Interpretation of wind components as compositional variables

The classical statistical study of the wind speed in the atmospheric surface layer is made generally from the analysis of the three habitual components that perform the wind data, that is, the component W-E, the component S-N and the vertical component, considering these components independent. When the goal of the study of these data is the Aeolian energy, so is when wind is studied from an energetic point of view and the squares of wind components can be considered as compositional variables. To do so, each component has to be divided by the module of the corresponding vector. In this work the theoretical analysis of the components of the wind as compositional data is presented and also the conclusions that can be obtained from the point of view of the practical applications as well as those that can be derived from the application of this technique in different conditions of weather

Assessing the Precision of Compositional Data in a Stratified Double Stage Cluster Sample: Application to the Swiss Earnings Structure Survey

Precision of released figures is not only an important quality feature of official statistics, it is also essential for a good understanding of the data. In this paper we show a case study of how precision could be conveyed if the multivariate nature of data has to be taken into account. In the official release of the Swiss earnings structure survey, the total salary is broken down into several wage components. We follow Aitchison's approach for the analysis of compositional data, which is based on logratios of components. We first present diferent multivariate analyses of the compositional data whereby the wage components are broken down by economic activity classes. Then we propose a number of ways to assess precision

Application of discriminant function analysis and change-point problem in dating volcanic ashes

The application of Discriminant function analysis (DFA) is not a new idea in the study of tephrochrology. In this paper, DFA is applied to compositional datasets of two different types of tephras from Mountain Ruapehu in New Zealand and Mountain Rainier in USA. The canonical variables from the analysis are further investigated with a statistical methodology of change-point problems in order to gain a better understanding of the change in compositional pattern over time. Finally, a special case of segmented regression has been proposed to model both the time of change and the change in pattern. This model can be used to estimate the age for the unknown tephras using Bayesian statistical calibration

Statistical Analysis of Eco-Toxicologic Data of Mammals from a Polluted Area

Several eco-toxicological studies have shown that insectivorous mammals, due to their feeding habits, easily accumulate high amounts of pollutants in relation to other mammal species. To assess the bio-accumulation levels of toxic metals and their in°uence on essential metals, we quantified the concentration of 19 elements (Ca, K, Fe, B, P, S, Na, Al, Zn, Ba, Rb, Sr, Cu, Mn, Hg, Cd, Mo, Cr and Pb) in bones of 105 greater white-toothed shrews (Crocidura russula) from a polluted (Ebro Delta) and a control (Medas Islands) area. Since chemical contents of a bio-indicator are mainly compositional data, conventional statistical analyses currently used in eco-toxicology can give misleading results. Therefore, to improve the interpretation of the data obtained, we used statistical techniques for compositional data analysis to define groups of metals and to evaluate the relationships between them, from an inter-population viewpoint. Hypothesis testing on the adequate balance-coordinates allow us to confirm intuition based hypothesis and some previous results. The main statistical goal was to test equal means of balance-coordinates for the two defined populations. After checking normality, one-way ANOVA or Mann-Whitney tests were carried out for the inter-group balances

Bayesian tools for zero counts in compositional data

The log-ratio methodology makes available powerful tools for analyzing compositional data. Nevertheless, the use of this methodology is only possible for those data sets without null values. Consequently, in those data sets where the zeros are present, a previous treatment becomes necessary. Last advances in the treatment of compositional zeros have been centered especially in the zeros of structural nature and in the rounded zeros. These tools do not contemplate the particular case of count compositional data sets with null values. In this work we deal with \count zeros" and we introduce a treatment based on a mixed Bayesian-multiplicative estimation. We use the Dirichlet probability distribution as a prior and we estimate the posterior probabilities. Then we apply a multiplicative modi¯cation for the non-zero values. We present a case study where this new methodology is applied. Key words: count data, multiplicative replacement, composition, log-ratio analysis

