A compositional data analysis package for R providing multiple approaches

”compositions” is a new R-package for the analysis of compositional and positive data. It contains four classes corresponding to the four different types of compositional and positive geometry (including the Aitchison geometry). It provides means for computation, plotting and high-level multivariate statistical analysis in all four geometries. These geometries are treated in an fully analogous way, based on the principle of working in coordinates, and the object-oriented programming paradigm of R. In this way, called functions automatically select the most appropriate type of analysis as a function of the geometry. The graphical capabilities include ternary diagrams and tetrahedrons, various compositional plots (boxplots, barplots, piecharts) and extensive graphical tools for principal components. Afterwards, ortion and proportion lines, straight lines and ellipses in all geometries can be added to plots. The package is accompanied by a hands-on-introduction, documentation for every function, demos of the graphical capabilities and plenty of usage examples. It allows direct and parallel computation in all four vector spaces and provides the beginner with a copy-and-paste style of data analysis, while letting advanced users keep the functionality and customizability they demand of R, as well as all necessary tools to add own analysis routines. A complete example is included in the appendix

A Factor analysis of hidrochemical composition of Llobregat river basin

Hydrogeological research usually includes some statistical studies devised to elucidate mean background state, characterise relationships among different hydrochemical parameters, and show the influence of human activities. These goals are achieved either by means of a statistical approach or by mixing models between end-members. Compositional data analysis has proved to be effective with the first approach, but there is no commonly accepted solution to the end-member problem in a compositional framework. We present here a possible solution based on factor analysis of compositions illustrated with a case study. We find two factors on the compositional bi-plot fitting two non-centered orthogonal axes to the most representative variables. Each one of these axes defines a subcomposition, grouping those variables that lay nearest to it. With each subcomposition a log-contrast is computed and rewritten as an equilibrium equation. These two factors can be interpreted as the isometric log-ratio coordinates (ilr) of three hidden components, that can be plotted in a ternary diagram. These hidden components might be interpreted as end-members. We have analysed 14 molarities in 31 sampling stations all along the Llobregat River and its tributaries, with a monthly measure during two years. We have obtained a bi-plot with a 57% of explained total variance, from which we have extracted two factors: factor G, reflecting geological background enhanced by potash mining; and factor A, essentially controlled by urban and/or farming wastewater. Graphical representation of these two factors allows us to identify three extreme samples, corresponding to pristine waters, potash mining influence and urban sewage influence. To confirm this, we have available analysis of diffused and widespread point sources identified in the area: springs, potash mining lixiviates, sewage, and fertilisers. Each one of these sources shows a clear link with one of the extreme samples, except fertilisers due to the heterogeneity of their composition. This approach is a useful tool to distinguish end-members, and characterise them, an issue generally difficult to solve. It is worth note that the end-member composition cannot be fully estimated but only characterised through log-ratio relationships among components. Moreover, the influence of each endmember in a given sample must be evaluated in relative terms of the other samples. These limitations are intrinsic to the relative nature of compositional data

Compositional Data Analysis with R

R from http://www.r-project.org/ is ‘GNU S’ – a language and environment for statistical computing and graphics. The environment in which many classical and modern statistical techniques have been implemented, but many are supplied as packages. There are 8 standard packages and many more are available through the cran family of Internet sites http://cran.r-project.org . We started to develop a library of functions in R to support the analysis of mixtures and our goal is a MixeR package for compositional data analysis that provides support for operations on compositions: perturbation and power multiplication, subcomposition with or without residuals, centering of the data, computing Aitchison’s, Euclidean, Bhattacharyya distances, compositional Kullback-Leibler divergence etc. graphical presentation of compositions in ternary diagrams and tetrahedrons with additional features: barycenter, geometric mean of the data set, the percentiles lines, marking and coloring of subsets of the data set, theirs geometric means, notation of individual data in the set . . . dealing with zeros and missing values in compositional data sets with R procedures for simple and multiplicative replacement strategy, the time series analysis of compositional data. We’ll present the current status of MixeR development and illustrate its use on selected data sets

Regression analysis of compositional data when both the dependent variable and independent variable are components

It is well known that regression analyses involving compositional data need special attention because the data are not of full rank. For a regression analysis where both the dependent and independent variable are components we propose a transformation of the components emphasizing their role as dependent and independent variables. A simple linear regression can be performed on the transformed components. The regression line can be depicted in a ternary diagram facilitating the interpretation of the analysis in terms of components. An exemple with time-budgets illustrates the method and the graphical features

