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Register a new userTowards a theory of statistical tree-shape analysis
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In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for tree-like shapes and study metrics on the shape space. This shape space has singularities, corresponding to topological transitions in the represented trees. We study two closely related metrics on the shape space, TED and QED. QED is a quotient Euclidean distance arising naturally from the shape space formulation, while TED is the classical tree edit distance. Using Gromovs metric geometry we gain new insight into the geometries defined by TED and QED. We show that the new metric QED has nice geometric properties which facilitate statistical analysis, such as existence and local uniqueness of geodesics and averages. TED, on the other hand, does not share the geometric advantages of QED, but has nice algorithmic properties. We provide a theoretical framework and experimental results on synthetic data trees as well as airway trees from pulmonary CT scans. This way, we effectively illustrate that our framework has both the theoretical and qualitative properties necessary to build a theory of statistical tree-shape analysis.
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In this position paper we suggest a possible metric approach to shape comparison that is based on a mathematical formalization of the concept of observer, seen as a collection of suitable operators acting on a metric space of functions. These functions represent the set of data that are accessible to the observer, while the operators describe the way the observer elaborates the data and enclose the invariance that he/she associates with them. We expose this model and illustrate some theoretical reasons that justify its possible use for shape comparison.
Modern microbiome compositional data are often high-dimensional and exhibit complex dependency among microbial taxa. However, existing approaches to analyzing microbiome compositional data either do not adequately account for the complex dependency or lack scalability to high-dimensionality, which presents challenges in appropriately incorporating the random effects in microbiome compositions in the resulting statistical analysis. We introduce a generative model called the logistic-tree normal (LTN) model to address this need. The LTN marries two popular classes of models -- the log-ratio normal (LN) and the Dirichlet-tree (DT) -- and inherits key benefits of each. LN models are flexible in characterizing covariance among taxa but lacks scalability to higher dimensions; DT avoids this issue through a tree-based binomial decomposition but incurs restrictive covariance. The LTN incorporates the tree-based decomposition as the DT does, but it jointly models the corresponding binomial probabilities using a (multivariate) logistic-normal distribution as in LN models. It therefore allows rich covariance structures as LN, along with computational efficiency realized through a Polya-Gamma augmentation on the binomial models at the tree nodes. Accordingly, Bayesian inference on LTN can readily proceed by Gibbs sampling. The LTN also allows common techniques for effective inference on high-dimensional data -- such as those based on sparsity and low-rank assumptions in the covariance structure -- to be readily incorporated. Depending on the goal of the analysis, LTN can be used either as a standalone model or embedded into more sophisticated hierarchical models. We demonstrate its use in estimating taxa covariance and in mixed-effects modeling. Finally, we carry out an extensive case study using an LTN-based mixed-effects model to analyze a longitudinal dataset from the DIABIMMUNE project.
When we use simulation to evaluate the performance of a stochastic system, the simulation often contains input distributions estimated from real-world data; therefore, there is both simulation and input uncertainty in the performance estimates. Ignoring either source of uncertainty underestimates the overall statistical error. Simulation uncertainty can be reduced by additional computation (e.g., more replications). Input uncertainty can be reduced by collecting more real-world data, when feasible. This paper proposes an approach to quantify overall statistical uncertainty when the simulation is driven by independent parametric input distributions; specifically, we produce a confidence interval that accounts for both simulation and input uncertainty by using a metamodel-assisted bootstrapping approach. The input uncertainty is measured via bootstrapping, an equation-based stochastic kriging metamodel propagates the input uncertainty to the output mean, and both simulation and metamodel uncertainty are derived using properties of the metamodel. A variance decomposition is proposed to estimate the relative contribution of input to overall uncertainty; this information indicates whether the overall uncertainty can be significantly reduced through additional simulation alone. Asymptotic analysis provides theoretical support for our approach, while an empirical study demonstrates that it has good finite-sample performance.
Item response theory (IRT) has become one of the most popular statistical models for psychometrics, a field of study concerned with the theory and techniques of psychological measurement. The IRT models are latent factor models tailored to the analysis, interpretation, and prediction of individuals behaviors in answering a set of measurement items that typically involve categorical response data. Many important questions of measurement are directly or indirectly answered through the use of IRT models, including scoring individuals test performances, validating a test scale, linking two tests, among others. This paper provides a review of item response theory, including its statistical framework and psychometric applications. We establish connections between item response theory and related topics in statistics, including empirical Bayes, nonparametric methods, matrix completion, regularized estimation, and sequential analysis. Possible future directions of IRT are discussed from the perspective of statistical learning.
Taking the Fourier integral theorem as our starting point, in this paper we focus on natural Monte Carlo and fully nonparametric estimators of multivariate distributions and conditional distribution functions. We do this without the need for any estimated covariance matrix or dependence structure between variables. These aspects arise immediately from the integral theorem. Being able to model multivariate data sets using conditional distribution functions we can study a number of problems, such as prediction for Markov processes, estimation of mixing distribution functions which depend on covariates, and general multivariate data. Estimators are explicit Monte Carlo based and require no recursive or iterative algorithms.
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