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Particle or grain size distributions often play an important role in understanding processes in the geosciences. Functional data analysis allows applying multivariate methods like principal components and discriminant analysis directly to such distributions. These are however often observed in the form of samples, and thus with a sampling error, i.e. each data point is a distribution, but one where the sampling error is present. This additional sampling error changes the properties of the multivariate variance and thus the value, number and direction of the principle components. The result of the principal component analysis becomes an artefact of the sampling error and can negatively affect the following data analysis. Our contribution presents how to compute this sampling error and how to confront it in the context of principal component analysis. We demonstrate the effect of the sampling error and the effectiveness of the correction with a simulated dataset. We show how the interpretability and reproducibility of the principal components improve and become independent of the selection of the basis. We also demonstrate how the correction improves interpretability of the results on a grain size distribution dataset from river sediments.

With large-scale database systems, statistical analysis of data, occurring in the form of probability distributions, becomes an important task in explorative data analysis. Nevertheless, due to specific properties of density functions, their proper statistical treatment of these data still represents a challenging task in functional data analysis. Namely, the usual (Formula presented.) metric does not fully accounts for the relative character of information, carried by density functions; instead, their geometrical features are captured by Bayes spaces of measures. The easiest possibility of expressing density functions in an (Formula presented.) space is to use centred logratio transformation, even though this results in functional data with a constant integral constraint that needs to be taken into account in further analysis. While theoretical background for reasonable analysis of density functions is already provided comprehensively by Bayes spaces themselves, preprocessing issues still need to be developed. The aim of this paper is to introduce optimal smoothing splines for centred logratio transformed density functions that take all their specific features into account and provide a concise methodology for reasonable preprocessing of raw (discretized) distributional observations. Theoretical developments are illustrated with a real-world data set from official statistics and with a simulation study.

This paper analyzes nonlinear contact problems of a large deformed beam on an elastic foundation. The beam model is governed by a nonlinear fourth-order differential equation developed by Gao (1996); while the elastic foundation model is assumed as Winkler’s type. Based on a decomposition method, the nonlinear variational inequality problem is able to be reformed as a min–max problem of a saddle Lagrangian. Therefore, by using mixed finite element method with independent discretization–interpolations for foundation and beam elements, the nonlinear contact problem in continuous space is eventually converted as a nonlinear mixed complementarity problem, which can be solved by combination of interior-point and Newton methods. Applications are illustrated by different boundary conditions. Results show that the nonlinear Gao beam is more stiffer than the Euler–Bernoulli beam.