{"title":"Design of High-Performance Computing System for Big Data Analytics","authors":"","doi":"10.52783/jas.v11i1.1437","DOIUrl":"https://doi.org/10.52783/jas.v11i1.1437","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70996190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-10DOI: 10.2140/ASTAT.2021.12.43
Andrew Frohmader, H. Volkmer
Wasserstein distances provide a metric on a space of probability measures. We consider the space $Omega$ of all probability measures on the finite set $chi = {1, dots ,n}$ where $n$ is a positive integer. 1-Wasserstein distance, $W_1(mu,nu)$ is a function from $Omega times Omega$ to $[0,infty)$. This paper derives closed form expressions for the First and Second moment of $W_1$ on $Omega times Omega$ assuming a uniform distribution on $Omega times Omega$.
沃瑟斯坦距离提供了一个概率度量空间的度量。我们考虑有限集合$chi = {1, dots ,n}$上所有概率测度的空间$Omega$，其中$n$是一个正整数。1-Wasserstein距离，$W_1(mu,nu)$是从$Omega times Omega$到$[0,infty)$的函数。本文导出了$Omega times Omega$上$W_1$的一阶矩和二阶矩在$Omega times Omega$上均匀分布的封闭表达式。
{"title":"1-Wasserstein distance on the standard simplex","authors":"Andrew Frohmader, H. Volkmer","doi":"10.2140/ASTAT.2021.12.43","DOIUrl":"https://doi.org/10.2140/ASTAT.2021.12.43","url":null,"abstract":"Wasserstein distances provide a metric on a space of probability measures. We consider the space $Omega$ of all probability measures on the finite set $chi = {1, dots ,n}$ where $n$ is a positive integer. 1-Wasserstein distance, $W_1(mu,nu)$ is a function from $Omega times Omega$ to $[0,infty)$. This paper derives closed form expressions for the First and Second moment of $W_1$ on $Omega times Omega$ assuming a uniform distribution on $Omega times Omega$.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74948639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-28DOI: 10.2140/astat.2022.13.19
Bohao Yao, R. Evans
In this paper, we explore some algebraic properties of linear structural equation modelsthat can be represented by an HTC-identifiable graph. In particular, we prove that all mixedgraphs are HTC-identifiable if and only if all the regression coefficients can be recovered fromthe covariance matrix using straightforward linear algebra operations. We also find a set ofpolynomials that generates the ideal that encompasses all the equality constraints of the modelon the cone of positive definite matrices. We further prove that this set of polynomials are theminimal generators of said ideal for a subset of HTC-identifiable graphs.
{"title":"Algebraic properties of HTC-identifiable\u0000graphs","authors":"Bohao Yao, R. Evans","doi":"10.2140/astat.2022.13.19","DOIUrl":"https://doi.org/10.2140/astat.2022.13.19","url":null,"abstract":"In this paper, we explore some algebraic properties of linear structural equation modelsthat can be represented by an HTC-identifiable graph. In particular, we prove that all mixedgraphs are HTC-identifiable if and only if all the regression coefficients can be recovered fromthe covariance matrix using straightforward linear algebra operations. We also find a set ofpolynomials that generates the ideal that encompasses all the equality constraints of the modelon the cone of positive definite matrices. We further prove that this set of polynomials are theminimal generators of said ideal for a subset of HTC-identifiable graphs.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77189640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-06DOI: 10.2140/astat.2022.13.57
Ali Zartash, Elina Robeva
In this work we study the estimation of the density of a totally positive random vector. Total positivity of the distribution of a random vector implies a strong form of positive dependence between its coordinates and, in particular, it implies positive association. Since estimating a totally positive density is a non-parametric problem, we take on a (modified) kernel density estimation approach. Our main result is that the sum of scaled standard Gaussian bumps centered at a min-max closed set provably yields a totally positive distribution. Hence, our strategy for producing a totally positive estimator is to form the min-max closure of the set of samples, and output a sum of Gaussian bumps centered at the points in this set. We can frame this sum as a convolution between the uniform distribution on a min-max closed set and a scaled standard Gaussian. We further conjecture that convolving any totally positive density with a standard Gaussian remains totally positive.
