Principal Component Analysis (PCA)
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Arxiv:1910.08883V3 [Stat.ML] 2 Apr 2021 in Real Data [8,9]
Nonpar MANOVA via Independence Testing Sambit Panda1;2, Cencheng Shen3, Ronan Perry1, Jelle Zorn4, Antoine Lutz4, Carey E. Priebe5 and Joshua T. Vogelstein1;2;6∗ Abstract. The k-sample testing problem tests whether or not k groups of data points are sampled from the same distri- bution. Multivariate analysis of variance (Manova) is currently the gold standard for k-sample testing but makes strong, often inappropriate, parametric assumptions. Moreover, independence testing and k-sample testing are tightly related, and there are many nonparametric multivariate independence tests with strong theoretical and em- pirical properties, including distance correlation (Dcorr) and Hilbert-Schmidt-Independence-Criterion (Hsic). We prove that universally consistent independence tests achieve universally consistent k-sample testing, and that k- sample statistics like Energy and Maximum Mean Discrepancy (MMD) are exactly equivalent to Dcorr. Empirically evaluating these tests for k-sample-scenarios demonstrates that these nonparametric independence tests typically outperform Manova, even for Gaussian distributed settings. Finally, we extend these non-parametric k-sample- testing procedures to perform multiway and multilevel tests. Thus, we illustrate the existence of many theoretically motivated and empirically performant k-sample-tests. A Python package with all independence and k-sample tests called hyppo is available from https://hyppo.neurodata.io/. 1 Introduction A fundamental problem in statistics is the k-sample testing problem. Consider the p p two-sample problem: we obtain two datasets ui 2 R for i = 1; : : : ; n and vj 2 R for j = 1; : : : ; m. Assume each ui is sampled independently and identically (i.i.d.) from FU and that each vj is sampled i.i.d. -
Correspondence Analysis
NCSS Statistical Software NCSS.com Chapter 430 Correspondence Analysis Introduction Correspondence analysis (CA) is a technique for graphically displaying a two-way table by calculating coordinates representing its rows and columns. These coordinates are analogous to factors in a principal components analysis (used for continuous data), except that they partition the Chi-square value used in testing independence instead of the total variance. For those of you who are new to CA, we suggest that you obtain Greenacre (1993). This is an excellent introduction to the subject, is very readable, and is suitable for self-study. If you want to understand the technique in detail, you should obtain this (paperback) book. Discussion We will explain CA using the following example. Suppose an aptitude survey consisting of eight yes or no questions is given to a group of tenth graders. The instructions on the survey allow the students to answer only those questions that they want to. The results of the survey are tabulated as follows. Aptitude Survey Results – Counts Question Yes No Total Q1 155 938 1093 Q2 19 63 82 Q3 395 542 937 Q4 61 64 125 Q5 1336 876 2212 Q6 22 14 36 Q7 864 354 1218 Q8 920 185 1105 Total 3772 3036 6808 Take a few moments to study this table and see what you can discover. The most obvious pattern is that many of the students did not answer all the questions. This makes response patterns between rows difficult to analyze. To solve this problem of differential response rates, we create a table of row percents (or row profiles as they are called in CA). -
Discussion Notes for Aristotle's Politics
Sean Hannan Classics of Social & Political Thought I Autumn 2014 Discussion Notes for Aristotle’s Politics BOOK I 1. Introducing Aristotle a. Aristotle was born around 384 BCE (in Stagira, far north of Athens but still a ‘Greek’ city) and died around 322 BCE, so he lived into his early sixties. b. That means he was born about fifteen years after the trial and execution of Socrates. He would have been approximately 45 years younger than Plato, under whom he was eventually sent to study at the Academy in Athens. c. Aristotle stayed at the Academy for twenty years, eventually becoming a teacher there himself. When Plato died in 347 BCE, though, the leadership of the school passed on not to