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

When zero doesn't mean it and other geomathematical mischief

There is almost not a case in exploration geology, where the studied data doesn’tincludes below detection limits and/or zero values, and since most of the geological dataresponds to lognormal distributions, these “zero data” represent a mathematicalchallenge for the interpretation.We need to start by recognizing that there are zero values in geology. For example theamount of quartz in a foyaite (nepheline syenite) is zero, since quartz cannot co-existswith nepheline. Another common essential zero is a North azimuth, however we canalways change that zero for the value of 360°. These are known as “Essential zeros”, butwhat can we do with “Rounded zeros” that are the result of below the detection limit ofthe equipment?Amalgamation, e.g. adding Na2O and K2O, as total alkalis is a solution, but sometimeswe need to differentiate between a sodic and a potassic alteration. Pre-classification intogroups requires a good knowledge of the distribution of the data and the geochemicalcharacteristics of the groups which is not always available. Considering the zero valuesequal to the limit of detection of the used equipment will generate spuriousdistributions, especially in ternary diagrams. Same situation will occur if we replace thezero values by a small amount using non-parametric or parametric techniques(imputation).The method that we are proposing takes into consideration the well known relationshipsbetween some elements. For example, in copper porphyry deposits, there is always agood direct correlation between the copper values and the molybdenum ones, but whilecopper will always be above the limit of detection, many of the molybdenum values willbe “rounded zeros”. So, we will take the lower quartile of the real molybdenum valuesand 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 theircorrelation 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, roundedzeros, correlation dependency

Indicator taxa of benthic diatom communities: a case study in Mediterranean streams

A key issue in the implementation of the Water Framework Directive is the classification of streams and rivers using biological quality parameters and type-specific reference conditions. Four groups of stream types were defined in NE Spain on the basis of 152 diatom samples by means of detrended correspondence analysis and classification techniques. Diatom analysis was restricted to epilithic taxa, and the sites included gradients ranging from near-natural streams to sites with poor ecological quality. The main gradient shows a clear separation of sites in relation to the degree of human influence: polluted streams (mainly located in the lowlands) differ from streams in mountainous areas and in the Pyrenees. A second gradient is related to physiographical features. Headwater streams can be distinguished by their catchment geology. The type-specific diatom taxa for the stream types studied were determined by using indicator species analysis (IndVal). The type-specific taxa from near-natural streams are coincident with the indicator taxa for high ecological status. Human impact reduced the typological heterogeneity of the diatom community composition. Overall, the diatom communities in NE Spain exhibit a regional distribution pattern that closely corresponds with that observed in river systems elsewhere. Physiographical differences are only evident in undisturbed sites, while nutrient enrichment and other human disturbances may mask the regional differences in the distribution of diatom communities

Geologia del Baix Ter

Article que analitza el valor i lasingularitat de les zones humides del Baix Ter, tenint en compte la descripció de les característiques geològiques de la Plana del Ter i dels diferents ambients morfodinàmics que s' hi troben

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Centralnotations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform.In this way very elaborated aspects of mathematical statistics can be understoodeasily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating,combination of likelihood and robust M-estimation functions are simple additions/perturbations in A2(Pprior). Weighting observations corresponds to a weightedaddition of the corresponding evidence.Likelihood based statistics for general exponential families turns out to have aparticularly easy interpretation in terms of A2(P). Regular exponential families formfinite dimensional linear subspaces of A2(P) and they correspond to finite dimensionalsubspaces formed by their posterior in the dual information space A2(Pprior).The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P.The discussion of A2(P) valued random variables, such as estimation functionsor likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

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

Compositions in life science data

The aim of this talk is to convince the reader that there are a lot of interesting statisticalproblems in presentday life science data analysis which seem ultimately connected withcompositional statistics.Key words: SAGE, cDNA microarrays, (1D-)NMR, virus quasispecies

