over a smooth, closed hypersurface is transformed into an equivalent volume integral defined in a sufficiently thin tubular neighborhood of the surface. The volumetric formulation makes it possible to use the simple trapezoidal rule on uniform Cartesian grids and relieves the need to use parameterization for developing quadrature. Consequently, typical point singularities in a layer potential extend along the surface's normal lines. We propose new higher-order corrections to the trapezoidal rule on the grid nodes around the singularities. This correction is based on local decompositions of the singularity and is dependent on the angle of approach to the singularity relative to the surface's principal curvature directions. The proposed decomposition, combined with the volumetric formulation, leads to a special quadrature error cancellation. literal Izzo, F., Runborg, O., & Tsai, R. Journal of Computational Physics 10.1016/j.jcp.2022.111193 IZZO2022111193 0021-9991 2022 Level set methods, Closest point projection, Boundary integral formulations, Singular integrals, Trapezoidal rules 111193 Corrected trapezoidal rules for singular implicit boundary integrals article-journal https://www.sciencedirect.com/science/article/pii/S0021999122002558 461 abstract collection-title container-title doi editor id issn issued page publisher title type url volume Publisher Summary This chapter discusses the trapezoidal rule integration formulas. The development of numerical integration formulas as such is not in the mainstream of the mathematical topics covered in the chapter. The modified trapezoidal rule uses functional values at points which lie midway between those points that apply for the trapezoidal rule. There is a certain class of transcendental functions defined by integrals which can be computed to high accuracy by use of these rules; with a view toward the applications, the chapter takes up this aspect of the subject. Algorithms for the evaluation of definite integrals by means of equally spaced data are closely related to the Euler-Maclaurin summation formula for which there is a vast literature. Comparison of the two formulations shows that the difference in accuracy is negligible. Mathematics in science and engineering The special functions and their approximations 10.1016/S0076-5392(09)60075-8 literal Luke, Y. L. luke1969214 0076-5392 1969 214-226 Elsevier Chapter XV trapezoidal rule integration formulas chapter https://www.sciencedirect.com/science/article/pii/S0076539209600758 53 abstract author container-title doi id issn issue issued page title type url volume There are many numerical integration methods that exist with its own advantage and disadvantage. This study focusses on the Trapezoidal rule, TR. The advantage of the Trapezoidal rule is that it is simple to use and can be applied to a wide range of functions. Since the Trapezoidal rule is an approximation rule, it will yield an error. In this study, we aim to reduce the error of the Trapezoidal rule can be reduced by applying mean averaging toward Trapezoidal rule called as Trapezoidal rule with mean averaging, TRMEAN. The arithmetic mean, geometric mean, Harmonic mean, Heronian mean, Contra harmonic, Root-mean-square mean, and Centroidal mean will be implemented. Hence, we will be experimented as examples before concluding the results. The results of this study showed that the error from TRMEAN is smaller as compared to TR. The findings indicate that TRMEAN presents a promising improvement in accuracy and reliability for this task over the original TR method. However, further research and validation are essential to firmly establish these conclusions and assess their applicability to diverse datasets and situations. literal Bujang, N., Nasir, M. A. S., & Ijam, H. M. AIP Conference Proceedings 10.1063/5.0294833 Bujang10.1063/5.0294833 0094-243X 1 2025-12 16392 Numerical integration of function using the modified trapezoidal rule and mean averaging method article-journal https://doi.org/10.1063/5.0294833 3338 author container-title doi id issued title type url literal Jarre, F. arXiv 10.48550/arXiv.2507.05083 jarre2025cubicsplinefunctionsrevisited 2025 Cubic spline functions revisited article-journal https://arxiv.org/abs/2507.05083 abstract author container-title doi id issn issued keyword page title type url volume In this paper a fourth order asymptotically optimal error bound