CODAPACK3D. A new version of Compositional Data Package

The statistical analysis of compositional data should be treated using logratios of parts, which are difficult to use correctly in standard statistical packages. For this reason a freeware package, named CoDaPack was created. This software implements most of the basic statistical methods suitable for compositional data. In this paper we describe the new version of the package that now is called CoDaPack3D. It is developed in Visual Basic for applications (associated with Excel©), Visual Basic and Open GL, and it is oriented towards users with a minimum knowledge of computers with the aim at being simple and easy to use. This new version includes new graphical output in 2D and 3D. These outputs could be zoomed and, in 3D, rotated. Also a customization menu is included and outputs could be saved in jpeg format. Also this new version includes an interactive help and all dialog windows have been improved in order to facilitate its use. To use CoDaPack one has to access Excel© and introduce the data in a standard spreadsheet. These should be organized as a matrix where Excel© rows correspond to the observations and columns to the parts. The user executes macros that return numerical or graphical results. There are two kinds of numerical results: new variables and descriptive statistics, and both appear on the same sheet. Graphical output appears in independent windows. In the present version there are 8 menus, with a total of 38 submenus which, after some dialogue, directly call the corresponding macro. The dialogues ask the user to input variables and further parameters needed, as well as where to put these results. The web site http://ima.udg.es/CoDaPack contains this freeware package and only Microsoft Excel© under Microsoft Windows© is required to run the software. Kew words: Compositional data Analysis, Software

A comparison of the alr and ilr transformations for kernel density estimation of compositional data

In a seminal paper, Aitchison and Lauder (1985) introduced classical kernel density estimation techniques in the context of compositional data analysis. Indeed, they gave two options for the choice of the kernel to be used in the kernel estimator. One of these kernels is based on the use the alr transformation on the simplex SD jointly with the normal distribution on RD-1. However, these authors themselves recognized that this method has some deficiencies. A method for overcoming these dificulties based on recent developments for compositional data analysis and multivariate kernel estimation theory, combining the ilr transformation with the use of the normal density with a full bandwidth matrix, was recently proposed in Martín-Fernández, Chacón and Mateu- Figueras (2006). Here we present an extensive simulation study that compares both methods in practice, thus exploring the finite-sample behaviour of both estimators

Studies in stoichiometry with compositional data

Geochemical data that is derived from the whole or partial analysis of various geologic materials represent a composition of mineralogies or solute species. Minerals are composed of structured relationships between cations and anions which, through atomic and molecular forces, keep the elements bound in specific configurations. The chemical compositions of minerals have specific relationships that are governed by these molecular controls. In the case of olivine, there is a well-defined relationship between Mn-Fe-Mg with Si. Balances between the principal elements defining olivine composition and other significant constituents in the composition (Al, Ti) have been defined, resulting in a near-linear relationship between the logarithmic relative proportion of Si versus (MgMnFe) and Mg versus (MnFe), which is typically described but poorly illustrated in the simplex. The present contribution corresponds to ongoing research, which attempts to relate stoichiometry and geochemical data using compositional geometry. We describe here the approach by which stoichiometric relationships based on mineralogical constraints can be accounted for in the space of simplicial coordinates using olivines as an example. Further examples for other mineral types (plagioclases and more complex minerals such as clays) are needed. Issues that remain to be dealt with include the reduction of a bulk chemical composition of a rock comprised of several minerals from which appropriate balances can be used to describe the composition in a realistic mineralogical framework. The overall objective of our research is to answer the question: In the cases where the mineralogy is unknown, are there suitable proxies that can be substituted? Kew words: Aitchison geometry, balances, mineral composition, oxides

Elemental analysis of New Zealand Nephrite Jade (pounamu) for forensic purposes

Pounamu (NZ jade), or nephrite, is a protected mineral in its natural form following the transfer of ownership back to Ngai Tahu under the Ngai Tahu (Pounamu Vesting) Act 1997. Any theft of nephrite is prosecutable under the Crimes Act 1961. Scientific evidence is essential in cases where origin is disputed. A robust method for discrimination of this material through the use of elemental analysis and compositional data analysis is required. Initial studies have characterised the variability within a given nephrite source. This has included investigation of both in situ outcrops and alluvial material. Methods for the discrimination of two geographically close nephrite sources are being developed. Key Words: forensic, jade, nephrite, laser ablation, inductively coupled plasma mass spectrometry, multivariate analysis, elemental analysis, compositional data analysis

