Clustering compositional data trajectories

Our essay aims at studying suitable statistical methods for the clustering ofcompositional data in situations where observations are constituted by trajectories ofcompositional data, that is, by sequences of composition measurements along a domain.Observed trajectories are known as “functional data” and several methods have beenproposed for their analysis.In particular, methods for clustering functional data, known as Functional ClusterAnalysis (FCA), have been applied by practitioners and scientists in many fields. To ourknowledge, FCA techniques have not been extended to cope with the problem ofclustering compositional data trajectories. In order to extend FCA techniques to theanalysis of compositional data, FCA clustering techniques have to be adapted by using asuitable compositional algebra.The present work centres on the following question: given a sample of compositionaldata trajectories, how can we formulate a segmentation procedure giving homogeneousclasses? 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 withthe smoothing of compositional data trajectories. In fact, an observed curve can bethought 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 dissimilaritybetween trajectories: we propose a metric that accounts for difference in both shape andlevel, and a metric accounting for differences in shape only.A simulation study is performed in order to evaluate the proposed methodologies, usingboth hierarchical and partitional clustering algorithm. The quality of the obtained resultsis assessed by means of several indices

Possible solution of some essential zero problems in compositional data analysis

One of the tantalising remaining problems in compositional data analysis lies in how to deal with data sets in which there are components which are essential zeros. By anessential zero we mean a component which is truly zero, not something recorded as zero simply because the experimental design or the measuring instrument has not been sufficiently sensitive to detect a trace of the part. Such essential zeros occur inmany compositional situations, such as household budget patterns, time budgets,palaeontological zonation studies, ecological abundance studies. Devices such as nonzero replacement and amalgamation are almost invariably ad hoc and unsuccessful insuch situations. From consideration of such examples it seems sensible to build up amodel in two stages, the first determining where the zeros will occur and the secondhow the unit available is distributed among the non-zero parts. In this paper we suggest two such models, an independent binomial conditional logistic normal model and a hierarchical dependent binomial conditional logistic normal model. The compositional data in such modelling consist of an incidence matrix and a conditional compositional matrix. Interesting statistical problems arise, such as the question of estimability of parameters, the nature of the computational process for the estimation of both the incidence and compositional parameters caused by the complexity of the subcompositional structure, the formation of meaningful hypotheses, and the devising of suitable testing methodology within a lattice of such essential zero-compositional hypotheses. The methodology is illustrated by application to both simulated and real compositional data

Monitoring procedures in environmental geochemistry and compositional data analysis theory

First discussion on compositional data analysis is attributable to Karl Pearson, in 1897. However, notwithstanding the recent developments on algebraic structure of the simplex, more than twenty years after Aitchison’s idea of log-transformations of closed data, scientific literature is again full of statistical treatments of this type of data by using traditional methodologies. This is particularly true in environmental geochemistry where besides the problem of the closure, the spatial structure (dependence) of the data have to be considered. In this work we propose the use of log-contrast values, obtained by asimplicial principal component analysis, as LQGLFDWRUV of given environmental conditions. The investigation of the log-constrast frequency distributions allows pointing out the statistical laws able togenerate the values and to govern their variability. The changes, if compared, for example, with the mean values of the random variables assumed as models, or other reference parameters, allow definingmonitors to be used to assess the extent of possible environmental contamination. Case study on running and ground waters from Chiavenna Valley (Northern Italy) by using Na+, K+, Ca2+, Mg2+, HCO3-, SO4 2- and Cl- concentrations will be illustrated

Quantifying rock fabrics: a test of autocorrelation of the spatial distribution of cristals

