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We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories. Asymptotic properties for a sample of $n$ subjects are investigated under mild conditions, as $n\to \infty$, and we obtain consistent estimation for the regression function. Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories. A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard $R^2$ to the functional case. The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index.Comment: Published at http://dx.doi.org/10.1214/009053605000000660 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A micro-/nanoscale hierarchical rough structure inspired by the underwater superaerophobicity of fish scales was fabricated by ablation of a silicon surface by a femtosecond laser. The resultant silicon surface showed superhydrophilicity in air and became superaerophobic after immersion in water. Additionally, inspired by the underwater superaerophilicity of lotus leaves, we showed that the polydimethylsiloxane surface after femtosecond laser ablation exhibits superhydrophobicity in air and becomes superaerophilic in water. The underwater superaerophobic surface showed excellent antibubble ability, whereas the underwater superaerophilic surface could absorb and capture air bubbles in a water medium. The experimental results revealed that the in-air superhydrophilic surface generally shows superaerophobicity in water and that the in-air superhydrophobic surface generally shows underwater superaerophilicity. An underwater superaerophobic porous aluminum sheet with through microholes was prepared, and this sheet was able to intercept underwater bubbles and further remove bubbles from water. In contrast, the underwater superaerophilic porous polytetrafluoroethylene sheet could allow the bubbles to pass through the sheet. We believe that these results are highly significant for providing guidance to researchers and engineers for obtaining excellent control of bubbles' behavior on a solid surface in a water medium.

In longitudinal data analysis one frequently encounters non-Gaussian data that are repeatedly collected for a sample of individuals over time. The repeated observations could be binomial, Poisson or of another discrete type or could be continuous. The timings of the repeated measurements are often sparse and irregular. We introduce a latent Gaussian process model for such data, establishing a connection to functional data analysis. The functional methods proposed are non-parametric and computationally straightforward as they do not involve a likelihood. We develop functional principal components analysis for this situation and demonstrate the prediction of individual trajectories from sparse observations. This method can handle missing data and leads to predictions of the functional principal component scores which serve as random effects in this model. These scores can then be used for further statistical analysis, such as inference, regression, discriminant analysis or clustering. We illustrate these non-parametric methods with longitudinal data on primary biliary cirrhosis and show in simulations that they are competitive in comparisons with generalized estimating equations and generalized linear mixed models. Copyright (c) 2008 Royal Statistical Society.

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