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Thesis Title: Robust Statistical Inference Through the Lens of Optimization
Advisor:
Dr. Yao Xie, School of Industrial and Systems Engineering, Georgia Tech
Committee Members:
Dr. George V. Moustakides, Department of Electrical and Computer Engineering, University of Patras
Dr. Arkadi Nemirovski, School of Industrial and Systems Engineering, Georgia Tech
Dr. Jianjun Shi, School of Industrial and Systems Engineering, Georgia Tech
Dr. Jeff Wu, School of Industrial and Systems Engineering, Georgia Tech
Date and Time: Wednesday, May 19, 2021 @ 12:30pm (EST)
Meeting URL (BlueJeans): https://bluejeans.com/4172449604
Meeting ID (BlueJeans): 4172449604
Abstract:
Robust statistical inference is an important and fundamental problem in modern data science. Many classical works in sequential analysis are designed for the case when we have full knowledge of the underlying data-generating distributions, such as the well- known Neyman-Pearson lemma for hypothesis testing and the cumulative sum (CUSUM) algorithm for sequential change-point detection. However, there are many cases when we do not have prior knowledge about the true distributions. In such cases, we need robust statistical methods that can guarantee the worst-case performance. Moreover, we also need algorithms that can be implemented efficiently in the online setting where data comes sequentially. This thesis tackles the robust statistical inference from three aspects.
Chapter 1 introduces the background and motivation for each topic as explained below. Chapter 2 reviews some preliminary and fundamental results in sequential change detection, distributionally robust optimization, and variational inequalities.
In Chapter 3, we consider the online monitoring of multivariate streaming data for changes that are characterized by an unknown subspace structure manifested in the covariance matrix. In particular, we consider the covariance structure changes from an identity matrix to an unknown spiked covariance model. We assume the post-change distribution is unknown, and propose two detection procedures: the Largest-Eigenvalue Shewhart chart and the Subspace-CUSUM detection procedure. We present theoretical approximations to the average run length and the expected detection delay for the Largest-Eigenvalue Shewhart chart, as well as the asymptotic optimality analysis for the Subspace-CUSUM procedure. The performance of the proposed methods is illustrated using simulation and real data for human gesture detection and seismic event detection.
In Chapter 4, we present a new non-parametric statistic, called the weighed l2 divergence, based on empirical distributions for sequential change detection. We start by constructing the weighed l2 divergence as a fundamental building block for two-sample tests and change detection. The proposed statistic is proved to attain the optimal sample complexity in the offline setting. We then study the sequential change detection using the weighed l2 divergence and characterize the fundamental performance metrics, including the average run length and the expected detection delay. We also present practical algorithms to find the optimal projection to handle high-dimensional data and the optimal weights, which is critical to quick detection since, in such settings, there are not many post-change samples. Simulation results and real data examples are provided to validate the good performance of the proposed method.
In Chapter 5, we consider a data-driven robust hypothesis test where the optimal test will minimize the worst-case performance regarding distributions close to the empirical distributions with respect to the Wasserstein distance. This leads to a new non-parametric hypothesis testing framework based on distributionally robust optimization, which is more robust when there are limited samples for one or both hypotheses. Such a scenario often arises from applications such as health care, online change-point detection, and anomaly detection. We study the computational and statistical properties of the proposed test by presenting a tractable convex reformulation of the original infinite-dimensional variational problem exploiting Wasserstein’s properties and characterizing the optimal radius for the uncertainty sets to control the generalization error. We also demonstrate the good performance of our method on synthetic and real data.
In Chapter 6, we introduce a new general modeling approach for multivariate discrete event data with categorical interacting marks, which we refer to as marked Bernoulli processes. In the proposed model, the probability of an event of a specific category to take place in a location may be influenced by past events at this and other locations. We do not restrict interactions to be positive or decaying over time as it is commonly adopted, allowing us to capture an arbitrary shape of influence from historical events, locations, and events of different categories. In our modeling, prior knowledge is incorporated by allowing general convex constraints on model parameters. We develop two parameter estimation procedures utilizing the constrained least square and maximum likelihood estimation. We discuss different applications of our approach and illustrate the performance of proposed recovery routines on synthetic examples and real-world data.