Aristotle, but to Plato’s nephew Speusippus. (As in the Republic, the stubborn reality of Plato’s family connections loomed large.) d. After living in Asia Minor from 347-343 BCE, Aristotle was invited by King Philip of Macedon to serve as the tutor for Philip’s son Alexander (yes, the Great). Aristotle taught Alexander for eight years, then returned to Athens in 335 BCE. There he founded his own school, the Lyceum. i. Aside: We should remember that these schools had substantial afterlives, not simply as ideas in texts, but as living sites of intellectual energy and exchange. The Academy lasted from 387 BCE until 83 BCE, then was re-founded as a ‘Neo-Platonic’ school in 410 CE. It was finally closed by Justinian in 529 CE. (Platonic philosophy was still being taught at Athens from 83 BCE through 410 CE, though it was not disseminated through a formalized Academy.) The Lyceum lasted from 334 BCE until 86 BCE, when it was abandoned as the Romans sacked Athens. -
A Correspondence Analysis of Child-Care Students' and Medical
International Education Journal Vol 5, No 2, 2004 http://iej.cjb.net 176 A Correspondence Analysis of Child-Care Students’ and Medical Students’ Knowledge about Teaching and Learning Helen Askell-Williams Flinders University School of Education [email protected] Michael J. Lawson Flinders University School of Education [email protected] This paper describes the application of correspondence analysis to transcripts gathered from focussed interviews about teaching and learning held with a small sample of child-care students, medical students and the students’ teachers. Seven dimensions emerged from the analysis, suggesting that the knowledge that underlies students’ learning intentions and actions is multi-dimensional and transactive. It is proposed that the multivariate, multidimensional, discovery approach of the correspondence analysis technique has considerable potential for data analysis in the social sciences. Teaching, learning, knowledge, correspondence analysis INTRODUCTION The purpose of this paper is to describe the application of correspondence analysis to rich text- based data derived from interviews with teachers and learners about their knowledge about teaching and learning. Correspondence analysis is a non-linear, multidimensional technique of multivariate descriptive analysis that “specialises in ‘discovering,’ through detailed analysis of a given data set” (Nishisato, 1994 p.7). A description of what teachers and learners know about teaching and learning will assist in developing the educational community’s understanding about teaching and learning. If researchers, designers and policy makers are well informed about teachers’ and learners’ knowledge, they will be better equipped to design and recommend educational programs that meet students’ learning needs. If teachers possess high quality knowledge about their own, and their students’, knowledge then they will be better equipped to design and deliver high quality teaching. -
449, 468 Across-Stage Inferencing 568, 57
xxxv Index 6 Cs 1254, 1258 Apriori-based Graph Mining (AGM) 13 6 Ps 1253 Apriori with Subgroup and Constraint (ASC) 948 Arabidopsis thaliana 205, 227 A ArM32 operations 458 ArnetMiner 331 abnormal alarm 2183 Arterial Pressure (AP) 902 academic theory 1678 ArTex with Resolutions (ARes) 1047 Access Methods (AM) 449, 468 arthrokinetics restrictions 653 across-stage inferencing 568, 575-576, 580, 583 Artificial Intelligence (AI) 991, 1262 Active Learning (AL) 67, 69 Artificial Neural Networks (ANN) 250, 299, 367, 704, ADaMSoft 1315 1114, 1573, 2253 Adaptive Modulation and Coding (AMC) 345 Art of War 477 Adaptive Neuro Fuzzy Inference System (ANFIS) Asset Management (AM) 1607 1114-1115, 1124, 2250, 2252, 2254-2255, 2268 Association Rule Mining (ARM) 28, 47, 108, 148, Additive White Gaussian Noise (AWGN) 2084 684, 856, 974, 989, 1740 advanced search 1314, 1317, 1338 ANGEL data 846-847 Adverse Drug Events (ADEs) 1942 approaches 1739-1740, 1743, 1748-1749 Agglomerative Hierarchical Clustering (AHC) 1435 association rules 57, 65, 587 aggregate constraint 1842 calendric 594 aggregate function 681-682, 686, 1434, 1842, 2028- maximal 503-509, 512-514 2029 quality measures 130 agrometeorological agents 1625 Association rules for Textures (ArTex) 1047 agrometeorological methods 1625-1626 Associative Classification (AC) 975 agrometeorological variables 1626 atomic condition 285-287 ambiguity-directed sampling 71 Attention Deficit Hyperactivity Disorder (ADHD) American Bureau of Shipping (ABS) 2175 1463 American National Election Studies (ANES) 30, 42 -
