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Title: Functional graphical modeling via Karhunen-Lo\`{e}ve expansions Authors:  Kuang-Yao Lee - Temple University (United States) [presenting]
Lexin Li - University of California Berkeley (United States)
Hongyu Zhao - Yale University (United States)
Dingjue Ji - Yale (United States)
Todd Constable - Yale (United States)
Abstract: Network estimation for multivariate functional data is becoming increasingly important in a wide variety of applications. The changes of graph structures can often be attributed to external variables such as other phenotypes observed in the data, or a time variable such as the subject's age. The latter gives rise to the problem of dynamic graphical modeling. Most existing methods focus on the random variable setting, and estimate the graph by aggregating samples, sometimes according to the diagnostic groups, but largely ignore the subject-level heterogeneity. We target graphical modeling of multivariate random functions, and treat the external variables as the conditioning set. We propose a new class of conditional functional graphical model that allows the graph links to vary along with the external variables. We develop two linear operators, the conditional precision operator and the conditional partial correlation operator, which generalize the precision matrix and the partial correlation matrix from the random variable setting to both the conditional and functional settings, and based on which we can estimate the conditional functional graph. We establish the error bounds of the corresponding estimators and the consistency of the conditional graph estimation. We demonstrate the efficacy of the proposed method through both simulations and a study of brain functional connectivity network.