A novel test of spatial independence of the distribution of crystals or phases in rocksbased on compositional statistics is introduced. It improves and generalizes the commonjoins-count statistics known from map analysis in geographic information systems.Assigning phases independently to objects in RD is modelled by a single-trial multinomialrandom function Z(x), where the probabilities of phases add to one and areexplicitly modelled as compositions in the K-part simplex SK. Thus, apparent inconsistenciesof the tests based on the conventional joins{count statistics and their possiblycontradictory interpretations are avoided. In practical applications we assume that theprobabilities of phases do not depend on the location but are identical everywhere inthe domain of de nition. Thus, the model involves the sum of r independent identicalmultinomial distributed 1-trial random variables which is an r-trial multinomialdistributed random variable. The probabilities of the distribution of the r counts canbe considered as a composition in the Q-part simplex SQ. They span the so calledHardy-Weinberg manifold H that is proved to be a K-1-affine subspace of SQ. This isa generalisation of the well-known Hardy-Weinberg law of genetics. If the assignmentof phases accounts for some kind of spatial dependence, then the r-trial probabilitiesdo not remain on H. This suggests the use of the Aitchison distance between observedprobabilities to H to test dependence. Moreover, when there is a spatial uctuation ofthe multinomial probabilities, the observed r-trial probabilities move on H. This shiftcan be used as to check for these uctuations. A practical procedure and an algorithmto perform the test have been developed. Some cases applied to simulated and realdata are presented.Key words: Spatial distribution of crystals in rocks, spatial distribution of phases,joins-count statistics, multinomial distribution, Hardy-Weinberg law, Hardy-Weinbergmanifold, Aitchison geometry

Geometry-based demosaicking

Demosaicking is a particular case of interpolation problems where, from a scalar image in which each pixel has either the red, the green or the blue component, we want to interpolate the full-color image. State-of-the-art demosaicking algorithms perform interpolation along edges, but these edges are estimated locally. We propose a level-set-based geometric method to estimate image edges, inspired by the image in-painting literature. This method has a time complexity of O(S) , where S is the number of pixels in the image, and compares favorably with the state-of-the-art algorithms both visually and in most relevant image quality measures.

Some last thoughts on compositional data analysis

One of the disadvantages of old age is that there is more past than future: this,however, may be turned into an advantage if the wealth of experience and, hopefully,wisdom gained in the past can be reflected upon and throw some light on possiblefuture trends. To an extent, then, this talk is necessarily personal, certainly nostalgic,but also self critical and inquisitive about our understanding of the discipline ofstatistics. A number of almost philosophical themes will run through the talk: searchfor appropriate modelling in relation to the real problem envisaged, emphasis onsensible balances between simplicity and complexity, the relative roles of theory andpractice, the nature of communication of inferential ideas to the statistical layman, theinter-related roles of teaching, consultation and research. A list of keywords might be:identification of sample space and its mathematical structure, choices betweentransform and stay, the role of parametric modelling, the role of a sample spacemetric, the underused hypothesis lattice, the nature of compositional change,particularly in relation to the modelling of processes. While the main theme will berelevance to compositional data analysis we shall point to substantial implications forgeneral multivariate analysis arising from experience of the development ofcompositional data analysis…

Compositional Data in Biomedical Research

Modern methods of compositional data analysis are not well known in biomedical research.Moreover, there appear to be few mathematical and statistical researchersworking on compositional biomedical problems. Like the earth and environmental sciences,biomedicine has many problems in which the relevant scienti c information isencoded in the relative abundance of key species or categories. I introduce three problemsin cancer research in which analysis of compositions plays an important role. Theproblems involve 1) the classi cation of serum proteomic pro les for early detection oflung cancer, 2) inference of the relative amounts of di erent tissue types in a diagnostictumor biopsy, and 3) the subcellular localization of the BRCA1 protein, and it'srole in breast cancer patient prognosis. For each of these problems I outline a partialsolution. However, none of these problems is \solved". I attempt to identify areas inwhich additional statistical development is needed with the hope of encouraging morecompositional data analysts to become involved in biomedical research

On some geometry and equivalence classes of normal form games

Equivalence classes of normal form games are defined using the geometryof correspondences of standard equilibiurm concepts like correlated, Nash,and robust equilibrium or risk dominance and rationalizability. Resultingequivalence classes are fully characterized and compared across differentequilibrium concepts for 2 x 2 games. It is argued that the procedure canlead to broad and game-theoretically meaningful distinctions of games aswell as to alternative ways of viewing and testing equilibrium concepts.Larger games are also briefly considered.

