Let (G) be a finite group, let (mathbb{C}G) be the complex group algebra of (G), and let (p in mathbb{C}G). In this paper, we show how to construct submodules(S) of (mathbb{C}G) of a fixed dimension with the property that the orthogonal projection of (p) onto (S) has maximal length. We then provide an example of how such submodules for the symmetric group (S_n) can be used to create new linear rank tests of uniformity in statistics for survey data that arises when respondents are asked to give a complete ranking of (n) items.
{"title":"Maximal Length Projections in Group Algebras with Applications to Linear Rank Tests of Uniformity","authors":"Anna E. Bargagliotti, Michael E. Orrison","doi":"10.18409/JAS.V9I1.59","DOIUrl":"https://doi.org/10.18409/JAS.V9I1.59","url":null,"abstract":"Let (G) be a finite group, let (mathbb{C}G) be the complex group algebra of (G), and let (p in mathbb{C}G). In this paper, we show how to construct submodules(S) of (mathbb{C}G) of a fixed dimension with the property that the orthogonal projection of (p) onto (S) has maximal length. We then provide an example of how such submodules for the symmetric group (S_n) can be used to create new linear rank tests of uniformity in statistics for survey data that arises when respondents are asked to give a complete ranking of (n) items.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44916870","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 : 2018-08-31DOI: 10.2140/astat.2020.11.155
N. Tran
We survey ways to define the analogue of the Gaussian measure in the tropical semiring.
我们探讨了在热带半环中定义高斯测度类似的方法。
{"title":"Tropical Gaussians: a brief survey","authors":"N. Tran","doi":"10.2140/astat.2020.11.155","DOIUrl":"https://doi.org/10.2140/astat.2020.11.155","url":null,"abstract":"We survey ways to define the analogue of the Gaussian measure in the tropical semiring.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72917909","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 : 2018-03-12DOI: 10.2140/astat.2020.11.125
Y. Tachibana, Y. Goto, Tamio Koyama, N. Takayama
The holonomic gradient method gives an algorithm to efficiently and accurately evaluate normalizing constants and their derivatives. It utilizes holonomic differential equations or holonomic difference equations. Y.Goto and K.Matsumoto gave a system of difference equations for the hypergeometric system of type (k, n). We apply their system to evaluate the normalizing constant and its derivatives of the conditional Poisson or multinomial distribution on two way contingency tables. The modular method in computer algebra has been used for an efficient and exact evaluation. We will also discuss on complexities of these algorithms and their implementation.
{"title":"Holonomic gradient method for two-way contingency tables","authors":"Y. Tachibana, Y. Goto, Tamio Koyama, N. Takayama","doi":"10.2140/astat.2020.11.125","DOIUrl":"https://doi.org/10.2140/astat.2020.11.125","url":null,"abstract":"The holonomic gradient method gives an algorithm to efficiently and accurately evaluate normalizing constants and their derivatives. It utilizes holonomic differential equations or holonomic difference equations. Y.Goto and K.Matsumoto gave a system of difference equations for the hypergeometric system of type (k, n). We apply their system to evaluate the normalizing constant and its derivatives of the conditional Poisson or multinomial distribution on two way contingency tables. The modular method in computer algebra has been used for an efficient and exact evaluation. We will also discuss on complexities of these algorithms and their implementation.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87707264","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}
E. Allman, H. Cervantes, R. Evans, Serkan Hocsten, Kaie Kubjas, Daniela Lemke, J. Rhodes, Piotr Zwiernik
The Expectation-Maximization (EM) algorithm is routinely used for the maximum likelihood estimation in the latent class analysis. However, the EM algorithm comes with no guarantees of reaching the global optimum. We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator. In particular, we characterize the boundary stratification of the binary latent class model with a binary hidden variable. For small models, such as for three binary observed variables, we show that this stratification allows exact computation of the maximum likelihood estimator. In this case we use simulations to study the maximum likelihood estimation attraction basins of the various strata. Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function. In particular, we compute the minimal primes of this ideal in the case of a binary latent class model with a binary or ternary hidden random variable.
