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The Oxford handbook of functional data analysis
\"As technology progresses, we are able to handle larger and larger datasets. At the same time, monitoring devices such as electronic equipment and sensors (for registering images, temperature, etc.) have become more and more sophisticated. This high-tech revolution offers the opportunity to observe phenomena in an increasingly accurate way by producing statistical units sampled over a finer and finer grid, with the measurement points so close that the data can be considered as observations varying over a continuum. Such continuous (or functional) data may occur in biomechanics (e.g. human movements), chemometrics (e.g. spectrometric curves), econometrics (e.g. the stock market index), geophysics (e.g. spatio-temporal events such as El Nino or time series of satellite images), or medicine (electro-cardiograms/electro-encephalograms). It is well known that standard multivariate statistical analyses fail with functional data. However, the great potential for applications has encouraged new methodologies able to extract relevant information from functional datasets. This Handbook aims to present a state of the art exploration of this high-tech field, by gathering together most of major advances in this area. Leading international experts have contributed to this volume with each chapter giving the key original ideas and comprehensive bibliographical information. The main statistical topics (classification, inference, factor-based analysis, regression modelling, resampling methods, time series, random processes) are covered in the setting of functional data. The twin challenges of the subject are the practical issues of implementing new methodologies and the theoretical techniques needed to expand the mathematical foundations and toolbox. The volume therefore mixes practical, methodological and theoretical aspects of the subject, sometimes within the same chapter. As a consequence, this book should appeal to a wide audience of engineers, practitioners and graduate students, as well as academic researchers, not only in statistics and probability but also in the numerous related application areas\"-- Provided by publisher.
Book
Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin).
We establish the following results:
The key tools behind our results
are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier
transforms for functions such that their Fourier transforms are of monotone type or lacunary series.
eBook
distance‐based framework for measuring functional diversity from multiple traits
A new framework for measuring functional diversity (FD) from multiple traits has recently been proposed. This framework was mostly limited to quantitative traits without missing values and to situations in which there are more species than traits, although the authors had suggested a way to extend their framework to other trait types. The main purpose of this note is to further develop this suggestion. We describe a highly flexible distance‐based framework to measure different facets of FD in multidimensional trait space from any distance or dissimilarity measure, any number of traits, and from different trait types (i.e., quantitative, semi‐quantitative, and qualitative). This new approach allows for missing trait values and the weighting of individual traits. We also present a new multidimensional FD index, called functional dispersion (FDis), which is closely related to Rao's quadratic entropy. FDis is the multivariate analogue of the weighted mean absolute deviation (MAD), in which the weights are species relative abundances. For unweighted presence–absence data, FDis can be used for a formal statistical test of differences in FD. We provide the “FD” R language package to easily implement our distance‐based FD framework.
Journal Article
Methods for Scalar-on-Function Regression
Recent years have seen an explosion of activity in the field of functional data analysis (FDA), in which curves, spectra, images and so on are considered as basic functional data units. A central problem in FDA is how to fit regression models with scalar responses and functional data points as predictors. We review some of the main approaches to this problem, categorising the basic model types as linear, non-linear and non-parametric. We discuss publicly available software packages and illustrate some of the procedures by application to a functional magnetic resonance imaging data set.
Journal Article
mFD: an R package to compute and illustrate the multiple facets of functional diversity
Functional diversity (FD), the diversity of organism attributes that relates to their interactions with the abiotic and biotic environment, has been increasingly used for the last two decades in ecology, biogeography and conservation. Yet, FD has many facets and their estimations are not standardized nor embedded in a single tool. mFD (multifaceted functional diversity) is an R package that uses matrices of species assemblages and species trait values as building blocks to compute most FD indices. mFD is firstly based on two functions allowing the user to summarize trait and assemblage data. Then it calculates trait-based distances between species pairs, informs the user whether species have to be clustered into functional entities and finally computes multidimensional functional space. To let the user choose the most appropriate functional space for computing multidimensional functional diversity indices, two mFD functions allow assessing and illustrating the quality of each functional space. Next, mFD provides 6 core functions to calculate 16 existing FD indices based on trait-based distances, functional entities or species position in a functional space. The mFD package also provides graphical functions based on the ggplot library to illustrate FD values through customizable and high-resolution plots of species distribution among functional entities or in a multidimensional space. All functions include internal validation processes to check for errors in data formatting which return detailed error messages. To facilitate the use of mFD framework, we built an associated website hosting five tutorials illustrating the use of all the functions step by step.
Journal Article
Embeddings of Decomposition Spaces
Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an
embedding between the two?
A decomposition space
We establish readily verifiable criteria which ensure the
existence of a continuous inclusion (“an embedding”)
In a nutshell, in order to apply the embedding results presented in this
article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved
coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings.
