arXiv subjects

Timothy W. Randolph

Publications and source records attributed to Timothy W. Randolph.

11 recordsLinked to original sources

arXiv2023

Generalized Matrix Decomposition Regression: Estimation and Inference for Two-way Structured Data

This paper studies high-dimensional regression with two-way structured data. To estimate the high-dimensional coefficient vector, we propose the generalized matrix decomposition regression (GMDR) to efficiently leverage any auxiliary information on row and column structures. The GMDR extends the principal component regression (PCR) to two-way structured data, but unlike PCR, the GMDR selects the components that are most predictive of the outcome, leading to more accurate prediction. For inference on regression coefficients of individual variables, we propose the generalized matrix decomposition inference (GMDI), a general high-dimensional inferential framework for a large family of estimators that include the proposed GMDR estimator. GMDI provides more flexibility for modeling relevant auxiliary row and column structures. As a result, GMDI does not require the true regression coefficients to be sparse; it also allows dependent and heteroscedastic observations. We study the theoretical properties of GMDI in terms of both the type-I error rate and power and demonstrate the effectiveness of GMDR and GMDI on simulation studies and an application to human microbiome data.

stat.ME↗

arXiv2020

A Sparsity Inducing Nuclear-Norm Estimator (SpINNEr) for Matrix-Variate Regression in Brain Connectivity Analysis

Classical scalar-response regression methods treat covariates as a vector and estimate a corresponding vector of regression coefficients. In medical applications, however, regressors are often in a form of multi-dimensional arrays. For example, one may be interested in using MRI imaging to identify which brain regions are associated with a health outcome. Vectorizing the two-dimensional image arrays is an unsatisfactory approach since it destroys the inherent spatial structure of the images and can be computationally challenging. We present an alternative approach - regularized matrix regression - where the matrix of regression coefficients is defined as a solution to the specific optimization problem. The method, called SParsity Inducing Nuclear Norm EstimatoR (SpINNEr), simultaneously imposes two penalty types on the regression coefficient matrix---the nuclear norm and the lasso norm---to encourage a low rank matrix solution that also has entry-wise sparsity. A specific implementation of the alternating direction method of multipliers (ADMM) is used to build a fast and efficient numerical solver. Our simulations show that SpINNEr outperforms other methods in estimation accuracy when the response-related entries (representing the brain's functional connectivity) are arranged in well-connected communities. SpINNEr is applied to investigate associations between HIV-related outcomes and functional connectivity in the human brain.

stat.ME↗

arXiv2018

Asymptotically Optimal Bounds for (t,2) Broadcast Domination on Finite Grids

Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The \emph{signal} that a tower vertex $T$ of signal strength $t$ supplies to a vertex $v$ is defined as $sig(T,v)=max(t-dist(T,v),0),$ where $dist(T,v)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a \emph{$(t,r)$ broadcast dominating set}, or simply a \emph{$(t,r)$ broadcast}, on $G$ as a set $\mathbb{T}\subseteq V$ such that the sum of all signals received at each vertex $v \in V$ from the set of towers $\mathbb{T}$ is at least $r$. The $(t,r)$ broadcast domination number of a finite graph $G$, denoted $γ_{t,r}(G)$, is the minimum cardinality over all $(t,r)$ broadcasts for $G$. Recent research has focused on bounding the $(t,r)$ broadcast domination number for the $m \times n$ grid graph $G_{m,n}$. In 2014, Grez and Farina bounded the $k$-distance domination number for grid graphs, equivalent to bounding $γ_{t,1}(G_{m,n})$. In 2015, Blessing et al. established bounds on $γ_{2,2}(G_{m,n})$, $γ_{3,2}(G_{m,n})$, and $γ_{3,3}(G_{m,n})$. In this paper, we take the next step and provide a tight upper bound on $γ_{t,2}(G_{m,n})$ for all $t>2$. We also prove the conjecture of Blessing et al. that their bound on $γ_{3,2}(G_{m,n})$ is tight for large values of $m$ and $n$.