Distributional equivalence and subcompositional coherence in the analysis of contingency tables, ratio-scale measurements and compositional data

We consider two fundamental properties in the analysis of two-way tables of positive data: the principle of distributional equivalence, one of the cornerstones of correspondence analysis of contingency tables, and the principle of subcompositional coherence, which forms the basis of compositional data analysis. For an analysis to be subcompositionally coherent, it suffices to analyse the ratios of the data values. The usual approach to dimension reduction in compositional data analysis is to perform principal component analysis on the logarithms of ratios, but this method does not obey the principle of distributional equivalence. We show that by introducing weights for the rows and columns, the method achieves this desirable property. This weighted log-ratio analysis is theoretically equivalent to spectral mapping , a multivariate method developed almost 30 years ago for displaying ratio-scale data from biological activity spectra. The close relationship between spectral mapping and correspondence analysis is also explained, as well as their connection with association modelling. The weighted log-ratio methodology is applied here to frequency data in linguistics and to chemical compositional data in archaeology.

Dynamic graphics of parametrically linked multivariate methods used in compositional data analysis

Many multivariate methods that are apparently distinct can be linked by introducing oneor more parameters in their definition. Methods that can be linked in this way arecorrespondence analysis, unweighted or weighted logratio analysis (the latter alsoknown as "spectral mapping"), nonsymmetric correspondence analysis, principalcomponent analysis (with and without logarithmic transformation of the data) andmultidimensional scaling. In this presentation I will show how several of thesemethods, which are frequently used in compositional data analysis, may be linkedthrough parametrizations such as power transformations, linear transformations andconvex linear combinations. Since the methods of interest here all lead to visual mapsof data, a "movie" can be made where where the linking parameter is allowed to vary insmall steps: the results are recalculated "frame by frame" and one can see the smoothchange from one method to another. Several of these "movies" will be shown, giving adeeper insight into the similarities and differences between these methods.

Geometry, petrology and growth of a shallow crustal laccolith : the Torres del Paine Mafic Complex (Patagonia)

