1st Edition

# Algebraic Statistics Computational Commutative Algebra in Statistics

178 Pages
by Chapman & Hall

184 Pages
by Chapman & Hall

Also available as eBook on:

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

It begins with an introduction to Gröbner bases and a thorough description of their applications to experimental design. A special chapter covers the binary case with new application to coherent systems in reliability and two level factorial designs. The work paves the way, in the last two chapters, for the application of computer algebra to discrete probability and statistical modelling through the important concept of an algebraic statistical model.

As the first book on the subject, Algebraic Statistics presents many opportunities for spin-off research and applications and should become a landmark work welcomed by both the statistical community and its relatives in mathematics and computer science.

INTRODUCTION
History and Motivation
Overview
Computer Algebra
Summary
ALGEBRAIC MODELS
Models
Polynomials and Polynomial Ideals
Term-Orderings
Division Algorithm
All Ideals Are Finitely Generated
Varieties and Equations
Gröbner Bases
Properties of Gröbner Basis
Elimination Theory
Polynomial Functions and Quotients by Ideals
Hilbert Function
Further Topics
THE DIRECT THEORY
Designs and Design Ideals
Computing the Gröbner basis of a design
Operations with Designs
Examples
Span of a Design
Models and Identifiability; Quotients
Examples
The Fan of an Experimental Design
Subsets and Sequential Algorithms
Regression Analysis
Other Topics
TWO-LEVEL DESIGNS. APPLICATION IN LOGIC AND RELIABILITY
The binary case: Boolean Representations
Reliability: Coherent Systems are Minimal Fan Designs
Two Level Factorial Design: Contrasts and Orthogonality
PROBABILITY AND STATISTICS
Random Variables on a Finite Support
Moments
Probability
Algebraic Representation of Exponentials
Generating Functions
Generating Functions and Exponential Models
Examples and Further Applications
Statistical Modelling
Likelihoods and Sufficient Statistics
A Ring of Random Variables
Score Function and Information

### Biography

Eva Riccomagno, Giovanni Pistone, Henry P. Wynn

"...authors have been the predominant contributors to the field.... for anyone who wants to learn about, and perhaps contribute to, the field, this monograph is undoubtedly the place to start."
-Biometrics, Vol. 57, No. 3, September 2001
"This very challenging monograph demonstrates how Gröbner bases may be used to represent experimental design, probability models and statistical models … The book points clearly to the future potential use of algebraic tools."

Short Book Reviews, Vol. 21, No. 2, August, 2001