Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium

Department Colloquium - Bill Ralph (Brock University)

Bill Ralph (Brock University)

Friday, February 14th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 128

Speaker: Bill Ralph (Brock University)

Title: Can Mathematics Recognize Great Art?

Abstract: Is there an objective truth hiding within great works of art that only mathematics can detect? In this talk, I'll present evidence for a mathematical aesthetic shared by many great artists across the centuries. We'll look at several striking works of art by artists ranging from Tintoretto to Picasso and use a new statistic to see that they are creating bell curves with their brushes. I’ll also show some of my own attempts to create visual art based on a variety of mathematically oriented techniques. Examples of my work can be viewed at www.billralph.com.

Bill Ralph is in the Faculty of Mathematics and Statistics at Brock University. His mathematical research began in algebraic topology with the study of exotic homology and cohomology theories and their connections with Banach Algebras. After that, he developed a transfer for finite group actions and studied a number that appears in the transfer that he calls the "coherence number" of the group. Lately, he has also been using the Hausdorff dimension of the orbits of dynamical systems to generate mathematical art. The following is an excerpt from the curator's notes from Prof. Ralph's Rodman Hall Museum show:
It is perhaps not surprising that some of the images have a painterly feel to them since the mixing of paint on the palette and the action of the brush on a surface are both processes that can be modeled as chaotic dynamical systems. In a sense, each image is a window into the intersection of the two great universes of mathematics and fine art.

Department Colloquium - Natalia Stepanova (Carleton University)

Natalia Stepanova (Carleton University)

Friday, February 7th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Natalia Stepanova (Carleton University)

Title: Goodness-of-fit tests based on sup-functionals of weighted empirical processes.

Abstract: A large class of goodness-of-fit test statistics based on sup-functionals of weighted empirical processes is proposed and studied. The weight functions employed are Erdos-Feller-Kolmogorov-Petrovski upper-class functions of a Brownian bridge. Based on the result of M. Csorgo, S. Csorgo, Horvath, and Mason obtained for this type of test statistics, we provide the asymptotic null distribution theory for the class of tests in hand, and present an algorithm for tabulating the limit distribution functions under the null hypothesis. A new family of nonparametric confidence bands is constructed for the true distribution function and it is found to perform very well. The results obtained, together with a new result on the convergence in distribution of the higher criticism statistic, introduced by Donoho and Jin, demonstrate the advantage of our approach over a common approach that utilizes a family of regularly varying weight functions. Furthermore, we show that, in various subtle problems of detecting sparse heterogeneous mixtures, the proposed test statistics achieve the detection boundary found by Ingster and, when distinguishing between the null and alternative hypotheses, perform optimally adaptively to unknown sparsity and size of the non-null effects. This is joint work with Tatjana Pavlenko (Sweden).

Natalia Stepanova is a Professor of Statistics in the School of Mathematics and Statistics at Carleton University. She has a Ph.D. degree in Statistics from St. Petersburg State University. From 2001--2003 Natalia Stepanova was a Postdoctoral Fellow at Queen's University, supervised by Prof. Boris Levit. Her research interests lie mainly in the area of nonparametric statistics, including high-dimensional statistical inference.

Department Colloquium - Dengdeng Yu (University of Toronto)

Dengdeng Yu (University of Toronto)

Thursday, February 6th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Dengdeng Yu (University of Toronto)

Title: Causal Inference with 2D Treatment: An Application to Alzheimer’s Studies

Abstract: Alzheimer's disease is a progressive form of dementia that causes problems with memory, thinking and behavior. It is important to identify the changes of certain brain regions that lead to behavioral deficits. In this paper, we study how hippocampal atrophy affects behavioral deficits using data from the Alzheimer's Disease Neuroimaging Initiative. The special features of the data include a 2D matrix-valued imaging treatment and more than $6$ million of potential genetic confounders, which bring significant challenges to causal inference. To address these challenges, we propose a novel two-step causal inference approach, which can naturally account for the 2D treatment structure while only adjusting for the necessary variables among the millions of covariates. Based on the analysis of the Alzheimer's Disease Neuroimaging Initiative dataset, we are able to identify important biomarkers that need to be accounted for in making causal inference and located the subregions of the hippocampus that may affect the behavioral deficits. We further evaluate our method using simulations and provide theoretical guarantees.

Dengdeng Yu is a postdoctoral fellow at the Department of Statistical Sciences at the University of Toronto and the Canadian Statistical Sciences Institute (CANSSI). He obtained his Ph.D. degree in Statistics from the University of Alberta in 2017. His research interests include high-dimensional and functional data analysis, neuroimaging and imaging genetics data analysis, and quantile regression.