CODAMAT: a modern analogue techinque for compositional data

The quantitative estimation of Sea Surface Temperatures from fossils assemblages is a fundamental issue in palaeoclimatic and paleooceanographic investigations. The Modern Analogue Technique, a widely adopted method based on direct comparison of fossil assemblages with modern coretop samples, was revised with the aim of conforming it to compositional data analysis. The new CODAMAT method was developed by adopting the Aitchison metric as distance measure. Modern coretop datasets are characterised by a large amount of zeros. The zero replacement was carried out by adopting a Bayesian approach to the zero replacement, based on a posterior estimation of the parameter of the multinomial distribution. The number of modern analogues from which reconstructing the SST was determined by means of a multiple approach by considering the Proxies correlation matrix, Standardized Residual Sum of Squares and Mean Squared Distance. This new CODAMAT method was applied to the planktonic foraminiferal assemblages of a core recovered in the Tyrrhenian Sea. Kew words: Modern analogues, Aitchison distance, Proxies correlation matrix, Standardized Residual Sum of Squares

Clustering compositional data trajectories

Our essay aims at studying suitable statistical methods for the clustering of compositional data in situations where observations are constituted by trajectories of compositional data, that is, by sequences of composition measurements along a domain. Observed trajectories are known as “functional data” and several methods have been proposed for their analysis. In particular, methods for clustering functional data, known as Functional Cluster Analysis (FCA), have been applied by practitioners and scientists in many fields. To our knowledge, FCA techniques have not been extended to cope with the problem of clustering compositional data trajectories. In order to extend FCA techniques to the analysis of compositional data, FCA clustering techniques have to be adapted by using a suitable compositional algebra. The present work centres on the following question: given a sample of compositional data trajectories, how can we formulate a segmentation procedure giving homogeneous classes? To address this problem we follow the steps described below. First of all we adapt the well-known spline smoothing techniques in order to cope with the smoothing of compositional data trajectories. In fact, an observed curve can be thought of as the sum of a smooth part plus some noise due to measurement errors. Spline smoothing techniques are used to isolate the smooth part of the trajectory: clustering algorithms are then applied to these smooth curves. The second step consists in building suitable metrics for measuring the dissimilarity between trajectories: we propose a metric that accounts for difference in both shape and level, and a metric accounting for differences in shape only. A simulation study is performed in order to evaluate the proposed methodologies, using both hierarchical and partitional clustering algorithm. The quality of the obtained results is assessed by means of several indices

Revisiting the compositional data. Some fundamental questions and new prospects in Archaeometry and Archaeology

In this paper we examine the problem of compositional data from a different starting point. Chemical compositional data, as used in provenance studies on archaeological materials, will be approached from the measurement theory. The results will show, in a very intuitive way that chemical data can only be treated by using the approach developed for compositional data. It will be shown that compositional data analysis is a particular case in projective geometry, when the projective coordinates are in the positive orthant, and they have the properties of logarithmic interval metrics. Moreover, it will be shown that this approach can be extended to a very large number of applications, including shape analysis. This will be exemplified with a case study in architecture of Early Christian churches dated back to the 5th-7th centuries AD

Measurement Quality in Indicators of Compositions: a Compositional Multitrait-Multimethod Approach

Compositional data, also called multiplicative ipsative data, are common in survey research instruments in areas such as time use, budget expenditure and social networks. Compositional data are usually expressed as proportions of a total, whose sum can only be 1. Owing to their constrained nature, statistical analysis in general, and estimation of measurement quality with a confirmatory factor analysis model for multitrait-multimethod (MTMM) designs in particular are challenging tasks. Compositional data are highly non-normal, as they range within the 0-1 interval. One component can only increase if some other(s) decrease, which results in spurious negative correlations among components which cannot be accounted for by the MTMM model parameters. In this article we show how researchers can use the correlated uniqueness model for MTMM designs in order to evaluate measurement quality of compositional indicators. We suggest using the additive log ratio transformation of the data, discuss several approaches to deal with zero components and explain how the interpretation of MTMM designs di ers from the application to standard unconstrained data. We show an illustration of the method on data of social network composition expressed in percentages of partner, family, friends and other members in which we conclude that the faceto-face collection mode is generally superior to the telephone mode, although primacy e ects are higher in the face-to-face mode. Compositions of strong ties (such as partner) are measured with higher quality than those of weaker ties (such as other network members)

