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AbstractConvex 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. [ABSTRACT FROM AUTHOR]

Systemic risk is a complex and multifaceted phenomenon that needs to be addressed from different perspectives. In this work we propose a Compositional Data (CoDa) approach to analyze the distribution of relative contributions to systemic risk associated with major European countries during the period 2008–2021. We represent systemic risk measures corresponding to those countries as percentage shares, or parts, of a compositional dataset and we perform a multivariate statistical analysis using specific CoDa procedures. The proposed approach sheds new light on some variability patterns and cross-country relationships that appear to be linked to the composition of systemic risk parts in the system. [ABSTRACT FROM AUTHOR]

The FeldsteinHorioka1980 study on investment flows through the correlation of domestic saving and investment concluded that liberalization of capital markets does not necessarily lead to a movement of capital looking for a better allocation of resources, as classical theory would suggest. Ever since, literature has been prolific regarding this "puzzle", with arguments for and against this conclusion. This paper aims to analyze the issue from a different perspective. In recent years, the stock markets of Chile, Colombia, Mexico and Peru joined the Latin American Integrated Market through an agreement that allows investors in any of the participating markets to invest in the others as if they were investing locally. Compositional methods are used to assess the hypothesis of a potential flow of capital between markets generated by the creation of the joint market. First, cross-sectional methods for compositional data were used to test the hypothesis. As a result, it was not possible to find a change in the composition of the investment in the four markets produced by the creation of the joint market. Secondly, vector autoregressive models were estimated and tested for structural breaks in the parameters. However, these models were not found to be informative. In conclusion, it was not possible to reject the Felstein-Horioka hypothesis, supporting the idea that liberalization is not enough to generate capital flows between markets.

Variational Bayesian inference is an important machine learning tool that finds application from statistics to robotics. The goal is to find an approximate probability density function (PDF) from a chosen family that is in some sense "closest" to the full Bayesian posterior. Closeness is typically defined through the selection of an appropriate loss functional such as the Kullback-Leibler (KL) divergence. In this paper, we explore a new formulation of variational inference by exploiting the fact that (most) PDFs are members of a Bayesian Hilbert space under careful definitions of vector addition, scalar multiplication, and an inner product. We show that, under the right conditions, variational inference based on KL divergence can amount to iterative projection, in the Euclidean sense, of the Bayesian posterior onto a subspace corresponding to the selected approximation family. We work through the details of this general framework for the specific case of the Gaussian approximation family and show the equivalence to another Gaussian variational inference approach. We furthermore discuss the implications for systems that exhibit sparsity, which is handled naturally in Bayesian space, and give an example of a high-dimensional robotic state estimation problem that can be handled as a result. We provide some preliminary examples of how the approach could be applied to non-Gaussian inference and discuss the limitations of the approach in detail to encourage follow-on work along these lines. [ABSTRACT FROM AUTHOR]

We study a partial linear model in which the response is compositional and the predictors include both compositional and Euclidean variables. We define a partial linear regression model under Aitchison geometry based on isometric log-ratio (ilr) transformation. An identification condition of linear parameters is provided in terms of expectations and conditional expectations of the response and covariates. An estimator based on the identification is developed and asymptotic properties of proposed estimators are derived. The proposed method can be implemented easily by using existing R packages such as np and compositions. The limiting distribution of the proposed estimator is provided with normal distribution in Euclidean space so that it is easy to use for inference. Also, some finite sample properties are presented via simulation studies. We also present election data as an illustrative example. [ABSTRACT FROM AUTHOR]

The Feldstein and Horioka (1980) study on investment flows through the correlation of domestic saving and investment concluded that liberalization of capital markets does not necessarily lead to a movement of capital looking for a better allocation of resources, as classical theory would suggest. Ever since, literature has been prolific regarding this "puzzle", with arguments for and against this conclusion. This paper aims to analyze the issue from a different perspective. In recent years, the stock markets of Chile, Colombia, Mexico and Peru joined the Latin American Integrated Market through an agreement that allows investors in any of the participating markets to invest in the others as if they were investing locally. Compositional methods are used to assess the hypothesis of a potential flow of capital between markets generated by the creation of the joint market. First, cross-sectional methods for compositional data were used to test the hypothesis. As a result, it was not possible to find a change in the composition of the investment in the four markets produced by the creation of the joint market. Secondly, vector autoregressive models were estimated and tested for structural breaks in the parameters. However, these models were not found to be informative. In conclusion, it was not possible to reject the Felstein-Horioka hypothesis, supporting the idea that liberalization is not enough to generate capital flows between markets. [ABSTRACT FROM AUTHOR]

