Search

We describe a set of network analysis methods based on the rows of the Krylov subspace matrix computed from a network adjacency matrix via power iteration using a non-random initial vector. We refer to these node-specific row vectors as Krylov subspace trajectories. While power iteration using a random initial starting vector is commonly applied to the network adjacency matrix to compute eigenvector centrality values, this application only uses the final vector generated after numerical convergence. Importantly, use of a random initial vector means that the intermediate results of power iteration are also random and lack a clear interpretation. To the best of our knowledge, use of intermediate power iteration results for network analysis has been limited to techniques that leverage just a single pre-convergence solution, e.g., Power Iteration Clustering. In this paper, we explore methods that apply power iteration with a non-random inital vector to the network adjacency matrix to generate Krylov subspace trajectories for each node. These non-random trajectories provide important information regarding network structure, node importance, and response to perturbations. We have created this short preprint in part to generate feedback from others in the network analysis community who might be aware of similar existing work.

This is the first in a series of papers presenting a new understanding of scattering amplitudes based on fundamentally combinatorial ideas in the kinematic space of the scattering data. We study the simplest theory of colored scalar particles with cubic interactions, at all loop orders and to all orders in the topological 't Hooft expansion. We find a novel formula for loop-integrated amplitudes, with no trace of the conventional sum over Feynman diagrams, but instead determined by a beautifully simple counting problem attached to any order of the topological expansion. These results represent a significant step forward in the decade-long quest to formulate the fundamental physics of the real world in a radically new language, where the rules of spacetime and quantum mechanics, as reflected in the principles of locality and unitarity, are seen to emerge from deeper mathematical structures.

We describe a family of iterative algorithms that involve the repeated execution of discrete and inverse discrete Fourier transforms. One interesting member of this family is motivated by the discrete Fourier transform uncertainty principle and involves the application of a sparsification operation to both the real domain and frequency domain data with convergence obtained when real domain sparsity hits a stable pattern. This sparsification variant has practical utility for signal denoising, in particular the recovery of a periodic spike signal in the presence of Gaussian noise. General convergence properties and denoising performance relative to existing methods are demonstrated using simulation studies. An R package implementing this technique and related resources can be found at https://hrfrost.host.dartmouth.edu/IterativeFT.

We present a novel approach for computing a variant of eigenvector centrality for multilayer networks with inter-layer constraints on node importance. Specifically, we consider a multilayer network defined by multiple edge-weighted, potentially directed, graphs over the same set of nodes with each graph representing one layer of the network and no inter-layer edges. As in the standard eigenvector centrality construction, the importance of each node in a given layer is based on the weighted sum of the importance of adjacent nodes in that same layer. Unlike standard eigenvector centrality, we assume that the adjacency relationship and the importance of adjacent nodes may be based on distinct layers. Importantly, this type of centrality constraint is only partially supported by existing frameworks for multilayer eigenvector centrality that use edges between nodes in different layers to capture inter-layer dependencies. For our model, constrained, layer-specific eigenvector centrality values are defined by a system of independent eigenvalue problems and dependent pseudo-eigenvalue problems, whose solution can be efficiently realized using an interleaved power iteration algorithm.

We present a novel technique for sparse principal component analysis. This method, named Eigenvectors from Eigenvalues Sparse Principal Component Analysis (EESPCA), is based on the formula for computing squared eigenvector loadings of a Hermitian matrix from the eigenvalues of the full matrix and associated sub-matrices. We explore two versions of the EESPCA method: a version that uses a fixed threshold for inducing sparsity and a version that selects the threshold via cross-validation. Relative to the state-of-the-art sparse PCA methods of Witten et al., Yuan & Zhang and Tan et al., the fixed threshold EESPCA technique offers an order-of-magnitude improvement in computational speed, does not require estimation of tuning parameters via cross-validation, and can more accurately identify true zero principal component loadings across a range of data matrix sizes and covariance structures. Importantly, the EESPCA method achieves these benefits while maintaining out-of-sample reconstruction error and PC estimation error close to the lowest error generated by all evaluated approaches. EESPCA is a practical and effective technique for sparse PCA with particular relevance to computationally demanding statistical problems such as the analysis of high-dimensional data sets or application of statistical techniques like resampling that involve the repeated calculation of sparse PCs.

