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Activity Number: 189
Type: Contributed
Date/Time: Monday, August 4, 2014 : 10:30 AM to 12:20 PM
Sponsor: IMS
Abstract #313462
Title: Covariance and Correlation for Multivariate Functional Data
Author(s): Alexander Petersen *+ and Hans-Georg Müller
Companies: University of California, Davis and University of California, Davis
Keywords: Functional Data ; Covariance ; Partial Correlation
Abstract:

Assume one has $p$-dimensional functional data for $n$ subjects, $\{X_{i1}, X_{i2},\ldots,X_{ip}\}$, $i = 1, \ldots n$ on a common domain. The design setting for these data is flexible, allowing even for sparse longitudinal data. Given the utility and influence of a covariance matrix in standard multivariate analysis, we define an overall $p\times p$ functional covariance matrix $\Sigma$ by first transforming the pointwise covariance matrices, integrating, and transforming back. Such transformations may be based on specialized metrics for the nonlinear space of covariance matrices. This yields a Fr\'echet-like mean matrix, where the $(k,l)$-th element is a scalar measurement of covariance between processes $X_k$ and $X_l$. This is then extended to a functional partial correlation matrix. Estimates are obtained through standard cross-covariance estimates for paired functional data and consistency results then depend on the observational design setting. Examples of application areas include dimension reduction through eigenanalysis of $\Sigma$ and the construction of graphical models via partial correlations.


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