Advances in sequencing technologies have enabled novel insights into microbial niche differentiation, from analyzing environmental samples to understanding human diseases and informing dietary studies. However, identifying the microbial taxa that differentiate these samples can be challenging. These issues stem from the compositional nature of 16S rRNA gene data (or, more generally, taxon or functional gene data); the changes in the relative abundance of one taxon influence the apparent abundances of the others. Here we acknowledge that inferring properties of individual bacteria is a difficult problem and instead introduce the concept of balances to infer meaningful properties of subcommunities, rather than properties of individual species. We show that balances can yield insights about niche differentiation across multiple microbial environments, including soil environments and lung sputum. These techniques have the potential to reshape how we carry out future ecological analyses aimed at revealing differences in relative taxonomic abundances across different samples. IMPORTANCE By explicitly accounting for the compositional nature of 16S rRNA gene data through the concept of balances, balance trees yield novel biological insights into niche differentiation. The software to perform this analysis is available under an open-source license and can be obtained at https://github.com/biocore/gneiss. Author Video: An author video summary of this article is available.

Association rule mining is a powerful data analytic technique used for extracting information from transaction databases with a collection of itemsets. The aim is to indicate what item goes with what item (ie, an association rule) in a set of collected transactions. It is extensively used in text analytics of text records or social media. Here we use Compositional Data analysis (CoDa) techniques to generate new visualizations and insights from association rule mining. These CoDa methods show the relationship between itemsets, their strength, and direction of dependency. Moreover, after expressing each association rule as a contingency table, we discuss two statistical tests to guide identification of the relevant rules by analyzing the relative importance of the elements of the table. As an example, we use these visualizations and statistical tests for investigating the association of negative mood emotions to various types of headache/migraine events. Data for those analysis comes from N1‐HeadacheTM, a digital platform where individual users record attacks and symptoms as well as their daily exposure to a list of potential factors. [ABSTRACT FROM AUTHOR]

Compositional data carry relative information. Hence, their statistical analysis has to be performed on coordinates with respect to a log-ratio basis. Frequently, the modeler is required to back-transform the estimates obtained with the modeling to have them in the original units such as euros, kg or mg/liter. Approaches for recovering original units need to be formally introduced and its properties explored. Here, we formulate and analyze the properties of two procedures: a simple approach consisting of adding a residual part to the composition and an approach based on the use of an auxiliary variable. Both procedures are illustrated using a geochemical data set where the original units are recovered when spatial models are applied. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Negative correlations between elements, molecules, or minerals can indicate a variety of geochemical processes, such as ion exchange, incongruent mineral precipitation/dissolution, and redox reactions. However, compositional data (those composed of relative parts) can also exhibit negative correlations simply due to displacement of one part by another, such as the addition of sodium chloride lowering the concentration of all ions other than sodium and chloride in a solution. Apart from this practical problem, the question is more general: how to address the relationships between components in data carrying relative information. For this purpose, the symmetric pivot coordinates were developed which allow for the identification of both positive and negative correlations between two parts in compositional data in terms of their relative dominance to the other parts. Accordingly, the symmetric pivot coordinate approach aggregates all of the logratios with those two parts of interest. This may not be desirable if data quality problems occur, because such parts would contribute the same weight to the coordinate as parts with good data quality. As a way out, the new method of weighted symmetric coordinates focusing on pairwise associations is proposed. In this approach, variables with large logratio variances are down-weighted to suppress their effect on the remaining variables, which is also demonstrated in a small simulation study. Finally, the weighted symmetric pivot coordinates are applied to chemistry data from a series of waste water samples from oil and gas wells produced from the lower Eagle Ford Group in the U.S. Gulf Coast Basin. In particular, strong negative correlations between ions are examined using this method to reveal processes which occur as a function of depth, including clay diagenesis, de-dolomitization, kerogen maturation, and sulfate and carbonate mineral saturation. [ABSTRACT FROM AUTHOR]

Compositional data refer to a vector with parts that are positive and subject to a constant-sum constraint. Examples of compositional data in the real world include a vector with each entry representing the weight of a stock in an investment portfolio, or the relative concentration of air pollutants in the environment. In this study, we developed a Convex Clustering approach for grouping Compositional data. Convex clustering is desirable because it provides a global optimal solution given its convex relaxations of hierarchical clustering. However, when directly applied to compositions, the clustering result offers little interpretability because it ignores the unit-sum constraint of compositional data. In this study, we discuss the clustering of compositional variables in the Aitchison framework with an isometric log-ratio (ilr) transformation. The objective optimization function is formulated as a combination of a L 2 -norm loss term and a L 1 -norm regularization term and is then efficiently solved using the alternating direction method of multipliers. Based on the numerical simulation results, the accuracy of clustering ilr-transformed data is higher than the accuracy of directly clustering untransformed compositional data. To demonstrate its practical use in real applications, the proposed method is also tested on several real-world datasets. [ABSTRACT FROM AUTHOR]

