Aitchison Geometry Research Articles

Efficient Convex PCA with Applications to Wasserstein GPCA and Ranked Data

Convex PCA, which was introduced by Bigot et al. modifies Euclidean PCA by restricting the data and the principal components to lie in a given convex subset of a Hilbert space. This setting arises naturally in many applications, including distributional data in the Wasserstein space of an interval, and ranked compositional data under the Aitchison geometry. Our contribution in this article is 3-fold. First, we present several new theoretical results including consistency as well as continuity and differentiability of the objective function in the finite dimensional case. Second, we develop a numerical implementation of finite dimensional convex PCA when the convex set is polyhedral, and show that this provides a natural approximation of Wasserstein GPCA. Third, we illustrate our results with two financial applications, namely distributions of stock returns ranked by size and the capital distribution curve, both of which are of independent interest in stochastic portfolio theory. Supplementary materials for this article are available online.

Lasso regression method for a compositional covariate regularised by the norm L1 pairwise logratio

Lasso regression methods include a penalty function expressed in terms of a norm defined in the space of model coefficients. The norm plays a key role as regards the way coefficients can become irrelevant in the model. For models with a compositional covariate, the norm should be coherent with the Aitchison geometry. The proposed method is based on a newly-defined compositional norm called L1 pairwise logratio. The novel approach allows one to construct an appropriate basis through a sequential binary partition for discriminating between balances that influence the response variable and those that have no effect. This generalised Lasso regression scheme is illustrated with the analysis of a geochemical data set.

Extending compositional data analysis from a graph signal processing perspective

Traditional methods for the analysis of compositional data consider the log-ratios between all different pairs of variables with equal weight, typically in the form of aggregated contributions. This is not meaningful in contexts where it is known that a relationship only exists between very specific variables (e.g. for metabolomic pathways), while for other pairs a relationship does not exist. Modeling absence or presence of relationships is done in graph theory, where the vertices represent the variables, and the connections refer to relations. This paper links compositional data analysis with graph signal processing, and it extends the Aitchison geometry to a setting where only selected log-ratios can be considered. The presented framework retains the desirable properties of scale invariance and compositional coherence. A real data example from bioinformatics underlines the usefulness of this approach.

Darwinian evolution as Brownian motion on the simplex: A geometric perspective on stochastic replicator dynamics

We prove that stochastic replicator dynamics can be interpreted as intrinsic Brownian motion on the simplex equipped with the Aitchison geometry. As an immediate consequence, we derive three approximation results in the spirit of Wong–Zakai approximation, Donsker’s invariance principle and a JKO-scheme. Using the Fokker–Planck equation and Wasserstein-contraction estimates, we also study the long time behavior of the stochastic replicator equation, as an example of a nongradient drift diffusion on the Aitchison simplex.

Coal elemental (compositional) data analysis with hierarchical clustering algorithms

The modes of occurrence for elements in coal are extremely important for deciphering geological process of coal formation and for anticipating the technological behavior and environmental and health impacts derived from coal utilization. Hierarchical clustering algorithm has been widely adopted to investigate the modes of occurrence of elements in coal. The traditional statistics (e.g., Pearson correlation, Euclidean distance) for the elemental data of coal may lead to misinterpretation because the elemental data of coal are of compositional nature and follow the rules of Aitchison geometry. This work applied log-ratio transformations in order to overcome this problem. Different hierarchical clustering algorithms with various data transformations can infer modes of occurrence for coal elements, but which algorithm is optimum deserves to be investigated. In this paper, we discuss four commonly used hierarchical clustering algorithms utilizing pivot coordinates and weighted symmetric pivot coordinates (WSPC), two types of log-ratio transformations, to infer modes of occurrence of elements in coal, based on published coal elemental data. Results showed that the Pearson correlation produces more meaningful results than the Euclidean distance in clustering rare earth elements and Y. WSPC produces more interpretable results than those from pivot coordinates transformed data for these coal elemental data. Compared with the single, complete, and centroid, the average-linkage algorithm is indeed the optimum.

Multi-Resolution Aitchison Geometry Image Denoising for Low-Light Photography.

In the low-photon imaging regime, noise in the image sensors is dominated by shot noise, best modeled statistically as Poisson distribution. In this work, we show that the Poisson likelihood function is very well matched with the Bayesian estimation of the "difference of log of contrast of pixel intensities." More specifically, our work is rooted in statistical compositional data analysis, whereby we reinterpret the Aitchison geometry as a multi-resolution analysis in the log-pixel domain. We demonstrate that the difference-log-contrast has wavelet-like properties that correspond well with the human visual system, while being robust to illumination variations. We derive a denoising technique based on an approximate conjugate prior for the latent Aitchison variable that gives rise to an explicit minimum mean squared error estimation. The resulting denoising technique preserves image contrast details that are arguably more meaningful to human vision than the pixel intensity values themselves.

A novel downscaling procedure for compositional data in the Aitchison geometry with application to soil texture data

In this work, we present a novel downscaling procedure for compositional quantities based on the Aitchison geometry. The method is able to naturally consider compositional constraints, i.e. unit-sum and positivity, accounting for the scale invariance and relative scale of these data. We show that the method can be used in a block sequential Gaussian simulation framework in order to assess the variability of downscaled quantities. Finally, to validate the method, we test it first in an idealized scenario and then apply it for the downscaling of digital soil maps on a more realistic case study. The digital soil maps for the realistic case study are obtained from SoilGrids, a system for automated soil mapping based on state-of-the-art spatial predictions methods.

Novel multivariate compositional data’s model for structurally analyzing sub-industrial energy consumption with economic data

Structural prediction and analysis of sub-industrial energy consumption with economic data across three industrial sectors are an important basis for reflecting the coordinated development relationship between energy consumption and industrial development. Empirically, the sub-industrial energy consumption and economic structure have numerous compositional data. The multivariate compositional data’s fractional gray multivariate model on the basis of the Simplex space and its algebraic system is proposed in this study aiming at the multi-dimensional small sample size with high uncertainties. First, the fractional accumulative generation operation sequence of multivariate compositional data is defined according to Aitchison geometry. Then, the novel model with the form of the compositional data vector is obtained. Second, the least square parameter estimation of the model is studied. A derived model is deduced and selected as the time-response expression of the model solution. Moreover, the gray wolf optimizer is introduced and designed to determine the optimal value of the fractional order. Detailed modeling procedures, including the computational steps and the intelligent optimization algorithm, have been clearly presented. Furthermore, 10-year economic structural data of 14 provinces in China are used to validate the effectiveness of the proposed model. The validation presents that the proposed model performs better in fitting, prediction, stability, and applicability, in comparison with the two other models in the Simplex space. Last, from the updated real-time datasets from 2008 to 2018, the proposed model is applied to analyze and forecast the sub-industrial energy consumption structure and industrial structure of Beijing. Results show that the proposed model presents high accuracy and is efficient in addressing the multivariate compositional data in some structural energy and economic issues.

A systematic approach for the comparison of PM10, PM2.5, and PM1 mass concentrations of characteristic environmental sites.

This study explores the use of a systematic approach in the comparison of simultaneous measurements of PM10, PM2.5, and PM1 mass concentrations using Aitchison geometry. Three case studies in three different Asian cities where the PM coarse, fine, and ultrafine size fraction prevail were investigated and the data was displayed using a dedicated triangular diagram. Simultaneous size-segregated PM measurements, for each case study, were assessed in terms of PM ratios and PM10 levels and were compared to similar measurements reported in literature. Non-central chi-squared distribution quantiles, for each case study, were evaluated and used to investigate the degree of similarity between simultaneous size-segregated PM ratios. Likewise, a comparative number k was used to show the proportion between PM10 levels. The issues relating to the location of the simultaneous size-segregated PM ratios on the triangular diagram were examined and the effects of the non-centrality parameter λ on PM comparison were indicated. The results show that the proposed systematic approach can estimate an explorative quantile (i.e., 2.5%) within which the simultaneous size-segregated PM measurements from one site can be compared with simultaneous size-segregated PM measurements from other sites reported in literature highlighting the existence of possible similarities or correspondences in the kind of sources influencing the PM.

