Algebraic Statistics: Computational Commutative Algebra in Statistics, 1st Edition (Hardback) book cover

Algebraic Statistics

Computational Commutative Algebra in Statistics, 1st Edition

By Giovanni Pistone, Eva Riccomagno, Henry P. Wynn

Chapman and Hall/CRC

184 pages

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pub: 2000-12-21
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Description

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.

Reviews

"…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

Table of Contents

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

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT029000
MATHEMATICS / Probability & Statistics / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis