## Covariate adjusted functional principal components analysis for longitudinal data.(English)Zbl 1183.62102

Summary: Classical multivariate principal component analysis has been extended to functional data and termed functional principal component analysis (FPCA). Most existing FPCA approaches do not accommodate covariate information, and it is the goal of this paper to develop two methods that do. In the first approach, both the mean and covariance functions depend on the covariate $$Z$$ and time scale $$t$$ while in the second approach only the mean function depends on the covariate $$Z$$. Both new approaches accommodate additional measurement errors and functional data sampled at regular time grids as well as sparse longitudinal data sampled at irregular time grids. The first approach to fully adjust both the mean and covariance functions adapts more to the data but is computationally more intensive than the approach to adjust the covariate effects on the mean function only. We develop general asymptotic theory for both approaches and compare their performance numerically through simulation studies and a data set.

### MSC:

 62H25 Factor analysis and principal components; correspondence analysis 62M15 Inference from stochastic processes and spectral analysis 62G20 Asymptotic properties of nonparametric inference 65C60 Computational problems in statistics (MSC2010)

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