{"title":"Convolutions of totally positive distributions with applications to kernel density estimation","authors":"Ali Zartash, Elina Robeva","doi":"10.2140/astat.2022.13.57","DOIUrl":"https://doi.org/10.2140/astat.2022.13.57","url":null,"abstract":"In this work we study the estimation of the density of a totally positive random vector. Total positivity of the distribution of a random vector implies a strong form of positive dependence between its coordinates and, in particular, it implies positive association. Since estimating a totally positive density is a non-parametric problem, we take on a (modified) kernel density estimation approach. Our main result is that the sum of scaled standard Gaussian bumps centered at a min-max closed set provably yields a totally positive distribution. Hence, our strategy for producing a totally positive estimator is to form the min-max closure of the set of samples, and output a sum of Gaussian bumps centered at the points in this set. We can frame this sum as a convolution between the uniform distribution on a min-max closed set and a scaled standard Gaussian. We further conjecture that convolving any totally positive density with a standard Gaussian remains totally positive.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86313875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-02DOI: 10.2140/ASTAT.2020.11.31
B. Sturmfels, S. Timme, Piotr Zwiernik
Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g. Toeplitz, sparse, trees) that are of interest in statistics. We study the maximum likelihood degree and its dual analogue, and we introduce a new software package LinearCovarianceModels.jl for solving the score equations. All local maxima can thus be computed reliably. In addition we identify several scenarios for which the estimator is a rational function.
{"title":"Estimating linear covariance models with numerical nonlinear algebra","authors":"B. Sturmfels, S. Timme, Piotr Zwiernik","doi":"10.2140/ASTAT.2020.11.31","DOIUrl":"https://doi.org/10.2140/ASTAT.2020.11.31","url":null,"abstract":"Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g. Toeplitz, sparse, trees) that are of interest in statistics. We study the maximum likelihood degree and its dual analogue, and we introduce a new software package LinearCovarianceModels.jl for solving the score equations. All local maxima can thus be computed reliably. In addition we identify several scenarios for which the estimator is a rational function.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77452546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-29DOI: 10.2140/astat.2022.13.41
Shaoxiong Hu, Hugo Maruri-Aguliar, Zixiang Ma
The LASSO is an attractive regularisation method for linear regression that combines variable selection with an efficient computation procedure. This paper is concerned with enhancing the performance of LASSO for square-free hierarchical polynomial models when combining validation error with a measure of model complexity. The measure of the complexity is the sum of Betti numbers of the model which is seen as a simplicial complex, and we describe the model in terms of components and cycles, borrowing from recent developments in computational topology. We study and propose an algorithm which combines statistical and topological criteria. This compound criterion would allow us to deal with model selection problems in polynomial regression models containing higher-order interactions. Simulation results demonstrate that the compound criteria produce sparser models with lower prediction errors than the estimators of several other statistical methods for higher order interaction models.
{"title":"Topological techniques in model selection","authors":"Shaoxiong Hu, Hugo Maruri-Aguliar, Zixiang Ma","doi":"10.2140/astat.2022.13.41","DOIUrl":"https://doi.org/10.2140/astat.2022.13.41","url":null,"abstract":"The LASSO is an attractive regularisation method for linear regression that combines variable selection with an efficient computation procedure. This paper is concerned with enhancing the performance of LASSO for square-free hierarchical polynomial models when combining validation error with a measure of model complexity. The measure of the complexity is the sum of Betti numbers of the model which is seen as a simplicial complex, and we describe the model in terms of components and cycles, borrowing from recent developments in computational topology. We study and propose an algorithm which combines statistical and topological criteria. This compound criterion would allow us to deal with model selection problems in polynomial regression models containing higher-order interactions. Simulation results demonstrate that the compound criteria produce sparser models with lower prediction errors than the estimators of several other statistical methods for higher order interaction models.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73196013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all $2$-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two ordinary double points and provide explicit expressions that allow one to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using $A$-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees.
{"title":"Maximum likelihood estimation of toric Fano varieties","authors":"Carlos Am'endola, Dimitra Kosta, Kaie Kubjas","doi":"10.2140/ASTAT.2020.11.5","DOIUrl":"https://doi.org/10.2140/ASTAT.2020.11.5","url":null,"abstract":"We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all $2$-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two ordinary double points and provide explicit expressions that allow one to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using $A$-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87343490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Stephen Fienberg (1942-2016) was a statistician whose career has been an inspiration for the engagement of statistics with social and scientific issues, and it is in this spirit that he helped steer algebraic statistics toward more of a mainstream. Many of his favorite topics in the area are covered in this special issue. We are grateful to all authors for contributing to this volume to honor him and his influence on the field. During the preparation of this issue, we also learned about the tragic killing of his widow, Joyce Fienberg, in the Tree of Life Synagogue in Pittsburgh. This issue is dedicated to their memory.
