Elizabeth Gross

San Jose State University | SJSU · Department of Mathematics

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to non-linearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data i...

Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euc...

The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph $G$ is an i...

Social networks and other large sparse data sets pose significant challenges for statistical inference, as many standard statistical methods for testing model fit are not applicable in such settings. Algebraic statistics offers a theoretically justified approach to goodness-of-fit testing that relies on the theory of Markov bases and is intimately...

Steady-state analysis of dynamical systems for biological networks gives rise to algebraic varieties in high-dimensional spaces whose study is of interest in their own right. We demonstrate this for the shuttle model of the Wnt signaling pathway. Here, the variety is described by a polynomial system in 19 unknowns and 36 parameters. It has degree 9...

Phylogenetic networks provide a means of describing the evolutionary history of sets of species believed to have undergone hybridization or gene flow during their evolution. The mutation process for a set of such species can be modeled as a Markov process on a phylogenetic network. Previous work has shown that a site-pattern probability distributio...

Inspired by the work of Amdeberhan, Can, and Moll on broken necklaces, we define a broken bracelet as a linear arrangement of marked and unmarked vertices and introduce a generalization called $n$-stars, which is a collection of $n$ broken bracelets whose final (unmarked) vertices are identified. Through these combinatorial objects, we provide a ne...

Stephen Fienberg’s affinity for contingency table problems and reinterpreting models with a fresh look gave rise to a new approach for hypothesis testing of network models that are linear exponential families. We outline his vision and influence in this fundamental problem, as well as generalizations to multigraphs and hypergraphs.

In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach...

The steady-state degree of a chemical reaction network is the number of complex steady-states, which is a measure of the algebraic complexity of solving the steady-state system. In general, the steady-state degree may be difficult to compute. Here, we give an upper bound to the steady-state degree of a reaction network by utilizing the underlying p...

Phylogenetic networks can represent evolutionary events that cannot be described by phylogenetic trees. These networks are able to incorporate reticulate evolutionary events such as hybridization, introgression, and lateral gene transfer. Recently, network-based Markov models of DNA sequence evolution have been introduced along with model-based met...

A foundational question in the theory of linear compartmental models is how to assess whether a model is identifiable -- that is, whether parameter values can be inferred from noiseless data -- directly from the combinatorics of the model. We completely answer this question for those models (with one input and one output) in which the underlying gr...

Modeling complex systems and data using the language of graphs and networks has become an essential topic across a range of different disciplines. Arguably, this network-based perspective derives is success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships be...

Many popular models from the networks literature can be viewed through a common lens. We describe it here and call the class of models log-linear ERGMs. It includes degree-based models, stochastic blockmodels, and combinations of these. Given the interest in combined node and block effects in network formation mechanisms, we introduce a general dir...

Chemical reaction networks are often used to model and understand biological processes such as cell signaling. Under the framework of chemical reaction network theory, a process is modeled with a directed graph and a choice of kinetics, which together give rise to a dynamical system. Under the assumption of mass action kinetics, the dynamical syste...

In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach...

Phylogenetic networks can represent evolutionary events that cannot be described by phylogenetic trees. These networks are able to incorporate reticulate evolutionary events such as hybridization, introgression, and lateral gene transfer. Recently, network-based Markov models of DNA sequence evolution have been introduced along with model-based met...

In systems and synthetic biology, much research has focused on the behavior and design of single pathways, while, more recently, experimental efforts have focused on how cross-talk (coupling two or more pathways) or inhibiting molecular function (isolating one part of the pathway) affects systems-level behavior. However, the theory for tackling the...

Phylogenetics is the study of the evolutionary relationships between organisms. One of the main challenges in the field is to take biological data for a group of organisms and to infer an evolutionary tree, a graph that represents these relationships. Developing practical and efficient methods for inferring phylogenetic trees has led to a number of...

Stephen Fienberg's affinity for contingency table problems and reinterpreting models with a fresh look gave rise to a new approach for hypothesis testing of network models that are linear exponential families. We outline his vision and influence in this fundamental problem, as well as generalizations to multigraphs and hypergraphs.

The steady-state degree of a chemical reaction network is the number of complex steady-states, which is a measure of the algebraic complexity of solving the steady-state system. In general, the steady-state degree may be difficult to compute. Here, we give an upper bound to the steady-state degree of a reaction network by utilizing the underlying p...

