Discrete and continuous time processes with an emphasis on Markov, Gaussian, and renewal processes. Martingales and further limit theorems. A variety of applications taken from some of the following areas are discussed in the context of stochastic modeling: Information Theory, Quantum Mechanics, Statistical Analyses of Stochastic Processes, Population Growth Models, Reliability, Queuing Models, Stochastic Calculus, Simulation (Monte Carlo Methods).

Data acquisition in the environmental, physical, and health sciences are increasingly spatial, and novel in the sense that specialized methods are required for analysis. This course will cover different types of spatial and spatiotemporal data and their analytic methods. Students will learn a variety of advanced techniques for analyzing geostatistical, areal, and point referenced data. Focus will be placed on visualizing spatial data, choosing the correct method for a specific research question, and communicating analytic results clearly and effectively.

A rigorous introduction to stochastic analysis and its applications. Topics include Brownian motion, continuous time martingale, stochastic integration, stochastic differential equations, diffusions, and further topics depending on the interests of the instructor.

Techniques for formulating data science models as optimization problems. Algorithms for solving data science problems including gradient-descent based algorithms and randomized algorithms. Emphasis on scalability and efficiency. Convergence analysis of algorithms. Coverage of both convex and nonconvex optimization.

Modern statistical methods and data analytics are increasingly informing decisions in law, business, medicine, and public life. While the use of statistics to understand social problems is not new, its pervasiveness in society and the scale of available data available opens up a host of new and/or salient moral problems including, for example, fairness, bias, privacy, equality, transparency, accountability, and accessibility. In this course, we will combine material from law and philosophy together with recent work in statistics and data science in order to gain a better understanding of how to intelligibly reason about these problems, and how to responsibly and creatively apply statistical methods to complex social problems. The course will be research/project based and the emphasis will be on using statistics to address complex social problems rather than on memorizing abstract ethical principles for handling or processing data.

The topics will vary year to year and give students the flexibility to examine a diverse range of subjects relevant to applied statistics and data science. This special topics course is repeatable for credit if taken with a different individual topic.

Statistical genetics is an important data science research area with direct impact on population health, and this course provides an introduction to its concepts and fundamentals. We start with an overview of genetic studies to have a general understanding of its goal and study design. We then introduce the basic genetic terminologies necessary for the ensuing discussion of the various statistical methods used for analyzing genetic data. The specific topics include population genetics, principles of inheritance, likelihood for pedigree data, aggregation, heritability, and segregation analyses, map and linkage analysis, population-based and family-based association studies and genome-wide association studies. The flow of the content generally follows that of the "The Fundamentals of Modern Statistical Genetics" by Laird and Lange, and additional materials will be provided. Participating students do not need formal training in genetics, but they are expected to have statistical knowledge at the level of STA303H1 Methods of Data Analysis II or equivalent.

This course will focus on principles and methods of applied statistical science. It is designed for MSc and PhD students in Statistics, and is required for the Applied Paper of the PhD comprehensive exams. The topics covered include: planning of studies, review of linear models, analysis of random and mixed effects models, model building and model selection, theory and methods for generalized linear models, and an introduction to nonparametric regression. Additional topics will be introduced as needed in the context of case studies in data analysis.

The goal of this course is to give an overview of some of the computational methods that are useful in statistics. The first part of the course will focus on basic algorithms, such as the Fast Fourier Transform (and related methods) and methods for generating random variables. The second part of the course will focus on numerical methods for linear algebra and optimization (for example, computing least squares estimates and maximum likelihood estimates). Along the way, you will learn some basic theory of numerical analysis (computational complexity, convergence rates of algorithms) and you will encounter some statistical methodology that you may not have seen in other courses.

This course will consider topics in statistics that have played a role in the development of techniques for data mining and machine learning. We will cover linear methods for regression and classification, nonparametric regression and classification methods, generalized additive models, aspects of model inference and model selection, model averaging, and tree-based methods.

This is a course designed for master's and PhD level students in statistics, mathematics, and other departments, who are interested in a rigorous, mathematical treatment of probability theory using measure theory. Specific topics to be covered include: probability measures, the extension theorem, random variables, distributions, expectations, laws of large numbers, Markov chains. Students should have a strong undergraduate background in Real Analysis, including calculus, sequences and series, elementary set theory, and epsilon-delta proofs.

This course introduces core statistical learning ideas through a computational framework based on (stochastic) gradient descent methods. Students will build on core probability and convergence concepts to learn how common modeling choices translate into practical estimation and prediction procedures. A range of supervised models (linear, generalized linear, and their flexible extensions) will be presented using a (stochastic) gradient descent approach with different losses (squared, logistic, quantile) and regularization.

We also introduce basic decision theory to connect predicted probabilities to actions, such as optimal classification thresholds. Model evaluation and model selection methods will be emphasized, as will careful comparison of models using appropriate performance metrics and diagnostic tools.