Department Colloquium - Gregory G. Smith (Queen's University)

Gregory G. Smith (Queen's University)

Friday, January 31st, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Gregory G. Smith (Queen's University)

Title: The geometry of closed subsets.

Abstract: How can we understand the spaces embedded in a fixed projective space? There are different ways to answer this question. After examining a few, we will focus on a geometric approach. Ultimately, we aim to determine when the natural parameter space (the so called Hilbert scheme) is smooth.

Gregory G. Smith is a Professor of Mathematics at Queen's University. His research interests include algebraic geometry, commutative algebra, computer algebra and combinatorics. He was elected a fellow of the Canadian Mathematical Society in 2018. He also received the 2012 Coxeter-James Prize and 2007 André Aisenstadt Prize.

Department Colloquium - Hok Kan Ling (Columbia University)

Hok Kan Ling (Columbia University)

Monday, January 27th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Hok Kan Ling (Columbia University)

Title: Shape-constrained Estimation and Testing.

Abstract: Shape-constrained inference has been gaining more attention recently. Such constraints are sometimes the direct consequence of the problem under investigation. In other times, they are used to replace parametric models while retaining qualitative shape properties that exist in problems from diverse disciplines. In this talk, I will first discuss the estimation of a monotone density in s-sample biased sampling models, which has been long missing in the literature due to certain non-standard nature of the problem. We established the asymptotic distribution of the maximum likelihood estimator (MLE) and the connection between this MLE and a Grenander-type estimator. In the second part of the talk, a nonparametric likelihood ratio test for the hypothesis testing problem on whether a random sample follows a distribution with a decreasing, k-monotone or log-concave density is proposed. The obtained test statistic has a surprisingly simple and universal asymptotic null distribution, which is Gaussian, instead of the well-known chi-square for generic likelihood ratio tests. We also established rates of convergence of the maximum likelihood estimator under weaker conditions than the existing literature that are of independent interest.

Hok Kan (Brian) Ling is a Ph.D. candidate in the Department of Statistics at Columbia University, working under the supervision of Dr. Zhiliang Ying. His research interests primarily lie in the areas of multivariate statistics, latent variable models, event history analysis, nonparametric estimation, semiparametric models and shape-restricted statistical inference.

Department Colloquium - Siliang Gong (University of Pennsylvania)

Siliang Gong (University of Pennsylvania)

Thursday, January 23rd, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Siliang Gong (University of Pennsylvania)

Title: Per-family Error Rate Control for Gaussian Graphical Models via Knockoffs.

Abstract: Driven by many real applications, the estimation of and inference for Gaussian Graphical Models (GGM) are fundamentally important and have attracted much research interest in the literature. However, it is still challenging to achieve overall error rate control when recovering the graph structures of GGM. In this paper, we propose a new multiple testing method for GGM using the knockoffs framework. Our method can control the overall finite-sample Per-Family Error Rate up to a probability error bound induced by the estimation errors of knockoff features. Numerical studies demonstrate that our method has competitive performance compared with existing methods. This is joint work with Qi Long and Weijie Su.

Siliang Gong is a postdoctoral fellow in the Department of Biostatistics at the University of Pennsylvania. She completed her Ph.D. in statistics at the University of North Carolina at Chapel Hill in 2018. She works on high-dimensional data analysis and statistical machine learning.

Department Colloquium - Kasun Fernando (University of Toronto)

Kasun Fernando (University of Toronto)

Friday, January 17th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Kasun Fernando (University of Toronto)

Title: Error terms in the Central Limit Theorem.

Abstract: Expressing the error terms in the Central Limit Theorem as an asymptotic expansion (commonly referred to as the Edgeworth expansion) goes back to Chebyshev. In the setting of sums of independent identically distributed (iid) random variables, sufficient conditions for the existence of such expansions have been extensively studied. However, there is almost no literature that describe this error when the expansions fail to exist. In this talk, I will discuss the case of sums of iid non-lattice random variables with $d+1$ atoms. It can shown that they never admit the Edgeworth expansion of order d. However, using tools from Homogeneous Dynamics, it can shown that for almost all parameters the Edgeworth expansion of order $d-1$ holds and the error of the order $d-1$ Edgeworth expansion is typically of order $n^{-d/2}$ but the order $n^{-d/2}$ terms have wild oscillations (to be made precise during the talk). This is a joint work with Dmitry Dolgopyat.

