Abstract
We propose a novel approach to the analysis of covariance operators making use of concentration inequalities. First, non-asymptotic confidence sets are constructed for such operators. Then, subsequent applications including a k sample test for equality of covariance, a functional data classifier, and an expectation-maximization style clustering algorithm are derived and tested on both simulated and phoneme data.
Highlights
We propose a novel approach to the analysis of covariance operators making use of concentration inequalities
Non-asymptotic confidence sets are constructed for such operators
Covariance operators are integral operators with the kernel function cf (s, t) = cov{f (s), f (t)} ∈ L2 (I × I). Such operators are characterized by the result that for f ∈ L2 (I), Σf is a covariance operator if and only if it is trace-class, self-adjoint, and compact on L2 (I) where the symmetry follows immediately from the definition and the finite trace norm comes from Parseval’s equality
Summary
Functional data spans many realms of application from medical imaging, Jiang et al (2016), to speech and linguistics, Pigoli et al (2014), to the movement of DNA molecules, Panaretos et al (2010). In the age of big data, if p-values less than 1/1000 are desired, this can become computationally intractable with permutation methods; see Fig. 1 We approach such inference for covariance operators by using a non-asymptotic concentration of measure approach, which can incorporate arbitrary norms. This has previously been used in nonparametric statistics and machine learning, sometimes under the name of ‘Rademacher complexities’ (Koltchinskii, 2001, 2006; Bartlett et al, 2002; Bartlett and Mendelson, 2003; Giné and Nickl, 2010; Arlot et al., 2010; Lounici and Nickl, 2011; Kerkyacharian et al, 2012; Fan, 2011). These methods are available in the R package fdcov (Cabassi and Kashlak, 2016)
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