for a new cubic interpolating spline function, denoted by Q-spline, is derived for the case that only function values at given points are used but not any derivative information. The bound seems to be stronger than earlier error bounds for cubic spline interpolation in such setting such as the not-a-knot spline. A brief analysis of the conditioning of the end conditions of cubic spline interpolation leads to a modification of the not-a-knot spline, and some numerical examples suggest that the interpolation error of this revised not-a-knot spline generally is comparable to the near optimal Q-spline and lower than for the not-a-knot spline when the mesh size is small. literal Jarre, F. Journal of Computational and Applied Mathematics 10.1016/j.cam.2025.117240 JARRE2026117240 0377-0427 2026 Cubic spline, Natural spline, Error estimate, Condition number 117240 Cubic spline functions revisited article-journal https://www.sciencedirect.com/science/article/pii/S037704272500754X 478 abstract author container-title doi id isbn issued page publisher publisher-place title type url This chapter reiterates the subject of the previous chapter. Instead of a rational cubic model, a polynomial cubic spline has been presented here for the same objective. A piecewise cubic spline has been introduced to preserve the shape of the data when it is convex, monotone or positive. The spline representation is interpolatory and applicable to the scalar valued data. The shape parameters, in the description of the cubic, have been constrained in such a way that they control the shape of the curve to avoid any noise. As far as visual smoothness is concerned, the curve scheme under discussion is GC1. Thus the continuity constraints have been relaxed from C1 to GC1. literal Sarfraz, M. Interactive curve modeling: With applications to computer graphics, vision and image processing 10.1007/978-1-84628-871-5_8 Sarfraz2008 978-1-84628-871-5 2008 157-172 Springer London London Visualization of shaped data by cubic spline interpolation chapter https://doi.org/10.1007/978-1-84628-871-5_8 abstract author container-title doi editor id isbn issued page publisher publisher-place title type In this article, the issues of digital processing and restoration of electroencephalogram (EEG) signals from biomedical signals are considered, the location of 21 sensors in the EEG apparatus along the brain, the naming of the sensors, their connection types, the use of bipolar coupling in the detection of disease symptoms, interpolation of received signals, disease symptoms. The processes of separating parts into scales have been studied. During the work, the B-spline function was selected as the most convenient mathematical model for digital processing of EEG signals, and the construction of the B-spline function was presented. Based on the constructed mathematical model, an algorithm for restoring the electroencephalogram signals by dividing the problematic parts into scales was developed, and the absolute error in the restoration of EEG signals was estimated. literal Abduganiev, M., Azimov, R., & Muydinov, L. Intelligent human computer interaction 10.1007/978-3-031-27199-1_3 literal Zaynidinov, H., Singh, M., Tiwary, U. S., & Singh, D. abd10.1007/978-3-031-27199-1_3 978-3-031-27199-1 2023 18-26 Springer Nature Switzerland Cham Digital processing algorithms of biomedical signals using cubic base splines paper-conference abstract author container-title doi id isbn issued page publisher publisher-place title type url In diesem Kapitel werden wir uns mit dem populärsten Kurvenschema befassen: C2-stetige kubische interpolierende Splines. Wir haben schon gesehen, daß Lagrange-Interpolation keine brauchbaren Resultate liefert. Wir haben aber andererseits gesehen, daß uns durch kubische B-Spline-Kurven ein mächtiges Werkzeug zum Modellieren an die Hand gegeben wurde; sie können auf einfache Weise komplexe Formen wiedergeben. Diese „Wiedergabe" geschieht als Approximationsprozeß, man verändert das Kontrollpolygon solange, bis die gewünschte Form gefunden ist. Wir werden sehen, wie man kubische Splines auch zur Interpolation benutzen kann, nämlich, eine Splinekurve zu finden, die durch einen gegebenen Satz von Punkten verläuft. Kubische Spline-Interpolation ist in die CAGD-Literatur von J. Ferguson \[184\] im Jahre 1964 eingeführt worden, während die zugrundeliegende Mathematik in der Approximationstheorie (siehe de Boor \[112\] oder Holladay \[266\]) entwickelt wurde. Eine Übersicht über die Geschichte der Splines ist bei Schumaker \[426\] zu finden. literal Farin, G. Kurven und flächen im computer aided geometric design: Eine praktische einführung 10.1007/978-3-663-10602-9_9 Farin1994 978-3-663-10602-9 1994 104-121 Vieweg+Teubner Verlag Wiesbaden Kubische spline-interpolation chapter https://doi.org/10.1007/978-3-663-10602-9_9 abstract author container-title doi id isbn issued page publisher publisher-place title type url Spline interpolation is an improvement over piecewise --- polynomial interpolation. It uses less information of the given function, yet furnishes smoother interpolates. The plan of this chapter is as follows: In Section6.2 we shall define the spline space Sm,$\tau$($\Delta$), and for a given function x (t) the spline and Lidstone --- spline interpolates Sm,$\tau$$\Delta$and LSm,2m−2$\Delta$x(t),respectively. Here, we shall also show that to acquire the bounds for `\textbardbl`{=latex} Dk(x− Sm,$\tau$$\Delta$) `\textbardbl`{=latex}∞ and `\textbardbl`{=latex} Dk(x− LSm,2m−2$\Delta$x) `\textbardbl`{=latex}∞ in terms of the derivatives of x(t) it is necessary to estimate several terms. While some of these terms can be estimated by the results of Chapter 5, other terms which require a different analysis for each m and $\tau$, need to be bounded. In Sections6.3,6.4 and 6.6 respectively, we shall consider the cases m = 2, $\tau$ = 2; m = 3, $\tau$ = 4 and m = 3, $\tau$ = 3. These cases correspond to cubic in the class C(2) \[a, b\], and quintic in the classes C(4)\[a, b\] and C(3)\[a, b\] spline interpolates. For the cases m = 3, $\tau$ = 4 and m = 3, $\tau$ = 3 in Sections 6.5 and 6.7 respectively, we shall discuss the construction of approximated quintic splines, and for these interpolates we will provide explicit error bounds in L∞ --- norm. For the cubic and quintic Lidstone --- spline interpolates error bounds in L∞ --- norm are obtained in Sections 6.8 and 6.9, respectively. In Section6.10 we shall extend Theorems 5.3.16 and 5.3.17 for the spline interpolate Sm,$\tau$$\Delta$x(t) literal Agarwal, R. P., & Wong, P. J. Y. Error inequalities in polynomial interpolation and their applications 10.1007/978-94-011-2026-5_6 Agarwal1993 978-94-011-2026-5 1993 281-362 Springer Netherlands Dordrecht Spline interpolation chapter https://doi.org/10.1007/978-94-011-2026-5_6 abstract author container-title doi id issn issue issued keyword page title title-short type url volume An algorithm for the construction of a cubic spline curve with relatively good shape that interpolates specified data points at some knots and passes through specified regions at some other knots is presented. The curve constructed by the algorithm has minimum energy on each of its components. This algorithm has applications in various fields, such as the reconstruction of natural phenomena where data points cannot be sampled exactly, or computer-aided modeling where some of the fitting points cannot be explicitly specified. literal Cheng, F., & Barsky, B. A. Computer-Aided Design 10.1016/0010-4485(91)90023-P CHENG1991700 0010-4485 10 1991 splines, interpolation, uncertain data 700-706 Interproximation: Interpolation and approximation using cubic spline curves Interproximation article-journal https://www.sciencedirect.com/science/article/pii/001044859190023P 23 abstract author container-title doi id issn issue issued keyword page title type url volume Evoked potentials and EEGs record punctuate electrical activity at electrode sites. To represent the overall potential distribution on the entire scalp it is necessary to interpolate between these sampled values. Surface splines are mathematical tools for interpolating functions of two variables. In comparison to the classical methods of interpolation, based on linear combination of the potentials of the 4 nearest electrodes, spline methods are smoother, give more precisely located extrema and converge faster toward the 'true' potential surface when the number of recording electrodes is increased. These advantages are at the expense of lengthier computation time. Résumé Les potentiels évoqués comme l'électroencéphalographie n'enregistrent l'activité électrique du scalp qu'en des points précis correspondant à l'emplacement des électrodes. Pour avoir une représentation globale de la distribution du potentiel