Experimental design on the simplex

Optimum experimental designs depend on the design criterion, the model andthe design region. The talk will consider the design of experiments for regressionmodels in which there is a single response with the explanatory variables lying ina simplex. One example is experiments on various compositions of glass such asthose considered by Martin, Bursnall, and Stillman (2001).Because of the highly symmetric nature of the simplex, the class of models thatare of interest, typically Scheff´e polynomials (Scheff´e 1958) are rather differentfrom those of standard regression analysis. The optimum designs are also ratherdifferent, inheriting a high degree of symmetry from the models.In the talk I will hope to discuss a variety of modes for such experiments. ThenI will discuss constrained mixture experiments, when not all the simplex is availablefor experimentation. Other important aspects include mixture experimentswith extra non-mixture factors and the blocking of mixture experiments.Much of the material is in Chapter 16 of Atkinson, Donev, and Tobias (2007).If time and my research allows, I would hope to finish with a few comments ondesign when the responses, rather than the explanatory variables, lie in a simplex.ReferencesAtkinson, A. C., A. N. Donev, and R. D. Tobias (2007). Optimum ExperimentalDesigns, with SAS. Oxford: Oxford University Press.Martin, R. J., M. C. Bursnall, and E. C. Stillman (2001). Further results onoptimal and efficient designs for constrained mixture experiments. In A. C.Atkinson, B. Bogacka, and A. Zhigljavsky (Eds.), Optimal Design 2000,pp. 225–239. Dordrecht: Kluwer.Scheff´e, H. (1958). Experiments with mixtures. Journal of the Royal StatisticalSociety, Ser. B 20, 344–360.1

Anàlisi geomètrica del Diaclasat de les Guilleries en el sector de la Vall del Ter entre els embassaments del Pasteral i Susqueda (Girona)

The Guilleries are a small and mountainous area located in the north-westem part of the Catalonian Coastal Ranges where metamorphic and igneous Paleozoic rocks are exposed. After the main hercynian folding this area was affected by a brittle deformation that is mainly manifested by the intrusion of a very large number of dykes of granodiorite and the development of a complex joint system. Trends of dykes indicate that their intrusion was related to a SE-NW extension, whose estimated value is 40% on an average. This extension seems to stand, although without any associated igneous event, with the development of NE-SW directed joints which make the main set. Five families more were developed later, onegently-dipping and fou upright; the latter trending roughly SE-NW, ENE-WSW, ESE-WNW and N-S. AU the joint sets appear in the metasedimentary Paleozoic rocks and in the hercynian intrusive bodies. Concerning the ages, joints that belong to the NE-SW and SE-NW directed sets and also those slightly dipping have been attributed to the late-hercynian times and all the other are considered to be later

Els clots de Sant Julià (Forallac, Baix Empordà): anàlisi geoarqueològic

Descripció de les roques dels Clots de Sant Julià (Forallac, Baix Empordà) i anàlisi arqueològica de les pedreres dels Clots

Aportaciones al conocimiento de la geología de Bac Grillera (prepirineo, Girona)

The Bac Grillera nappe (Prepirenean of Girona) shows a stratigraphic section composed of two unconformi ty-bounded units. The lower unit is composed by rethian (marly limestones) and liassic (brechoid marly limestones and limestones). The upper unit (sandstones and marly limestones) belongs to the upper Cretaceous (Campanian)

Les Gavarres: la personalitat geológica

Descripció litològica i morfològica del Massís de les Gavarres

Geological and Geomorphological Cartography at Scale 1: 10.000: a Tool of Management for Towns

Geological and geomorphological mapping at scale 1:10.000 besides from being an important source of scientific information it is also a necessary tool for municipal organs in order to make proper decisions when dealing with geo-environmental problems concerning integral territorial development. In this work, detailed information is given on the contents of such maps, their social and economical application, and a balance of the investment and gains that derives from them

Costa Brava central y meridional

Esta excursión, centrada en el extremo septentrional de la costa mediterránea de la península ibérica, tiene como objetivo fundamental la visita de algunos elementos geológicos puntuales que destacan por su especial interés, y que constituyen parte del patrimonio natural de esta zona. Además, a lo largo del itinerario se ofrece una visión general de la constitución geológica de este sector costero