Due to boundedness and sum constraint, compositional data are often transformed by logratio transformation and their transformed data are put into traditional binary classification or discriminant analysis. However, it may be problematic to directly apply traditional multivariate approaches to the transformed data because class distributions are not Gaussian and Bayes decision boundary are not polynomial on the transformed space. In this study, we propose to use flexible classification approaches to transformed data for compositional data classification. Empirical studies using synthetic and real examples demonstrate that flexible approaches outperform traditional multivariate classification or discriminant analysis. [ABSTRACT FROM AUTHOR]

The log-ratio approach to compositional data (CoDa) analysis has now entered a mature phase. The principles and statistical tools introduced by J. Aitchison in the eighties have proven successful in solving a number of applied problems. The algebraic–geometric structure of the sample space, tailored to those principles, was developed at the beginning of the millennium. Two main ideas completed the J. Aitchison's seminal work: the conception of compositions as equivalence classes of proportional vectors, and their representation in the simplex endowed with an interpretable Euclidean structure. These achievements allowed the representation of compositions in meaningful coordinates (preferably Cartesian), as well as orthogonal projections compatible with the Aitchison distance introduced two decades before. These ideas and concepts are reviewed up to the normal distribution on the simplex and the associated central limit theorem. Exploratory tools, specifically designed for CoDa, are also reviewed. To illustrate the adequacy and interpretability of the sample space structure, a new inequality index, based on the Aitchison norm, is proposed. Most concepts are illustrated with an example of mean household gross income per capita in Spain. [ABSTRACT FROM AUTHOR]

Compositional 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. [ABSTRACT FROM AUTHOR]

This work focuses on the characterization of the central tendency of a sample of compositional data. It provides newresults about theoretical properties of means and covariance functions for compositional data, with an axiomatic perspective. Original results that shed new light on geostatistical modeling of compositional data are presented. As a first result, it is shown that the weighted arithmetic mean is the only central tendency characteristic satisfying a small set of axioms, namely continuity, reflexivity, and marginal stability. Moreover, this set of axioms also implies that the weights must be identical for all parts of the composition. This result has deep consequences for spatial multivariate covariance modeling of compositional data. In a geostatistical setting, it is shown as a second result that the proportional model of covariance functions (i.e., the product of a covariance matrix and a single correlation function) is the only model that provides identical kriging weights for all components of the compositional data. As a consequence of these two results, the proportional model of covariance function is the only covariance model compatible with reflexivity and marginal stability. [ABSTRACT FROM AUTHOR]

ABSTRACT Advances in sequencing technologies have enabled novel insights into microbial niche differentiation, from analyzing environmental samples to understanding human diseases and informing dietary studies. However, identifying the microbial taxa that differentiate these samples can be challenging. These issues stem from the compositional nature of 16S rRNA gene data (or, more generally, taxon or functional gene data); the changes in the relative abundance of one taxon influence the apparent abundances of the others. Here we acknowledge that inferring properties of individual bacteria is a difficult problem and instead introduce the concept of balances to infer meaningful properties of subcommunities, rather than properties of individual species. We show that balances can yield insights about niche differentiation across multiple microbial environments, including soil environments and lung sputum. These techniques have the potential to reshape how we carry out future ecological analyses aimed at revealing differences in relative taxonomic abundances across different samples. IMPORTANCE By explicitly accounting for the compositional nature of 16S rRNA gene data through the concept of balances, balance trees yield novel biological insights into niche differentiation. The software to perform this analysis is available under an open-source license and can be obtained at https://github.com/biocore/gneiss . Author Video: An author video summary of this article is available.