math.CO↗

arXiv2018

Incorporating Deep Features in the Analysis of Tissue Microarray Images

Tissue microarray (TMA) images have been used increasingly often in cancer studies and the validation of biomarkers. TACOMA---a cutting-edge automatic scoring algorithm for TMA images---is comparable to pathologists in terms of accuracy and repeatability. Here we consider how this algorithm may be further improved. Inspired by the recent success of deep learning, we propose to incorporate representations learnable through computation. We explore representations of a group nature through unsupervised learning, e.g., hierarchical clustering and recursive space partition. Information carried by clustering or spatial partitioning may be more concrete than the labels when the data are heterogeneous, or could help when the labels are noisy. The use of such information could be viewed as regularization in model fitting. It is motivated by major challenges in TMA image scoring---heterogeneity and label noise, and the cluster assumption in semi-supervised learning. Using this information on TMA images of breast cancer, we have reduced the error rate of TACOMA by about 6%. Further simulations on synthetic data provide insights on when such representations would likely help. Although we focus on TMAs, learnable representations of this type are expected to be applicable in other settings.

cs.CV↗

arXiv2018

(k,p)-Planarity: A Relaxation of Hybrid Planarity

We present a new model for hybrid planarity that relaxes existing hybrid representations. A graph $G = (V,E)$ is $(k,p)$-planar if $V$ can be partitioned into clusters of size at most $k$ such that $G$ admits a drawing where: (i) each cluster is associated with a closed, bounded planar region, called a cluster region; (ii) cluster regions are pairwise disjoint, (iii) each vertex $v \in V$ is identified with at most $p$ distinct points, called \emph{ports}, on the boundary of its cluster region; (iv) each inter-cluster edge $(u,v) \in E$ is identified with a Jordan arc connecting a port of $u$ to a port of $v$; (v) inter-cluster edges do not cross or intersect cluster regions except at their endpoints. We first tightly bound the number of edges in a $(k,p)$-planar graph with $p<k$. We then prove that $(4,1)$-planarity testing and $(2,2)$-planarity testing are NP-complete problems. Finally, we prove that neither the class of $(2,2)$-planar graphs nor the class of $1$-planar graphs contains the other, indicating that the $(k,p)$-planar graphs are a large and novel class.

cs.DS↗

arXiv2017

Computing upper bounds for optimal density of $(t,r)$ broadcasts on the infinite grid

The domination number of a finite graph $G$ with vertex set $V$ is the cardinality of the smallest set $S\subseteq V$ such that for every vertex $v\in V$ either $v\in S$ or $v$ is adjacent to a vertex in $S$. A set $S$ satisfying these conditions is called a dominating set. In 2015 Blessing, Insko, Johnson, and Mauretour introduced $(t,r)$ broadcast domination, a generalization of graph domination parameterized by the nonnegative integers $t$ and $r$. In this setting, we say that the signal a vertex $v\in V$ receives from a tower of strength $t$ located at vertex $T$ is defined by $sig(v,T)=max(t-dist(v,T),0)$. Then a $(t,r)$ broadcast dominating set on $G$ is a set $S\subseteq V$ such that the sum of all signal received at each vertex $v \in V$ is at least $r$. In this paper, we consider $(t,r)$ broadcasts of the infinite grid and present a Python program to compute upper bounds on the minimal density of a $(t,r)$ broadcast on the infinite grid. These upper bounds allow us to construct counterexamples to a conjecture by Blessing et al. that the $(t,r)$ and $(t+1, r+2)$ broadcasts are equal whenever $t,r\geq 1$.

math.CO↗

arXiv2017

Optimal $(t,r)$ Broadcasts On the Infinite Grid

Let $G=(V,E)$ be a graph and $t,r$ be positive integers. The signal that a vertex $v$ receives from a tower of signal strength $t$ located at vertex $T$ is defined as $sig(v,T)=max(t-dist(v,T),0)$, where $dist(v,T)$ denotes the distance between the vertices $v$ and $T$. In 2015 Blessing, Insko, Johnson, and Mauretour defined a $(t,r)$ broadcast dominating set, or simply a $(t,r)$ broadcast, on $G$ as a set $\mathbb{T}\subseteq V$ such that the sum of all signal received at each vertex $v \in V$ is at least $r$. We say that $\mathbb{T}$ is optimal if $|\mathbb{T}|$ is minimal among all such sets $\mathbb{T}$. The cardinality of an optimal $(t,r)$ broadcast on a finite graph $G$ is called the $(t,r)$ broadcast domination number of $G$. The concept of $(t,r)$ broadcast domination generalizes the classical problem of domination on graphs. In fact, the $(2,1)$ broadcasts on a graph $G$ are exactly the dominating sets of $G$. In their paper, Blessing et al. considered $(t,r)\in\{(2,2),(3,1),(3,2),(3,3)\}$ and gave optimal $(t,r)$ broadcasts on $G_{m,n}$, the grid graph of dimension $m\times n$, for small values of $m$ and $n$. They also provided upper bounds on the optimal $(t,r)$ broadcast numbers for grid graphs of arbitrary dimensions. In this paper, we define the density of a $(t,r)$ broadcast, which allows us to provide optimal $(t,r)$ broadcasts on the infinite grid graph for all $t\geq2$ and $r=1,2$, and bound the density of the optimal $(t,3)$ broadcast for all $t\geq2$. In addition, we give a family of counterexamples to the conjecture of Blessing et al. that the optimal $(t,r)$ and $(t+1, r+2)$ broadcasts are identical for all $t\geq1$ and $r\geq1$ on the infinite grid.

