Algebraic Statistics: Computational Commutative Algebra in Statistics

by Pistone; Giovanni

• ISBN13: 9781584882046
• ISBN10: 1584882042
• eBook ISBN(s): 9781040062128
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-12-21
• Publisher: Chapman & Hall/
• More Book Details

Summary

Written by pioneers in this exciting new field, "Algebraic Statistics" introduces the application of polynomial algebra to experimental design, discrete probability, and statistics. (Mathematics)

Author Biography

Giovanni Pistone is a Professor of Mathematical Statistics in the Mathematics Department at the Politecnico, Torino, Italy.

Table of Contents

List of figures	ix
List of tables	xi
Preface	xiii
Notation	xv
Introduction	1	(14)
Outline	1	(4)
Computer Algebra	5	(5)
An example: the 2 3-1 fractional factorial design	10	(5)
Algebraic models	15	(28)
Models	16	(1)
Polynomial ideals	17	(2)
Term-orderings	19	(3)
Division algorithm	22	(1)
Hilbert basis theorem	23	(2)
Varieties and equations	25	(2)
Grobner bases	27	(2)
Properties of a Grobner basis	29	(2)
Elimination theory	31	(2)
Polynomial functions and quotients by ideals	33	(2)
Hilbert function	35	(1)
Further topics	36	(7)
Grobner bases in experimental design	43	(32)
Designs and design ideals	43	(1)
Computing the Grobner basis of a design	44	(3)
Operations with designs	47	(1)
Examples	48	(2)
Span of a design	50	(3)
Models and identifiability: quotients	53	(1)
Confounding of models	54	(2)
Further examples	56	(4)
The fan of an experimental design	60	(3)
Minimal and maximal fan designs	63	(2)
Hilbert functions and fans for graded ordering	65	(1)
Subsets and algorithms	66	(5)
Regression analysis	71	(1)
Non-polynomial models	72	(3)
Two-level factors: logic, reliability, design	75	(20)
The binary case: Boolean representations	75	(3)
Grobner bases and Boolean ideals	78	(2)
Logic and learning	80	(1)
Reliability: coherent systems as minimal fan designs	81	(2)
Inclusion-exclusion and tube theory	83	(7)
Two-level factorial design: contrasts and orthogonality	90	(5)
Probability	95	(24)
Random variables on a finite support	96	(1)
The ring of random variables	97	(2)
Matrix representation of ≈(D,K)	99	(2)
Uniform probability	101	(2)
Probability densities	103	(3)
Image probability and marginalisation	106	(2)
Conditional expectation	108	(3)
Algebraic representation of exponentials	111	(2)
Exponential form of a probability	113	(6)
Statistical modeling	119	(36)
Introduction	119	(1)
Statistical models	120	(8)
Generating functions and exponential submodels	128	(3)
Likelihoods and sufficient statistics	131	(4)
Score function and information	135	(1)
Estimation: lattice case	136	(2)
Finitely generated cumulants	138	(1)
Estimating functions	139	(1)
An extended example	140	(9)
Orthogonality and toric ideals	149	(6)
References	155	(4)
Index	159