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. [ABSTRACT FROM AUTHOR]

Guava response to liming and fertilization can be monitored by tissue testing. Tissue nutrient signature is often diagnosed against nutrient concentration standards. However, this approach has been criticized for not considering nutrient interactions and to generate numerical biases as a result of data redundancy, scale dependency and non-normal distribution. Techniques of compositional data analysis can control those biases by balancing groups of nutrients, such as those involved in liming and fertilization. The sequentially arranged and orthonormal isometric log ratios (ilr) or balances avoid numerical bias inherent to compositional data. The objectives were to relate tissue nutrient balances with the production of "Paluma" guava orchards differentially limed and fertilized, and to adjust the current patterns of nutrient balance with the range of more productive guava trees. It was conducted one experiment of 7-yr of liming and three experiments of 3-yr with N, P and K trials in 'Paluma' orchards on an Oxisol. Plant N, P, K, Ca and Mg were monitored yearly. It was selected the [N, P, K | Ca, Mg], [N, P | K], [N | P] and [Ca | Mg] balances to set apart the effects of liming (Ca-Mg) and fertilizers (N-K) on macronutrient balances. Liming largely influenced nutrient balances of guava in the Oxisol while fertilization was less influential. The large range of guava yields and nutrient balances allowed defining balance ranges and comparing them with the critical ranges of nutrient concentration values currently used in Brazil and combined into ilr coordinates.A resposta da goiabeira à calagem e à adubação pode ser monitorada por análises de tecido vegetal. O perfil nutricional é definido em relação a padrões de teores de nutrientes. No entanto, os teores de nutrientes-padrão são constantemente criticados por não considerarem as interações que ocorrem entre nutrientes e por gerarem tendências numéricas, decorrentes da redundância dos dados, da dependência de escala e da distribuição não normal. As técnicas de análise composicional de dados podem controlar esses dados tendenciosos, equilibrando os grupos de nutrientes, tais como os envolvidos na calagem e na adubação. A utilização das relações log isométricas (ilr) ortonormais, sequencialmente dispostas, evita tendências numéricas inerentes aos dados de composição. Os objetivos do trabalho foram relacionar o balanço de nutrientes dos tecidos vegetais com a produção de goiabeiras em pomares de 'Paluma' diferentemente corrigidos e adubados, e ajustar os atuais padrões de nutrientes com a faixa de equilíbrio das goiabeiras mais produtivas. Um experimento de calagem de sete anos e três, experimentos de três anos com doses de N, P2O5 e K2O, foram conduzidos em pomares de goiabeiras 'Paluma' em um Latossolo Vermelho-Amarelo. Os teores de N, P, K, Ca e Mg na planta foram monitorados anualmente. Selecionaram-se os balanços [N, P, K | Ca, Mg], [N, P | K], [N | P] e [Ca | Mg] para separar os efeitos da calagem (Ca-Mg) e dos fertilizantes (N-K) nos balanços de macronutrientes. Os balanços foram mais influenciados pela calagem do que pela fertilização. A produtividade das goiabeiras e seu balanço nutricional permitiram a definição de faixas de equilíbrio de nutrientes e sua validação com as faixas de concentrações críticas atualmente utilizadas no Brasil e combinadas em coordenadas ilr.

Compositional data, as they typically appear in geochemistry in terms of concentrations of chemical elements in soil samples, need to be expressed in log-ratio coordinates before applying the traditional statistical tools if the relative structure of the data is of primary interest. There are different possibilities for this purpose, like centered log-ratio coefficients, or isometric log-ratio coordinates. In both the approaches, geometric means of the compositional parts are involved, and it is unclear how measurement errors or detection limit problems affect their presentation in coordinates. This problem is investigated theoretically by making use of the theory of error propagation. Due to certain limitations of this approach, the effect of error propagation is also studied by means of simulations. This allows to provide recommendations for practitioners on the amount of error and on the expected distortion of the results, depending on the purpose of the analysis. [ABSTRACT FROM AUTHOR]

For the exploratory analysis of three-way data, the Tucker3 is one of the most applied models to study three-way arrays when the data are quadrilinear. When the data consist of vectors of positive values summing to a unit, as in the case of compositional data, this model should consider the specific problems that compositional data analysis brings. The main purpose of this paper is to describe how to do a Tucker3 analysis of compositional data, and to show the relationships between the loading matrices when different preprocessing procedures are used. [ABSTRACT FROM PUBLISHER]

Learning distances that are specifically designed to compare histograms in the probability simplex has recently attracted the attention of the machine learning community. Learning such distances is important because most machine learning problems involve bags of features rather than simple vectors. Ample empirical evidence suggests that the Euclidean distance in general and Mahalanobis metric learning in particular may not be suitable to quantify distances between points in the simplex. We propose in this paper a new contribution to address this problem by generalizing a family of embeddings proposed by Aitchison (J R Stat Soc 44:139-177, ) to map the probability simplex onto a suitable Euclidean space. We provide algorithms to estimate the parameters of such maps by building on previous work on metric learning approaches. The criterion we study is not convex, and we consider alternating optimization schemes as well as accelerated gradient descent approaches. These algorithms lead to representations that outperform alternative approaches to compare histograms in a variety of contexts. [ABSTRACT FROM AUTHOR]