Forest Policy Scenario Analysis: Sensitivity of Songbird Community to Changes in Forest Cover Amount and Configuration

Changes in mature forest cover amount, composition, and configuration can be of significant consequence to wildlife populations. The response of wildlife to forest patterns is of concern to forest managers because it lies at the heart of such competing approaches to forest planning as aggregated vs. dispersed harvest block layouts. In this study, we developed a species assessment framework to evaluate the outcomes of forest management scenarios on biodiversity conservation objectives. Scenarios were assessed in the context of a broad range of forest structures and patterns that would be expected to occur under natural disturbance and succession processes. Spatial habitat models were used to predict the effects of varying degrees of mature forest cover amount, composition, and configuration on habitat occupancy for a set of 13 focal songbird species. We used a spatially explicit harvest scheduling program to model forest management options and simulate future forest conditions resulting from alternative forest management scenarios, and used a process-based fire-simulation model to simulate future forest conditions resulting from natural wildfire disturbance. Spatial pattern signatures were derived for both habitat occupancy and forest conditions, and these were placed in the context of the simulated range of natural variation. Strategic policy analyses were set in the context of current Ontario forest management policies. This included use of sequential time-restricted harvest blocks (created for Woodland caribou (Rangifer tarandus) conservation) and delayed harvest areas (created for American marten (Martes americana atrata) conservation). This approach increased the realism of the analysis, but reduced the generality of interpretations. We found that forest management options that create linear strips of old forest deviate the most from simulated natural patterns, and had the greatest negative effects on habitat occupancy, whereas policy options that specify deferment and timing of harvest for large blocks helped ensure the stable presence of an intact mature forest matrix over time. The management scenario that focused on maintaining compositional targets best supported biodiversity objectives by providing the composition patterns required by the 13 focal species, but this scenario may be improved by adding some broad-scale spatial objectives to better maintain large blocks of interior forest habitat through time.

Comparison of pollinators and natural enemies: a meta-analysis of landscape and local effects on abundance and richness in crops

To manage agroecosystems for multiple ecosystem services, we need to know whether the management of one service has positive, negative, or no effects on other services. We do not yet have data on the interactions between pollination and pest-control services. However, we do have data on the distributions of pollinators and natural enemies in agroecosystems. Therefore, we compared these two groups of ecosystem service providers, to see if the management of farms and agricultural landscapes might have similar effects on the abundance and richness of both. In a meta-analysis, we compared 46 studies that sampled bees, predatory beetles, parasitic wasps, and spiders in fields, orchards, or vineyards of food crops. These studies used the proximity or proportion of non-crop or natural habitats in the landscapes surrounding these crops (a measure of landscape complexity), or the proximity or diversity of non-crop plants in the margins of these crops (a measure of local complexity), to explain the abundance or richness of these beneficial arthropods. Compositional complexity at both landscape and local scales had positive effects on both pollinators and natural enemies, but different effects on different taxa. Effects on bees and spiders were significantly positive, but effects on parasitoids and predatory beetles (mostly Carabidae and Staphylinidae) were inconclusive. Landscape complexity had significantly stronger effects on bees than it did on predatory beetles and significantly stronger effects in non-woody rather than in woody crops. Effects on richness were significantly stronger than effects on abundance, but possibly only for spiders. This abundance-richness difference might be caused by differences between generalists and specialists, or between arthropods that depend on non-crop habitats (ecotone species and dispersers) and those that do not (cultural species). We call this the ‘specialist-generalist’ or ‘cultural difference’ mechanism. If complexity has stronger effects on richness than abundance, it might have stronger effects on the stability than the magnitude of these arthropod-mediated ecosystem services. We conclude that some pollinators and natural enemies seem to have compatible responses to complexity, and it might be possible to manage agroecosystems for the benefit of both. However, too few studies have compared the two, and so we cannot yet conclude that there are no negative interactions between pollinators and natural enemies, and no trade-offs between pollination and pest-control services. Therefore, we suggest a framework for future research to bridge these gaps in our knowledge.