1. ABSTRACTS - RÉSUMÉSSCIENTIFIC ABSTRACT - ENGLISH VERSIONGeometry, petrology and growth of a shallow crustal laccolith: the Torres del Paine Mafi c Complex (Patagonia)The Torres del Paine intrusive complex (TPIC) is a composite mafic-granitic intrusion, ~70km2, belonging to a chain of isolated Miocene plutons in southern Patagonia. Their position is intermediate between the Mesozoic-Cenozoic calc-alkaline subduction related Patagonian batholith in the West and the late Cenozoic alkaline basaltic back-arc related plateau lavas in the East. The Torres del Paine complex formed during an important reconfiguration of the Patagonian geodynamic setting, with a migration of magmatism from the arc to the back-arc, possibly related to the Chile ridge subductionThe complex intruded the flysch of the Cretaceous Cerro Toro and Punta Barrosa Formations during the Miocene, creating a well-defined narrow contact aureole of 200-400 m width.In its eastern part, the Torres del Paine intrusive complex is a laccolith, composed of a succession of hornblende-gabbro to diorite sills at its base, with a total thickness of ~250m, showing brittle contacts with the overlying granitic sills, that form spectacular cliffs of more than 1000m. This laccolith is connected, in the western part, to its feeding system, with vertical alternating sheets of layered gabbronorite and Hbl-gabbro, surrounded and percolated by diorites. ID-TIMS U-Pb on zircons on feeder zone (FZ) gab- bros yield 12.593±0.009Ma and 12.587±0.009Ma, which is identifcal within error to the oldest granite dated so far by Michel et al. (2008). In contrast, the laccolith mafic complex is younger than than the youngest granite (12.50±0.02Ma), and has been emplaced from 12.472±0.009Ma to 12.431 ±0.006Ma, by under-accretion beneath the youngest granite at the interface with previously emplaced mafic sills.The gabbronorite crystallization sequence in the feeder zone is dominated by olivine, plagioclase, clinopyroxene and orthopyroxene, while amphibole forms late interstitial crystals. The crystallization sequence is identical in Hornblende-gabbro from the feeder zone, with higher modal hornblende. Gabbronorite and Hornblende-gabbro both display distinct Eu and Sr positive anomalies. In the laccolith, a lower Hornblende-gabbro crystallized in sills and evolved to a high alkali shoshonitic series. The Al203, Ti02, Na20, K20, Ba and Sr composition of these gabbros is highly variable and increases up to ~50wt% Si02. The lower hornblende-gabbro is characterized by kaersutite anhedral cores with inclusions of olivine, clino- and orthopyroxene and rare apatite and An70 plagioclase. Trace element modelling indicates that hornblende and clinopyroxene are in equilibrium with a liquid whose composition is similar to late basaltic trachyandesitic dikes that cut the complex. The matrix in the lower hornblende gabbro is composed of normally zoned oligoclase, Magnesio-hornblende, biotite, ilmenite and rare quartz and potassium feldspar. This assemblage crystallized in-situ from a Ba and Sr-depleted melts. In contrast, the upper Hbl-gabbro is high-K calc-alkaline. Poikilitic pargasite cores have inclusions of euhedral An70 plagioclase inclusions, and contain occasionally clinopyroxene, olivine and orthopyroxene. The matrix composition is identical to the lower hornblende-gabbro and similar to the diorite. Diorite bulk rock compositions show the same mineralogy but different modal proportions relative to hornblende-gabbrosThe Torres del Paine Intrusive Complex isotopic composition is 87Sr/86Sr=0.704, 143Nd/144Nd=0.5127, 206Pb/204Pb=18.70 and 207Pb/204Pb=15.65. Differentiated dioritic and granitic units may be linked to the gabbroic cumulates series, with 20-50% trapped interstitial melt, through fractionation of olivine-bearing gabbronorite or hornblende-gabbro fractionation The relative homogeneity of the isotopic compositions indicate that only small amounts of assimilation occurred. Two-pyroxenes thermometry, clinopyroxene barometry and amphibole-plagioclase thermometry was used to estimate pressure and temperature conditions. The early fractionation of ultramafic cumulates occurs at mid to lower crustal conditions, at temperatures exceeding 900°C. In contrast, the TPIC emplacement conditions have been estimated to ~0.7±0.5kbar and 790±60°C.Based on field and microtextural observations and geochemical modelling, fractionation of basaltic-trachyandesitic liquids at intermediate to lower crustal levels, has led to the formation of the Torres del Paine granites. Repetitive replenishment of basaltic trachy- andesitic liquid in crustal reservoirs led to mixed magmas that will ascend via the feeder zone, and crystallize into a laccolith, in the form of successive dioritic and gabbroic sills. Dynamic fractionation during emplacement concentrated hornblende rich cumulates in the center of individual sills. Variable degrees.of post-emplacement compaction led to the expulsion of felsic liquids that preferentially concentrated at the top of the sills. Incremental sills amalgamation of the entire Torres del Paine Intrusive Complex