We propose a functional version of extended redundancy analysis that examines directional relationships among several sets of multivariate variables. As in extended redundancy analysis, the proposed method posits that a weighed composite of each set of exogenous variables influences a set of endogenous variables. It further considers endogenous and/or exogenous variables functional, varying over time, space, or other continua. Computationally, the method reduces to minimizing a penalized least-squares criterion through the adoption of a basis function expansion approach to approximating functions. We develop an alternating regularized least-squares algorithm to minimize this criterion. We apply the proposed method to real datasets to illustrate the empirical feasibility of the proposed method.

We propose functional multiple-set canonical correlation analysis for exploring associations among multiple sets of functions. The proposed method includes functional canonical correlation analysis as a special case when only two sets of functions are considered. As in classical multiple-set canonical correlation analysis, computationally, the method solves a matrix eigen-analysis problem through the adoption of a basis expansion approach to approximating data and weight functions. We apply the proposed method to functional magnetic resonance imaging (fMRI) data to identify networks of neural activity that are commonly activated across subjects while carrying out a working memory task.

Functional principal component analysis (FPCA) and functional multiple-set canonical correlation analysis (FMCCA) are data reduction techniques for functional data that are collected in the form of smooth curves or functions over a continuum such as time or space. In FPCA, low-dimensional components are extracted from a single functional dataset such that they explain the most variance of the dataset, whereas in FMCCA, low-dimensional components are obtained from each of multiple functional datasets in such a way that the associations among the components are maximized across the different sets. In this paper, we propose a unified approach to FPCA and FMCCA. The proposed approach subsumes both techniques as special cases. Furthermore, it permits a compromise between the techniques, such that components are obtained from each set of functional data to maximize their associations across different datasets, while accounting for the variance of the data well. We propose a single optimization criterion for the proposed approach, and develop an alternating regularized least squares algorithm to minimize the criterion in combination with basis function approximations to functions. We conduct a simulation study to investigate the performance of the proposed approach based on synthetic data. We also apply the approach for the analysis of multiple-subject functional magnetic resonance imaging data to obtain low-dimensional components of blood-oxygen level-dependent signal changes of the brain over time, which are highly correlated across the subjects as well as representative of the data. The extracted components are used to identify networks of neural activity that are commonly activated across the subjects while carrying out a working memory task.

This paper develops the estimation method of mean and covariance functions of functional data with additional covariate information. With the strength of both local linear smoothing modeling and general weighing scheme, we are able to explicitly characterize the mean and covariance functions with incorporating covariate for irregularly spaced and sparsely observed longitudinal data, as typically encountered in engineering technology or biomedical studies, as well as for functional data which are densely measured. Theoretically, we establish the uniform convergence rates of the estimators in the general weighing scheme. Monte Carlo simulation is conducted to investigate the finite-sample performance of the proposed approach. Two applications including the children growth data and white matter tract dataset obtained from Alzheimer's Disease Neuroimaging Initiative study are also provided.

Many real-world problems require to optimize trajectories under constraints. Classical approaches are often based on optimal control methods but require an exact knowledge of the underlying dynamics and constraints, which could be challenging or even out of reach. In view of this, we leverage data-driven approaches to design a new end-to-end framework which is dynamics-free for optimized and realistic trajectories. Trajectories are here decomposed on function basis, trading the initial infinite dimension problem on a multivariate functional space for a parameter optimization problem. Then a maximum a posteriori approach which incorporates information from data is used to obtain a new penalized optimization problem. The penalized term narrows the search on a region centered on data and includes estimated features of the problem. We apply our data-driven approach to two settings in aeronautics and sailing routes optimization. The developed approach is implemented in the Python library PyRotor.

One of the most fundamental tasks in non-life insurance, done on regular basis, is risk reserving assessment analysis, which amounts to predict stochastically the overall loss reserves to cover possible claims. The most common reserving methods are based on different parametric approaches using aggregated data structured in the run-off triangles. In this paper, we propose a rather non-parametric approach, which handles the underlying loss development triangles as functional profiles and predicts the claim reserve distribution through permutation bootstrap. Three competitive functional-based reserving techniques, each with slightly different scope, are presented; their theoretical and practical advantages – in particular, effortless implementation, robustness against outliers, and wide-range applicability – are discussed. Theoretical justifications of the methods are derived as well. An evaluation of the empirical performance of the designed methods and a full-scale comparison with standard (parametric) reserving techniques are carried on several hundreds of real run-off triangles against the known real loss outcomes. An important objective of the paper is also to promote the idea of natural usefulness of the functional reserving methods among the reserving practitioners.

The aging of population is perhaps the most important problem that developed countries must face in the near future. Dependency can be seen as a consequence of the process of gradual aging. In a health context, this contingency is defined as a lack of autonomy in performing basic activities of daily living that requires the care of another person or significant help. In Europe in general and in Spain in particular, this phenomena represents a problem with economic, political, social and demographic implications. The prevalence of dependency in the population, as well as its intensity and evolution over the course of a person’s life are issues of greatest importance that should be addressed. The aim of this work is the estimation of life expectancy free of dependency (LEFD) based on functional trajectories to enhance the regular estimation of health expectancy. Using information from the Spanish survey EDAD 2008, we estimate the number of years spent free of dependency for disabled people according to gender, dependency degree (moderate, severe, major) and the earlier or later onset of dependency compared to a central trend. The main findings are as follows: first, we show evidence that to estimate LEFD ignoring the information provided by the functional trajectories may lead to non-representative LEFD estimates; second, in general, dependency-free life expectancy is higher for women than for men. However, its intensity is higher in women with later onset on dependency; Third, the loss of autonomy is higher (and more abrupt) in men than in women. Finally, the diversity of patterns observed at later onset of dependency tends to a dependency extreme-pattern in both genders.

We study the estimation of the mean function of a continuous-time stochastic process andits derivatives. The covariance function of the process is assumed to be nonparametric andto satisfy mild smoothness conditions. Assuming that n independent realizationsof the process are observed at a sampling design of size N generated by a positivedensity, we derive the asymptotic bias and variance of the local polynomial estimator asn,Nincrease to infinity. We deduce optimal sampling densities, optimal bandwidths, andpropose a new plug-in bandwidth selection method. We establish the asymptotic performanceof the plug-in bandwidth estimator and we compare, in a simulation study, its performancefor finite sizes n,N to the cross-validation and the optimalbandwidths. A software implementation of the plug-in method is available in the Renvironment.