東華大學圖書館 |

語系: 繁體中文

說明(常見問題)

回圖書館首頁

手機版館藏查詢

登入

回首頁

切換: 標籤 | MARC模式 | ISBD

Algebraic Complexity in Statistics u...

Gross, Elizabeth.

FindBook

Google Book

Amazon

博客來

Algebraic Complexity in Statistics using Combinatorial and Tensor Methods.

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Algebraic Complexity in Statistics using Combinatorial and Tensor Methods./
作者:	Gross, Elizabeth.
面頁冊數:	110 p.
附註:	Source: Dissertation Abstracts International, Volume: 75-03(E), Section: B.
Contained By:	Dissertation Abstracts International75-03B(E).
標題:	Mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3604239
ISBN:	9781303596391

Algebraic Complexity in Statistics using Combinatorial and Tensor Methods.
Gross, Elizabeth.

Algebraic Complexity in Statistics using Combinatorial and Tensor Methods. - 110 p.

Source: Dissertation Abstracts International, Volume: 75-03(E), Section: B.

Thesis (Ph.D.)--University of Illinois at Chicago, 2013.

Fundamental questions in statistical modeling ask about the best methods for model selection, goodness-of-fit testing, and estimation of parameters. For example, given a collection of aligned DNA sequences from a group of extant species, how can we decide which evolutionary tree best describes the species' ancestral history, or, given a sparse high-dimensional contingency table, how can we perform goodness-of-fit testing when exact tests are infeasible? In questions such as these, combinatorics, commutative algebra and algebraic geometry play a leading role. We explore such questions for specific classes of models, e.g. toric models, phylogenetic models, and variance components models, and tackle the algebraic complexity problems that lie at the root of them. We begin our exploration by studying toric ideals of hypergraphs, algebraic objects that are used for goodness-of-fit testing for log-linear models. In this study, we use the combinatorics of hypergraphs to give degree bounds on the generators of the ideals, give sufficiency conditions of when a binomial in the ideal is indispensable, show that the ideal of Tan( P P1)n is generated by quadratics and cubics in cumulant coordinates, and recover a well-known complexity theorem in algebraic statistics due to De Loera and Onn. Second, we explore phylogenetic models by viewing the models as sets of tensors with bounded rank. We show that the variety of 4 x 4 x 4 complex-valued tensors with border rank at most 4 is ddefined by polynomials of degree 5, 6, and 9. This variety corresponds to the 4-state general Markov model on the claw tree K1,3 and its defining polynomials can be used in model selection. This result also gives further evidence that the phylogenetic ideal of the model can be generated by polynomials of degree 9 and less. Finally, we look at the algebraic complexity of maximum likelihood estimation for variance components models, where we give explicit formulas for the ML and REML degree of the random effects model for the one-way layout and give examples of multimodal likelihood surfaces.

ISBN: 9781303596391Subjects--Topical Terms:

515831
Mathematics.

Algebraic Complexity in Statistics using Combinatorial and Tensor Methods.
LDR:02990nam a2200289 4500 001 1965427
005 20141022133323.5
008 150210s2013 ||||||||||||||||| ||eng d
020 $a 9781303596391
035 $a (MiAaPQ)AAI3604239
035 $a AAI3604239
040 $a MiAaPQ $c MiAaPQ
100 1 $a Gross, Elizabeth. $3 2102084
245 1 0 $a Algebraic Complexity in Statistics using Combinatorial and Tensor Methods.
300 $a 110 p.
500 $a Source: Dissertation Abstracts International, Volume: 75-03(E), Section: B.
500 $a Adviser: Shmuel Friedland.
502 $a Thesis (Ph.D.)--University of Illinois at Chicago, 2013.
520 $a Fundamental questions in statistical modeling ask about the best methods for model selection, goodness-of-fit testing, and estimation of parameters. For example, given a collection of aligned DNA sequences from a group of extant species, how can we decide which evolutionary tree best describes the species' ancestral history, or, given a sparse high-dimensional contingency table, how can we perform goodness-of-fit testing when exact tests are infeasible? In questions such as these, combinatorics, commutative algebra and algebraic geometry play a leading role. We explore such questions for specific classes of models, e.g. toric models, phylogenetic models, and variance components models, and tackle the algebraic complexity problems that lie at the root of them. We begin our exploration by studying toric ideals of hypergraphs, algebraic objects that are used for goodness-of-fit testing for log-linear models. In this study, we use the combinatorics of hypergraphs to give degree bounds on the generators of the ideals, give sufficiency conditions of when a binomial in the ideal is indispensable, show that the ideal of Tan( P P1)n is generated by quadratics and cubics in cumulant coordinates, and recover a well-known complexity theorem in algebraic statistics due to De Loera and Onn. Second, we explore phylogenetic models by viewing the models as sets of tensors with bounded rank. We show that the variety of 4 x 4 x 4 complex-valued tensors with border rank at most 4 is ddefined by polynomials of degree 5, 6, and 9. This variety corresponds to the 4-state general Markov model on the claw tree K1,3 and its defining polynomials can be used in model selection. This result also gives further evidence that the phylogenetic ideal of the model can be generated by polynomials of degree 9 and less. Finally, we look at the algebraic complexity of maximum likelihood estimation for variance components models, where we give explicit formulas for the ML and REML degree of the random effects model for the one-way layout and give examples of multimodal likelihood surfaces.
590 $a School code: 0799.
650 4 $a Mathematics. $3 515831
650 4 $a Applied Mathematics. $3 1669109
650 4 $a Statistics. $3 517247
690 $a 0405
690 $a 0364
690 $a 0463
710 2 $a University of Illinois at Chicago. $b Mathematics. $3 2094775
773 0 $t Dissertation Abstracts International $g 75-03B(E).
790 $a 0799
791 $a Ph.D.
792 $a 2013
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3604239