Kasun Fernando is a postdoctoral fellow in the Department of Mathematics at the University of Toronto. He completed his Ph.D. in 2018 at the University of Maryland, College Park. His research is primarily focused on possible extensions of this theory of asymptotic expansions to more general settings that are not included in the classical theory, including the case of random variables arising as observations of chaotic dynamical systems.

Department Colloquium - Yi Xiong (Simon Fraser University)

Yi Xiong (Simon Fraser University)

Thursday, January 16th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Yi Xiong (Simon Fraser University)

Title: Statistical Issues in Forest Fire Control.

Abstract: This talk presents statistical issues arising from forest fire (wildfire) control with a particular focus on studying the duration times in the presence of missing origin. A new methodology is proposed to tackle the issue of missing time origin with the aid of available longitudinal measures.  I present an intuitive and easy-to-implement estimator for the distribution together with a method to conduct semi-parametric regression analysis. The estimation procedure is also extended to accommodate the spatial correlation in the data. A collection of wildfire records from Alberta, Canada is used for illustration and motivation. The finite-sample performances of proposed approaches are examined via simulation. On-going work and future directions to overcome other challenges of making inference on the underlying wildfire process will be discussed.

Yi Xiong is a Ph.D. student in the Department of Statistics at Simon Fraser University, under the supervision of Dr. Joan Hu and Dr. John Braun. She is interested in developing statistical methods to analyze complex data including missing data, censored lifetime data and spatio-temporal data.

Department Colloquium - Michael Gallaugher (McMaster University)

Michael Gallaugher (McMaster University)

Monday, January 13th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Michael Gallaugher (McMaster University)

Title: Clustering and Classification of Three-Way Data.

Abstract: Clustering and classification is the process of finding and analyzing underlying group structure in heterogenous data and is fundamental to computational statistics and machine learning. In the past, relatively simple techniques could be used for clustering; however, with data becoming increasingly complex, these methods are oftentimes not advisable, and in some cases not possible. One such such example is the analysis of three-way data where each data point is represented as a matrix instead of a traditional vector. Examples of three-way include greyscale images and multivariate longitudinal data. In this talk, recent methods for clustering three-way data will be presented including high-dimensional and skewed three-way data. Both simulated and real data will be used for illustration and future directions and extensions will be discussed.

Michael Gallaugher is a Ph.D. candidate in the Department of Mathematics and Statistics at McMaster University, working under the supervision of Dr. Paul D. McNicholas. His research interests lie in the area of clustering and classification which aims to find underlying group structure in heterogenous data.

Department Colloquium - Dimitris Koukoulopoulos (U Montreal)

Dimitris Koukoulopoulos (Universite de Montreal)

Friday, January 10th, 2020

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Dimitris Koukoulopoulos (Université de Montréal)

Title: On the Duffin-Schaeffer conjecture.

Abstract: Given any real number $\alpha$, Dirichlet proved that there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q|\le 1/q^2$. Can we get closer to $\alpha$ than that? For certain "quadratic irrationals" such as $\alpha=\sqrt{2}$ the answer is no. However, Khinchin proved that if we exclude such thin sets of numbers, then we can do much better. More precisely, let $(\Delta_q)_{q=1}^\infty$ be a sequence of error terms such that $q^2\Delta_q$ decreases. Khinchin showed that if the series $\sum_{q=1}^\infty q\Delta_q$ diverges, then almost all $\alpha$ (in the Lebesgue sense) admit infinitely many reduced rational approximations $a/q$ such that $|\alpha-a/q|\le \Delta_q$. Conversely, if the series $\sum_{q=1}^\infty q\Delta_q$ converges, then almost no real number is well-approximable with the above constraints. In 1941, Duffin and Schaeffer set out to understand what is the most general Khinchin-type theorem that is true, i.e., what happens if we remove the assumption that $q^2\Delta_q$ decreases. In particular, they were interested in choosing sequences $(\Delta_q)_{q=1}^\infty$ supported on sparse sets of integers. They came up with a general and simple criterion for the solubility of the inequality $|\alpha-a/q|\le\Delta_q$. In this talk, I will explain the conjecture of Duffin-Schaeffer as well as the key ideas in recent joint work with James Maynard that settles it.

Dimitris Koukoulopoulos is an Associate Professor of Mathematics at the Université de Montréal. He received his PhD from the University of Illinois in 2010. He works in analytic number theory, especially probabilistic and multiplicative aspects of the subject. Among his accolades, he was the cowinner of the 2013 Paul R. Halmos - Lester R. Ford Award. He is the author of the recent book The Distribution of Prime Numbers, published by the AMS.

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