Statistical inference is concerned with using the evidence, available from observed data, to draw inferences about an unknown probability measure. A variety of theoretical approaches have been developed to address this problem and these can lead to quite different inferences. A natural question is then concerned with how one determines and validates appropriate statistical methodology in a given problem. The course considers this larger statistical question. This involves a discussion of topics such as model specification and checking, the likelihood function and likelihood inferences, repeated sampling criteria, loss (utility) functions and optimality, prior specification and checking, Bayesian inferences, principles, and axioms, etc. The overall goal of the course is to leave students with an understanding of the different approaches to the theory of statistical inference while developing a critical point-of-view.

This course presents mathematical foundations for learning, prediction, and decision making. Unlike in traditional statistical learning, however, our focus will be on notions of optimality that do not rely on stochastic modeling assumptions on data. A primary focus will be on learning from data to compete with a class of baselines predictors/strategies, often referred to as experts. A secondary focus will be on the ability to adapt to the presence or absence of statistical patterns, without presuming at the outset that such patterns will arise. Topics include: regret; prediction with expert advice; the role of the loss function in tight bounds; online classification; online linear and convex optimization; regularization; bandit problems/decisions with limited feedback; minimax optimality and adaptivity; relationships with statistical learning.

The course will focus on generalized linear models (GLM) and related methods, such as generalized additive model involving nonparametric regression, generalized estimating equations (GEE), and generalized linear mixed models (GLMM) for longitudinal data. This course is designed for master's and PhD students in Statistics, and is required for the applied paper of the PhD comprehensive exams in Statistics. We deal with a class of statistical models that generalizes classical linear models to include many other models that have been found useful in statistical analysis, especially in biomedical applications. The course is a mixture of theory and applications and includes computer projects featuring R (S+) or/and SAS programming. Topics: Brief review of likelihood theory, fundamental theory of generalized linear models, iterated weighted least squares, binary data and logistic regression, epidemiological study designs, counts data and log-linear models, models with constant coefficient of variation, quasi-likelihood, generalized additive models involving nonparametric smoothing, generalized estimating equations (GEE), and generalized linear mixed models (GLMM) for longitudinal data.

An overview of methods and problems in the analysis of time series data. Topics include: descriptive methods, filtering and adjustment, spectral estimation, bivariate time series models. The course will cover the following topics: Theory of stationary processes, linear processes; Elements of inference in time domain with applications; Spectral representation of stationary processes; Elements of inference in frequency domain with applications; Theory of prediction (forecasting) with applications > ARMA processes, inference, and forecasting; Non-stationarity and seasonality, ARIMA, and SARIMA processes. Further topics, time permitting: multivariate models; GARCH models; state-space models.

This course introduces some advanced time series topics beyond the stationary ARMA models and techniques which are relevant in time series applications in the big data era. Our focus will be on non-stationary and nonlinear time series analyses from a nonparametric statistical perspective. Topics include: the concept of unit root and cointegrated processes; unit root testing; ARCH/GARCH models for stochastic volatility; kernel regression; nonparametric smoothing in the time/state domain; nonparametric spectral density estimation; linear and nonlinear state-space models. Some associated computational skills in real data applications will be introduced as well.

This course aims to introduce model selection methods for survival time and recurrent event data analysis. Topics include parametric models for lifetime and recurrent event data, regression models, parametric, semiparametric and nonparametric inference, goodness-of-fit and model selection. With applications to statistics, actuarial science, biostatistics, and engineering.

This is a follow-up course to STA2111H, designed for master's and PhD level students in statistics, mathematics, and other departments, who are interested in a rigorous, mathematical treatment of probability theory using measure theory. Specific topics to be covered include: weak convergence, characteristic functions, central limit theorems, the Radon-Nykodym Theorem, Lebesgue Decomposition, conditional probability and expectation, martingales, and Kolmogorov's Existence Theorem.

This course builds on Statistical Learning I to study modern machine learning methods from a statistical perspective. The course covers classical machine learning approaches and unsupervised learning, then develops probabilistic and representation learning frameworks used in contemporary large-scale systems. Gradient-based likelihood optimization for complex models, variational inference, stochastic variational inference, Bayesian neural networks for uncertainty estimation, and representation learning will be covered. Large language models are introduced as autoregressive probabilistic models trained by maximum likelihood. The course emphasizes reproducible computation with robust model evaluation.