When zero doesn't mean it and other geomathematical mischief

There is almost not a case in exploration geology, where the studied data doesn’t includes below detection limits and/or zero values, and since most of the geological data responds to lognormal distributions, these “zero data” represent a mathematical challenge for the interpretation. We need to start by recognizing that there are zero values in geology. For example the amount of quartz in a foyaite (nepheline syenite) is zero, since quartz cannot co-exists with nepheline. Another common essential zero is a North azimuth, however we can always change that zero for the value of 360°. These are known as “Essential zeros”, but what can we do with “Rounded zeros” that are the result of below the detection limit of the equipment? Amalgamation, e.g. adding Na2O and K2O, as total alkalis is a solution, but sometimes we need to differentiate between a sodic and a potassic alteration. Pre-classification into groups requires a good knowledge of the distribution of the data and the geochemical characteristics of the groups which is not always available. Considering the zero values equal to the limit of detection of the used equipment will generate spurious distributions, especially in ternary diagrams. Same situation will occur if we replace the zero values by a small amount using non-parametric or parametric techniques (imputation). The method that we are proposing takes into consideration the well known relationships between some elements. For example, in copper porphyry deposits, there is always a good direct correlation between the copper values and the molybdenum ones, but while copper will always be above the limit of detection, many of the molybdenum values will be “rounded zeros”. So, we will take the lower quartile of the real molybdenum values and establish a regression equation with copper, and then we will estimate the “rounded” zero values of molybdenum by their corresponding copper values. The method could be applied to any type of data, provided we establish first their correlation dependency. One of the main advantages of this method is that we do not obtain a fixed value for the “rounded zeros”, but one that depends on the value of the other variable. Key words: compositional data analysis, treatment of zeros, essential zeros, rounded zeros, correlation dependency

Compositional Time Series: An Application

The composition of the labour force is an important economic factor for a country. Often the changes in proportions of different groups are of interest. I this paper we study a monthly compositional time series from the Swedish Labour Force Survey from 1994 to 2005. Three models are studied: the ILR-transformed series, the ILR-transformation of the compositional differenced series of order 1, and the ILRtransformation of the compositional differenced series of order 12. For each of the three models a VAR-model is fitted based on the data 1994-2003. We predict the time series 15 steps ahead and calculate 95 % prediction regions. The predictions of the three models are compared with actual values using MAD and MSE and the prediction regions are compared graphically in a ternary time series plot. We conclude that the first, and simplest, model possesses the best predictive power of the three models

Compositional evolution with mass transfer in closed systems

Evolution of compositions in time, space, temperature or other covariates is frequent in practice. For instance, the radioactive decomposition of a sample changes its composition with time. Some of the involved isotopes decompose into other isotopes of the sample, thus producing a transfer of mass from some components to other ones, but preserving the total mass present in the system. This evolution is traditionally modelled as a system of ordinary di erential equations of the mass of each component. However, this kind of evolution can be decomposed into a compositional change, expressed in terms of simplicial derivatives, and a mass evolution (constant in this example). A rst result is that the simplicial system of di erential equations is non-linear, despite of some subcompositions behaving linearly. The goal is to study the characteristics of such simplicial systems of di erential equa- tions such as linearity and stability. This is performed extracting the compositional dif ferential equations from the mass equations. Then, simplicial derivatives are expressed in coordinates of the simplex, thus reducing the problem to the standard theory of systems of di erential equations, including stability. The characterisation of stability of these non-linear systems relays on the linearisation of the system of di erential equations at the stationary point, if any. The eigenvelues of the linearised matrix and the associated behaviour of the orbits are the main tools. For a three component system, these orbits can be plotted both in coordinates of the simplex or in a ternary diagram. A characterisation of processes with transfer of mass in closed systems in terms of stability is thus concluded. Two examples are presented for illustration, one of them is a radioactive decay

Mixing compositions and other scales

Theory of compositional data analysis is often focused on the composition only. However in practical applications we often treat a composition together with covariables with some other scale. This contribution systematically gathers and develop statistical tools for this situation. For instance, for the graphical display of the dependence of a composition with a categorical variable, a colored set of ternary diagrams might be a good idea for a first look at the data, but it will fast hide important aspects if the composition has many parts, or it takes extreme values. On the other hand colored scatterplots of ilr components could not be very instructive for the analyst, if the conventional, black-box ilr is used. Thinking on terms of the Euclidean structure of the simplex, we suggest to set up appropriate projections, which on one side show the compositional geometry and on the other side are still comprehensible by a non-expert analyst, readable for all locations and scales of the data. This is e.g. done by defining special balance displays with carefully- selected axes. Following this idea, we need to systematically ask how to display, explore, describe, and test the relation to complementary or explanatory data of categorical, real, ratio or again compositional scales. This contribution shows that it is sufficient to use some basic concepts and very few advanced tools from multivariate statistics (principal covariances, multivariate linear models, trellis or parallel plots, etc.) to build appropriate procedures for all these combinations of scales. This has some fundamental implications in their software implementation, and how might they be taught to analysts not already experts in multivariate analysis

