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To reduce the dimensionality of the functional covariate, functional principal component analysis plays a key role, however, there is uncertainty on the number of principal components. Model averaging addresses this uncertainty by taking a weighted average of the prediction obtained from a set of candidate models. In this paper, we develop an optimal model averaging approach that selects the weights by minimizing a $K$-fold cross-validation criterion. We prove the asymptotic optimality of the selected weights in terms of minimizing the excess final prediction error, which greatly improves the usual asymptotic optimality in terms of minimizing the final prediction error in the literature. When the true regression relationship belongs to the set of candidate models, we provide the consistency of the averaged estimators. Numerical studies indicate that in most cases the proposed method performs better than other model selection and averaging methods, especially for extreme quantiles., Comment: Any comments are welcome

With the advance of modern technology, more and more data are being recorded continuously during a time interval or intermittently at several discrete time points. They are both examples of "functional data", which have become a prevailing type of data. Functional Data Analysis (FDA) encompasses the statistical methodology for such data. Broadly interpreted, FDA deals with the analysis and theory of data that are in the form of functions. This paper provides an overview of FDA, starting with simple statistical notions such as mean and covariance functions, then covering some core techniques, the most popular of which is Functional Principal Component Analysis (FPCA). FPCA is an important dimension reduction tool and in sparse data situations can be used to impute functional data that are sparsely observed. Other dimension reduction approaches are also discussed. In addition, we review another core technique, functional linear regression, as well as clustering and classification of functional data. Beyond linear and single or multiple index methods we touch upon a few nonlinear approaches that are promising for certain applications. They include additive and other nonlinear functional regression models, such as time warping, manifold learning, and dynamic modeling with empirical differential equations. The paper concludes with a brief discussion of future directions., Comment: 47 pages

Motivated by the need for accurate traffic flow prediction in transportation management, we propose a functional data method to analyze traffic flow patterns and predict future traffic flow. In this study we approach the problem by sampling traffic flow trajectories from a mixture of stochastic processes. The proposed functional mixture prediction approach combines functional prediction with probabilistic functional classification to take distinct traffic flow patterns into account. The probabilistic classification procedure, which incorporates functional clustering and discrimination, hinges on subspace projection. The proposed methods not only assist in predicting traffic flow trajectories, but also identify distinct patterns in daily traffic flow of typical temporal trends and variabilities. The proposed methodology is widely applicable in analysis and prediction of longitudinally recorded functional data., Comment: Published in at http://dx.doi.org/10.1214/12-AOAS595 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)