Research Article the Use of Multiple Correspondence Analysis to Explore Associations Between Categories of Qualitative Variables in Healthy Ageing
Hindawi Publishing Corporation Journal of Aging Research Volume 2013, Article ID 302163, 12 pages http://dx.doi.org/10.1155/2013/302163 Research Article The Use of Multiple Correspondence Analysis to Explore Associations between Categories of Qualitative Variables in Healthy Ageing Patrício Soares Costa,1,2 Nadine Correia Santos,1,2 Pedro Cunha,1,2,3 Jorge Cotter,1,2,3 and Nuno Sousa1,2 1 Life and Health Sciences Research Institute (ICVS), School of Health Sciences, University of Minho, 4710-057 Braga, Portugal 2 ICVS/3B’s, PT Government Associate Laboratory, Guimaraes,˜ 4710-057 Braga, Portugal 3 Centro Hospital Alto Ave, EPE, 4810-055 Guimaraes,˜ Portugal Correspondence should be addressed to Nuno Sousa; [email protected] Received 30 June 2013; Revised 23 August 2013; Accepted 30 August 2013 Academic Editor: F. Richard Ferraro Copyright © 2013 Patr´ıcio Soares Costa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The main focus of this study was to illustrate the applicability of multiple correspondence analysis (MCA) in detecting and representing underlying structures in large datasets used to investigate cognitive ageing. Principal component analysis (PCA) was used to obtain main cognitive dimensions, and MCA was used to detect and explore relationships between cognitive, clinical, physical, and lifestyle variables. Two PCA dimensions were identified (general cognition/executive function and memory), and two MCA dimensions were retained. Poorer cognitive performance was associated with older age, less school years, unhealthier lifestyle indicators, and presence of pathology. -
On the Meaning and Use of Kurtosis
Psychological Methods Copyright 1997 by the American Psychological Association, Inc. 1997, Vol. 2, No. 3,292-307 1082-989X/97/$3.00 On the Meaning and Use of Kurtosis Lawrence T. DeCarlo Fordham University For symmetric unimodal distributions, positive kurtosis indicates heavy tails and peakedness relative to the normal distribution, whereas negative kurtosis indicates light tails and flatness. Many textbooks, however, describe or illustrate kurtosis incompletely or incorrectly. In this article, kurtosis is illustrated with well-known distributions, and aspects of its interpretation and misinterpretation are discussed. The role of kurtosis in testing univariate and multivariate normality; as a measure of departures from normality; in issues of robustness, outliers, and bimodality; in generalized tests and estimators, as well as limitations of and alternatives to the kurtosis measure [32, are discussed. It is typically noted in introductory statistics standard deviation. The normal distribution has a kur- courses that distributions can be characterized in tosis of 3, and 132 - 3 is often used so that the refer- terms of central tendency, variability, and shape. With ence normal distribution has a kurtosis of zero (132 - respect to shape, virtually every textbook defines and 3 is sometimes denoted as Y2)- A sample counterpart illustrates skewness. On the other hand, another as- to 132 can be obtained by replacing the population pect of shape, which is kurtosis, is either not discussed moments with the sample moments, which gives or, worse yet, is often described or illustrated incor- rectly. Kurtosis is also frequently not reported in re- ~(X i -- S)4/n search articles, in spite of the fact that virtually every b2 (•(X i - ~')2/n)2' statistical package provides a measure of kurtosis. -
Univariate and Multivariate Skewness and Kurtosis 1
