Comparing correspondence analysis and the log-ratio alternative for representing categorical data

We compare correspondance análisis to the logratio approach based on compositional data. We also compare correspondance análisis and an alternative approach using Hellinger distance, for representing categorical data in a contingency table. We propose a coefficient which globally measures the similarity between these approaches. This coefficient can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions to the difference between the two representations. These three methods of representation can produce quite similar results. One illustrative example is given

Comparing correspondence analysis and the log-ratio alternative for representing categorical data

We compare correspondance análisis to the logratio approach based on compositional data. We also compare correspondance análisis and an alternative approach using Hellinger distance, for representing categorical data in a contingency table. We propose a coefficient which globally measures the similarity between these approaches. This coefficient can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions to the difference between the two representations. These three methods of representation can produce quite similar results. One illustrative example is given

Log-ratio of silica to aluminium counts (ln(Si/Al)) from ODP site 108-658

Growing evidence suggests that the low atmospheric CO2 concentration of the ice ages resulted from enhanced storage of CO2 in the ocean interior, largely as a result of changes in the Southern Ocean1. Early in the most recent deglaciation, a reduction in North Atlantic overturning circulation seems to have driven CO2 release from the Southern Ocean**2, 3, 4, 5, but the mechanism connecting the North Atlantic and the Southern Ocean remains unclear. Biogenic opal export in the low-latitude ocean relies on silicate from the underlying thermocline, the concentration of which is affected by the circulation of the ocean interior. Here we report a record of biogenic opal export from a coastal upwelling system off the coast of northwest Africa that shows pronounced opal maxima during each glacial termination over the past 550,000 years. These opal peaks are consistent with a strong deglacial reduction in the formation of silicate-poor glacial North Atlantic intermediate water**2 (GNAIW). The loss of GNAIW allowed mixing with underlying silicate-rich deep water to increase the silicate supply to the surface ocean. An increase in westerly-wind-driven upwelling in the Southern Ocean in response to the North Atlantic change has been proposed to drive the deglacial rise in atmospheric CO2 (refs 3, 4). However, such a circulation change would have accelerated the formation of Antarctic intermediate water and sub-Antarctic mode water, which today have as little silicate as North Atlantic Deep Water and would have thus maintained low silicate concentrations in the Atlantic thermocline. The deglacial opal maxima reported here suggest an alternative mechanism for the deglacial CO2 release**5, 6. Just as the reduction in GNAIW led to upward silicate transport, it should also have allowed the downward mixing of warm, low-density surface water to reach into the deep ocean. The resulting decrease in the density of the deep Atlantic relative to the Southern Ocean surface promoted Antarctic overturning, which released CO2 to the atmosphere.

Multivariate ARIMA Compositional Time Series Analysis

A compositional time series is obtained when a compositional data vector is observed atdifferent points in time. Inherently, then, a compositional time series is a multivariatetime 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, atrawl through the statistical literature reveals that research in the field is very much in itsinfancy and that many theoretical and empirical issues still remain to be addressed. Anyappropriate statistical methodology for the analysis of compositional time series musttake into account the constraints which are not allowed for by the usual statisticaltechniques available for analysing multivariate time series. One general approach toanalyzing compositional time series consists in the application of an initial transform tobreak the positive and unit sum constraints, followed by the analysis of the transformedtime series using multivariate ARIMA models. In this paper we discuss the use of theadditive log-ratio, centred log-ratio and isometric log-ratio transforms. We also presentresults from an empirical study designed to explore how the selection of the initialtransform affects subsequent multivariate ARIMA modelling as well as the quality ofthe forecasts

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 modelsbetween 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 thatlay 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 hiddencomponents, 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 totalvariance, 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. Graphicalrepresentation 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 analysisof 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, exceptfertilisers 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 areintrinsic to the relative nature of compositional data

Diagnosis of the nutrient compositional space of fruit crops

Tissue analysis is a useful tool for the nutrient management of fruit orchards. The mineral composition of diagnostic tissues expressed as nutrient concentration on a dry weight basis has long been used to assess the status of 'pure' nutrients. When nutrients are mixed and interact in plant tissues, their proportions or concentrations change relatively to each other as a result of synergism, antagonism, or neutrality, hence producing resonance within the closed space of tissue composition. Ternary diagrams and nutrient ratios are early representations of interacting nutrients in the compositional space. Dual and multiple interactions were integrated by the Diagnosis and Recommendation Integrated System (DRIS) into nutrient indexes and by Compositional Nutrient Diagnosis into centered log ratios (CND-clr). DRIS has some computational flaws such as using a dry matter index that is not a part as well as nutrient products (e.g. NxCa) instead of ratios. DRIS and CND-clr integrate all possible nutrient interactions without defining an ad hoc interactive model. They diagnose D components while D-1 could be diagnosed in the D-compositional Hilbert space. The isometric log ratio (ilr) coordinates overcome these problems using orthonormal binary nutrient partitions instead of dual ratios. In this study, it is presented a nutrient interactive model as well as computation methods for DRIS and CND-clr and CND-ilr coordinates (CND-ilr) using leaf analytical data from an experimental apple orchard in Southwestern Quebec, Canada. It was computed the Aitchison and Mahalanobis distances across ilr coordinates as measures of nutrient imbalance. The effect of changing nutrient concentrations on ilr coordinates are simulated to identify the ones contributing the most to nutrient imbalance.

Nutrient signature of Quebec (Canada) cranberry (Vaccinium macrocarpon Ait.)

Fertilizer recommendations for cranberry crops are guided by plant and soil tests. However, critical tissue concentration ranges used for diagnostic purposes are inherently biased by nutrient interactions and physiological age. Compositional data analysis using isometric log ratios (ilr) of nutrients as well as time detrending can avoid numerical biases. The objective was to derive unbiased nutrient signature standards for cranberry in Quebec and compare those standards to literature data. Field trials were conducted during 3 consecutive years with varying P treatments at six commercial sites in Quebec. Leaf tissues were analyzed for N, P, K, Ca, Mg, B, Cu, Zn, Mn and Fe. The analytical results were transformed into ilr nutrient balances of parts and groups of parts. High-yield reference ilr values were computed for cranberry yielding greater than 35 Mg ha-1. Many cranberry fields appeared to be over-supplied with K and either under-supplied with Mn or over-supplied with Fe as shown by their imbalanced [K | Ca, Mg] and [Mn | Fe] ratios. Nutrient concentration ranges from Maine and Wisconsin, USA, were combined into ilr values to generate ranges of balances. It was found that these nutrient ranges were much too broad for application in Quebec or outside the Quebec ranges for the [Ca | Mg] and the [Mn | Fe] balances, that were lower compared to those of high yielding cranberry crops in Quebec.