sur le scalp, il est nécessaire d'interpoler entre ces points de mesure. Les surfaces splines permettent d'effectuer de telles interpolations bi-dimensionnelles. Par comparaison avec la méthode d'interpolation couramment utilisée basée sur la combinaison linéaire des potentiels des 4 plus proches électrodes la méthode des splines donne des surfaces plus régulières, les extrêmes sont mieux localisés et quand on augmente le nombre d'électrodes, les surfaces obtenues convergent plus rapidement vers la 'vraie' surface. En contrepartie le calcul de ces fonctions est plus long. literal Perrin, F., Pernier, J., Bertnard, O., Giard M. H., & Echallier, J. F. Electroencephalography and Clinical Neurophysiology 10.1016/0013-4694(87)90141-6 PERRIN198775 0013-4694 1 1987 interpolation, spline 75-81 Mapping of scalp potentials by surface spline interpolation article-journal https://www.sciencedirect.com/science/article/pii/0013469487901416 66 abstract author container-title doi id issn issue issued keyword page title type url volume This paper is a review of the main interpolation methods applicable to 3-dimensional EEG mapping. The use of simple statistical comparison methods on recorded EEG maps allowed us to evaluate the qualities of interpolation methods belonging to 3 mathematical families (barycentric, polynomial, spline). A combination of a 3-dimensional representation of EEG maps and a reliable interpolation method makes it possible to obtain better spatial resolution than with standard planar mapping. literal Soufflet, L., Toussaint, M., Luthringer, R., Gresser, J., Minot, R., & Macher, J. P. Electroencephalography and Clinical Neurophysiology 10.1016/0013-4694(91)90204-H SOUFFLET1991393 0013-4694 5 1991 EEG, 3D interpolation, Image synthesis 393-402 A statistical evaluation of the main interpolation methods applied to 3-dimensional EEG mapping article-journal https://www.sciencedirect.com/science/article/pii/001346949190204H 79 author container-title doi id issued keyword page title type volume literal Jiang, H., & Zhao, Y. 2009 sixth international conference on fuzzy systems and knowledge discovery 10.1109/FSKD.2009.591 5360654 2009 Interpolation;Spline;Isosurfaces;Boundary conditions;Software algorithms;Convergence;Hardware;Costs;Biomedical image processing;Rendering (computer graphics);Cubic spline;MC;interpolation 80-83 The study of interpolation algorithm based on cubic spline in marching cubes method paper-conference 5 abstract author container-title doi id issn issued keyword page title type url volume We express the interpolating cubic splines of class C2 in their new, explicit forms. We construct the desired forms, the spline's Hermitian and B-spline representations for both equidistant and arbitrary nodes. During this process we demonstrate an innovative way to compute the inverse of a special class of tridiagonal matrices. Afterward, we propose the corresponding interpolating spline based linear regression models with easily interpretable coefficients suitable for smoothing data of complex structures. literal Török, C., Hudák, J., Pristaš, V., & Antoni, L. Applied Mathematics and Computation 10.1016/j.amc.2025.129411 TOROK2025129411 0096-3003 2025 Data smoothing, Cubic splines, Interpolation, Linear regression 129411 Explicit forms of interpolating cubic splines and data smoothing article-journal https://www.sciencedirect.com/science/article/pii/S0096300325001389 500 abstract author container-title doi id issn issue issued keyword page title type url volume An algorithm for computing the cubic spline interpolation coefficients without solving the matrix equation involved is presented in this paper. It requires only O(n) multiplication or division operations for computing the inverse of the matrix compared to O(n2) or larger number of operations in the Gauss elimination method. literal Moon, B. S. Applied Mathematics and Computation 10.1016/S0096-3003(99)00178-2 MOON2001251 0096-3003 2 2001 Explicit solution, Cubic spline interpolation, B-splines, Inverse matrix 251-255 An explicit solution for the cubic spline interpolation for functions of a single variable article-journal https://www.sciencedirect.com/science/article/pii/S0096300399001782 117 abstract author container-title doi id issn issued keyword page title type url volume Traditional end conditions for cubic spline interpolation consist of values, the first or the