Running head: UNIVARIATE AND MULTIVARIATE SKEWNESS AND KURTOSIS 1 Univariate and Multivariate Skewness and Kurtosis for Measuring Nonnormality: Prevalence, Influence and Estimation Meghan K. Cain, Zhiyong Zhang, and Ke-Hai Yuan University of Notre Dame Author Note This research is supported by a grant from the U.S. Department of Education (R305D140037). However, the contents of the paper do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government. Correspondence concerning this article can be addressed to Meghan Cain ([email protected]), Ke-Hai Yuan ([email protected]), or Zhiyong Zhang ([email protected]), Department of Psychology, University of Notre Dame, 118 Haggar Hall, Notre Dame, IN 46556. UNIVARIATE AND MULTIVARIATE SKEWNESS AND KURTOSIS 2 Abstract Nonnormality of univariate data has been extensively examined previously (Blanca et al., 2013; Micceri, 1989). However, less is known of the potential nonnormality of multivariate data although multivariate analysis is commonly used in psychological and educational research. Using univariate and multivariate skewness and kurtosis as measures of nonnormality, this study examined 1,567 univariate distriubtions and 254 multivariate distributions collected from authors of articles published in Psychological Science and the American Education Research Journal. We found that 74% of univariate distributions and 68% multivariate distributions deviated from normal distributions. In a simulation study using typical values of skewness and kurtosis that we collected, we found that the resulting type I error rates were 17% in a t-test and 30% in a factor analysis under some conditions. Hence, we argue that it is time to routinely report skewness and kurtosis along with other summary statistics such as means and variances. -
A Pragmatic Stylistic Framework for Text Analysis
International Journal of Education ISSN 1948-5476 2015, Vol. 7, No. 1 A Pragmatic Stylistic Framework for Text Analysis Ibrahim Abushihab1,* 1English Department, Alzaytoonah University of Jordan, Jordan *Correspondence: English Department, Alzaytoonah University of Jordan, Jordan. E-mail: [email protected] Received: September 16, 2014 Accepted: January 16, 2015 Published: January 27, 2015 doi:10.5296/ije.v7i1.7015 URL: http://dx.doi.org/10.5296/ije.v7i1.7015 Abstract The paper focuses on the identification and analysis of a short story according to the principles of pragmatic stylistics and discourse analysis. The focus on text analysis and pragmatic stylistics is essential to text studies, comprehension of the message of a text and conveying the intention of the producer of the text. The paper also presents a set of standards of textuality and criteria from pragmatic stylistics to text analysis. Analyzing a text according to principles of pragmatic stylistics means approaching the text’s meaning and the intention of the producer. Keywords: Discourse analysis, Pragmatic stylistics Textuality, Fictional story and Stylistics 110 www.macrothink.org/ije International Journal of Education ISSN 1948-5476 2015, Vol. 7, No. 1 1. Introduction Discourse Analysis is concerned with the study of the relation between language and its use in context. Harris (1952) was interested in studying the text and its social situation. His paper “Discourse Analysis” was a far cry from the discourse analysis we are studying nowadays. The need for analyzing a text with more comprehensive understanding has given the focus on the emergence of pragmatics. Pragmatics focuses on the communicative use of language conceived as intentional human action. -
Mathematics: Analysis and Approaches First Assessments for SL and HL—2021
International Baccalaureate Diploma Programme Subject Brief Mathematics: analysis and approaches First assessments for SL and HL—2021 The Diploma Programme (DP) is a rigorous pre-university course of study designed for students in the 16 to 19 age range. It is a broad-based two-year course that aims to encourage students to be knowledgeable and inquiring, but also caring and compassionate. There is a strong emphasis on encouraging students to develop intercultural understanding, open-mindedness, and the attitudes necessary for them LOMA PROGRA IP MM to respect and evaluate a range of points of view. B D E I DIES IN LANGUA STU GE ND LITERATURE The course is presented as six academic areas enclosing a central core. Students study A A IN E E N D N DG two modern languages (or a modern language and a classical language), a humanities G E D IV A O L E I W X S ID U IT O T O G E U or social science subject, an experimental science, mathematics and one of the creative IS N N C N K ES TO T I A U CH E D E A A A L F C T L Q O H E S O R I I C P N D arts. Instead of an arts subject, students can choose two subjects from another area. P G E A Y S A E R S It is this comprehensive range of subjects that makes the Diploma Programme a O S E A Y H T demanding course of study designed to prepare students effectively for university entrance. -