{"title":"Maximum likelihood estimation of the Latent Class Model through model boundary decomposition","authors":"E. Allman, H. Cervantes, R. Evans, Serkan Hocsten, Kaie Kubjas, Daniela Lemke, J. Rhodes, Piotr Zwiernik","doi":"10.18409/JAS.V10I1.75","DOIUrl":"https://doi.org/10.18409/JAS.V10I1.75","url":null,"abstract":"The Expectation-Maximization (EM) algorithm is routinely used for the maximum likelihood estimation in the latent class analysis. However, the EM algorithm comes with no guarantees of reaching the global optimum. We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator. In particular, we characterize the boundary stratification of the binary latent class model with a binary hidden variable. For small models, such as for three binary observed variables, we show that this stratification allows exact computation of the maximum likelihood estimator. In this case we use simulations to study the maximum likelihood estimation attraction basins of the various strata. Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function. In particular, we compute the minimal primes of this ideal in the case of a binary latent class model with a binary or ternary hidden random variable.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44495722","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}
The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for nite discrete random vectors and the Gaussian random vector. The precision space of some cubature rules/designed experiments are determined.
{"title":"Cubature rules and expected value of some complex functions","authors":"C. Fassino, Eva Riccomagno, M. Rogantin","doi":"10.18409/JAS.V10I1.72","DOIUrl":"https://doi.org/10.18409/JAS.V10I1.72","url":null,"abstract":"The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for nite discrete random vectors and the Gaussian random vector. The precision space of some cubature rules/designed experiments are determined.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49098100","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 compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.
{"title":"Mixtures and products in two graphical models","authors":"A. Seigal, Guido Montúfar","doi":"10.18409/JAS.V9I1.90","DOIUrl":"https://doi.org/10.18409/JAS.V9I1.90","url":null,"abstract":"We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42221942","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 derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size.
{"title":"On Exchangeability in Network Models","authors":"S. Lauritzen, A. Rinaldo, Kayvan Sadeghi","doi":"10.18409/JAS.V10I1.73","DOIUrl":"https://doi.org/10.18409/JAS.V10I1.73","url":null,"abstract":"We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size. ","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44442112","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 : 2017-08-28DOI: 10.2140/astat.2021.12.115
É. Czabarka, L. Székely, Z. Toroczkai, Shanise Walker
The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously is a relevant problem in data driven modeling of networks. Here we formulate common generalizations of this problem and the Exact Matching Problem, and solve them with an algebraic Monte-Carlo algorithm that runs in polynomial time if the number of partition classes is bounded.
{"title":"An algebraic Monte-Carlo algorithm for the partition adjacency matrix realization problem","authors":"É. Czabarka, L. Székely, Z. Toroczkai, Shanise Walker","doi":"10.2140/astat.2021.12.115","DOIUrl":"https://doi.org/10.2140/astat.2021.12.115","url":null,"abstract":"The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously is a relevant problem in data driven modeling of networks. Here we formulate common generalizations of this problem and the Exact Matching Problem, and solve them with an algebraic Monte-Carlo algorithm that runs in polynomial time if the number of partition classes is bounded.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77656776","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 consider the lattice, $mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(cdot)$, on $mathcal{L}$. We derive from the supermodularity of $n(cdot)$ some generalized Fr'echet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $n(cdot)$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fr'echet inequalities for multidimensional contingency tables.
{"title":"Generalized Fréchet Bounds for Cell Entries in Multidimensional Contingency Tables","authors":"Caroline Uhler, D. Richards","doi":"10.18409/JAS.V10I1.71","DOIUrl":"https://doi.org/10.18409/JAS.V10I1.71","url":null,"abstract":"We consider the lattice, $mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(cdot)$, on $mathcal{L}$. We derive from the supermodularity of $n(cdot)$ some generalized Fr'echet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $n(cdot)$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fr'echet inequalities for multidimensional contingency tables.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48577406","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 : 2017-06-13DOI: 10.2140/ASTAT.2020.11.99
R. Mulas, N. Tran
In the past few years, the study of receptive field codes has been of large interest to mathematicians. Here we give a complete characterization of receptive field codes realizable by connected receptive fields and we give the minimal embedding dimensions of these codes. In particular, we show that all connected codes are realizable in dimension at most 3. To our knowledge, this is the first family of receptive field codes for which the exact characterization and minimal embedding dimension is known.
{"title":"Minimal embedding dimensions of connected neural codes","authors":"R. Mulas, N. Tran","doi":"10.2140/ASTAT.2020.11.99","DOIUrl":"https://doi.org/10.2140/ASTAT.2020.11.99","url":null,"abstract":"In the past few years, the study of receptive field codes has been of large interest to mathematicians. Here we give a complete characterization of receptive field codes realizable by connected receptive fields and we give the minimal embedding dimensions of these codes. In particular, we show that all connected codes are realizable in dimension at most 3. To our knowledge, this is the first family of receptive field codes for which the exact characterization and minimal embedding dimension is known.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78736773","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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