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

Compositional analysis of bivariate discrete probabilities

A joint distribution of two discrete random variables with finite support can be displayed as a two way table of probabilities adding to one. Assume that this table has n rows and m columns and all probabilities are non-null. This kind of table can be seen as an element in the simplex of n · m parts. In this context, the marginals are identified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclidean elements of the Aitchison geometry of the simplex can also be translated into the table of probabilities: subspaces, orthogonal projections, distances. Two important questions are addressed: a) given a table of probabilities, which is the nearest independent table to the initial one? b) which is the largest orthogonal projection of a row onto a column? or, equivalently, which is the information in a row explained by a column, thus explaining the interaction? To answer these questions three orthogonal decompositions are presented: (1) by columns and a row-wise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independent two-way tables and fully dependent tables representing row-column interaction. An important result is that the nearest independent table is the product of the two (row and column)-wise geometric marginal tables. A corollary is that, in an independent table, the geometric marginals conform with the traditional (arithmetic) marginals. These decompositions can be compared with standard log-linear models. Key words: balance, compositional data, simplex, Aitchison geometry, composition, orthonormal basis, arithmetic and geometric marginals, amalgam, dependence measure, contingency table

Compositional data and Simpson's paradox

Simpson's paradox, also known as amalgamation or aggregation paradox, appears when dealing with proportions. Proportions are by construction parts of a whole, which can be interpreted as compositions assuming they only carry relative information. The Aitchison inner product space structure of the simplex, the sample space of compositions, explains the appearance of the paradox, given that amalgamation is a nonlinear operation within that structure. Here we propose to use balances, which are specific elements of this structure, to analyse situations where the paradox might appear. With the proposed approach we obtain that the centre of the tables analysed is a natural way to compare them, which avoids by construction the possibility of a paradox. Key words: Aitchison geometry, geometric mean, orthogonal projection

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

A unified approach for representing rows and columns in contingency tables

By using suitable parameters, we present a uni¯ed aproach for describing four methods for representing categorical data in a contingency table. These methods include: correspondence analysis (CA), the alternative approach using Hellinger distance (HD), the log-ratio (LR) alternative, which is appropriate for compositional data, and the so-called non-symmetrical correspondence analysis (NSCA). We then make an appropriate comparison among these four methods and some illustrative examples are given. Some approaches based on cumulative frequencies are also linked and studied using matrices. Key words: Correspondence analysis, Hellinger distance, Non-symmetrical correspondence analysis, log-ratio analysis, Taguchi inertia

Intrinsic test of independence in contingency tables

A condition needed for testing nested hypotheses from a Bayesian viewpoint is that the prior for the alternative model concentrates mass around the small, or null, model. For testing independence in contingency tables, the intrinsic priors satisfy this requirement. Further, the degree of concentration of the priors is controlled by a discrete parameter m, the training sample size, which plays an important role in the resulting answer regardless of the sample size. In this paper we study robustness of the tests of independence in contingency tables with respect to the intrinsic priors with different degree of concentration around the null, and compare with other “robust” results by Good and Crook. Consistency of the intrinsic Bayesian tests is established. We also discuss conditioning issues and sampling schemes, and argue that conditioning should be on either one margin or the table total, but not on both margins. Examples using real are simulated data are given

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

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

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

A new distribution on the simplex containing the Dirichlet family

The Dirichlet family owes its privileged status within simplex distributions to easyness of interpretation and good mathematical properties. In particular, we recall fundamental properties for the analysis of compositional data such as closure under amalgamation and subcomposition. From a probabilistic point of view, it is characterised (uniquely) by a variety of independence relationships which makes it indisputably the reference model for expressing the non trivial idea of substantial independence for compositions. Indeed, its well known inadequacy as a general model for compositional data stems from such an independence structure together with the poorness of its parametrisation. In this paper a new class of distributions (called Flexible Dirichlet) capable of handling various dependence structures and containing the Dirichlet as a special case is presented. The new model exhibits a considerably richer parametrisation which, for example, allows to model the means and (part of) the variance-covariance matrix separately. Moreover, such a model preserves some good mathematical properties of the Dirichlet, i.e. closure under amalgamation and subcomposition with new parameters simply related to the parent composition parameters. Furthermore, the joint and conditional distributions of subcompositions and relative totals can be expressed as simple mixtures of two Flexible Dirichlet distributions. The basis generating the Flexible Dirichlet, though keeping compositional invariance, shows a dependence structure which allows various forms of partitional dependence to be contemplated by the model (e.g. non-neutrality, subcompositional dependence and subcompositional non-invariance), independence cases being identified by suitable parameter configurations. In particular, within this model substantial independence among subsets of components of the composition naturally occurs when the subsets have a Dirichlet distribution