math.CO↗

arXiv2017

Kernel-Penalized Regression for Analysis of Microbiome Data

The analysis of human microbiome data is often based on dimension-reduced graphical displays and clustering derived from vectors of microbial abundances in each sample. Common to these ordination methods is the use of biologically motivated definitions of similarity. Principal coordinate analysis, in particular, is often performed using ecologically defined distances, allowing analyses to incorporate context-dependent, non-Euclidean structure. Here we describe how to take a step beyond ordination plots and incorporate this structure into high-dimensional penalized regression models. Within this framework, the estimate of a regression coefficient vector is obtained via the joint eigen properties of multiple similarity matrices, or kernels. This allows for multivariate regression models to incorporate both a matrix of microbial abundances and, for instance, a matrix of phylogenetically-informed similarities between the abundance profiles. Further, we show how this regression framework can be used to address the compositional nature of multivariate predictors comprised of relative abundances; that is, vectors whose entries sum to a constant. We illustrate this regression framework with several simulations using data from two recent studies on the gut and vaginal microbiome. We conclude with an application to our own data, where we also incorporate a significance test for the estimated coefficients that represent associations between microbial abundance and a response.

stat.AP↗

arXiv2014

Longitudinal Functional Models with Structured Penalties

This paper addresses estimation in a longitudinal regression model for association between a scalar outcome and a set of longitudinally-collected functional covariates or predictor curves. The framework consists of estimating a time-varying coefficient function that is modeled as a linear combination of time-invariant functions but having time-varying coefficients. The estimation procedure exploits the equivalence between penalized least squares estimation and a linear mixed model representation. The process is empirically evaluated with several simulations and it is applied to analyze the neurocognitive impairment of HIV patients and its association with longitudinally-collected magnetic resonance spectroscopy curves.

stat.AP↗

arXiv2011

Analyzing LC-MS/MS data by spectral count and ion abundance: two case studies

In comparative proteomics studies, LC-MS/MS data is generally quantified using one or both of two measures: the spectral count, derived from the identification of MS/MS spectra, or some measure of ion abundance derived from the LC-MS data. Here we contrast the performance of these measures and show that ion abundance is the more sensitive. We also examine how the conclusions of a comparative analysis are influenced by the manner in which the LC-MS/MS data is `rolled up' to the protein level, and show that divergent conclusions obtained using different rollups can be informative. Our analysis is based on two publicly available reference data sets, BIATECH-54 and CPTAC, which were developed for the purpose of assessing methods used in label-free differential proteomic studies. We find that the use of the ion abundance measure reveals properties of both data sets not readily apparent using the spectral count.

stat.AP↗

arXiv2011

Structured penalties for functional linear models---partially empirical eigenvectors for regression

One of the challenges with functional data is incorporating spatial structure, or local correlation, into the analysis. This structure is inherent in the output from an increasing number of biomedical technologies, and a functional linear model is often used to estimate the relationship between the predictor functions and scalar responses. Common approaches to the ill-posed problem of estimating a coefficient function typically involve two stages: regularization and estimation. Regularization is usually done via dimension reduction, projecting onto a predefined span of basis functions or a reduced set of eigenvectors (principal components). In contrast, we present a unified approach that directly incorporates spatial structure into the estimation process by exploiting the joint eigenproperties of the predictors and a linear penalty operator. In this sense, the components in the regression are `partially empirical' and the framework is provided by the generalized singular value decomposition (GSVD). The GSVD clarifies the penalized estimation process and informs the choice of penalty by making explicit the joint influence of the penalty and predictors on the bias, variance, and performance of the estimated coefficient function. Laboratory spectroscopy data and simulations are used to illustrate the concepts.

stat.AP↗