Advances in self-organizing maps for their application to compositional data

A self-organizing map (SOM) is a non-linear projection of a D-dimensional data set, where the distance among observations is approximately preserved on to a lower dimensional space. The SOM arranges multivariate data based on their similarity to each other by allowing pattern recognition leading to easier interpretation of higher dimensional data. The SOM algorithm allows for selection of different map topologies, distances and parameters, which determine how the data will be organized on the map. In the particular case of compositional data (such as elemental, mineralogical, or maceral abundance), the sample space is governed by Aitchison geometry and extra steps are required prior to their SOM analysis. Following the principle of working on log-ratio coordinates, the simplicial operations and the Aitchison distance, which are appropriate elements for the SOM, are presented. With this structure developed, a SOM using Aitchison geometry is applied to properly interpret elemental data from combustion products (bottom ash, fly ash, and economizer fly ash) in a Wyoming coal-fired power plant. Results from this effort provide knowledge about the differences between the ash composition in the coal combustion process.

Evidence functions: a compositional approach to information

The discrete case of Bayes’ formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes’ formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.

General approach to coordinate representation of compositional tables

AbstractCompositional tables can be considered a continuous counterpart to the well‐known contingency tables. Their cells, which generally contain positive real numbers rather than just counts, carry relative information about relationships between two factors. Hence, compositional tables can be seen as a generalization of (vector) compositional data. Due to their relative character, compositions are commonly expressed in orthonormal coordinates using a sequential binary partition prior to being further processed by standard statistical tools. Unfortunately, the resulting coordinates do not respect the two‐dimensional nature of compositional tables. Information about relationship between factors is thus not well captured. The aim of this paper is to present a general system of orthonormal coordinates with respect to the Aitchison geometry, which allows for an analysis of the interactions between factors in a compositional table. This is achieved using logarithms of odds ratios, which are also widely used in the context of contingency tables.

Linear Association in Compositional Data Analysis

With compositional data ordinary covariation indexes, designed for real random variables, fail to describe dependence. There is a need for compositional alternatives to covariance and correlation. Based on the Euclidean structure of the simplex, called Aitchison geometry, compositional association is identied to a linear restriction of the sample space when a log-contrast is constant. In order to simplify interpretation, a sparse and simple version of compositional association is dened in terms of balances which are constant across the sample. It is called b-association. This kind of association of compositional variables is extended to association between groups of compositional variables. In practice, exact b-association seldom occurs, and measures of degree of b-association are reviewed based on those previously proposed. Also, some techniques for testing b-association are studied. These techniques are applied to available oral microbiome data to illustrate both their advantages and diculties. Both testing and measurements of b-association appear to be quite sensible to heterogeneities in the studied populations and to outliers.

Weighted Pivot Coordinates for Compositional Data and Their Application to Geochemical Mapping

The log ratio methodology converts compositional data, such as concentrations of chemical elements in a rock, from their original Aitchison geometry to interpretable real orthonormal coordinates, thereby allowing meaningful statistical processing and visualization. However, it must be taken into account that the original concentrations can be flawed by detection limit or imprecision problems that can severely affect the resulting coordinates. This paper aims to construct such orthonormal log ratio coordinates, called weighted pivot coordinates, that capture the relevant relative information about an original component and treat the redundant information in a controlled manner. Theoretical developments are supported by a thorough simulation study. Weighted pivot coordinates are then applied to the geochemical mapping of catchment outlet sediments from the National Geochemical Survey of Australia illustrating their advantage over possible alternatives.

Changing the Reference Measure in the Simplex and its Weighting Effects

Standard analysis of compositional data under the assumption that the Aitchison geometry holds assumes a uniform distribution as reference measure of the space. Weighting of parts can be done changing the reference measure. The changes that appear in the algebraic-geometric structure of the simplex are analysed, as a step towards understanding the implications for elementary statistics of random compositions. Some of the standard tools in exploratory analysis of compositional data analysis, such as center, variation matrix and biplots are studied in some detail, although further research is still needed. The main conclusion is that down-weighting some parts is approaching the geometry of the corresponding subcomposition, thus preserving a kind of coherence between standard and down-weighted analyses.

