A geometric approach to visualization of variability in functional data

We propose a new method for the construction and visualization of boxplot-type displays for functional data. We use a recent functional data analysis framework, based on a representation of functions called square-root slope functions, to decompose observed variation in functional data into three main components: amplitude, phase, and vertical translation. We then construct separate displays for each component, using the geometry and metric of each representation space, based on a novel definition of the median, the two quartiles, and extreme observations. The outlyingness of functional data is a very complex concept. Thus, we propose to identify outliers based on any of the three main components after decomposition. We provide a variety of visualization tools for the proposed boxplot-type displays including surface plots. We evaluate the proposed method using extensive simulations and then focus our attention on three real data applications including exploratory data analysis of sea surface temperature functions, electrocardiogram functions and growth curves

Leaf Morphology, Taxonomy and Geometric Morphometrics: A Simplified Protocol for Beginners

Taxonomy relies greatly on morphology to discriminate groups. Computerized geometric morphometric methods for quantitative shape analysis measure, test and visualize differences in form in a highly effective, reproducible, accurate and statistically powerful way. Plant leaves are commonly used in taxonomic analyses and are particularly suitable to landmark based geometric morphometrics. However, botanists do not yet seem to have taken advantage of this set of methods in their studies as much as zoologists have done. Using free software and an example dataset from two geographical populations of sessile oak leaves, we describe in detailed but simple terms how to: a) compute size and shape variables using Procrustes methods; b) test measurement error and the main levels of variation (population and trees) using a hierachical design; c) estimate the accuracy of group discrimination; d) repeat this estimate after controlling for the effect of size differences on shape (i.e., allometry). Measurement error was completely negligible; individual variation in leaf morphology was large and differences between trees were generally bigger than within trees; differences between the two geographic populations were small in both size and shape; despite a weak allometric trend, controlling for the effect of size on shape slighly increased discrimination accuracy. Procrustes based methods for the analysis of landmarks were highly efficient in measuring the hierarchical structure of differences in leaves and in revealing very small-scale variation. In taxonomy and many other fields of botany and biology, the application of geometric morphometrics contributes to increase scientific rigour in the description of important aspects of the phenotypic dimension of biodiversity. Easy to follow but detailed step by step example studies can promote a more extensive use of these numerical methods, as they provide an introduction to the discipline which, for many biologists, is less intimidating than the often inaccessible specialistic literature

Probabilistic modeling of flood characterizations with parametric and minimum information pair-copula model

This paper highlights the usefulness of the minimum information and parametric pair-copula construction (PCC) to model the joint distribution of flood event properties. Both of these models outperform other standard multivariate copula in modeling multivariate flood data that exhibiting complex patterns of dependence, particularly in the tails. In particular, the minimum information pair-copula model shows greater flexibility and produces better approximation of the joint probability density and corresponding measures have capability for effective hazard assessments. The study demonstrates that any multivariate density can be approximated to any degree of desired precision using minimum information pair-copula model and can be practically used for probabilistic flood hazard assessment

High-dimensional modeling of spatial and spatio-temporal conditional extremes using INLA and the SPDE approach

The conditional extremes framework allows for event-based stochastic modeling of dependent extremes, and has recently been extended to spatial and spatio-temporal settings. After standardizing the marginal distributions and applying an appropriate linear normalization, certain non-stationary Gaussian processes can be used as asymptotically-motivated models for the process conditioned on threshold exceedances at a fixed reference location and time. In this work, we adopt a Bayesian perspective by implementing estimation through the integrated nested Laplace approximation (INLA), allowing for novel and flexible semi-parametric specifications of the Gaussian mean function. By using Gauss-Markov approximations of the Mat\'ern covariance function (known as the Stochastic Partial Differential Equation approach) at a latent stage of the model, likelihood-based inference becomes feasible even with thousands of observed locations. We explain how constraints on the spatial and spatio-temporal Gaussian processes, arising from the conditioning mechanism, can be implemented through the latent variable approach without losing the computationally convenient Markov property. We discuss tools for the comparison of models via their posterior distributions, and illustrate the flexibility of the approach with gridded Red Sea surface temperature data at over 6,000 observed locations. Posterior sampling is exploited to study the probability distribution of cluster functionals of spatial and spatio-temporal extreme episodes

Stochastic processes for graphs, extreme values and their causality: inference, asymptotic theory and applications

This thesis provides some theoretical and practical statistical inference tools for multivariate stochastic processes to better understand the behaviours and properties present in the data. In particular, we focus on the modelling of graphs, that is a family of nodes linked together by a collection of edges, and extreme values, that are values above a high threshold to have their own dynamics compared to the typical behaviour of the process. We develop an ensemble of statistical models, statistical inference methods and their asymptotic study to ensure the good behaviour of estimation schemes in a wide variety of settings. We also devote a chapter to the formulation of a methodology based on pre-existing theory to unveil the causal dependency structure behind high-impact events.Open Acces

Probabilistic models for wind loads and reliability analysis

Probabilistic models and surrogate solution are developed to characterize wind loads acting on the structures and assess the corresponding structural responses of linear systems, respectively. These developments can be regarded as essential building blocks in the prediction of structural reliability subject to extreme wind. In this dissertation, we first review the commonly-used probabilistic models in literature and benchmark these models on a test example to illustrate their properties and examine their advantages and disadvantages. It is shown that the approximations based on these existing models on the extreme estimates exhibit large discrepancies. A new probabilistic model is then proposed to overcome this limitation. The model utilized Markov process whose finite dimensional distribution is characterized in terms of copulas. Finally, an efficient and accurate surrogate model is presented as an alternative to the traditional Monte Carlo method to evaluate the structural responses. The responses are approximated by translation processes whose second-moment properties and marginal distribution are obtained from linear random vibration theory and moment equations. The statements in this dissertation are supported by theoretical arguments and numerical examples

Population Genetic Structure is Unrelated to Shell Shape, Thickness and Organic Content in European Populations of the Soft-Shell Clam Mya Arenaria.

The soft-shell clam Mya arenaria is one of the most ancient invaders of European coasts and is present in many coastal ecosystems, yet little is known about its genetic structure in Europe. We collected 266 samples spanning a latitudinal cline from the Mediterranean to the North Sea and genotyped them at 12 microsatellite loci. In parallel, geometric morphometric analysis of shell outlines was used to test for associations between shell shape, latitude and genotype, and for a selection of shells we measured the thickness and organic content of the granular prismatic (PR), the crossed-lamellar (CL) and the complex crossed-lamellar (CCL) layers. Strong population structure was detected, with Bayesian cluster analysis identifying four groups located in the Mediterranean, Celtic Sea, along the continental coast of the North Sea and in Scotland. Multivariate analysis of shell shape uncovered a significant effect of collection site but no associations with any other variables. Shell thickness did not vary significantly with either latitude or genotype, although PR thickness and calcification were positively associated with latitude, while CCL thickness showed a negative association. Our study provides new insights into the population structure of this species and sheds light on factors influencing shell shape, thickness and microstructure