has lasted for ~160ka.RESUME SCIENTIFIQUE - VERSION FRANÇAISEGéométrie, pétrologie et croissance d'un laccolite peu profond : Le complexe ma- fique du Torres del Paine (Patagonie)Le Complexe Intrusif du Torres del Paine (CITP) est une intrusion bimodale, d'environ 70km2, appartenant à une chaîne de plutons Miocènes isolés, dans le sud de la Patago-nie. Leur position est intermédiaire entre le batholite patagonien calco-alcalin, à l'Ouest, mis en place au Mesozoïque-Cenozoïque dans un contexte de subduction, et les basal-tes andésitiques et trachybasaltes alcalins de plateau, plus jeune, à l'Est, lié à l'ouverture d'un arrière-arc.A son extrémité Est, le CITP est une succession de sills de gabbro à Hbl et de diorite, sur une épaisseur de ~250m, avec des évidences de mélange. Les contacts avec les sills de granite au-dessus, formant des parois de plus de 1000m, sont cassants. Ce laccolite est connecté, dans sa partie Ouest, à une zone d'alimentation, avec des intrusions sub-ver- ticales de gabbronorite litée et de gabbro à Hbl, en alternance. Celles-ci sont traversées et entourées par des diorites. Les zircons des gabbros de la zone d'alimentation, datés par ID-TIMS, ont cristallisés à 12.593±0.009Ma et 12.587±0.009Ma, ce qui correspond au plus vieux granite daté à ce jour par Michel et al. (2008). A l'inverse, les roches manques du laccolite se sont mises en place entre 12.472±0.009Ma et 12.431 ±0.006Ma, par sous-plaquage successifs à l'interface avec le granite le plus jeune daté à ce jour (12.50±0.02Ma).La séquence de cristallisation des gabbronorites est dominée par Ol, Plg, Cpx et Opx, alors que la Hbl est un cristal interstitiel. Elle est identique dans les gabbros à Hbl de la zone d'alimentation, avec ~30%vol de Hbl. Les gabbros de la zone d'alimentation montrent des anomalies positives en Eu et Sr distinctes. Dans le laccolite, le gabbro à Hbl inférieur évolue le long d'une série shoshonitique, riche en éléments incompatibles. Sa concentration en Al203, Ti02, Na20, K20, Ba et Sr est très variable et augmente rapide-ment jusqu'à ~50wt% Si02. Il est caractérisé par la présence de coeurs résorbés de kaer- sutite, entourés de Bt, et contenant des inclusions d'OI, Cpx et Opx, ou alors d'Ap et de rares Plg (An70). Hbl et Cpx ont cristallisés à partir d'un liquide de composition similaire aux dykes trachy-andesite basaltique du CITP. La matrice, cristallisée in-situ à partir d'un liquide pauvre en Ba et Sr, est composée d'oligoclase zoné de façon simple, de Mg-Hbl, Bt, llm ainsi que de rares Qtz et KF. Le gabbro à Hbl supérieur, quant à lui, appartient à une suite chimique calco-alcaline riche en K. Des coeurs poecilitiques de pargasite con-tiennent de nombreuses inclusions de Plg (An70) automorphe, ainsi que des Ol, Cpx et Opx. La composition de la matrice est identique à celle des gabbros à Hbl inférieurs et toutes deux sont similaires à la minéralogie des diorites. Les analyses sur roches totales de diorites montrent la même variabilité que celles de gabbros à Hbl, mais avec une ten-eur en Si02 plus élevée.La composition isotopique des liquides primitifs du CITP a été mesurée à 87Sr/86Sr=0.704, 143Nd/144Nd=0.5127, 206Pb/204Pb=18.70 et 207Pb/204Pb=15.65. Les granites et diorites différenciés peuvent être reliés à des cumulais gabbronoritiques (F=0.74 pour les granites et F=1-0.5 pour les diorites) et gabbroïques à Hbl (fractionnement supplémentaire pour les granites, avec F=0.3). La cristallisation de 20 à 50%vol de liquide interstitiel piégé dans les gabbros du CITP explique leur signature géochimique. Seules de faibles quantités de croûte continentale ont été assimilées. La température et la pression de fractionnement ont été estimées, sur la base des thermobaromètres Opx-Cpx, Hbl-Plg et Cpx, à plus de 900°C et une profondeur correspondant à la croûte inférieure-moyenne. A l'inverse, les conditions de cristallisation de la matrice des gabbros et diorites du laccolite ont été estimées à 790±60°C et ~0.7±0.5kbar.Je propose que les liquides felsiques du CITP se soient formés par cristallisation frac-tionnée en profondeur des assemblages minéralogiques observés dans les gabbros du CITP, à partir d'un liquide trachy-andesite basaltique. La percolation de magma dans les cristaux accumulés permet la remontée du mélange à travers la zone d'alimentation, vers le laccolite, où des sills se mettent en place successivement. L'amalgamation de sills dans le CITP a duré ~160ka.Le CITP s'est formé durant une reconfiguration importante du contexte géodynamique en Patagonie, avec un changement du magmatisme d'arc vers un volcanisme d'arrière- arc. Ce changement est certainement lié à la subduction de la ride du Chili.RESUME GRAND PUBLIC - VERSION FRANÇAISEGéométrie, pétrologie et croissance d'une chambre magmatique peu profonde : Le complexe mafique du Torres del Paine (Patagonie)Le pourtour de l'Océan Pacifique est caractérisé par une zone de convergence de plaques tectoniques, appelée zone de subduction, avec le plongement de croûte océa-nique sous les Andes dans le cas de la Patagonie. De nombreux volcans y sont associés, formant la ceinture de