“Representation of Samples of Density Functions and Regression for Random Objects”

Dr. Alex Petersen, University of CA at Davis

Date: Tuesday, February 02, 2016

Time: 3:30 PM – 4:30 PM

Location: Engineering Hall Room 106B1

Sponsor: Department of Statistics

Abstract:

In the first part of this talk, we will discuss challenges associated with the analysis of samples of one-dimensional density functions.  Due to their inherent constraints, densities do not live in a vector space and therefore commonly used Hilbert space based methods of functional data analysis are not appropriate.  To address this problem, we introduce a transformation approach, mapping probability densities to a Hilbert space of functions through a continuous and invertible map. Basic methods of functional data analysis, such as the construction of functional modes of variation, functional regression or classification, are then implemented by using representations of the densities in this linear space.  Transformations of interest include log quantile density and log hazard transformations, among others.  Rates of convergence are derived, taking into account the necessary preprocessing step of density estimation.  The proposed methods are illustrated through applications in brain imaging.

The second part of the talk will address the more general problem of analyzing complex data that are non-Euclidean and specifically do not lie in a vector space.  To address the need for statistical methods for such data, we introduce the concept of Fr\’echet regression. This is a general approach to regression when responses are complex random objects in a metric space and predictors are in $\mathcal{R}^p$. We develop generalized versions of both global least squares regression and local weighted least squares smoothing.  We derive asymptotic rates of convergence for the corresponding sample based fitted regressions to the population targets under suitable regularity conditions by applying empirical process methods.  Illustrative examples include responses that consist of probability distributions and correlation matrices, and we demonstrate the proposed Fr\’echet regression for demographic and brain imaging data.

http://illinois.edu/calendar/detail/1439?eventId=33075438&calMin=201602&cal=20160201&skinId=13335

“Novel Statistical Frameworks for Analysis of Structured Sequential Data”

Dr. Abhra Sarkar, Duke University

Date: Thursday, February 04, 2016

Time: 3:30 PM – 4:30 PM

Location: Engineering Hall Room 106B1

Sponsor: Department of Statistics

Abstract:

We are developing a broad array of novel statistical frameworks for analyzing complex sequential data sets. Our research is primarily motivated by a collaboration with neuroscientists trying to understand the neurological, genetic and evolutionary basis of human communication using bird and mouse models. The data sets comprise structured sequences of syllables or `songs’ produced by animals from different genotypes under different experimental conditions. The primary goal is then to elucidate the roles of different genotypes and experimental conditions on animal vocalization behaviors and capabilities. We have developed novel statistical methods based on first order Markovian dynamics that help answer these important scientific queries. First order dynamics is, however, insufficiently flexible to learn complex serial dependency structures and systematic patterns in the vocalizations, an important secondary goal in these studies. To this end, we have developed a sophisticated nonparametric Bayesian approach to higher order Markov chains building on probabilistic tensor factorization techniques. Our proposed method is of very broad utility, with applications not limited to analysis of animal vocalizations, and provides new insights into the serial dependency structures of many previously analyzed sequential data sets arising from diverse application areas. Our method has appealing theoretical properties and practical advantages, and achieves substantial gains in performance compared to previously existing methods. Our research also paves the way to advanced automated methods for more sophisticated dynamical systems, including higher order hidden Markov models that can accommodate more general data types.

http://illinois.edu/calendar/detail/1439?eventId=33075444&calMin=201602&cal=20160201&skinId=13335

“Computationally efficient high-dimensional variable selection via Bayesian procedures”

Dr. Yun Yang, University of California at Berkeley

Date: Tuesday, January 26, 2016

Time: 3:30 PM – 4:30 PM

Location: Engineering Hall Room 106B1

Sponsor: Department of Statistics

Abstract:

Variable selection is fundamental in many high-dimensional statistical problems with sparsity structures. Much of the literature is based on optimization methods, where penalty terms are incorporated that yield both convex and non-convex optimization problems. In this talk, I will take a Bayesian point of view on high-dimensional regression, by placing a prior on the model space and performing the necessary integration so as to obtain a posterior distribution. In particular, I will show that a Bayesian approach can consistently select all relevant covariates under relatively mild conditions from a frequentist point of view.

Although Bayesian procedures for variable selection are provably effective and easy to implement, it has been suggested by many statisticians that Markov Chain Monte Carlo (MCMC) algorithms for sampling from its posterior need a long time to converge, as sampling from an exponentially large number of sub-models is an intrinsically hard problem. Surprisingly, our work shows that this plausible “exponentially many model” argument is misleading. By introducing a truncated sparsity prior for variable selection, we provide a set of conditions that guarantee the rapid mixing of a particular Metropolis-Hastings algorithm. The number of iterations for this Markov chain to reach stationarity is linear in the number of covariates up to a logarithmic factor.

http://illinois.edu/calendar/detail/1439?eventId=33075436&calMin=201601&cal=20160125&skinId=13335

“Robust causal inference with continuous exposures”

Dr. Edward Kennedy, University of Pennsylvania

Date: Thursday, January 28, 2016

Time: 3:30 PM – 4:30 PM

Location: Engineering Hall Room 106B1

Sponsor: Department of Statistics

Abstract:

Continuous treatments (e.g., doses) arise often in practice, but standard causal effect estimators are limited: they either employ parametric models for the effect curve, or else do not allow for doubly robust covariate adjustment. Double robustness allows one of two nuisance estimators to be misspecified, and is important for protecting against model misspecification as well as reducing sensitivity to the curse of dimensionality. In this work we develop a novel approach for causal dose-response curve estimation that is doubly robust without requiring any parametric assumptions, and which naturally incorporates general off-the-shelf machine learning. We derive asymptotic properties for a kernel-based version of our approach and propose a method for data-driven bandwidth selection. The methods are illustrated via simulation and in a study of the effect of hospital nurse staffing on excess readmissions penalties.

http://illinois.edu/calendar/detail/1439?eventId=33075449&calMin=201601&cal=20160125&skinId=13335

“High dimensional spatio-temporal modeling with matrix variate distributions”

Dr. Shuheng Zhou, University of Michigan, Ann Arbor

Date: Thursday, December 03, 2015

Time: 3:30 PM – 4:30 PM

Location: 269 Everitt

Sponsor: Department of Statistics

Abstract:

In the first part of this talk, I will discuss new methods for estimating the graphical structures and underlying parameters, namely, the row and column covariance and inverse covariance matrices from the matrix variate data. Under sparsity conditions, we show that one is able to recover the graphs and covariance matrices with a single random matrix from the matrix variate normal distribution. Our method extends, with suitable adaptation, to the general setting where replicates are available.

In the second part of talk, I will discuss an errors-in-variables model where the covariates in the data matrix are contaminated with random noise. Under sparsity and restrictive eigenvalue type of conditions, we show that one is able to recover a sparse vector $\beta \in \mathbb{R}^m$ from the model given a single observation matrix X and the response vector y. This model is significantly different from those analyzed in the literature in the sense that we allow the measurement error for each covariate to be a dependent vector across its n observations. Such error structures appear in the science literature, for example, when modeling the trial-to-trial fluctuations in response strength shared across a set of neurons.

We provide a real-data example and simulation evidence showing that we can recover graphical structures as well as estimating the precision matrices and the regression coefficients for these two classes of problems.

http://illinois.edu/calendar/detail/1439?eventId=32723391&calMin=201511&cal=20151130&skinId=13335