Mechanisms of inverse agonist action at D-2 dopamine receptors

1 Mechanisms of inverse agonist action at the D-2(short) dopamine receptor have been examined. 2 Discrimination of G-protein-coupled and -uncoupled forms of the receptor by inverse agonists was examined in competition ligand-binding studies versus the agonist [H-3]NPA at a concentration labelling both G-protein-coupled and -uncoupled receptors. 3 Competition of inverse agonists versus [H-3] NPA gave data that were fitted best by a two-binding site model in the absence of GTP but by a one-binding site model in the presence of GTP. K-i values were derived from the competition data for binding of the inverse agonists to G-protein-uncoupled and -coupled receptors. K-coupled and K-uncoupled were statistically different for the set of compounds tested ( ANOVA) but the individual values were different in a post hoc test only for (+)-butaclamol. 4 These observations were supported by simulations of these competition experiments according to the extended ternary complex model. 5 Inverse agonist efficacy of the ligands was assessed from their ability to reduce agonist-independent [S-35]GTPγ S binding to varying degrees in concentration-response curves. Inverse agonism by (+)-butaclamol and spiperone occurred at higher potency when GDP was added to assays, whereas the potency of (-)-sulpiride was unaffected. 6 These data show that some inverse agonists ((+)-butaclamol, spiperone) achieve inverse agonism by stabilising the uncoupled form of the receptor at the expense of the coupled form. For other compounds tested, we were unable to define the mechanism.

Mechanisms of agonist action at D-2 dopamine receptors

In this study, we investigated the biochemical mechanisms of agonist action at the G protein-coupled D-2 dopamine receptor expressed in Chinese hamster ovary cells. Stimulation of guanosine 5'-O-(3-[S-35]thio) triphosphate ([S-35]GTPgammaS) binding by full and partial agonists was determined at different concentrations of [S-35]GTPgammaS (0.1 and 10 nM) and in the presence of different concentrations of GDP. At both concentrations of [S-35]GTPgammaS, increasing GDP decreased the [S-35]GTPgammaS binding observed with maximally stimulating concentrations of agonist, with partial agonists exhibiting greater sensitivity to the effects of GDP than full agonists. The relative efficacy of partial agonists was greater at the lower GDP concentrations. Concentration-response experiments were performed for a range of agonists at the two [S-35]GTPgammaS concentrations and with different concentrations of GDP. At 0.1 nM [S-35]GTPgammaS, the potency of both full and partial agonists was dependent on the GDP concentration in the assays. At 10 nM [S-35]GTPgammaS, the potency of full agonists exhibited a greater dependence on the GDP concentration, whereas the potency of partial agonists was virtually independent of GDP. We concluded that at the lower [S-35]GTPgammaS concentration, the rate-determining step in G protein activation is the binding of [S-35]GTPgammaS to the G protein. At the higher [S-35]GTPgammaS concentration, for full agonists, [S-35]GTPgammaS binding remains the slowest step, whereas for partial agonists, another (GDP-independent) step, probably ternary complex breakdown, becomes rate-determining.

Analysis of molecular determinants of affinity and relative efficacy of a series of R- and S-2-(dipropylamino)tetralins at the 5-HT1A serotonin receptor

