Statistical Analysis with Missing Data

Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

https://www.springer.com/productFlyer_978-0-387-22356-8.pdf?SGWID=0-0-1297-34952457-0

Combinatorial Commutative Algebra

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active "network community" and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online "networking communities" such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data.

Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.

https://dash.harvard.edu/bitstream/handle/1/4645865/Goldenberg_NetSurvey.pdf?sequence=1

A Survey of Statistical Network Models

Polynomials in one variable Grobner bases of zero-dimensional ideals Bernstein's theorem and fewnomials Resultants Primary decomposition Polynomial systems in economics Sums of squares Polynomial systems in statistics Tropical algebraic geometry Linear partial differential equations with constant coefficients Bibliography Index.

/pdf/solving-systems-of-polynomial-equations-3xfr3xgaue.pdf

Solving systems of polynomial equations

Preface Part I. Introduction to the Four Themes: 1. Statistics L. Pachter and B. Sturmfels 2. Computation L. Pachter and B. Sturmfels 3. Algebra L. Pachter and B. Sturmfels 4. Biology L. Pachter and B. Sturmfels Part II. Studies on the Four Themes: 5. Parametric inference R. Mihaescu 6. Polytope propagation on graphs M. Joswig 7. Parametric sequence alignment C. Dewey and K. Woods 8. Bounds for optimal sequence alignment S. Elizalde 9. Inference functions S. Elizalde 10. Geometry of Markov chains E. Kuo 11. Equations defining hidden Markov models N. Bray and J. Morton 12. The EM algorithm for hidden Markov models I. B. Hallgrimsdottir, A. Milowski and J. Yu 13. Homology mapping with Markov random fields A. Caspi 14. Mutagenetic tree models N. Beerenwinkel and M. Drton 15. Catalog of small trees M. Casanellas, L. Garcia and S. Sullivant 16. The strand symmetric model M. Casanellas and S. Sullivant 17. Extending statistical models from trees to splits graphs D. Bryant 18. Small trees and generalized neighbor-joining M. Contois and D. Levy 19. Tree construction using Singular Value Decomposition N. Eriksson 20. Applications of interval methods to phylogenetics R. Sainudiin and R. Yoshida 21. Analysis of point mutations in vertebrate genomes J. Al-Aidroos and S. Snir 22. Ultra-conserved elements in vertebrate genomes M. Drton, N. Eriksson and G. Leung Index.

/pdf/algebraic-statistics-for-computational-biology-2k0yvw9brc.pdf

Algebraic Statistics for Computational Biology

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses, the maximum likelihood degree equals the top Chern class of a sheaf of logarithmic differential forms. Exact formulae in terms of degrees and Newton polytopes are given for polynomials with generic coefficients.

The maximum likelihood degree

Given a model in algebraic statistics and data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. Applications include models specified by rank conditions on matrices and the Jukes–Cantor models of phylogenetics. The maximum likelihood degree of a generic complete intersection is also determined.

Serkan Hosten

Papers

The maximum likelihood degree

Solving the Likelihood Equations

Gröbner bases and polyhedral geometry of reducible and cyclic models

A finiteness theorem for Markov bases of hierarchical models

Primary decomposition of lattice basis ideals