A MATLAB Toolbox for Circular Statistics
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Binomials, Contingency Tables, Chi-Square Tests
Binomials, contingency tables, chi-square tests Joe Felsenstein Department of Genome Sciences and Department of Biology Binomials, contingency tables, chi-square tests – p.1/16 Confidence limits on a proportion To work out a confidence interval on binomial proportions, use binom.test. If we want to see whether 0.20 is too high a probability of Heads when we observe 5 Heads out of 50 tosses, we use binom.test(5, 50, 0.2) which gives probability of 5 or fewer Heads as 0.09667, so a test does not exclude 0.20 as the Heads probability. > binom.test(5,50,0.2) Exact binomial test data: 5 and 50 number of successes = 5, number of trials = 50, p-value = 0.07883 alternative hypothesis: true probability of success is not equal to 0.2 95 percent confidence interval: 0.03327509 0.21813537 sample estimates: probability of success 0.1 Binomials, contingency tables, chi-square tests – p.2/16 Confidence intervals and tails of binomials Confidence limits for p if there are 5 Heads out of 50 tosses p = 0.0332750 is 0.025 p = 0.21813537 is 0.025 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Binomials, contingency tables, chi-square tests – p.3/16 Testing equality of binomial proportions How do we test whether two coins have the same Heads probability? This is hard, but there is a good approximation, the chi-square test. You set up a 2 × 2 table of numbers of outcomes: Heads Tails Coin #1 15 25 Coin #2 9 31 In fact the chi-square test can test bigger tables: R rows by C columns. -
1 Introduction
TR:UPM-ETSINF/DIA/2015-1 1 Univariate and bivariate truncated von Mises distributions Pablo Fernandez-Gonzalez, Concha Bielza, Pedro Larra~naga Department of Artificial Intelligence, Universidad Polit´ecnica de Madrid Abstract: In this article we study the univariate and bivariate truncated von Mises distribution, as a generalization of the von Mises distribution (Jupp and Mardia (1989)), (Mardia and Jupp (2000)). This implies the addition of two or four new truncation parameters in the univariate and, bivariate cases, respectively. The re- sults include the definition, properties of the distribution and maximum likelihood estimators for the univariate and bivariate cases. Additionally, the analysis of the bivariate case shows how the conditional distribution is a truncated von Mises dis- tribution, whereas the marginal distribution that generalizes the distribution intro- duced in Singh (2002). From the viewpoint of applications, we test the distribution with simulated data, as well as with data regarding leaf inclination angles (Bowyer and Danson. (2005)) and dihedral angles in protein chains (Murzin AG (1995)). This research aims to assert this probability distribution as a potential option for modelling or simulating any kind of phenomena where circular distributions are applicable. Key words and phrases: Angular probability distributions, Directional statistics, von Mises distribution, Truncated probability distributions. 1 Introduction The von Mises distribution has received undisputed attention in the field of di- rectional statistics (Jupp and Mardia (1989)) and in other areas like supervised classification (Lopez-Cruz et al. (2013)). Thanks to desirable properties such as its symmetry, mathematical tractability and convergence to the wrapped nor- mal distribution (Mardia and Jupp (2000)) for high concentrations, it is a viable option for many statistical analyses. -
Package 'Rfast'
Package ‘Rfast’ May 18, 2021 Type Package Title A Collection of Efficient and Extremely Fast R Functions Version 2.0.3 Date 2021-05-17 Author Manos Papadakis, Michail Tsagris, Marios Dimitriadis, Stefanos Fafalios, Ioannis Tsamardi- nos, Matteo Fasiolo, Giorgos Borboudakis, John Burkardt, Changliang Zou, Kleanthi Lakio- taki and Christina Chatzipantsiou. Maintainer Manos Papadakis <[email protected]> Depends R (>= 3.5.0), Rcpp (>= 0.12.3), RcppZiggurat LinkingTo Rcpp (>= 0.12.3), RcppArmadillo SystemRequirements C++11 BugReports https://github.com/RfastOfficial/Rfast/issues URL https://github.com/RfastOfficial/Rfast Description A collection of fast (utility) functions for data analysis. Column- and row- wise means, medians, variances, minimums, maximums, many t, F and G-square tests, many re- gressions (normal, logistic, Poisson), are some of the many fast functions. Refer- ences: a) Tsagris M., Papadakis M. (2018). Taking R to its lim- its: 70+ tips. PeerJ Preprints 6:e26605v1 <doi:10.7287/peerj.preprints.26605v1>. b) Tsagris M. and Pa- padakis M. (2018). Forward regression in R: from the extreme slow to the extreme fast. Jour- nal of Data Science, 16(4): 771--780. <doi:10.6339/JDS.201810_16(4).00006>. License GPL (>= 2.0) NeedsCompilation yes Repository CRAN Date/Publication 2021-05-17 23:50:05 UTC R topics documented: Rfast-package . .6 All k possible combinations from n elements . 10 Analysis of covariance . 11 1 2 R topics documented: Analysis of variance with a count variable . 12 Angular central Gaussian random values simulation . 13 ANOVA for two quasi Poisson regression models . 14 Apply method to Positive and Negative number . -
Binomial Test Models and Item Difficulty
Binomial Test Models and Item Difficulty Wim J. van der Linden Twente University of Technology In choosing a binomial test model, it is impor- have characteristic functions of the Guttman type. tant to know exactly what conditions are imposed In contrast, the stochastic conception allows non- on item difficulty. In this paper these conditions Guttman items but requires that all characteristic are examined for both a deterministic and a sto- functions must intersect at the same point, which chastic conception of item responses. It appears implies equal classically defined difficulty. The that they are more restrictive than is generally beta-binomial model assumes identical char- understood and differ for both conceptions. When acteristic functions for both conceptions, and this the binomial model is applied to a fixed examinee, also implies equal difficulty. Finally, the compound the deterministic conception imposes no conditions binomial model entails no restrictions on item diffi- on item difficulty but requires instead that all items culty. In educational and psychological testing, binomial models are a class of models increasingly being applied. For example, in the area of criterion-referenced measurement or mastery testing, where tests are usually conceptualized as samples of items randomly drawn from a large pool or do- main, binomial models are frequently used for estimating examinees’ mastery of a domain and for de- termining sample size. Despite the agreement among several writers on the usefulness of binomial models, opinions seem to differ on the restrictions on item difficulties implied by the models. Mill- man (1973, 1974) noted that in applying the binomial models, items may be relatively heterogeneous in difficulty. -
Bitest — Binomial Probability Test
Title stata.com bitest — Binomial probability test Description Quick start Menu Syntax Option Remarks and examples Stored results Methods and formulas Reference Also see Description bitest performs exact hypothesis tests for binomial random variables. The null hypothesis is that the probability of a success on a trial is #p. The total number of trials is the number of nonmissing values of varname (in bitest) or #N (in bitesti). The number of observed successes is the number of 1s in varname (in bitest) or #succ (in bitesti). varname must contain only 0s, 1s, and missing. bitesti is the immediate form of bitest; see [U] 19 Immediate commands for a general introduction to immediate commands. Quick start Exact test for probability of success (a = 1) is 0.4 bitest a = .4 With additional exact probabilities bitest a = .4, detail Exact test that the probability of success is 0.46, given 22 successes in 74 trials bitesti 74 22 .46 Menu bitest Statistics > Summaries, tables, and tests > Classical tests of hypotheses > Binomial probability test bitesti Statistics > Summaries, tables, and tests > Classical tests of hypotheses > Binomial probability test calculator 1 2 bitest — Binomial probability test Syntax Binomial probability test bitest varname== #p if in weight , detail Immediate form of binomial probability test bitesti #N #succ #p , detail by is allowed with bitest; see [D] by. fweights are allowed with bitest; see [U] 11.1.6 weight. Option Advanced £ £detail shows the probability of the observed number of successes, kobs; the probability of the number of successes on the opposite tail of the distribution that is used to compute the two-sided p-value, kopp; and the probability of the point next to kopp. -
Jise.Org/Volume31/N1/Jisev31n1p72.Html
Journal of Information Volume 31 Systems Issue 1 Education Winter 2020 Experiences in Using a Multiparadigm and Multiprogramming Approach to Teach an Information Systems Course on Introduction to Programming Juan Gutiérrez-Cárdenas Recommended Citation: Gutiérrez-Cárdenas, J. (2020). Experiences in Using a Multiparadigm and Multiprogramming Approach to Teach an Information Systems Course on Introduction to Programming. Journal of Information Systems Education, 31(1), 72-82. Article Link: http://jise.org/Volume31/n1/JISEv31n1p72.html Initial Submission: 23 December 2018 Accepted: 24 May 2019 Abstract Posted Online: 12 September 2019 Published: 3 March 2020 Full terms and conditions of access and use, archived papers, submission instructions, a search tool, and much more can be found on the JISE website: http://jise.org ISSN: 2574-3872 (Online) 1055-3096 (Print) Journal of Information Systems Education, Vol. 31(1) Winter 2020 Experiences in Using a Multiparadigm and Multiprogramming Approach to Teach an Information Systems Course on Introduction to Programming Juan Gutiérrez-Cárdenas Faculty of Engineering and Architecture Universidad de Lima Lima, 15023, Perú [email protected] ABSTRACT In the current literature, there is limited evidence of the effects of teaching programming languages using two different paradigms concurrently. In this paper, we present our experience in using a multiparadigm and multiprogramming approach for an Introduction to Programming course. The multiparadigm element consisted of teaching the imperative and functional paradigms, while the multiprogramming element involved the Scheme and Python programming languages. For the multiparadigm part, the lectures were oriented to compare the similarities and differences between the functional and imperative approaches. For the multiprogramming part, we chose syntactically simple software tools that have a robust set of prebuilt functions and available libraries. -
Arxiv:1911.00962V1 [Cs.CV] 3 Nov 2019 Though the Label of Pose Itself Is Discrete
Conservative Wasserstein Training for Pose Estimation Xiaofeng Liu1;2†∗, Yang Zou1y, Tong Che3y, Peng Ding4, Ping Jia4, Jane You5, B.V.K. Vijaya Kumar1 1Carnegie Mellon University; 2Harvard University; 3MILA 4CIOMP, Chinese Academy of Sciences; 5The Hong Kong Polytechnic University yContribute equally ∗Corresponding to: [email protected] Pose with true label Probability of Softmax prediction Abstract Expected probability (i.e., 1 in one-hot case) σ푁−1 푡 푠 푡푗∗ of each pose ( 푖=0 푠푖 = 1) 푗∗ 3 푠 푠 2 푠3 2 푠푗∗ 푠푗∗ This paper targets the task with discrete and periodic 푠1 푠1 class labels (e:g:; pose/orientation estimation) in the con- Same Cross-entropy text of deep learning. The commonly used cross-entropy or 푠0 푠0 regression loss is not well matched to this problem as they 푠푁−1 ignore the periodic nature of the labels and the class simi- More ideal Inferior 푠푁−1 larity, or assume labels are continuous value. We propose to Distribution 푠푁−2 Distribution 푠푁−2 incorporate inter-class correlations in a Wasserstein train- Figure 1. The limitation of CE loss for pose estimation. The ing framework by pre-defining (i:e:; using arc length of a ground truth direction of the car is tj ∗. Two possible softmax circle) or adaptively learning the ground metric. We extend predictions (green bar) of the pose estimator have the same proba- the ground metric as a linear, convex or concave increasing bility at tj ∗ position. Therefore, both predicted distributions have function w:r:t: arc length from an optimization perspective. the same CE loss. -
Contributions to Directional Statistics Based Clustering Methods
University of California Santa Barbara Contributions to Directional Statistics Based Clustering Methods A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Statistics & Applied Probability by Brian Dru Wainwright Committee in charge: Professor S. Rao Jammalamadaka, Chair Professor Alexander Petersen Professor John Hsu September 2019 The Dissertation of Brian Dru Wainwright is approved. Professor Alexander Petersen Professor John Hsu Professor S. Rao Jammalamadaka, Committee Chair June 2019 Contributions to Directional Statistics Based Clustering Methods Copyright © 2019 by Brian Dru Wainwright iii Dedicated to my loving wife, Carmen Rhodes, without whom none of this would have been possible, and to my sons, Max, Gus, and Judah, who have yet to know a time when their dad didn’t need to work from early till late. And finally, to my mother, Judith Moyer, without your tireless love and support from the beginning, I quite literally wouldn’t be here today. iv Acknowledgements I would like to offer my humble and grateful acknowledgement to all of the wonderful col- laborators I have had the fortune to work with during my graduate education. Much of the impetus for the ideas presented in this dissertation were derived from our work together. In particular, I would like to thank Professor György Terdik, University of Debrecen, Faculty of Informatics, Department of Information Technology. I would also like to thank Professor Saumyadipta Pyne, School of Public Health, University of Pittsburgh, and Mr. Hasnat Ali, L.V. Prasad Eye Institute, Hyderabad, India. I would like to extend a special thank you to Dr Alexander Petersen, who has held the dual role of serving on my Doctoral supervisory commit- tee as well as wearing the hat of collaborator. -
Recent Advances in Directional Statistics
Recent advances in directional statistics Arthur Pewsey1;3 and Eduardo García-Portugués2 Abstract Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio- temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed. Keywords: Classification; Clustering; Dimension reduction; Distributional -
