A Kronecker-based Covariance Specification for Spatially Continuous Multivariate Data

We propose a covariance specification for modeling spatially continuous multivariate data. This model is based on a reformulation of Kronecker’s product of covariance matrices for Gaussian random fields. We illustrate the case with the Matérn function used for specifying marginal covariances. The structure holds for other choices of covariance functions with parameters varying in their usual domains, which makes the estimation process more accessible. The reduced computational time and flexible generalization for increasing number of variables, make it an attractive alternative for modelling spatially continuous data. Theoretical results for the likelihood function and the derivatives of the covariance matrix are presented. The proposed model is fitted to the literature’s soil250 dataset, and adequacy measures, forecast errors and estimation times are compared with the ones obtained based on classical models. Furthermore, the model is fitted to the classic meuse dataset to illustrate the model’s flexibility in a four-variate analysis. A simulation study is performed considering different parametric scenarios to evaluate the asymptotic properties of the maximum likelihood estimators. The satisfactory results, its simpler structure and the reduced estimation time make the proposed model a candidate approach for multivariate analysis of spatial data.

An Integrated NMR, LC-DAD-MS, LC-QTOF Metabolomic Characterization of Sartoria hedysaroides: Correlation of Antioxidant and Enzyme Inhibitory Activity with Chemical Composition by Multivariate Data Analysis

Sartoria hedysaroides Boiss and Heldr. (Fabaceae) is an endemic plant of Turkey that has received little scientific consideration so far. In the present study, the chemical profiles of extracts from the aerial part and roots of S. hedysaroides obtained using solvents with different polarities were analyzed combining integrated NMR, LC-DAD-MSn, and LC-QTOF methods. In vitro antioxidant and enzyme inhibitory activities were evaluated, and the results were combined with chemical data using multivariate approaches. Phenolic acids, flavonoids, ellagitannins, and coumarins were identified and quantified in the extracts of aerial part and roots. Methanolic extract of S. hedysaroides aerial part showed the highest phenolic content and the highest antioxidant activity and cupric ion reducing antioxidant capacity. Dichloromethane extract of S. hedysaroides roots showed the highest inhibition of butyryl cholinesterase, while methanolic extract of S. hedysaroides aerial part was the most active tyrosinase inhibitor. Multivariate data analysis allowed us to observe a good correlation between phenolic compounds, especially caffeoylquinic derivatives and flavonoids and the antioxidant activity of extracts. Acetylcholinesterase inhibition was correlated with the presence of caffeoylquinic acids and coumarins. Overall, the present study appraised the biological potential of understudied S. hedysaroides, and provided a comprehensive approach combining metabolomic characterization of plant material and multivariate data analysis for the correlation of chemical data with results from multi-target biological assays.

Discrimination of 11 Malaysian Durian Cultivars Based on Sulfur-Containing Volatiles and Esters Using Multivariate Data Analysis

There are reports documenting the volatile oils of several durian cultivars in Malaysia. However, there is limited information on the rapid discrimination of the durian cultivars based on the composition of the total volatiles and individual volatile compounds. Thus, the present work aims to discriminate 11 Malaysian durian cultivars based on their volatile compositions using multivariate data analysis. Sulfur-containing volatiles are the major volatiles in D175 (Udang Merah), D88 (Darling), D13 (Golden Bun), DXO (D24 Special), D17 (Green Bamboo), D2 (Dato Nina), and D168 (Hajah Hasmah) durian cultivars, while esters are predominant in D99 (Kop Kecil), D24 (Bukit Merah), and D160 (Musang Queen) durian cultivars. D197 (Musang King) cultivar has an almost equal composition of sulfur-containing volatiles and esters. In the ester predominated volatile durian oil, ethyl 2-methylbutanoate and propyl 2-methylbutanoate are the major volatile compounds, while the durian cultivars with predominant sulfur-containing volatiles mainly contain diethyl disulfide, diethyl trisulfide, and 3,5-dimethyl-1,2,4-trithiolane. The durian cultivars were clustered into 8 clusters using principal component analysis, with 3 clusters consisting of 2 cultivars, and with the remaining cultivars clustered individually. The highly sought-after durian cultivars, D160 and D197, were clustered into one. Hierarchal clustering analysis identified the distinct compounds which discriminate every durian cultivar.

Investigation of the Variability of Alkaloids in Buxus sempervirens L. Using Multivariate Data Analysis of LC/MS Profiles

Buxus sempervirens L. is a common ornamental plant in southern and central Europe, and has been used ethopharmacologically against a wide variety of diseases due to it containing nor-triterpene alkaloids of the nor-cycloartane type. Recently, we demonstrated the interesting antiprotozoal potential of some of these compounds. To characterize the temporal variability in the alkaloid profile of two different varieties and their leaves and twigs, 30 different extracts of B. sempervirens were evaluated by Ultra High Performance Liquid Chromatography/positive Mode-Electrospray Ionization Quadrupole Time-of-Flight-Tandem Mass Spectrometry (UHPLC/+ESI-QqTOF-MS/MS). The analytical profiles were thoroughly investigated by various methods of multivariate data analysis (MVDA). A principal component analysis (PCA) model elucidates the seasonal variation in the phytochemical composition of B. sempervirens var. arborescens and suffruticosa along with differences between the varieties. Analysis of a volcano plot illustrated the differences between the two organs, the leaf and twig. Eighteen compounds were highlighted by the models as constituents of the plant characteristic for a season, variety or organ. These compounds were dereplicated based on their chromatographic and +ESI-QqTOF-MS and –MS/MS data. In addition, mass spectral fragmentation pathways for already known alkaloids as well as new natural products could be postulated for the first time. In conclusion, the MVDA models give detailed information on the temporal variability in the alkaloid profile of two different varieties and their organs (leaf vs. twig) of B. sempervirens. Thus, the results of this study allow, e.g., the identification of characteristic compounds for the different varieties, plant organs, seasons, and the optimal harvesting time for the isolation of particular Buxus-alkaloids of interest for subsequent studies.

On New Types of Multivariate Trigonometric Copulas

Copulas are useful functions for modeling multivariate distributions through their univariate marginal distributions and dependence structures. They have a wide range of applications in all fields of science that deal with multivariate data. While there is a plethora of copulas, those based on trigonometric functions, especially in dimensions greater than two, have received much less attention. They are, however, of interest because of the properties of oscillation and periodicity of the trigonometric functions, which can appear in certain models of correlation of natural phenomena. In order to fill this gap, this paper introduces and investigates two new types of “multivariate trigonometric copulas”. Their main theoretical properties are studied, and some perspectives for applications are sketched for future work. In particular, we show that the proposed copulas are symmetric, not associative, with no orthant dependence, and with copula densities that have wide oscillations, which remains an uncommon property in the field. The expressions of their multivariate Spearman’s rho are also determined. Furthermore, the first type of the proposed copulas has the interesting feature of having a multivariate Spearman’s rho equal to 0 for all of the dimensions. Some graphic evidence supports the findings. Some mathematical formulas involving the product of n trigonometric functions may be of independent interest.