A Log-linear model of grain size influence on the geochemistry of sediments

Sediment composition is mainly controlled by the nature of the source rock(s), and chemical (weathering) and physical processes (mechanical crushing, abrasion, hydrodynamic sorting) during alteration and transport. Although the factors controlling these processes are conceptually well understood, detailed quantification of compositional changes induced by a single process are rare, as are examples where the effects of several processes can be distinguished. The present study was designed to characterize the role of mechanical crushing and sorting in the absence of chemical weathering. Twenty sediment samples were taken from Alpine glaciers that erode almost pure granitoid lithologies. For each sample, 11 grain-size fractions from granules to clay (ø grades <-1 to >9) were separated, and each fraction was analysed for its chemical composition. The presence of clear steps in the box-plots of all parts (in adequate ilr and clr scales) against ø is assumed to be explained by typical crystal size ranges for the relevant mineral phases. These scatter plots and the biplot suggest a splitting of the full grain size range into three groups: coarser than ø=4 (comparatively rich in SiO2, Na2O, K2O, Al2O3, and dominated by “felsic” minerals like quartz and feldspar), finer than ø=8 (comparatively rich in TiO2, MnO, MgO, Fe2O3, mostly related to “mafic” sheet silicates like biotite and chlorite), and intermediate grains sizes (4≤ø <8; comparatively rich in P2O5 and CaO, related to apatite, some feldspar). To further test the absence of chemical weathering, the observed compositions were regressed against three explanatory variables: a trend on grain size in ø scale, a step function for ø≥4, and another for ø≥8. The original hypothesis was that the trend could be identified with weathering effects, whereas each step function would highlight those minerals with biggest characteristic size at its lower end. Results suggest that this assumption is reasonable for the step function, but that besides weathering some other factors (different mechanical behavior of minerals) have also an important contribution to the trend. Key words: sediment, geochemistry, grain size, regression, step function

An alternative method for dating unknown tephras based on a segmented regression model

In CoDaWork’05, we presented an application of discriminant function analysis (DFA) to 4 different compositional datasets and modelled the first canonical variable using a segmented regression model solely based on an observation about the scatter plots. In this paper, multiple linear regressions are applied to different datasets to confirm the validity of our proposed model. In addition to dating the unknown tephras by calibration as discussed previously, another method of mapping the unknown tephras into samples of the reference set or missing samples in between consecutive reference samples is proposed. The application of these methodologies is demonstrated with both simulated and real datasets. This new proposed methodology provides an alternative, more acceptable approach for geologists as their focus is on mapping the unknown tephra with relevant eruptive events rather than estimating the age of unknown tephra. Kew words: Tephrochronology; Segmented regression

Statistical evidences of cyclic changes in volcanic gas chemistry composition by inverse modelling

The identification of compositional changes in fumarolic gases of active and quiescent volcanoes is one of the most important targets in monitoring programs. From a general point of view, many systematic (often cyclic) and random processes control the chemistry of gas discharges, making difficult to produce a convincing mathematical-statistical modelling. Changes in the chemical composition of volcanic gases sampled at Vulcano Island (Aeolian Arc, Sicily, Italy) from eight different fumaroles located in the northern sector of the summit crater (La Fossa) have been analysed by considering their dependence from time in the period 2000-2007. Each intermediate chemical composition has been considered as potentially derived from the contribution of the two temporal extremes represented by the 2000 and 2007 samples, respectively, by using inverse modelling methodologies for compositional data. Data pertaining to fumaroles F5 and F27, located on the rim and in the inner part of La Fossa crater, respectively, have been used to achieve the proposed aim. The statistical approach has allowed us to highlight the presence of random and not random fluctuations, features useful to understand how the volcanic system works, opening new perspectives in sampling strategies and in the evaluation of the natural risk related to a quiescent volcano

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

Application of compositional data analysis to geochemical data of marine sediments

In an earlier investigation (Burger et al., 2000) five sediment cores near the Rodrigues Triple Junction in the Indian Ocean were studied applying classical statistical methods (fuzzy c-means clustering, linear mixing model, principal component analysis) for the extraction of endmembers and evaluating the spatial and temporal variation of geochemical signals. Three main factors of sedimentation were expected by the marine geologists: a volcano-genetic, a hydro-hydrothermal and an ultra-basic factor. The display of fuzzy membership values and/or factor scores versus depth provided consistent results for two factors only; the ultra-basic component could not be identified. The reason for this may be that only traditional statistical methods were applied, i.e. the untransformed components were used and the cosine-theta coefficient as similarity measure. During the last decade considerable progress in compositional data analysis was made and many case studies were published using new tools for exploratory analysis of these data. Therefore it makes sense to check if the application of suitable data transformations, reduction of the D-part simplex to two or three factors and visual interpretation of the factor scores would lead to a revision of earlier results and to answers to open questions . In this paper we follow the lines of a paper of R. Tolosana- Delgado et al. (2005) starting with a problem-oriented interpretation of the biplot scattergram, extracting compositional factors, ilr-transformation of the components and visualization of the factor scores in a spatial context: The compositional factors will be plotted versus depth (time) of the core samples in order to facilitate the identification of the expected sources of the sedimentary process. Kew words: compositional data analysis, biplot, deep sea sediments