Profile transformations for reciprocal averaging and singular value decomposition

IF 1 4区 数学 Q3 STATISTICS & PROBABILITY
Ting-Wu Wang, Eric J. Beh, Rosaria Lombardo, Ian W. Renner
{"title":"Profile transformations for reciprocal averaging and singular value decomposition","authors":"Ting-Wu Wang, Eric J. Beh, Rosaria Lombardo, Ian W. Renner","doi":"10.1007/s00180-024-01517-x","DOIUrl":null,"url":null,"abstract":"<p>Power transformations of count data, including cell frequencies of a contingency table, have been well understood for nearly 100 years, with much of the attention focused on the square root transformation. Over the past 15 years, this topic has been the focus of some new insights into areas of correspondence analysis where two forms of power transformation have been discussed. One type considers the impact of raising the joint proportions of the cell frequencies of a table to a known power while the other examines the power transformation of the relative distribution of the cell frequencies. While the foundations of the graphical features of correspondence analysis rest with the numerical algorithms like reciprocal averaging, and other analogous techniques, discussions of the role of power transformations in reciprocal averaging have not been described. Therefore, this paper examines this link where a power transformation is applied to the cell frequencies of a two-way contingency table. In doing so, we show that reciprocal averaging can be performed under such a transformation to obtain row and column scores that provide the maximum association between the variables and the greatest discrimination between the categories. Finally, we discuss the connection between performing reciprocal averaging and singular value decomposition under this type of power transformation. The <span>R</span> function, <span>powerRA.exe</span> is included in the Appendix and performs reciprocal averaging of a power transformation of the cell frequencies of a two-way contingency table.</p>","PeriodicalId":55223,"journal":{"name":"Computational Statistics","volume":"17 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00180-024-01517-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

Abstract

Power transformations of count data, including cell frequencies of a contingency table, have been well understood for nearly 100 years, with much of the attention focused on the square root transformation. Over the past 15 years, this topic has been the focus of some new insights into areas of correspondence analysis where two forms of power transformation have been discussed. One type considers the impact of raising the joint proportions of the cell frequencies of a table to a known power while the other examines the power transformation of the relative distribution of the cell frequencies. While the foundations of the graphical features of correspondence analysis rest with the numerical algorithms like reciprocal averaging, and other analogous techniques, discussions of the role of power transformations in reciprocal averaging have not been described. Therefore, this paper examines this link where a power transformation is applied to the cell frequencies of a two-way contingency table. In doing so, we show that reciprocal averaging can be performed under such a transformation to obtain row and column scores that provide the maximum association between the variables and the greatest discrimination between the categories. Finally, we discuss the connection between performing reciprocal averaging and singular value decomposition under this type of power transformation. The R function, powerRA.exe is included in the Appendix and performs reciprocal averaging of a power transformation of the cell frequencies of a two-way contingency table.

Abstract Image

用于倒数平均和奇异值分解的轮廓变换
近 100 年来,人们对计数数据(包括或然率表中的单元频率)的幂变换已经有了很好的理解,其中大部分注意力都集中在平方根变换上。在过去的 15 年里,这个话题成为了对应分析领域一些新见解的焦点,其中有两种形式的幂变换得到了讨论。一种是考虑将表格中单元格频率的联合比例提高到已知幂的影响,另一种是研究单元格频率相对分布的幂变换。虽然对应分析图形特征的基础是倒数平均等数值算法和其他类似技术,但关于幂变换在倒数平均中的作用的讨论却未曾涉及。因此,本文在对双向或然表的单元频率进行幂变换时,对这一联系进行了研究。在此过程中,我们证明了在这种变换下可以进行往复平均,从而获得行和列分数,使变量之间的关联度最大,类别之间的区分度最大。最后,我们讨论了在这种幂变换下进行倒数平均和奇异值分解之间的联系。附录中包含了 R 函数 powerRA.exe,它可以对双向或然表的单元频率进行幂变换的倒数平均。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Computational Statistics
Computational Statistics 数学-统计学与概率论
CiteScore
2.90
自引率
0.00%
发文量
122
审稿时长
>12 weeks
期刊介绍: Computational Statistics (CompStat) is an international journal which promotes the publication of applications and methodological research in the field of Computational Statistics. The focus of papers in CompStat is on the contribution to and influence of computing on statistics and vice versa. The journal provides a forum for computer scientists, mathematicians, and statisticians in a variety of fields of statistics such as biometrics, econometrics, data analysis, graphics, simulation, algorithms, knowledge based systems, and Bayesian computing. CompStat publishes hardware, software plus package reports.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信