Compositional data analysis usually deals with relative information between parts where the total (abundances, mass, amount, etc.) is unknown or uninformative. This article addresses the question of what to do when the total is known and is of interest. Tools used in this case are reviewed and analysed, in particular the relationship between the positive orthant of D-dimensional real space, the product space of the real line times the D-part simplex, and their Euclidean space structures. The first alternative corresponds to data analysis taking logarithms on each component, and the second one to treat a log-transformed total jointly with a composition describing the distribution of component amounts. Real data about total abundances of phytoplankton in an Australian river motivated the present study and are used for illustration. [ABSTRACT FROM PUBLISHER]

Sources of uncertainty in the boundaries of structures in medical images have motivated the use of probabilistic labels in segmentation applications. An important component in many medical image segmentation tasks is the use of a shape model, often generated by applying statistical techniques to training data. Standard statistical techniques (e.g., principal component analysis) often assume data lies in an unconstrained vector space, but probabilistic labels are constrained to the unit simplex. If these statistical techniques are used directly on probabilistic labels, relative uncertainty information can be sacrificed. A standard method for facilitating analysis of probabilistic labels is to map them to a vector space using the LogOdds transform. However, the LogOdds transform is asymmetric in one of the labels, which skews results in some applications. The isometric log-ratio (ILR) transform is a symmetrized version of the LogOdds transform, and is so named as it is an isometry between the Aitchison geometry, the inherent geometry of the simplex, and standard Euclidean geometry. We explore how to interpret the Aitchison geometry when applied to probabilistic labels in medical image segmentation applications. We demonstrate the differences when applying the LogOdds transform or the ILR transform to probabilistic labels prior to statistical analysis. Specifically, we show that statistical analysis of ILR transformed data better captures the variability of anatomical shapes in cases where multiple different foreground regions share boundaries (as opposed to foreground-background boundaries). [ABSTRACT FROM AUTHOR]

General systems are frequently decomposable into parts and these parts can evolve in time or space, a frequent occurrence in the field of Geosciences. In most cases, fitting models to forecast future states of the system is a goal of the analysis. Modelling interactions between parts may also be of common interest. The system can be analysed from different points of view; the traditional one consists in modelling each part of the system in time. Alternatively, modelling the evolution of the parts as proportions is proposed herein and attention is centred on the compositional evolution. The compositions are expressed in orthogonal coordinates (ilr) and then modelled using first-order differential equations with constant coefficients. Simple models are shown to be very flexible, including many of the standard growth curve models. The models are fitted using regression techniques on the integrated coordinates. The use and interpretation of these differential models is illustrated with several examples: a simulated example; urban waste in Catalonia (Spain); oil production and reserves; and growth of a luzonite crystal. [ABSTRACT FROM AUTHOR]

Data collected for the investigation of the environmental and ecological characteristics of a river basin are often in the form of a large three-way array; hence, a particular version of the Tucker model could be applied to gather more information contained in such complex geochemical systems. Indeed, when the data are in compositional form, more attention must be given to the analysis of the numerical data. Recently, the Tucker3 model has been proposed to analyze compositional data characterized by a three-way structure. In this work, a particular version of the Tucker model, known as the weighted principal component analysis, was used to analyze water samples collected from the Arno river (Tuscany, central Italy) in order to evaluate the method's effectiveness. Several graphical displays have been developed to allow an accurate and complete interpretation of results. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The sample space of compositional data, a simplex, induces a different kind of geometry, known as Aitchison geometry, with the Euclidean space property. For this reason, the standard statistical analysis is not meaningful here, and this is also true for measures of location and covariance. The measure of location, called centre, is the best linear unbiased estimator of the central tendency of the distribution of a random composition with respect to the geometry on the simplex (Pawlowsky-Glahn and Egozcue in Stoch Envir Res Risk Ass, 15:384-398, 2001; Math Geol, 34:259-274, 2002). Its covariance structure is described through a variation matrix, which induces the so called total variation as a measure of dispersion. The aim of the paper is to show that its sample counterpart has theoretical properties, corresponding to the standard multivariate case, like unbiasedness and convergence in probability. Moreover, its distribution in the case of normality on the simplex is developed. [ABSTRACT FROM AUTHOR]