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

Inference of distributional parameters from compositional samples containing nondetects

Low concentrations of elements in geochemical analyses have the peculiarity of being compositional data and, for a given level of significance, are likely to be beyond the capabilities of laboratories to distinguish between minute concentrations and complete absence, thus preventing laboratories from reporting extremely low concentrations of the analyte. Instead, what is reported is the detection limit, which is the minimum concentration that conclusively differentiates between presence and absence of the element. A spatially distributed exhaustive sample is employed in this study to generate unbiased sub-samples, which are further censored to observe the effect that different detection limits and sample sizes have on the inference of population distributions starting from geochemical analyses having specimens below detection limit (nondetects). The isometric logratio transformation is used to convert the compositional data in the simplex to samples in real space, thus allowing the practitioner to properly borrow from the large source of statistical techniques valid only in real space. The bootstrap method is used to numerically investigate the reliability of inferring several distributional parameters employing different forms of imputation for the censored data. The case study illustrates that, in general, best results are obtained when imputations are made using the distribution best fitting the readings above detection limit and exposes the problems of other more widely used practices. When the sample is spatially correlated, it is necessary to combine the bootstrap with stochastic simulation

Multivariate ARIMA Compositional Time Series Analysis

A compositional time series is obtained when a compositional data vector is observed at different points in time. Inherently, then, a compositional time series is a multivariate time series with important constraints on the variables observed at any instance in time. Although this type of data frequently occurs in situations of real practical interest, a trawl through the statistical literature reveals that research in the field is very much in its infancy and that many theoretical and empirical issues still remain to be addressed. Any appropriate statistical methodology for the analysis of compositional time series must take into account the constraints which are not allowed for by the usual statistical techniques available for analysing multivariate time series. One general approach to analyzing compositional time series consists in the application of an initial transform to break the positive and unit sum constraints, followed by the analysis of the transformed time series using multivariate ARIMA models. In this paper we discuss the use of the additive log-ratio, centred log-ratio and isometric log-ratio transforms. We also present results from an empirical study designed to explore how the selection of the initial transform affects subsequent multivariate ARIMA modelling as well as the quality of the forecasts

Robust Factor Analysis for Compositional Data

Factor analysis as frequent technique for multivariate data inspection is widely used also for compositional data analysis. The usual way is to use a centered logratio (clr) transformation to obtain the random vector y of dimension D. The factor model is then y = Λf + e (1) with the factors f of dimension k < D, the error term e, and the loadings matrix Λ. Using the usual model assumptions (see, e.g., Basilevsky, 1994), the factor analysis model (1) can be written as Cov(y) = ΛΛT + ψ (2) where ψ = Cov(e) has a diagonal form. The diagonal elements of ψ as well as the loadings matrix Λ are estimated from an estimation of Cov(y). Given observed clr transformed data Y as realizations of the random vector y. Outliers or deviations from the idealized model assumptions of factor analysis can severely effect the parameter estimation. As a way out, robust estimation of the covariance matrix of Y will lead to robust estimates of Λ and ψ in (2), see Pison et al. (2003). Well known robust covariance estimators with good statistical properties, like the MCD or the S-estimators (see, e.g. Maronna et al., 2006), rely on a full-rank data matrix Y which is not the case for clr transformed data (see, e.g., Aitchison, 1986). The isometric logratio (ilr) transformation (Egozcue et al., 2003) solves this singularity problem. The data matrix Y is transformed to a matrix Z by using an orthonormal basis of lower dimension. Using the ilr transformed data, a robust covariance matrix C(Z) can be estimated. The result can be back-transformed to the clr space by C(Y ) = V C(Z)V T where the matrix V with orthonormal columns comes from the relation between the clr and the ilr transformation. Now the parameters in the model (2) can be estimated (Basilevsky, 1994) and the results have a direct interpretation since the links to the original variables are still preserved. The above procedure will be applied to data from geochemistry. Our special interest is on comparing the results with those of Reimann et al. (2002) for the Kola project data

Reintroduction of the Eurasian otter (Lutra lutra) in Muga and Fluvià basins (north-eastern Spain): viability, development, monitoring and trends of the new population

Aquesta tesi es basa en el programa de reintroducció de la llúdriga eurasiàtica (Lutra lutra) a les conques dels rius Muga i Fluvià (Catalunya) durant la segona meitat dels 1990s. Els objectius de la tesi foren demostrar la viabilitat de la reintroducció, demostrar l'èxit de la mateixa, estudiar aspectes ecològics i etològics de l'espècie, aprofitant l'oportunitat única de gaudir d'una població "de disseny" i determinar les probabilitats de supervivència de la població a llarg termini. La reintroducció de la llúdriga a les conques dels rius Muga i Fluvià va reeixir, doncs l'àrea geogràfica ocupada efectivament es va incrementar fins a un 64% d'estacions positives a l'hivern 2001-02. La troballa de tres exemplars adults nascuts a l'àrea de reintroducció és una altra prova que valida l'èxit del programa. La densitat d'exemplars calculada a través dels censos visuals ha resultat baixa (0.04-0.11 llúdrigues/km), però s'aproxima al que hom pot esperar en els primers estadis d'una població reintroduïda, encara poc nombrosa però distribuïda en una gran àrea. La mortalitat post-alliberament va ser del 22% un any després de l'alliberament, similar o inferior a la d'altres programes de reintroducció de llúdrigues reeixits. La mortalitat va ser deguda principalment a atropellaments (56%). El patró d'activitat de les llúdrigues reintroduïdes va esdevenir principalment nocturn i crepuscular, amb una escassa activitat diürna. Les seves àrees vitals van ser del mateix ordre (34,2 km) que les calculades en d'altres estudis realitzats a Europa. La longitud mitjana de riu recorreguda per una llúdriga durant 24 hores va ser de 4,2 km per les femelles i 7,6 km pels mascles. Durant el període de radioseguiment dues femelles van criar i els seus moviments van poder ser estudiats amb deteniment. La resposta de la nova població de llúdrigues a les fluctuacions estacionals en la disponibilitat d'aigua, habitual a les regions mediterrànies, va consistir en la concentració en una àrea menor durant el període de sequera estival, a causa de l'increment de trams secs, inhabitables per la llúdriga per la manca d'aliment, fet que va provocar expansions i contraccions periòdiques en l'àrea de distribució. La persistència a llarg termini de la població reintroduïda va ser estudiada mitjançant una Anàlisi de Viabilitat Poblacional (PVA). El resultat va ser un baix risc d'extinció de la població en els propers 100 anys i la majoria dels escenaris simulats (65%) van assolir el criteri d'un mínim de 90% de probabilitat de supervivència. Del model poblacional construït es dedueix que un punt clau per assegurar la viabilitat de la població reintroduïda és la reducció de la mortalitat accidental. A l'àrea d'estudi, els atropellaments causen més del 50% de la mortalitat i aquesta pot ser reduïda mitjançant la construcció de passos de fauna, el tancament lateral d'alguns trams de carretera perillosos i el control de la velocitat en algunes vies. El projecte de reintroducció ha posat a punt un protocol per a la captura, maneig i alliberament de llúdrigues salvatges, que pot contenir informació útil per a programes similars. També ha suposat una oportunitat única d'estudiar una població dissenyada artificialment i poder comparar diversos mètodes per estimar la distribució i la densitat de poblacions de llúdrigues. Per últim, la reintroducció portada a terme a les conques dels rius Muga i Fluvià ha aconseguit crear una nova població de llúdrigues, que persisteix en el temps, que es reprodueix regularment i que es dispersa progressivament, fins i tot a noves conques fluvials.