second derivatives of interpolated functions on the boundary interpolation knots. The not-a-knot end condition proposed by de Boor (1985) is a kind of end condition of cubic spline interpolation for the practical application without the requirements of the derivatives at the end knots. However, a significant disadvantage of such end condition is that there is a sharp decrease in the accuracy of the interpolation at boundary intervals. In this paper, by changing the locations of two spline knots in not-a-knot end condition, we propose the optimal arrangement of shifted spline knots for cubic spline interpolation. The proposed scheme leads to an approximately 3.4 times increasing in the accuracy of the interpolation compared to the de Boor's not-a-knot end condition. Furthermore, we also present the optimal end conditions of cubic spline interpolation to approximate the first and the second derivatives of interpolated functions. The representative examples show the effectiveness and the superiority of the proposed method. literal Sun, M., Lan, L., Zhu C. G., & Lei, F. Journal of Computational and Applied Mathematics 10.1016/j.cam.2022.115039 SUN2023115039 0377-0427 2023 Cubic spline interpolation, End conditions, Not-a-knot end condition, Interpolation error estimation 115039 Cubic spline interpolation with optimal end conditions article-journal https://www.sciencedirect.com/science/article/pii/S0377042722006379 425 abstract author container-title doi id issue issued page publisher title type volume Resampling of digitized electroencephalographic data allows changing the sampling rate with minimal distortion of the signal. Useful applications of the procedure include compatibility among diverse hardware and software and the customization of data analysis. The natural cubic spline interpolation procedure is introduced in a discursive fashion. A formal presentation is provided in the appendix. literal Congedo, M., Özen, C., & Sherlin, L. Journal of Neurotherapy 10.1300/J184v06n04_08 Congedo01092002 4 2002 73-80 Routledge Notes on EEG resampling by natural cubic spline interpolation article-journal 6 abstract author container-title doi id issue issued page title type volume Electroencephalogram (EEG) signal is usually suffered from motion artifacts, generated randomly during signal acquisition timings. These artifacts sturdily affect the investigation and therefore, diagnosis of neural diseases from EEG signal. The artifact removal may cause loss of important information from the signal. Therefore, it is required to remove the motion artifacts and simultaneously preserve the desired information, which makes EEG artifact removal a vital task. Enhanced Empirical Mode Decomposition (EEMD) is the most widespread method used for artifact removal, as it is a data-driven based feature extraction method. In this research work the efficiency of various EEMD with different interpolation based artifact removal method have been compared. The EEMD is used to convert input single channel EEG signal to a multichannel signal, and in order to remove the randomness of motion artifact, CCA and DWT filtering were used successively. The performance of different interpolation based artifact removal methods have evaluated and results indicate that the proposed algorithm is suitable for use as a supplement to algorithms currently in use as it offers improvements in DSNR and various other performance parameters. literal Roy, V., & Shukla, S. Wireless Personal Communications 10.1007/s11277-017-4846-3 roy2017 4 2017 6441-6451 Effective EEG motion artifacts elimination based on comparative interpolation analysis article-journal 97 abstract author container-title doi id issn issue issued page title type url volume In this paper we derive necessary optimality conditions for an interpolating spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are monotone. To this end optimal control theory for state-restricted optimal control problems is applied. The necessary conditions yield a complete characterization of the optimal spline. In the case of two or three interpolation knots, which we call thelocalcase, the optimality conditions are treated analytically. They reduce to polynomial equations which can very easily be solved numerically. These results are used for the construction of a numerical algorithm for the optimal monotone spline in the general (global) case via Newton's method. Here, the local optimal spline serves as a favourable initial estimation for the additional grid points of the optimal spline. Some numerical examples are presented which are constructed by FORTRAN and MATLAB programs. literal Fredenhagen, S., Oberle, H. J., & Opfer, G. Journal of Approximation Theory 10.1006/jath.1998.3247 FREDENHAGEN1999182 0021-9045 2 1999 182-201 On the construction of optimal monotone cubic spline interpolations article-journal https://www.sciencedirect.com/science/article/pii/S0021904598932476 96 abstract author container-title doi editor id isbn issued page publisher publisher-place title type url Interpolation is the art of reading between the lines of a mathematical table. It can be used to express nonelementary functions approximately in terms of the four basic arithmetic operations, thus making them accessible to computer evaluation. literal Rutishauser, H. Lectures on numerical mathematics 10.1007/978-1-4612-3468-5_6 literal Gutknecht, M. Rutishauser1990 978-1-4612-3468-5 1990 128-174 Birkhäuser Boston Boston, MA Interpolation chapter https://doi.org/10.1007/978-1-4612-3468-5_6 abstract author container-title doi id isbn issued page publisher publisher-place title type url Experiments usually produce a discrete set of data points. If additional data points are needed, for instance, to draw a continuous curve or to change the sampling frequency of audio or video signals, interpolation methods are necessary. But interpolation is also helpful to develop more sophisticated numerical methods for the calculation of numerical derivatives or integrals. Polynomial interpolation is discussed in large detail together with its drawbacks. The methods by Lagrange and Newton are discussed, as well as the Neville method, which allow efficient determination and evaluation of the interpolating polynomial. For interpolation over a larger range, larger number of data spline interpolation is very useful which does not show the oscillatory behavior characteristic of polynomial interpolation. In a computer experiment both these approaches are compared. Multivariate interpolation is a necessary tool to process multidimensional data sets, for instance, for image processing. A computer experiment compares bilinear interpolation and bicubic spline interpolation. literal Scherer, P. O.J. Computational physics: Simulation of classical and quantum systems 10.1007/978-3-642-13990-1_2 Scherer2010 978-3-642-13990-1 2010 15-27 Springer Berlin, Heidelberg Interpolation chapter https://doi.org/10.1007/978-3-642-13990-1_2 abstract author container-title doi id issue issued page title type url volume Newton's interpolation is a classical polynomial interpolation approach and plays a significant role in numerical analysis and image processing. The interpolation function of most classical approaches is unique to the given data. In this paper, univariate and bivariate parameterized Newton-type polynomial interpolation methods are introduced. In order to express the divided differences tables neatly, the multiplicity of the points can be adjusted by introducing new parameters. Our new polynomial interpolation can be constructed only based on divided differences with one or multiple parameters which satisfy the interpolation conditions. We discuss the interpolation algorithm, theorem, dual interpolation, and information matrix algorithm. Since the proposed novel interpolation functions are parametric, they are not unique to the interpolation data. Therefore, its value in the interpolant region can be adjusted under unaltered interpolant data through the parameter values. Our parameterized Newton-type polynomial interpolating functions have a simple and explicit mathematical representation, and the proposed algorithms are simple and easy to calculate. Various numerical examples are given to demonstrate the efficiency of our method. literal Zou, L., Song, L., Wang, X., Weise, T., Chen, Y., & Zhang, C. Mathematical Problems in Engineering 10.1155/2020/9020541 https://doi.org/10.1155/2020/9020541 1 2020 9020541 A new approach to newton-type polynomial interpolation with parameters article-journal https://onlinelibrary.wiley.com/doi/abs/10.1155/2020/9020541 2020 abstract author chapter-number container-title doi id isbn issued keyword page publisher title type url Summary This chapter presents the information of interpolation and approximation