Palliative medicine is an interdisciplinary specialty focusing on improving quality of life (QOL) for patients with serious illness and their families. Palliative care programs are available or under development at over 80% of large US hospitals (300+ beds). Palliative care clinical trials present unique analytic challenges relative to evaluating the palliative care treatment efficacy which is to improve patients diminishing QOL as disease progresses towards end of life (EOL). A unique feature of palliative care clinical trials is that patients will experience decreasing QOL during the trial despite potentially beneficial treatment. Often longitudinal QOL and survival data are highly correlated which, in the face of censoring, makes it challenging to properly analyze and interpret longitudinal QOL trajectory. To address these issues, we propose a novel semiparametric statistical approach to jointly model longitudinal QOL and survival data. There are two sub-models in our approach: a semiparametric mixed effects model for longitudinal QOL and a Cox model for survival. We use regression splines method to estimate the nonparametric curves and AIC to select knots. We assess the model through simulation and application to establish a novel modeling approach that could be applied in future palliative care treatment research trials.

Motivation: Gene set testing is typically performed in a supervised context to quantify the association between groups of genes and a clinical phenotype. In many cases, however, a gene set-based interpretation of genomic data is desired in the absence of a phenotype variable. Although methods exist for unsupervised gene set testing, they predominantly compute enrichment relative to clusters of the genomic variables with performance strongly dependent on the clustering algorithm and number of clusters. Results: We propose a novel method, spectral gene set enrichment (SGSE), for unsupervised competitive testing of the association between gene sets and empirical data sources. SGSE first computes the statistical association between gene sets and principal components (PCs) using our principal component gene set enrichment (PCGSE) method. The overall statistical association between each gene set and the spectral structure of the data is then computed by combining the PC-level p-values using the weighted Z-method with weights set to the PC variance scaled by Tracey-Widom test p-values. Using simulated data, we show that the SGSE algorithm can accurately recover spectral features from noisy data. To illustrate the utility of our method on real data, we demonstrate the superior performance of the SGSE method relative to standard cluster-based techniques for testing the association between MSigDB gene sets and the variance structure of microarray gene expression data. Availability: http://cran.r-project.org/web/packages/PCGSE/index.html Contact: rob.frost@dartmouth.edu or jason.h.moore@dartmouth.edu

Motivation: Although principal component analysis (PCA) is widely used for the dimensional reduction of biomedical data, interpretation of PCA results remains daunting. Most existing methods attempt to explain each principal component (PC) in terms of a small number of variables by generating approximate PCs with few non-zero loadings. Although useful when just a few variables dominate the population PCs, these methods are often inadequate for characterizing the PCs of high-dimensional genomic data. For genomic data, reproducible and biologically meaningful PC interpretation requires methods based on the combined signal of functionally related sets of genes. While gene set testing methods have been widely used in supervised settings to quantify the association of groups of genes with clinical outcomes, these methods have seen only limited application for testing the enrichment of gene sets relative to sample PCs. Results: We describe a novel approach, principal component gene set enrichment (PCGSE), for computing the statistical association between gene sets and the PCs of genomic data. The PCGSE method performs a two-stage competitive gene set test using the correlation between each gene and each PC as the gene-level test statistic with flexible choice of both the gene set test statistic and the method used to compute the null distribution of the gene set statistic. Using simulated data with simulated gene sets and real gene expression data with curated gene sets, we demonstrate that biologically meaningful and computationally efficient results can be obtained from a simple parametric version of the PCGSE method that performs a correlation-adjusted two-sample t-test between the gene-level test statistics for gene set members and genes not in the set. Availability: http://cran.r-project.org/web/packages/PCGSE/index.html Contact: rob.frost@dartmouth.edu or jason.h.moore@dartmouth.edu