These
sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of
We also prove a
The resulting embedding theory is illustrated by applications
to
eBook
Electronic Properties and COsub.2-Selective Adsorption of Clusters: A Density Functional Theory Study
In this study, we investigated the electronic properties and selective adsorption for CO[sub.2] of nickel boride clusters (NiB)[sub.n], (n = 1~10) using the first principles method. We optimized the structures of the clusters and analyzed their stability based on binding energy per atom. It was observed that (NiB)[sub.n] clusters adopt 3D geometries from n = 4, which were more stable compared to the plane clusters. The vertical electron affinity, vertical ionization energy, chemical potential, and highest occupied molecular orbital (HOMO)–lowest unoccupied molecular orbital (LUMO) gap were calculated. Our results revealed that (NiB)[sub.6] and (NiB)[sub.10], with high chemical potential, exhibit a higher affinity for CO[sub.2] adsorption due to a charge delivery channel that forms along the Ni→B→CO[sub.2] path. Notably, (NiB)[sub.10] demonstrated a more practical CO[sub.2] desorption temperature, as well as a broader window for the selective adsorption of CO[sub.2] over N[sub.2]. The density of states analysis showed that the enhanced CO[sub.2] adsorption on (NiB)[sub.10] can be attributed to the synergistic effect between Ni and B, which provides more active sites for CO[sub.2] adsorption and promotes the electron transfer from the surface to the CO[sub.2] molecule. Our theoretical results imply that (NiB)[sub.10] should be a promising candidate for CO[sub.2] capture.
Journal Article
On the Efficiency of the Density Functional Theory Basis Sets
The basis set issue has always been one of the most important factors of accuracy in the quantum chemical calculations of NMR chemical shifts. In a previous paper, we developed new pecS-n (n = 1, 2) basis sets purposed for the calculations of the NMR chemical shifts of the nuclei of the most popular NMR-active isotopes of 1–2 row elements and successfully approbated these on the DFT calculations of chemical shifts in a limited series of small molecules. In this paper, we demonstrate the performance of the pecS-n (n = 1, 2) basis sets on the calculations of as much as 713 [sup.1]H and 767 [sup.13]C chemical shifts of 23 biologically active natural products with complicated stereochemical structures, carried out using the GIAO-DFT(PBE0) approach. We also proposed new alternative contraction schemes for our basis sets characterized by less contraction depth of the p-shell. New contraction coefficients have been optimized with the property-energy consistent (PEC) method. The accuracies of the pecS-n (n = 1, 2) basis sets of both the original and newly contracted forms were assessed on massive benchmark calculations of proton and carbon chemical shifts of a vast variety of natural products. It was found that less contracted pecS-n (n = 1, 2) basis sets provide no noticeable improvement in accuracy. These calculations represent the most austere test of our basis sets as applied to routine calculations of the NMR chemical shifts of real-life compounds.
Journal Article
How many dimensions are needed to accurately assess functional diversity? A pragmatic approach for assessing the quality of functional spaces
Aim: Functional diversity is a key facet of biodiversity that is increasingly being measured to quantify its changes following disturbance and to understand its effects on ecosystem functioning. Assessing the functional diversity of assemblages based on species traits requires the building of a functional space (dendrogram or multidimensional space) where indices will be computed. However, there is still no consensus on the best method for measuring the quality of functional spaces. Innovation: Here we propose a framework for evaluating the quality of a functional space (i.e. the extent to which it is a faithful representation of the initial functional trait values). Using simulated dataseis, we analysed the influence of the number and type of functional traits used and of the number of species studied on the identity and quality of the best functional space. We also tested whether the quality of the functional space affects functional diversity patterns in local assemblages, using simulated datasets and a real study case. Main conclusions: The quality of functional space strongly varied between situations. Spaces having at least four dimensions had the highest quality, while functional dendrograms and two-dimensional functional spaces always had a low quality. Importantly, we showed that using a poor-quality functional space could led to a biased assessment of functional diversity and false ecological conclusions. Therefore, we advise a pragmatic approach consisting of computing all the possible functional spaces and selecting the most parsimonious one.
Journal Article
A Systemic Insight into Exohedral Actinides and Endohedral Borospherenes: AnBsub.m and An@Bsub.n
A series of exohedral actinide borospherenes, An&B[sub.m], and endohedral borospherenes, An@B[sub.n] (An=U, Np, Pu; m = 28, 32, 34, 36, 38, 40; n = 36, 38, 40), have been characterized by density functional theory calculations. The electronic structures, chemical bond topological properties and spectra have been systematically investigated. It was found that An@B[sub.n] is more stable than An&B[sub.n] in terms of structure and energy, and UB[sub.36] in an aqueous solution is the most stable molecular in this research. The IR and UV-vis spectra of An&B[sub.m] and An@B[sub.n] are computationally predicted to facilitate further experimental investigations. Charge-transfer spectroscopy decomposes the total UV-Vis absorption curve into the contributions of different excitation features, allowing insight into what form of electronic excitation the UV–Vis absorption peak is from the perspective of charge transfer between the An atoms and borospherenes.
Journal Article