1 Factors influencing agonist affinity and relative efficacy have been studied for the 5-HT1A serotonin receptor using membranes of CHO cells expressing the human form of the receptor and a series of R-and S-2-(dipropylamino)tetralins (nonhydroxylated and monohydroxylated (5-OH, 6-OH, 7-OH, 8-OH) species). 2 Ligand binding studies were used to determine dissociation constants for agonist binding to the 5HT(1A) receptor: (a) K-i values for agonists were determined in competition versus the binding of the agonist [H-3]-8-OH DPAT. Competition data were all fitted best by a one-binding site model. (b) K-i values for agonists were also determined in competition versus the binding of the antagonist [H-3]-NAD-199. Competition data were all fitted best by a two-binding site model, and agonist affinities for the higher (K-h) and lower affinity (K-1) sites were determined. 3 The ability of the agonists to activate the 5-HT1A receptor was determined using stimulation of [S-35]-GTPgammaS binding. Maximal effects of agonists (E-max) and their potencies (EC50) were determined from concentration/response curves for stimulation of [S-35]-GTPgammaS binding. 4 K-1/K-h determined from ligand binding assays correlated with the relative efficacy (relative Em) of agonists determined in [S-35]-GTPgammaS binding assays. There was also a correlation between K-1/K-h and K-1/EC50 for agonists determined from ligand binding and [S-35]-GTPgammaS binding assays. 5 Simulations of agonist binding and effect data were performed using the Ternary Complex Model in order to assess the use of K-1/K-h for predicting the relative efficacy of agonists. British Journal of Pharmacology (2003) 138, 1129-1139. doi: 10. 1038/sj.bjp.705085.

Equilibrium studies on mixed ligand complex formation of Co(II), Ni(II), Cu(II) and Zn(II) with N-(2-hydroxybenzyl)-L-histidine (H(2)hb-L-his) and typical N,N donor ligands: Crystal structure of [Ni(hb-L-his)(bipyridine)]center dot H2O complex

Equilibrium study on complex formation of Co(II), Ni(II), Cu(II) and Zn(II), hereafter M(II), with the quadridentate (O-, N, O-, N) donor ligand, N-(2-hydroxybenzyl)-L-histidine (H(2)hb-L-his, hereafter H2L), in the absence and in the presence of typical (N, N) donor bidentate ligands, 1,10 phenanthroline(phen), 2, 2'-bipyridine(bipy), ethylenediamine(en), hereafter B, in aqueous solution at 25 +/- 1 degrees C was done at a fixed ionic strength, I = 0.1 mol dm(-3) (NaNO3) by combined pH-metric, UV-Vis and EPR measurements provide evidence for the formation of mononuclear and dinuclear binary and mixed ligand complexes of the types: M(L), M(L)(2)(2-), M-2(L)(2+), M-2(H-1L)(+), M(L)(B), (B)M(H-1L)M(B)(+). The imidazole moiety of the ligand is found to act as a bridging bidentate ligand in the dinuclear M-2(L)(2+), M-2(H-1L)(+) and (B)M(H-1L)M(B)(+) complexes, using its N-3 atom and N1-H deprotonated moiety. Stability constants of the complexes provide evidence of discrimination of Cu(II) from the other M(II) ions by this ligand. Solid complexes: [Ni(L)(H2O)(2)] (1), [Cu(L)(H2O)] (2), and [Ni(L)(bipy)] (.) H2O (3) have been isolated and characterized by various physicochemical studies. Single crystal X-ray diffraction of the ternary complex, 3, shows an octahedral [(O-,N,N,O-)(N,N)] geometry with extensive pi-pi stacking of the aromatic rings and H-bonding with imidazole (N1-H), secondary amino N-atom, the lattice H2O molecule, and the carboxylate and phenolate O-atoms. (c) 2006 Elsevier B.V. All rights reserved.

Self-assembly of two-component gels: stoichiometric control and component selection

Two-component systems capable of self-assembling into soft gel-phase materials are of considerable interest due to their tunability and versatility. This paper investigates two-component gels based on a combination of a L-lysine-based dendron and a rigid diamine spacer (1,4-diaminobenzene or 1,4-diaminocyclohexane). The networked gelator was investigated using thermal measurements, circular dichroism, NMR spectroscopy and small angle neutron scattering (SANS) giving insight into the macroscopic properties, nanostructure and molecular-scale organisation. Surprisingly, all of these techniques confirmed that irrespective of the molar ratio of the components employed, the "solid-like" gel network always consisted of a 1:1 mixture of dendron/diamine. Additionally, the gel network was able to tolerate a significant excess of diamine in the "liquid-like" phase before being disrupted. In the light of this observation, we investigated the ability of the gel network structure to evolve from mixtures of different aromatic diamines present in excess. We found that these two-component gels assembled in a component-selective manner, with the dendron preferentially recognising 1,4-diaminobenzene (>70%). when similar competitor diamines (1,2- and 1,3-diaminobenzene) are present. Furthermore, NMR relaxation measurements demonstrated that the gel based oil 1,4-diaminobenzene was better able to form a selective ternary complex with pyrene than the gel based oil 1,4-diaminocyclohexane, indicative of controlled and selective pi-pi interactions within a three-component assembly. As such, the results ill this paper demonstrate how component selection processes in two-component gel systems call control hierarchical self-assembly.