Cronbach, L. J. (1951). Coefficient Alpha and the Internal Structure of Tests. Psychometrika, 16, 297-334 (28,307 Citations in Google Scholar As of 4/1/2016)
Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334 (28,307 citations in Google Scholar as of 4/1/2016). By far the most cited article in Psychometrika is Lee Cronbach’s famous study of coefficient alpha. I will address four questions: What makes the article special? Why do people cite the article more than other special articles? Was the article also influential beyond psychology? How does alpha fit in present-day psychometrics? What makes the article special? In the first half of the 20 th century, psychometricians distinguished two practically useful types of reliability, which were coefficients of stability and coefficients of equivalence. Coefficients of stability are correlations between two test scores obtained with the same test on two different time points, administered with a time interval that serves the psychologist’s purpose. The coefficient provides information about the stability of the attribute across the time interval. Coefficients of equivalence are correlations between two test scores based on different sets of items intended to measure the same attribute, and express the degree to which the different sets are interchangeable. Cronbach mentioned two additional coefficient types. One type is the coefficient of equivalence and stability that refers to the correlation between test scores on different sets of items, one set administered first and the other set administered after a useful time interval. The other type is the coefficient of precision , referring to the correlation between the scores on the same test administered twice in one session to the same group of persons. The latter coefficient is hypothetical and not realizable in practice. -
Multivariate Chemometrics As a Strategy to Predict the Allergenic Nature of Food Proteins
S S symmetry Article Multivariate Chemometrics as a Strategy to Predict the Allergenic Nature of Food Proteins Miroslava Nedyalkova 1 and Vasil Simeonov 2,* 1 Department of Inorganic Chemistry, Faculty of Chemistry and Pharmacy, University of Sofia, 1 James Bourchier Blvd., 1164 Sofia, Bulgaria; [email protected]fia.bg 2 Department of Analytical Chemistry, Faculty of Chemistry and Pharmacy, University of Sofia, 1 James Bourchier Blvd., 1164 Sofia, Bulgaria * Correspondence: [email protected]fia.bg Received: 3 September 2020; Accepted: 21 September 2020; Published: 29 September 2020 Abstract: The purpose of the present study is to develop a simple method for the classification of food proteins with respect to their allerginicity. The methods applied to solve the problem are well-known multivariate statistical approaches (hierarchical and non-hierarchical cluster analysis, two-way clustering, principal components and factor analysis) being a substantial part of modern exploratory data analysis (chemometrics). The methods were applied to a data set consisting of 18 food proteins (allergenic and non-allergenic). The results obtained convincingly showed that a successful separation of the two types of food proteins could be easily achieved with the selection of simple and accessible physicochemical and structural descriptors. The results from the present study could be of significant importance for distinguishing allergenic from non-allergenic food proteins without engaging complicated software methods and resources. The present study corresponds entirely to the concept of the journal and of the Special issue for searching of advanced chemometric strategies in solving structural problems of biomolecules. Keywords: food proteins; allergenicity; multivariate statistics; structural and physicochemical descriptors; classification 1.