An Introduction to Algebraic Statistics with Tensors

• Κύριοι συγγραφείς: Bocci, Cristiano (Συγγραφέας, http://id.loc.gov/vocabulary/relators/aut), Chiantini, Luca (http://id.loc.gov/vocabulary/relators/aut)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Cham : Springer International Publishing : Imprint: Springer, 2019.
• Έκδοση: 1st ed. 2019.
• Σειρά: La Matematica per il 3+2, 118
• Θέματα: Statistics .; Applied mathematics.; Engineering mathematics.; Algebraic geometry.; Statistical Theory and Methods.; Applications of Mathematics.; Algebraic Geometry.

 This book provides an introduction to various aspects of Algebraic Statistics with the principal aim of supporting Master's and PhD students who wish to explore the algebraic point of view regarding recent developments in Statistics. The focus is on the background needed to explore the connections among discrete random variables. The main objects that encode these relations are multilinear matrices, i.e., tensors. The book aims to settle the basis of the correspondence between properties of tensors and their translation in Algebraic Geometry. It is divided into three parts, on Algebraic Statistics, Multilinear Algebra, and Algebraic Geometry. The primary purpose is to describe a bridge between the three theories, so that results and problems in one theory find a natural translation to the others. This task requires, from the statistical point of view, a rather unusual, but algebraically natural, presentation of random variables and their main classical features. The third part of the book can be considered as a short, almost self-contained, introduction to the basic concepts of algebraic varieties, which are part of the fundamental background for all who work in Algebraic Statistics. 

 PART I: Algebraic Statistics -- 1 Systems of Random Variables and Distributions -- 2 Basic Statistics -- 3 Statistical models -- 4 Complex projective algebraic statistics -- 5 Conditional independence -- PART II: Multilinear Algebra -- 6 Tensors -- 7 Symmetric tensors -- 8 Marginalisation and attenings -- PART III: Commutative Algebra and Algebraic Geometry -- 9 Elements of Projective Algebraic Geometry -- 10 Projective maps and the Chow's Theorem -- 11 Dimension Theory -- 12 Secant varieties -- 13 Groebner bases.