2014 Fall

Date: 2014-12-12

Time: 15:30-16:30

Location: BURN 1205

Abstract:

We will discuss the detection of pattern in images and graphs from a high-dimensional Gaussian measurement. This problem is relevant to many applications including detecting anomalies in sensor and computer networks, large-scale surveillance, co-expressions in gene networks, disease outbreaks, etc. Beyond its wide applicability, structured Normal means detection serves as a case study in the difficulty of balancing computational complexity with statistical power. We will begin by discussing the detection of active rectangles in images and sensor grids. We will develop an adaptive scan test and determine its asymptotic distribution. We propose an approximate algorithm that runs in nearly linear time but achieves the same asymptotic distribution as the naive, quadratic run-time algorithm. We will move on to the more general problem of detecting a well-connected active subgraph within a graph in the Normal means context. Because the generalized likelihood ratio test is computationally infeasible, we propose approximate algorithms and study their statistical efficiency. One such algorithm that we develop is the graph Fourier scan statistic, whose statistical performance is characterized by the spectrum of the graph Laplacian. Another relaxation that we have developed is the Lovasz extended scan statistic (LESS), which is based on submodular optimization and the performance is described using electrical network theory. We also introduce the spanning tree wavelet basis over graphs, a localized basis that reflects the topology of the graph. For each of these tests we compare their statistical guarantees to an information theoretic lower bound.

Date: 2014-12-05

Time: 15:30-16:30

Location: BURN 1205

Abstract:

Copula model selection is an important problem because similar but differing copula models can offer different conclusions surrounding the dependence structure of random variables. Chen & Fan (2005) proposed a model selection method involving a statistical hypothesis test. The hypothesis test attempts to take into account the randomness of the AIC and other likelihood-based model selection methods for finite samples. Performance of the test compared to the more common approach of AIC is illustrated in a series of simulations.

Date: 2014-11-28

Time: 15:30-16:30

Location: BURN 1205

Abstract:

Model-based clustering via finite mixture models is a popular clustering method for finding hidden structures in data. The model is often assumed to be a finite mixture of multivariate normal distributions; however, flexible extensions have been developed over recent years. This talk demonstrates some methods employed in unsupervised, semi-supervised, and supervised classification that include skew-normal and skew-t mixture models. Both real and simulated data sets are used to demonstrate the efficacy of these techniques.

Date: 2014-11-21

Time: 15:30-16:30

Location: BURN 1205

Abstract:

Many statistical estimators are based on solving nonconvex programs. Although the practical performance of such methods is often excellent, the associated theory is frequently incomplete, due to the potential gaps between global and local optima. In this talk, we present theoretical results that apply to all local optima of various regularized M-estimators, where both loss and penalty functions are allowed to be nonconvex. Our theory covers a broad class of nonconvex objective functions, including corrected versions of the Lasso for error-in-variables linear models; regression in generalized linear models using nonconvex regularizers such as SCAD and MCP; and graph and inverse covariance matrix estimation. Under suitable regularity conditions, our theory guarantees that any local optimum of the composite objective function lies within statistical precision of the true parameter vector. This result closes the gap between theory and practice for these methods.

Date: 2014-11-20

Time: 16:00-17:00

Location: CRM 1360 (U. de Montréal)

Abstract:

Statistical models in which the ambient dimension is of the same order or larger than the sample size arise frequently in different areas of science and engineering. Although high-dimensional models of this type date back to the work of Kolmogorov, they have been the subject of intensive study over the past decade, and have interesting connections to many branches of mathematics (including concentration of measure, random matrix theory, convex geometry, and information theory). In this talk, we provide a broad overview of the general area, including vignettes on phase transitions in high-dimensional graph recovery, and randomized approximations of convex programs.

Date: 2014-11-14

Time: 15:30-16:30

Location: BURN 1205

Abstract:

In the parametric setting, the notion of a likelihood function forms the basis for the development of tests of hypotheses and estimation of parameters. Tests in connection with the analysis of variance stem entirely from considerations of the likelihood function. On the other hand, non- parametric procedures have generally been derived without any formal mechanism and are often the result of clever intuition. In this talk, we propose a more formal approach for deriving tests involving the use of ranks. Specifically, we define a likelihood function motivated by characteristics of the ranks of the data and demonstrate that this leads to well-known tests of hypotheses. We also point to various areas of further exploration.

Date: 2014-11-07

Time: 15:30-16:30

Location: BURN 1205

Abstract:

We approach the problem of shape constrained regression from a Bayesian perspective. A B-spline basis is used to model the regression function. The smoothness of the regression function is controlled by the order of the B-splines and the shape is controlled by the shape of an associated control polygon. Controlling the shape of the control polygon reduces to some inequality constraints on the spline coefficients. Our approach enables us to take into account combinations of shape constraints and to localize each shape constraint on a given interval. The performances of our method is investigated through a simulation study. Applications to real data sets from the food industry and Global Warming are provided.

Date: 2014-10-31

Time: 15:30-16:30

Location: BURN 1205

Abstract:

A flexible approach is proposed for risk aggregation. The model consists of a tree structure, bivariate copulas, and marginal distributions. The construction relies on a conditional independence assumption whose implications are studied. Selection the tree structure, estimation and model validation are illustrated using data from a Canadian property and casualty insurance company.

Speaker

Marie-Pier Côté is a PhD student in the Department of Mathematics and Statistics at McGill University.

Date: 2014-10-24

Time: 15:30-16:30

Location: BURN 1205

Abstract:

Next generation sequencing (NGS) is a technology revolutionizing genetics and biology. Compared with the old Sanger sequencing method, the throughput is astounding and has fostered a slew of innovative sequencing applications. Unfortunately, the error rates are also higher, complicating many downstream analyses. For example, de novo assembly of genomes is less accurate and slower when reads include many errors. We develop a probabilistic model for NGS reads that can detect and correct errors without a reference genome and while flexibly modeling and estimating the error properties of the sequencing machine. It uses a penalized likelihood to enforce our prior belief that the kmer spectrum (collection of k-length strings observed in the reads) generated from a genome is sparse when k is sufficiently large. The model formalizes core ideas that are used in many ad hoc algorithmic approaches to error correction. We show our method can detect and remove more errors from sequencing reads than existing methods. Though our method carries a higher computational burden than the best algorithmic approaches, the probabilistic approach is extensible, flexible, and well-positioned to support downstream statistical analysis of the increasing volume of sequence data.

Date: 2014-10-17

Time: 15:30-16:30

Location: BURN 1205

Abstract:

In the recent past, electronic health records and distributed data networks emerged as a viable resource for medical and scientific research. As the use of confidential patient information from such sources become more common, maintaining privacy of patients is of utmost importance. For a binary disease outcome of interest, we show that the techniques of specimen pooling could be applied for analysis of large and/or distributed data while respecting patient privacy. I will review the pooled analysis for a binary outcome and then show how it can be used for distributed data. Aggregate level data are passed from the nodes of the network to the analysis center and can be used very easily with logistic regression for estimation of disease odds ratio associated with a set of categorical or continuous covariates. Pooling approach allows for consistent estimation of the parameters of logistic regression that can include confounders. Additionally, since the individual covariate values can be accessed within a network, effect modifiers can be accommodated and consistently estimated. Since pooling effectively reduces the size of the dataset by creating pools or sets of individual, the resulting dataset can be analyzed much more quickly as compared to an original dataset that is too big as compared to computing environment.