Stephen Fienberg（1942-2016）是一位统计学家，他的职业生涯激励了统计学与社会和科学问题的互动，正是本着这种精神，他帮助将代数统计学推向了主流。本期特刊报道了他在该领域最喜欢的许多话题。我们感谢所有为纪念他和他在该领域的影响力而为本卷做出贡献的作者。在本期的准备过程中，我们还了解到了他的遗孀Joyce Fienberg在匹兹堡生命之树犹太教堂不幸遇害的情况。这个问题是专门为他们的记忆。
{"title":"Stephen Fienberg's influence on algebraic statistics","authors":"Sonja Petrović, A. Slavkovic, R. Yoshida","doi":"10.18409/JAS.V10I1.100","DOIUrl":"https://doi.org/10.18409/JAS.V10I1.100","url":null,"abstract":"Stephen Fienberg (1942-2016) was a statistician whose career has been an inspiration for the engagement of statistics with social and scientific issues, and it is in this spirit that he helped steer algebraic statistics toward more of a mainstream. Many of his favorite topics in the area are covered in this special issue. We are grateful to all authors for contributing to this volume to honor him and his influence on the field. \u0000During the preparation of this issue, we also learned about the tragic killing of his widow, Joyce Fienberg, in the Tree of Life Synagogue in Pittsburgh. This issue is dedicated to their memory.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47156748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tools of algebraic statistics combined with MCMC algorithms have been used in contingency table analysis for model selection and model fit testing of log-linear models. However, this approach has not been considered so far for association models, which are special log-linear models for tables with ordinal classification variables. The simplest association model for two-way tables, the uniform (U) association model, has just one parameter more than the independence model and is applicable when both classification variables are ordinal. Less parsimonious are the row (R) and column (C) effect association models, appropriate when at least one of the classification variables is ordinal. Association models have been extended for multidimensional contingency tables as well. Here, we adjust algebraic methods for association models analysis and investigate their eligibility, focusing mainly on two-way tables. They are implemented in the statistical software R and illustrated on real data tables. Finally the algebraic model fit and selection procedure is assessed and compared to the asymptotic approach in terms of a simulation study.
{"title":"Inference for Ordinal Log-Linear Models Based on Algebraic Statistics","authors":"T. M. Pham, M. Kateri","doi":"10.18409/JAS.V10I1.74","DOIUrl":"https://doi.org/10.18409/JAS.V10I1.74","url":null,"abstract":"Tools of algebraic statistics combined with MCMC algorithms have been used in contingency table analysis for model selection and model fit testing of log-linear models. However, this approach has not been considered so far for association models, which are special log-linear models for tables with ordinal classification variables. The simplest association model for two-way tables, the uniform (U) association model, has just one parameter more than the independence model and is applicable when both classification variables are ordinal. Less parsimonious are the row (R) and column (C) effect association models, appropriate when at least one of the classification variables is ordinal. Association models have been extended for multidimensional contingency tables as well. Here, we adjust algebraic methods for association models analysis and investigate their eligibility, focusing mainly on two-way tables. They are implemented in the statistical software R and illustrated on real data tables. Finally the algebraic model fit and selection procedure is assessed and compared to the asymptotic approach in terms of a simulation study.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42333846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-08DOI: 10.2140/astat.2021.12.97
Thomas Kahle, Frank Röttger, R. Schwabe
Optimal design theory for nonlinear regression studies local optimality on a given design space. We identify designs for the Bradley--Terry paired comparison model with small undirected graphs and prove that every saturated D-optimal design is represented by a path. We discuss the case of four alternatives in detail and derive explicit polynomial inequality descriptions for optimality regions in parameter space. Using these regions, for each point in parameter space we can prescribe a D-optimal design.
{"title":"The semialgebraic geometry of saturated\u0000optimal designs for the Bradley–Terry model","authors":"Thomas Kahle, Frank Röttger, R. Schwabe","doi":"10.2140/astat.2021.12.97","DOIUrl":"https://doi.org/10.2140/astat.2021.12.97","url":null,"abstract":"Optimal design theory for nonlinear regression studies local optimality on a given design space. We identify designs for the Bradley--Terry paired comparison model with small undirected graphs and prove that every saturated D-optimal design is represented by a path. We discuss the case of four alternatives in detail and derive explicit polynomial inequality descriptions for optimality regions in parameter space. Using these regions, for each point in parameter space we can prescribe a D-optimal design.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86935878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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