Phylogenetics is the study of the evolutionary relationships between organisms. One of the main challenges in the field is to take biological data for a group of organisms and to infer an evolutionary tree, a graph that represents these relationships. Developing practical and efficient methods for inferring phylogenetic trees has lead to a number o...

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we develop algebraic methods to determine if a code is convex. Specifically, we u...

In systems and synthetic biology, much research has focused on the behavior and design of single pathways, while, more recently, experimental efforts have focused on how cross-talk (coupling two or more pathways) or inhibiting molecular function (isolating one part of the pathway) affects systems-level behavior. However, the theory for tackling the...

This work focuses on the question of how identifiability of a mathematical model, that is, whether parameters can be recovered from data, is related to identifiability of its submodels. We look specifically at linear compartmental models and investigate when identifiability is preserved after adding or removing model components. In particular, we e...

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use t...

In this paper we study group-based Markov models of evolution and their mixtures. In the algebreo-geometric setting, group-based phylogenetic tree models correspond to toric varieties, while their mixtures correspond to secant and join varieties. Determining properties of these secant and join varieties can aid both in model selection and establish...

This work addresses the problem of identifiability, that is, the question of whether parameters can be recovered from data, for linear compartment models. Using standard differential algebra techniques, the question of whether a given model is generically locally identifiable is equivalent to asking whether the Jacobian matrix of a certain coeffici...

Phylogenetic networks are becoming increasingly popular in phylogenetics since they have the ability to describe a wider range of evolutionary events than their tree counterparts. In this paper, we study Markov models on phylogenetic networks and their associated geometry. We restrict our attention to large-cycle networks, networks with a single un...

Multiple root estimation problems in statistical inference arise in many contexts in the literature. In the context of maximum likelihood estimation, the existence of multiple roots causes uncertainty in the computation of maximum likelihood estimators using hill-climbing algorithms, and consequent difficulties in the resulting statistical inferenc...

We introduce the package PhylogeneticTrees for Macaulay2 which allows users to compute phylogenetic invariants for group-based tree models. We provide some background information on phylogenetic algebraic geometry and show how the package PhylogeneticTrees can be used to calculate a generating set for a phylogenetic ideal as well as a lower bound f...

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are...

Supplementary Material for: Numerical algebraic geometry for model selection and its application to the life sciences

A neural code $\mathcal{C}$ is a collection of binary vectors of a given length n that record the co-firing patterns of a set of neurons. Our focus is on neural codes arising from place cells, neurons that respond to geographic stimulus. In this setting, the stimulus space can be visualized as subset of $\mathbb{R}^2$ covered by a collection $\math...

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging wooing to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data...

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging wooing to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data...

Given a statistical model, the maximum likelihood degree is the number of complex solutions to the likelihood equations for generic data. In this paper, we consider discrete algebraic statistical models and study the solutions to the likelihood equations when the data are no longer generic, but instead contain zeros. Focusing on sampling and model...

Numerical algebraic geometry is the field of computational mathematics concerning the numerical solution of polynomial systems of equations. Bertini, a popular software package for computational applications of this field, includes implementations of a variety of algorithms based on polynomial homotopy continuation. The Macaulay2 package Bertini.m2...

This Macaulay2 package provides an interface to PHCpack, a general-purpose polynomial system solver that uses homotopy continuation. The main method is a numerical blackbox solver which is implemented for all Laurent systems. The package also provides a fast mixed volume computation, the ability to filter solutions, homotopy path tracking, and a nu...

Associated to any hypergraph is a toric ideal encoding the algebraic relations among its edges. We study these ideals and the combinatorics of their minimal generators, and derive general degree bounds for both uniform and non-uniform hypergraphs in terms of balanced hypergraph bicolorings, separators, and splitting sets. In turn, this provides com...

Most statistical software packages implement numerical strategies for computation of maximum likelihood estimates in random effects models. Little is known, however, about the algebraic complexity of this problem. For the one-way layout with random effects and unbalanced group sizes, we give formulas for the algebraic degree of the likelihood equat...

The Macaulay2 package PHCpack.m2 provides an interface to PHCpack, a general-purpose polynomial system solver that uses homotopy continuation. The main method is a numerical blackbox solver which is implemented for all Laurent systems. The package also provides a fast mixed volume computation, the ability to filter solutions, homotopy path tracking...

We show that the irreducible variety of 4 x 4 x 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg-Manivel polynomials), and of degree 9 (the symmetrization conditions).