Chapter 2 Parameter Estimation for Wrapped Distributions
ABSTRACT RAVINDRAN, PALANIKUMAR. Bayesian Analysis of Circular Data Using Wrapped Dis- tributions. (Under the direction of Associate Professor Sujit K. Ghosh). Circular data arise in a number of different areas such as geological, meteorolog- ical, biological and industrial sciences. We cannot use standard statistical techniques to model circular data, due to the circular geometry of the sample space. One of the com- mon methods used to analyze such data is the wrapping approach. Using the wrapping approach, we assume that, by wrapping a probability distribution from the real line onto the circle, we obtain the probability distribution for circular data. This approach creates a vast class of probability distributions that are flexible to account for different features of circular data. However, the likelihood-based inference for such distributions can be very complicated and computationally intensive. The EM algorithm used to compute the MLE is feasible, but is computationally unsatisfactory. Instead, we use Markov Chain Monte Carlo (MCMC) methods with a data augmentation step, to overcome such computational difficul- ties. Given a probability distribution on the circle, we assume that the original distribution was distributed on the real line, and then wrapped onto the circle. If we can unwrap the distribution off the circle and obtain a distribution on the real line, then the standard sta- tistical techniques for data on the real line can be used. Our proposed methods are flexible and computationally efficient to fit a wide class of wrapped distributions. Furthermore, we can easily compute the usual summary statistics. We present extensive simulation studies to validate the performance of our method. -
A Compendium of Conjugate Priors
A Compendium of Conjugate Priors Daniel Fink Environmental Statistics Group Department of Biology Montana State Univeristy Bozeman, MT 59717 May 1997 Abstract This report reviews conjugate priors and priors closed under sampling for a variety of data generating processes where the prior distributions are univariate, bivariate, and multivariate. The effects of transformations on conjugate prior relationships are considered and cases where conjugate prior relationships can be applied under transformations are identified. Univariate and bivariate prior relationships are verified using Monte Carlo methods. Contents 1 Introduction Experimenters are often in the position of having had collected some data from which they desire to make inferences about the process that produced that data. Bayes' theorem provides an appealing approach to solving such inference problems. Bayes theorem, π(θ) L(θ x ; : : : ; x ) g(θ x ; : : : ; x ) = j 1 n (1) j 1 n π(θ) L(θ x ; : : : ; x )dθ j 1 n is commonly interpreted in the following wayR. We want to make some sort of inference on the unknown parameter(s), θ, based on our prior knowledge of θ and the data collected, x1; : : : ; xn . Our prior knowledge is encapsulated by the probability distribution on θ; π(θ). The data that has been collected is combined with our prior through the likelihood function, L(θ x ; : : : ; x ) . The j 1 n normalized product of these two components yields a probability distribution of θ conditional on the data. This distribution, g(θ x ; : : : ; x ) , is known as the posterior distribution of θ. Bayes' j 1 n theorem is easily extended to cases where is θ multivariate, a vector of parameters. -
A General Approach for Obtaining Wrapped Circular Distributions Via Mixtures
UC Santa Barbara UC Santa Barbara Previously Published Works Title A General Approach for obtaining wrapped circular distributions via mixtures Permalink https://escholarship.org/uc/item/5r7521z5 Journal Sankhya, 79 Author Jammalamadaka, Sreenivasa Rao Publication Date 2017 Peer reviewed eScholarship.org Powered by the California Digital Library University of California 1 23 Your article is protected by copyright and all rights are held exclusively by Indian Statistical Institute. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Sankhy¯a:TheIndianJournalofStatistics 2017, Volume 79-A, Part 1, pp. 133-157 c 2017, Indian Statistical Institute ! AGeneralApproachforObtainingWrappedCircular Distributions via Mixtures S. Rao Jammalamadaka University of California, Santa Barbara, USA Tomasz J. Kozubowski University of Nevada, Reno, USA Abstract We show that the operations of mixing and wrapping linear distributions around a unit circle commute, and can produce a wide variety of circular models. In particular, we show that many wrapped circular models studied in the literature can be obtained as scale mixtures of just the wrapped Gaussian and the wrapped exponential distributions, and inherit many properties from these two basic models.