This course is part one of a two-course sequence that introduces graduate students to computational methods designed specifically for statistical inference. This course will cover methods for optimization and simulation methods in several contexts. Optimization methods are introduced in order to conduct likelihood-based inference, while simulation techniques are used for studying the performance of a given statistical model and to conduct Bayesian analysis. Covered topics include gradient-based optimization algorithms (Newton method, Fisher scoring), the Expectation-Maximization (EM) algorithm and its variants (ECM, MCEM, etc), basic simulation principles and techniques for model analysis (cross-validation independent replications, etc.), Monte Carlo and Markov chain Monte Carlo algorithms (accept-reject, importance sampling Metropolis-Hastings and Gibbs samplers, adaptive MCMC, Approximate Bayesian computation, consensus Monte Carlo, subsampling MCMC, etc.). Particular emphasis will be placed on modern developments that address situations in which the Bayesian analysis is conducted when data are massive or the likelihood is intractable. The focus of the course is on correct usage of these methods rather than the detailed study of underlying theoretical arguments.

The course will discuss the technical side of statistical methods focusing on two key aspects: optimization and implementation. The first part of the course will introduce necessary background for understanding and devising algorithms for modern statistical methodology. It will cover core concepts and tools from convex optimization such as convexity of sets and functions, Lagrange multipliers method, Newton's method, proximal gradient descent, coordinate descent, alternating direction method of multipliers. In addition, it will include the review of key topics in linear algebra such as matrix and vector norms, quadratic forms and positive semidefinite matrices, matrix calculus (gradient, Hessian, and determinant), matrix decompositions (QR, Cholesky, eigen, and singular value). The second part of the course will focus on topics from statistical methodology with an emphasis on computational aspects. The covered concepts will include model assessment and selection (bias-variance trade-off, cross-validation, and bootstrap), feature selection (penalized generalized linear models, elastic net, group and fused lasso, least angle regression), dimension reduction (principal component analysis, independent component analysis, factor analysis), data compression (k-means, hierarchical, and spectral clustering). The course will involve a significant practical component, which will include labs and coding assignments where students will master their skills in implementing statistical optimization algorithms.

This course is designed to provide graduate students with experience in statistical consulting. Students are active participants in research projects brought to the Statistical Consulting Service (SCS) of the Department of Statistics. The course is offered over the two sessions, Fall (September to December) and Winter (January to April). The overall workload is approximately equivalent to a half graduate course and students receive a half credit.

Students are not expected to have had any experience as consultants. The purpose of the course is to provide this experience so that graduates will be better able to function in such an environment when they have completed the course. The course also provides students with the opportunity to become familiar with statistical software packages such as The SAS System. There is supervision and assistance to novice consultants.

An overview of theory and methods in the analysis of survival data. Topics include survival distributions and their applications, parametric and non-parametric methods, proportional hazards regression, and extensions to competing risks and multistate modelling.

Parametric distributions and transformations, insurance coverage modifications, limits and deductibles, models for claim frequency and severity, models for aggregate claims, stop-loss insurance, risk measures.

Consult the instructor for further details.

This course is an introduction to the stochastic models used in Finance and Actuarial Science. Students will be exposed to the basics of stochastic calculus, particularly focusing on Brownian motions and simple stochastic differential equations. The role that martingales play in the pricing of derivative instruments will be investigated. Some exotic equity derivative products will be explored together with stochastic models for interest rates.

This course focuses on financial mathematics and its applications to a broad array of financial derivative contracts, with emphasis on continuous time stochastic methods and their applications in modern quantitative finance. Students are expected to have a solid foundation in probability theory, calculus, ordinary and partial differential equations, and a basic familiarity with financial instruments.

Tentative topics include, but are not limited to: the no-arbitrage principle and the fundamental theorem of asset pricing; binomial pricing models and their continuous-time limits; Stochastic differential equations; Ito's Lemma; Girsanov's Theorem, the Black-Scholes model; sensitivity measures (Greeks) and hedging strategies; pricing of European, American, Asian, barrier, and other path-dependent options; short-rate models and interest rate derivatives; and stochastic volatility models.

Limited fluctuation or American credibility, on a full and partial basis. Greatest accuracy or European credibility, predictive distributions and the Bayesian premium, credibility premiums including the Buhlmann and Buhlmann-Straub models, empirical Bayes nonparametric and semi-parametric parameter estimation. Simulation, random numbers, discrete and continuous random variable generation, discrete event simulation, statistical analysis of simulated data and validation techniques.

This course introduces the principles and practices of applied data science and machine learning in the context of finance and insurance. Topics include data extraction from structured and unstructured sources (text, images, audio, and geospatial data), data cleaning, integration, and transformation, feature engineering, and exploratory analysis. Students will learn frameworks for model development and monitoring, including handling missing data, categorical encoding, variable and model selection, and lifecycle management.

The course also includes an introduction to Generalized Linear Models (GLMs). The course places emphasis on data preparation and the overall model lifecycle, providing the foundation for model-building techniques covered in Data Science and Machine Learning II. Reproducible, code-driven workflows using Python and SQL are emphasized, supported by case studies from financial analytics and insurance applications.