Foundations of data science (Springfield, Mo.)最新文献

We present an application of topological data analysis (TDA) to discrete optimization problems, which we show can improve the performance of the 2-opt local search method for the traveling salesman problem by simply applying standard Vietoris-Rips construction to a data set of trials. We then construct a simplicial complex which is specialized for this sort of simulated data set, determined by a stochastic matrix with a steady state vector begin{document}$ (P,pi) $end{document}. When begin{document}$ P $end{document} is induced from a random walk on a finite metric space, this complex exhibits similarities with standard constructions such as Vietoris-Rips on the data set, but is not sensitive to outliers, as sparsity is a natural feature of the construction. We interpret the persistent homology groups in several examples coming from random walks and discrete optimization, and illustrate how higher dimensional Betti numbers can be used to classify connected components, i.e. zero dimensional features in higher dimensions.

The p-persistent q-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the q-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.

Control-type particle filters have been receiving increasing attention over the last decade as a means of obtaining sample based approximations to the sequential Bayesian filtering problem in the nonlinear setting. Here we analyse one such type, namely the feedback particle filter and a recently proposed approximation of the associated gain function based on diffusion maps. The key purpose is to provide analytic insights on the form of the approximate gain, which are of interest in their own right. These are then used to establish a roadmap to obtaining well-posedness and convergence of the finite begin{document}$ N $end{document} system to its mean field limit. A number of possible future research directions are also discussed.

Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a unified multiscale paradigm to encompass TDA and geometric analysis. In PSG theory, families of persistent Laplacian matrices (PLMs) corresponding to various topological dimensions are constructed via a filtration to sample a given dataset at multiple scales. The harmonic spectra from the null spaces of PLMs offer the same topological invariants, namely persistent Betti numbers, at various dimensions as those provided by PH, while the non-harmonic spectra of PLMs give rise to additional geometric analysis of the shape of the data. In this work, we develop an open-source software package, called highly efficient robust multidimensional evolutionary spectra (HERMES), to enable broad applications of PSGs in science, engineering, and technology. To ensure the reliability and robustness of HERMES, we have validated the software with simple geometric shapes and complex datasets from three-dimensional (3D) protein structures. We found that the smallest non-zero eigenvalues are very sensitive to data abnormality.

The disparity in the impact of COVID-19 on minority populations in the United States has been well established in the available data on deaths, case counts, and adverse outcomes. However, critical metrics used by public health officials and epidemiologists, such as a time dependent viral reproductive number (begin{document}$ R_t $end{document}), can be hard to calculate from this data especially for individual populations. Furthermore, disparities in the availability of testing, record keeping infrastructure, or government funding in disadvantaged populations can produce incomplete data sets. In this work, we apply ensemble data assimilation techniques which optimally combine model and data to produce a more complete data set providing better estimates of the critical metrics used by public health officials and epidemiologists. We employ a multi-population SEIR (Susceptible, Exposed, Infected and Recovered) model with a time dependent reproductive number and age stratified contact rate matrix for each population. We assimilate the daily death data for populations separated by ethnic/racial groupings using a technique called Ensemble Smoothing with Multiple Data Assimilation (ESMDA) to estimate model parameters and produce an begin{document}$R_t(n)$end{document} for the begin{document}$n^{th}$end{document} population. We do this with three distinct approaches, (1) using the same contact matrices and prior begin{document}$R_t(n)$end{document} for each population, (2) assigning contact matrices with increased contact rates for working age and older adults to populations experiencing disparity and (3) as in (2) but with a time-continuous update to begin{document}$R_t(n)$end{document}. We make a study of 9 U.S. states and the District of Columbia providing a complete time series of the pandemic in each and, in some cases, identifying disparities not otherwise evident in the aggregate statistics.