Synthesis, structure and solution chemistry of quaternary oxovanadium(V) complexes incorporating hydrazone ligands

[VIVO(acac)(2)] reacts with an equimolar amount of benzoyl hydrazone of 2-hydroxyacetophenone (H2L1) or 5-chloro-2-hydroxyacetophenone (H2L2) in the presence of excess pyridine (py) in methanol to produce the quaternary [(VO)-O-V(L-1)(OCH3)(py)] (1) and [(VO)-O-V(L-2)(OCH3)(py)] (2) complexes, respectively, while under similar condition, the benzoyl hydrazones of 2-hydroxy-5-methylacetophenone (H2L3) and 2-hydroxy-5-methoxyacetophenone (H2L4) afforded only the methoxy bridged dimeric [(VO)-O-V(L-3/L-4)(OCH3)](2) complexes. The X-ray structural analysis of 1 and 2 indicates that the geometry around the metal is distorted octahedral where the three equatorial positions are occupied by the phenolate-O, enolate-O and the imine-N of the fully deprotonated hydrazone ligand in its enolic form and the fourth one by a methoxide-O atom. An oxo-O and a pyridine-N atom occupy two axial positions. Quaternary complexes exhibit one quasi-reversible one-electron reduction peak near 0.25 V versus SCE in CH2Cl2 and they decompose appreciably to the corresponding methoxy bridged dimeric complex in CDCl3 solution as indicated by their H-1 NMR spectra. These quaternary VO3+ complexes are converted to the corresponding V2O34+-complexes simply on refluxing them in acetone and to the VO2+-complexes on reaction with KOH in methanol. An equimolar amount of 8-hydroxyquinoline (Hhq) converts these quaternary complexes to the ternary [(VO)-O-V(L)(hq)] complexes in CHCl3. (C) 2009 Elsevier B. V. All rights reserved.

Anion-directed template synthesis and hydrolysis of mono-condensed Schiff base of 1,3-pentanediamine and o-hydroxyacetophenone in Ni-II and Cu-II complexes

Bis(o-hydroxyacetophenone)nickel(II) dihydrate, on reaction with 1,3-pentanediamine, yields a bis-chelate complex [NiL2]center dot 2H(2)O (1) of mono-condensed tridentate Schiff base ligand HL {2-[1-(3-aminopentylimino)ethyl]phenol}. The Schiff base has been freed from the complex by precipitating the Nil, as a dimethylglyoximato complex. HL reacts smoothly with Ni(SCN)(2)center dot 4H(2)O furnishing the complex [NiL(NCS)] (2) and with CuCl2 center dot 2H(2)O in the presence of NaN3 or NH4SCN producing [CuL(N-3)](2) (3) or [CuL(NCS)] (4). On the other hand, upon reaction with Cu(ClO4)(2)center dot 6H(2)O and Cu(NO3)(2)center dot 3H(2)O, the Schiff base undergoes hydrolysis to yield ternary complexes [Cu(hap)(pn)(H2O)]ClO4 (5) and [Cu(hap)(pn)(H2O)]NO3 (6), respectively (Hhap = o-hydroxyacetophenone and pn = 1,3-pentanediamine). The ligand HL undergoes hydrolysis also on reaction with Ni(ClO4)(2)center dot 6H(2)O or Ni(NO3)(2)center dot 6H(2)O to yield [Ni(hap)(2)] (7). The structures of the complexes 2, 3, 5, 6, and 7 have been confirmed by single-crystal X-ray analysis. In complex 2, Ni-II possesses square-planar geometry, being coordinated by the tridentate mono-negative Schiff base, L and the isothiocyanate group. The coordination environment around Cu-II in complex 3 is very similar to that in complex 2 but here two units are joined together by end-on, axial-equatorial azide bridges to result in a dimer in which the geometry around Cu-II is square pyramidal. In both 5 and 6, the Cu-II atoms display the square-pyramidal environment; the equatorial sites being coordinated by the two amine groups of 1,3-pentanediamine and two oxygen atoms of o-hydroxyacetophenone. The axial site is coordinated by a water molecule. Complex 7 is a square-planar complex with the Ni atom bonded to four oxygen atoms from two hap moieties. The mononuclear units of 2 and dinuclear units of 3 are linked by strong hydrogen bonds to form a one-dimensional network. The mononuclear units of 5 and 6 are joined together to form a dimer by very strong hydrogen bonds through the coordinated water molecule. These dimers are further involved in hydrogen bonding with the respective counteranions to form 2-D net-like open frameworks. ((C) Wiley-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2008).