Unit-sum Constraint Research Articles

A Multivariate Mixture Regression Model for Constrained Responses

Compositional data are vectors typically representing proportions of a whole, that is, those whose elements are strictly positive and subject to a unit-sum constraint. The increasing number of fields where this type of data arises makes the development of proper statistical tools an important issue. From a regression perspective, whenever the multivariate response is a compositional vector, a proper model that accounts for the unit-sum constraint is the well-established Dirichlet regression model. However, there are significant drawbacks mainly due to the limited flexibility of the Dirichlet distribution. The aim of this contribution is to introduce a new multivariate regression model for constrained responses, that is based on the extended flexible Dirichlet distribution (which is a structured mixture with Dirichlet distributed components). The new model is obtained by adopting a novel reparameterization which allows for, among other things, the presence of suitably designed cluster-specific regression patterns. It is shown to provide considerably greater flexibility and better performance than the standard Dirichlet regression model. In particular, from theoretical analysis, intensive simulation studies in many challenging scenarios, as well as from a real data application, it emerges that the new regression model can handle several issues affecting the Dirichlet regression, such as the presence of outliers, latent groups, multi-modality, and positive correlations.

Bayesian Variable Shrinkage and Selection in Compositional Data Regression: Application to Oral Microbiome

Microbiome studies generate multivariate compositional responses, such as taxa counts, which are strictly non-negative, bounded, residing within a simplex, and subject to unit-sum constraint. In presence of covariates (which can be moderate to high dimensional), they are popularly modeled via the Dirichlet-Multinomial (D-M) regression framework. In this paper, we consider a Bayesian approach for estimation and inference under a D-M compositional framework, and present a comparative evaluation of some state-of-the-art continuous shrinkage priors for efficient variable selection to identify the most significant associations between available covariates, and taxonomic abundance. Specifically, we compare the performances of the horseshoe and horseshoe+ priors (with the benchmark Bayesian lasso), utilizing Hamiltonian Monte Carlo techniques for posterior sampling, and generating posterior credible intervals. Our simulation studies using synthetic data demonstrate excellent recovery and estimation accuracy of sparse parameter regime by the continuous shrinkage priors. We further illustrate our method via application to a motivating oral microbiome data generated from the NYC-Hanes study. RStan implementation of our method is made available at the GitHub link: (https://github.com/dattahub/compshrink).

Unveiling and Unraveling Aggregation and Dispersion Fallacies in Group MCDM

Priorities in multi-criteria decision-making (MCDM) convey the relevance preference of one criterion over another, which is usually reflected by imposing the non-negativity and unit-sum constraints. The processing of such priorities is different than other unconstrained data, but this point is often neglected by researchers, which results in fallacious statistical analysis. This article studies three prevalent fallacies in group MCDM along with solutions based on compositional data analysis to avoid misusing statistical operations. First, we use a compositional approach to aggregate the priorities of a group of DMs and show that the outcome of the compositional analysis is identical to the normalized geometric mean, meaning that the arithmetic mean should be avoided. Furthermore, a new aggregation method is developed, which is a robust surrogate for the geometric mean. We also discuss the errors in computing measures of dispersion, including standard deviation and distance functions. Discussing the fallacies in computing the standard deviation, we provide a probabilistic criteria ranking by developing proper Bayesian tests, where we calculate the extent to which a criterion is more important than another. Finally, we explain the errors in computing the distance between priorities, and a clustering algorithm is specially tailored based on proper distance metrics.

Clustering compositional data using Dirichlet mixture model.

A model-based clustering method for compositional data is explored in this article. Most methods for compositional data analysis require some kind of transformation. The proposed method builds a mixture model using Dirichlet distribution which works with the unit sum constraint. The mixture model uses a hard EM algorithm with some modification to overcome the problem of fast convergence with empty clusters. This work includes a rigorous simulation study to evaluate the performance of the proposed method over varied dimensions, number of clusters, and overlap. The performance of the model is also compared with other popular clustering algorithms often used for compositional data analysis (e.g. KMeans, Gaussian mixture model (GMM) Gaussian Mixture Model with Hard EM (Hard GMM), partition around medoids (PAM), Clustering Large Applications based on Randomized Search (CLARANS), Density-Based Spatial Clustering of Applications with Noise (DBSCAN) etc.) for simulated data as well as two real data problems coming from the business and marketing domain and physical science domain, respectively. The study has shown promising results exploiting different distributional patterns of compositional data.

Decomposing structural decomposition: The role of changes in individual industry shares

Decomposition studies typically present structural change effect – such as the impact of changes in industry output shares on energy demand – as a single, aggregate contribution. However, if the purpose is to identify the role of individual sectors, an alternative approach is required. The generalised Divisia index method (GDIM) proposed by Vaninsky (2014) allows to decompose the structural change effect by attributing its parts to changes in individual shares. At the same time, GDIM takes into account interdependence of individual shares, stemming from the unit-sum constraint. In this paper we propose an arguably simpler and intuitive formulation of such a decomposition, building on the Harrison, Horridge, and Pearson (2000) decomposition method. This formulation also provides a rationale for the interpretation of individual share effects. We demonstrate that unravelling the pattern of structural change in a formal way can lead to identifying a few expanding or contracting activities as key contributors to change in energy use. This point is supported by an illustrative application, in which we decompose changes in electricity and heat demand in the European Union in the years 2000–2014.