for solving problems of mathematical analysis. It contains exercises and solutions that present an introduction to key concepts, a calculus review, and an updated primer on Lagrange interpolation, Newton interpolation and divided differences, interpolation error, Muller's method and inverse quadratic interpolation, Hermite interpolation, piecewise polynomial interpolation, Splines, tension Splines, and least squares concepts in approximation. The chapter features new and updated material reflecting new trends and applications in numerical methods and analysis. It is the perfect for upper-level undergraduate students in mathematics, science, and engineering courses, as well as for courses in the social sciences, medicine, and business with numerical methods and analysis components. literal Epperson, J. F. 4 An introduction to numerical methods and analysis 10.1002/9781119604570.ch4 doi:10.1002/9781119604570.ch4 9781119604570 2021 divided differences, Hermite interpolation, interpolation error, inverse quadratic interpolation, Lagrange interpolation, Muller's method, Newton interpolation, piecewise polynomial interpolation, tension Splines 101-148 John Wiley & Sons, Ltd Interpolation and approximation chapter https://onlinelibrary.wiley.com/doi/abs/10.1002/9781119604570.ch4 abstract author container-title doi id issue issued title type volume The Newton interpolation approach is developed for approximation of linear functionals. It is shown that in numerical interpolation and numerical differentiation, the Newton interpolation approach is more efficient than solving the Vandermonde systems. literal Tsao, N.-K. Numerische Mathematik 10.1007/BF01389317 tsao1977 1 1977 Newton interpolation is efficient for approximation of linear functionals article-journal 29 accessed author container-title id issn issued page publisher title type url volume 2026-04-08 literal Cayley , A. Philosophical Transactions of the Royal Society of London 35c3b3c9-7c44-327c-a4e2-a2ca3c5baca3 725331 1858 17-37 The Royal Society A memoir on the theory of matrices article-journal http://www.jstor.org/stable/108649 148 abstract author container-title doi id issn issue issued keyword page title type volume In his pioneering papers of 1845 and 1846, Arthur Cayley (1821--1895) introduced several approaches to invariant theory, the most prominent being the method of hyperdeterminant derivation. This article discusses these papers in the light of Cayley's unpublished correspondence with George Boole, who exercised considerable influence on Cayley at this formative stage of invariant theory. In the 1850s Cayley put forward a new synthesis for invariant theory framed in terms of partial differential equations. In this period he published his memoirs on quantics, the first seven of which appeared in quick succession. This article examines the background of these memoirs and makes use of unpublished correspondence with Cayley's lifelong friend, J. J. Sylvester. literal Crilly T. Historia Mathematica 10.1016/0315-0860(86)90091-1 CRILLY1986241 0315-0860 3 1986 hyperdeterminants, George Boole, partial differential equations, quantics, J. J. Sylvester, multilinear forms 241-254 The rise of cayley's invariant theory (1841--1862) article-journal 13 author container-title id issued page title type url volume literal Turing, A. M. The Quarterly Journal of Mechanics and Applied Mathematics alan1948 1948 287-308 Rounding-off errors in matrix processes article-journal https://turing.academicwebsite.com/publications/20-rounding-off-errors-in-matrix-processes 1 author doi edition id issued publisher publisher-place title title-short type literal Saff, E. B., & Snider, A. D. 10.1007/978-3-031-97222-5 2 saff2025 2025 Springer Cham Matrix fundamentals: From equation solving to signal processing Matrix fundamentals book author collection-title doi edition id issued publisher publisher-place title title-short type literal Gentle, J. E. Springer texts in statistics 10.1007/978-3-031-42144-0 3 gentle2024 2024 Springer Cham Matrix algebra: Theory, computations and applications in statistics Matrix algebra book abstract author container-title doi id isbn issued page publisher publisher-place title type url In Chapter 1 we used matrices and vectors as simple storage devices. In this chapter matrices and vectors take on a life of their own. We develop the arithmetic of matrices and vectors. Much of