feu. Mais seuls quelques pourcents de tout le magma traversant la croûte terrestre parviennent à la surface et la majeure partie cristallise en profondeur, dans des chambres magmatiques. Quelles est leur forme, croissance, cristallisation et durée de vie ? Le complexe magmatique du Torres del Paine représente l'un des meilleurs endroits au monde pour répondre à ces questions. Il se situe au sud de la Patagonie, formant un massif de 70km2. Des réponses peuvent être trouvées à différentes échelles, variant de la montagne à des minéraux de quelques 1000ème de millimètres.Il est possible de distinguer trois types de roches : des gabbros et des diorites sur une épaisseur de 250m, surmontées par des parois de granite de plus de 1000m. Les contacts entre ces roches sont tous horizontaux. Entre granites et gabbro-diorite, le contact est net, indiquant que le second magma s'est mis en place au contact avec un magma plus ancien, totalement solidifié. Entre gabbros et diorites, les contacts sont diffus, souvent non-linéaires, indiquant à l'inverse la mise en contact de magmas encore partiellement liquides. Dans la partie Ouest de cette chambre magmatique, les contacts entre roches sont verticaux. Il s'agit certainement du lieu de remplissage de la chambre magmatique.Lors du refroidissement d'un magma, différents cristaux vont se former. Leur stabilité et leur composition varient en fonction de la pression, de la température ou de la chimie du magma. La séquence de cristallisation peut être définie sur la base d'observations microscopiques et de la composition chimique des minéraux. Différents gabbros sont ainsi distingués : le gabbro à la base est riche en hornblende, d'une taille de ~5mm, sans inclusion de plagioclase mais avec des cristaux d'olivine, clinopyroxene et orthopyroxene inclus ; le gabbro supérieur est lui-aussi riche en hornblende (~5mm), avec les mêmes inclusions additionnées de plagioclase. Ces cristaux se sont formés à une température supérieure à 900°C et une profondeur correspondant à la croûte moyenne ou inférieure. Les minéraux plus fin, se trouvant hors des cristaux de hornblende des deux gabbros, sont similaires à ceux des diorites : plagioclase, biotite, hornblende, apatite, quartz et feldspath alcalin. Ces minéraux sont caractéristiques des granites. Ils ont cristallisé à ~790°C et ~2km de profondeur.La cristallisation des minéraux et leur extraction du magma par gravité provoque un changement progressif de la composition de ce dernier. Ainsi, après extraction d'olivine et d'orthopyroxene riches en Mg, de clinopyroxene riche en Ca, de plagioclase riche en Ca et Al et d'hornblende riche en Ca, Al et Mg, le liquide final sera appauvri en ces élé-ments. Un lien peut ainsi être proposé entre les diorites dont la composition est proche du liquide de départ, les granites dont la composition est similaire au liquide final, et les gabbros dont la minéralogie correspond aux minéraux extraits.L'utilisation de zircons, un minéral riche en U dont les atomes se transforment en Pb par décomposition radioactive au cours de millions d'années, permet de dater le refroidissement des roches qui les contiennent. Ainsi, il a été observé que les roches de la zone d'alimentation, à l'Ouest du complexe magmatique, ont cristallisés il y a 12.59±0.01 Ma, en même temps que les granites les plus vieux, se trouvant au sommet de la chambre magmatique, datés par Michel et al. (2008). Les deux roches pourraient donc avoir la même origine. A l'inverse, les gabbros et diorites de la chambre magmatique ont cristallisé entre 12.47±0.01Ma et 12.43±0.01Ma, les roches les plus vieilles étant à la base.En comparant la composition des roches du Torres del Paine avec celles d'autres en-tités géologiques de Patagonie, les causes du magmatisme peuvent être recherchées. A l'Ouest, on trouve en effet des intrusions granitiques, plus anciennes, caractéristiques de zones de convergence de plaque tectonique, alors qu'à l'Est, des laves basaltiques plus jeunes sont caractéristiques d'une dynamique d'extension. Sur la base des compositions chimiques des roches de ces différentes entités, l'évolution progressive de l'une à l'autre a pu être démontrée. Elle est certainement due à l'arrivée d'une dorsale océanique (zone d'extension crustale et de création de croûte océanique par la remontée de magma) dans la zone de subduction, le long des Andes.Je propose que, dans un premier temps, des magmas granitiques sont remontés dans la chambre magmatique, laissant d'importants volumes de cristaux dans la croûte pro-fonde. Dans un second épisode, les cristaux formés en profondeur ont été transportés à travers la croûte continentale, suite au mélange avec un nouveau magma injecté. Ces magmas chargés de cristaux ont traversé la zone d'alimentation avant de s'injecter dans la chambre magmatique. Différents puises ont été distingués, injectés dans la chambre magmatique du sommet à la base concernant les granites, puis à la base du granite le plus jeune pour les gabbros et diorites. Le complexe magmatique du Torres del Paine s'est construit sur une période totale de 160'000±20'000 ans.