Compositional Tables Analysis in Coordinates

AbstractCompositional tables – a continuous counterpart to the contingency tables – carry relative information about relationships between row and column factors; thus, for their analysis, only ratios between cells of a table are informative. Consequently, the standard Euclidean geometry should be replaced by the Aitchison geometry on the simplex that enables decomposition of the table into its independent and interactive parts. The aim of the paper is to find interpretable coordinate representation for independent and interaction tables (in sense of balances and odds ratios of cells, respectively), where further statistical processing of compositional tables can be performed. Theoretical results are applied to real‐world problems from a health survey and in macroeconomics.

Independence in Contingency Tables Using Simplicial Geometry

Frequently, contingency tables are generated in a multinomial sampling. Multinomial probabilities are then organized in a table assigning probabilities to each cell. A probability table can be viewed as an element in the simplex. The Aitchison geometry of the simplex identifies independent probability tables as a linear subspace. An important consequence is that, given a probability table, the nearest independent table is obtained by orthogonal projection onto the independent subspace. The nearest independent table is identified as that obtained by the product of geometric marginals, which do not coincide with the standard marginals, except in the independent case. The original probability table is decomposed into orthogonal tables, the independent and the interaction tables. The underlying model is log-linear, and a procedure to test independence of a contingency table, based on a multinomial simulation, is developed. Its performance is studied on an illustrative example.

Simplicial principal component analysis for density functions in Bayes spaces

Probability density functions are frequently used to characterize the distributional properties of large-scale database systems. As functional compositions, densities primarily carry relative information. As such, standard methods of functional data analysis (FDA) are not appropriate for their statistical processing. The specific features of density functions are accounted for in Bayes spaces, which result from the generalization to the infinite dimensional setting of the Aitchison geometry for compositional data. The aim is to build up a concise methodology for functional principal component analysis of densities. A simplicial functional principal component analysis (SFPCA) is proposed, based on the geometry of the Bayes space B2 of functional compositions. SFPCA is performed by exploiting the centred log-ratio transform, an isometric isomorphism between B2 and L2 which enables one to resort to standard FDA tools. The advantages of the proposed approach with respect to existing techniques are demonstrated using simulated data and a real-world example of population pyramids in Upper Austria.

Modeling Compositional Time Series with Vector Autoregressive Models

Multivariate time series describing relative contributions to a total (like proportional data) are called compositional time series. They need to be transformed first to the usual Euclidean geometry before a time series model is fitted. It is shown how an appropriate transformation can be chosen, resulting in coordinates with respect to the Aitchison geometry of compositional data. Using vector autoregressive models, the standard approach based on raw data is compared with the compositional approach based on transformed data. The results from the compositional approach are consistent with the relative nature of the observations, while the analysis of the raw data leads to several inconsistencies and artifacts. The compositional approach is extended to the case when also the total of the compositional parts is of interest. Moreover, a concise methodology for an interpretation of the coordinates in the transformed space together with the corresponding statistical inference (like hypotheses testing) is provided. Copyright © 2015 John Wiley & Sons, Ltd.

The Use of Robust Factor Analysis of Compositional Geochemical Data for the Recognition of the Target Area in Khusf 1:100000 Sheet, South Khorasan, Iran

The closed nature of geochemical data has been proven in many studies. Compositional data have special properties that mean that standard statistical methods cannot be used to analyse them. These data imply a particular geometry called Aitchison geometry in the simplex space. For analysis, the dataset must first be opened by the various transformations provided. One of the most popular of the applied transformations is the log-ratio transform. The main purpose of this research is to identify the anomalous area in the Khusf 1:100000 sheet which is located in the western part of Birjand, South Khorasan province. To achieve the goal, a dataset of 652 stream sediments geochemically analysed for 20 elements was collected. In practice, the geochemical data were first opened by CLR transformation and then the range correlation coefficient (RCC) ratio was calculated and mapped. In consequence, the robust factor analysis for compositional data was used to separate the elements, mostly in the high-value regions obtained by the method of RCC. Finally, the priority of anomalies was specified using weighted catchment analysis. The above procedures led to the recognition of some anomaly zones for elements of Cu, Bi, Sb, Ni and Cr in the study area. Such results can be useful for designing an appropriate exploratory plan for semi-detailed and detailed exploration steps.