Date of Graduation

7-2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Qingyang Zhang

Committee Member

Avishek Chakraborty

Second Committee Member

Jyotishka Datta

Keywords

Categorical Data, Distance Correlation, Graph-Based Multivariate Test, Learning Networks

Abstract

We study the use of distance correlation for statistical inference on categorical data, especially the induction of probability networks. Szekely et al. first defined distance correlation for continuous variables in [42], and Zhang translated the concept into the categorical setting in [57] by defining dCor(X,Y) for categorical variables X = (x1,...,xI) and Y = (y1,...,yJ) where P(X=xi)=[pi]i and P(Y=yi)=[pi]j with the formula [Please open the document]

Part I of the dissertation covers the background we need to understand this formula, and prepares us to analyze the properties and performance of its applications.

Part II then presents the main results of the dissertation, applying distance correlation to learn the structure of probability networks with categorical nodes. We cover in detail how the distance correlation measure may be combined with search methods based on graphical models to induce network structure. This leads to our empirical results obtained by enhancing the INeS software library [6]. These results involve experiments using six data sets such as the Danish Jersey cattle blood type determination data and the ALARM network; in terms of accuracy metrics such as edges missed from the true network, induction with distance correlation achieves higher accuracy relative on average than does induction with existing measures such as mutual information and chi-squared. We conclude Part II by connecting to earlier joint work with Zhang in [58] on the use of conditional distance covariance for conditional independence and homogeneity tests in large sparse three-way tables. The simulation studies in this work offer another source of intuition for why distance correlation may be able to recover network structure more accurately than traditional measures.

In Part III, we end the dissertation by discussing another application of graphical models, in this case to the derivation of a graph-based multivariate test. The test statistic is computationally cheap, and proven to converge to a chi-squared distribution with favorable asymptotics. We present empirical results in which we use the test to analyze the roles of various oncogenic and suppressor pathways in tumor progression.

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