Related papers: Experiments with the Census
We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…
Sample surveys are widely used to obtain information about totals, means, medians, and other parameters of finite populations. In many applications, similar information is desired for subpopulations such as individuals in specific…
We consider the statistical analysis of trajectories on Riemannian manifolds that are observed under arbitrary temporal evolutions. Past methods rely on cross-sectional analysis, with the given temporal registration, and consequently may…
Tests of equality of copulas between two samples are introduced and studied using the empirical Bernstein copula process. Three statistics are proposed and their asymptotic properties are established. Besides, a subsampling Bernstein…
This paper investigates a statistical procedure for testing the equality of two independently estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors, that is,…
We consider the relation for the stochastic equilibrium states between the reduced system on a random slow manifold and the original system. This provides a theoretical basis for the reduction about sophisti- cated detailed models by the…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
Understanding the higher-order interactions within network data is a key objective of network science. Surveys of metadata triangles (or patterned 3-cycles in metadata-enriched graphs) are often of interest in this pursuit. In this work, we…
In this monograph, we prove an asymptotic approximation for integrals of probability densities over sets in finite dimensional euclidean space, which are far away from the origin (asymptotic sets). We use this approximation to investigate…
This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density…
Research on manifold learning within a density ridge estimation framework has shown great potential in recent work for both estimation and de-noising of manifolds, building on the intuitive and well-defined notion of principal curves and…
A common observation in data-driven applications is that high-dimensional data have a low intrinsic dimension, at least locally. In this work, we consider the problem of point estimation for manifold-valued data. Namely, given a finite set…
Interesting data often concentrate on low dimensional smooth manifolds inside a high dimensional ambient space. Random projections are a simple, powerful tool for dimensionality reduction of such data. Previous works have studied bounds on…
We develop differentially private methods for estimating various distributional properties. Given a sample from a discrete distribution $p$, some functional $f$, and accuracy and privacy parameters $\alpha$ and $\varepsilon$, the goal is to…
Topological aspects of interfaces are studied by comparing quantitatively the evolving three-color patterns in three different models, such as the three-state voter, Potts and extended voter models. The statistical analysis of some…
This preprint is an exploration in how a single mathematical idea - entropy - can be applied to redistricting in a number of ways. It's meant to be read not so much as a call to action for entropy, but as a case study illustrating one of…
We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions. Multiscale CTFs quantify variation of the data about every point in the data…
Often, high dimensional data lie close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds. The homology groups of a manifold are important topological invariants that provide an algebraic…
Fine resolution estimates of demographic and socioeconomic attributes are crucial for planning and policy development. While several efforts have been made to produce fine-scale gridded population estimates, socioeconomic features are…
Survey sampling theory and methods are introduced. Sampling designs and estimation methods are carefully discussed as a textbook for survey sampling. Topics includes Horvitz-Thompson estimation, simple random sampling, stratified sampling,…