what we do is motivated by a desire to extend the ideas of ordinary arithmetic to matrices. literal Shores, T. S. Applied linear algebra and matrix analysis 10.1007/978-3-319-74748-4_2 Shores2018 978-3-319-74748-4 2018 65-180 Springer International Publishing Cham MATRIX ALGEBRA chapter https://doi.org/10.1007/978-3-319-74748-4_2 abstract author container-title doi id isbn issued page publisher publisher-place title type url Wir betrachten das Problem, zu einer invertierbaren Matrix \$\$A \\in \\mathbb {R}\^{n\\times n}\$\$A∈Rn`\texttimes`{=latex}nund einem Vektor \$\$b \\in \\mathbb {R}\^n\$\$b∈Rneinen Vektor \$\$x \\in \\mathbb {R}\^n\$\$x∈Rnmit \$\$A \\, x = b\$\$Ax=bzu bestimmen; kurz: Wir lösen das lineare Gleichungssystem \$\$A x = b\$\$Ax=b. Formal erhält man die Lösung durch \$\$x = A\^{-1} b\$\$x=A-1b. Aber die Berechnung von \$\$A\^{-1}\$\$A-1ist bei einer großen Matrix A aufwendig. Die Cramer'sche Regel (siehe Rezept in Abschn. 12.3) ist aus numerischer Sicht zur Berechnung der Lösung x ungeeignet. Tatsächlich liefert das Gauß'sche Eliminationsverfahren, das wir auch in Kap. 9 zur händischen Lösung eines LGS empfohlen haben, eine Zerlegung der Koeffizientenmatrix A, mit deren Hilfe es möglich ist, ein Gleichungssystem der Form \$\$A \\, x = b\$\$Ax=bmit invertierbarem A zu lösen. Diese sogenannte \$\$L\\, R\$\$LR-Zerlegung ist zudem numerisch gutartig. Gleichungssysteme mit bis zu etwa 10000 Zeilen und Unbekannten lassen sich auf diese Weise vorteilhaft lösen. Für größere Gleichungssysteme sind iterative Lösungsverfahren zu bevorzugen (siehe Kap. 71). literal Karpfinger, C. Höhere mathematik in rezepten: Begriffe, sätze und zahlreiche beispiele in kurzen lerneinheiten 10.1007/978-3-662-63305-2_11 Karpfinger2022a 978-3-662-63305-2 2022 107-117 Springer Berlin, Heidelberg L r-zerlegung einer matrix chapter https://doi.org/10.1007/978-3-662-63305-2_11 abstract author container-title doi id isbn issued page publisher publisher-place title type url We have already used matrices to solve systems of linear equations: Matrices have been a helpful tool here to represent linear systems of equations economically and clearly. Matrices also serve as a tool in other, manifold ways. This is one reason to consider matrices in their own right, and to clearly illustrate and practice all kinds of manipulations that are possible with them: We will add, multiply, multiply, exponentiate, transpose, and invert matrices. But everything in order. literal Karpfinger, C. Calculus and linear algebra in recipes: Terms, phrases and numerous examples in short learning units 10.1007/978-3-662-65458-3_10 Karpfinger2022b 978-3-662-65458-3 2022 87-100 Springer Berlin, Heidelberg Calculating with matrices chapter https://doi.org/10.1007/978-3-662-65458-3_10 abstract author container-title doi id issn issue issued keyword page title type url volume Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method --- which Euler did not recommend, which Legendre called "ordinary," and which Gauss called "common" --- is now named after Gauss: "Gaussian" elimination. Gauss's name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least-squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices. Zusammenfassung In Aufzeichnungen, die Newton lieber nicht der Veröffentlichung preisgegeben hätte, beschreibt er den Prozess für die Lösung von simultanen Gleichungen, den spätere Autoren speziell für lineare Gleichungen anwandten. Diese Methode --- welche Euler nicht empfahl, welche Legendre "ordinaire" nannte, und welche Gauß "gewöhnlich" nannte --- wird nun nach Gauß benannt: Gaußsches Eliminationsverfahren. Die Verbindung des Gaußschen Namens mit Elimination wurde dadurch hervorgebracht, dass professionelle Rechner eine Notation übernahmen, die Gauß speziell für seine eigenen Berechnungen der kleinsten Quadrate ersonnen hatte, welche zuließ, das Elimination als eine Sequenz von arithmetischen Rechenoperationen betrachtet wurde, die wiederholt für Handrechnungen optimisiert wurden und schließlich durch Matrizen beschrieben wurden. literal Grcar, J. F. Historia Mathematica 10.1016/j.hm.2010.06.003 GRCAR2011163 0315-0860 2 2011 Algebra before 1800, Gaussian elimination, Human computers, Least squares method, Mathematics education 163-218 How ordinary elimination became gaussian elimination article-journal https://www.sciencedirect.com/science/article/pii/S0315086010000376 38