Biplots of compositional data

The singular value decomposition and its interpretation as alinear biplot has proved to be a powerful tool for analysing many formsof multivariate data. Here we adapt biplot methodology to the specifficcase of compositional data consisting of positive vectors each of whichis constrained to have unit sum. These relative variation biplots haveproperties relating to special features of compositional data: the studyof ratios, subcompositions and models of compositional relationships. Themethodology is demonstrated on a data set consisting of six-part colourcompositions in 22 abstract paintings, showing how the singular valuedecomposition can achieve an accurate biplot of the colour ratios and howpossible models interrelating the colours can be diagnosed.

Latitudinal gradient effect on the wing geometry of Auca coctei (Guérin)(Lepidoptera, Nymphalidae)

Latitudinal gradient effect on the wing geometry of Auca coctei (Guérin) (Lepidoptera, Nymphalidae). When the environmental conditions change locally, the organisms and populations may also change in response to the selection pressure, so that the development of individuals may become affected in different degrees. There have been only a few studies in which the patterns of wing morphology variation have been looked into along a latitudinal gradient by means of geometric morphometrics. The aim of this work was to assess the morphologic differentiation of wing among butterfly populations of the species Auca coctei. For this purpose, 9 sampling locations were used which are representative of the distribution range of the butterfly and cover a wide latitudinal range in Chile. The wing morphology was studied in a total of 202 specimens of A. coctei (150 males and 52 females), based on digitization of 17 morphologic landmarks. The results show variation of wing shape in both sexes; however, for the centroid size there was significant variation only in females. Females show smaller centroid size at higher latitudes, therefore in this study the Bergmann reverse rule is confirmed for females of A. coctei. Our study extends morphologic projections with latitude, suggesting that wing variation is an environmental response from diverse origins and may influence different characteristics of the life history of a butterfly.

Compositional and structural variabilities of Mg-rich iron oxide spinels from tuffite

Maghemite (γFe2O3) from tuffite is exceptionally rich in Mg, relatively to most of those reportedly found in other mafic lithosystems. To investigate in detail the compositional and structural variabilities of this natural magnetic iron oxide, sets of crystals were isolated from samples collected at different positions in a tuffite weathering mantle. These sets of crystal were individually powdered and studied by X-ray diffractometry, Mössbauer spectroscopy, magnetization measurements and chemical analysis. Lattice parameter of the cubic cell (a0) was found to vary from 0.834(1) to 0.8412(1) nm. Lower a0-values are characteristic of maghemite whereas higher ones are related to a magnetite precursor. FeO content ranges up to 17 mass % and spontaneous magnetization ranges from 8 to 32 J T-1 kg-1. Zero-field room temperature Mössbauer spectra are rather complex, indicating that the hyperfine field distributions due to Fe3+ and mixed valence Fe3+/2+ overlap. The structural variabilities of the (Mg, Ti)-rich iron oxide spinels is essentially related to the range of chemical composition of its precursor (Mg, Ti)-rich magnetite, and probably to the extent to which it has been oxidized during transformation in soil.