Modeling Longitudinal Microbiome Compositional Data: A Two-Part Linear Mixed Model with Shared Random Effects

Longitudinal microbiome studies have been widely used to unveil the dynamics in the complex host-microbial ecosystems. Modeling the longitudinal microbiome compositional data, which is semi-continuous in nature, is challenging in several aspects: the overabundance of zeros, the heavy skewness of non-zero values that are bounded in (0, 1), and the dependence between the binary and non-zero parts. To deal with these challenges, we first extended the work of Chen and Li [1] and proposed a two-part zero-inflated Beta regression model with shared random effects (ZIBR-SRE), which characterize the dependence between the binary and the continuous parts. Besides, the microbiome compositional data have unit-sum constraint, indicating the existence of negative correlations among taxa. As ZIBR-SRE models each taxon separately, it does not satisfy the sum-to-one constraint. We then proposed a two-part linear mixed model (TPLMM) with shared random effects to formulate the log-transformed standardized relative abundances rather than the original ones. Such transformation is called “additive logistic transformation”, initially developed for cross-sectional compositional data. We extended it to analyze the longitudinal microbiome compositions and showed that the unit-sum constraint can be automatically satisfied under the TPLMM framework. Model performances of TPLMM and ZIBR-SRE were compared with existing methods in simulation studies. Under settings adopted from real data, TPLMM had the best performance and is recommended for practical use. An oral microbiome application further showed that TPLMM and ZIBR-SRE estimated a strong correlation structure in the binary and the continuous parts, suggesting models without accounting for this dependence would lead to biased inferences.

Convex clustering method for compositional data modeling

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.

Identification of microbial interaction network: zero-inflated latent Ising model based approach

BackgroundThroughout their lifespans, humans continually interact with the microbial world, including those organisms which live in and on the human body. Research in this domain has revealed the extensive links between the human-associated microbiota and health. In particular, the microbiota of the human gut plays essential roles in digestion, nutrient metabolism, immune maturation and homeostasis, neurological signaling, and endocrine regulation. Microbial interaction networks are frequently estimated from data and are an indispensable tool for representing and understanding the conditional correlation between the microbes. In this high-dimensional setting, zero-inflation and unit-sum constraint for relative abundance data pose challenges to the reliable estimation of microbial interaction networks.Methods and ResultsTo identify the microbial interaction network, the zero-inflated latent Ising (ZILI) model is proposed which assumes the distribution of relative abundance relies only on finite latent states and provides a novel way to solve issues induced by the unit-sum and zero-inflation constrains. A two-step algorithm is proposed for the model selection of ZILI. ZILI is evaluated through simulated data and subsequently applied to an infant gut microbiota dataset from New Hampshire Birth Cohort Study. The results are compared with results from Gaussian graphical model (GGM) and dichotomous Ising model (DIS). Providing ZILI is the true data-generating model, the simulation studies show that the two-step algorithm can identify the graphical structure effectively and is robust to a range of parameter settings. For the infant gut microbiota dataset, the final estimated networks from GGM and ZILI turn out to have significant overlap in which the ZILI tends to select the sparser network than those from GGM. From the shared subnetwork, a hub taxon Lachnospiraceae is identified whose involvement in human disease development has been discovered recently in literature.ConclusionsConstrains induced by relative abundance of microbiota such as zero inflation and unit sum render the conditional correlation analysis unreliable for conventional methods such as GGM. The proposed optimal categoricalization based ZILI model provides an alternative yet elegant way to deal with these difficulties. The results from ZILI have reasonable biological interpretation. This model can also be used to study the microbial interaction in other body parts.

Large Covariance Estimation for Compositional Data Via Composition-Adjusted Thresholding

ABSTRACTHigh-dimensional compositional data arise naturally in many applications such as metagenomic data analysis. The observed data lie in a high-dimensional simplex, and conventional statistical methods often fail to produce sensible results due to the unit-sum constraint. In this article, we address the problem of covariance estimation for high-dimensional compositional data and introduce a composition-adjusted thresholding (COAT) method under the assumption that the basis covariance matrix is sparse. Our method is based on a decomposition relating the compositional covariance to the basis covariance, which is approximately identifiable as the dimensionality tends to infinity. The resulting procedure can be viewed as thresholding the sample centered log-ratio covariance matrix and hence is scalable for large covariance matrices. We rigorously characterize the identifiability of the covariance parameters, derive rates of convergence under the spectral norm, and provide theoretical guarantees on support recovery. Simulation studies demonstrate that the COAT estimator outperforms some existing optimization-based estimators. We apply the proposed method to the analysis of a microbiome dataset to understand the dependence structure among bacterial taxa in the human gut.

Bio-CoDa: A new analysis class to ensure accurate and precise monoclonal antibody CQA estimation and control.

Monoclonal antibody production processes control critical quality attributes (CQAs), which are the measures that provide proof of a product's identity and quality. Critical decisions rely on the accuracy and precision of these measures, as well as their appropriate statistical treatment. Many measures require special attention. For example, the charge heterogeneity CQA measured by ionic exchange chromatography reports proportions or percentages of the total integrated peak area of known species. Since proportions sum to a constant (1 or 100%), they fall into a special class of data called compositional data that have a unit sum constraint and therefore an inherent correlation. However, these measures are often analyzed assuming independence which is incorrect. Estimating statistics with incorrect assumptions can lead to inferential failures (e.g., shelf life failures), or can lead to missing important structural patterns in the data. Presented here is a new class of analysis methods for CQAs compositional data called Biologic Compositional Data Analysis (Bio-CoDa). The method is based on the elegant solution to analysis issues by Aitchison (1986). An introduction to the Bio-CoDa methods with rational is presented as well as examples demonstrating its strengths. Biotechnol. Bioeng. 2017;114: 2001-2010. © 2017 Wiley Periodicals, Inc.

Composite likelihood approach to the regression analysis of spatial multivariate ordinal data and spatial compositional data with exact zero values

In many environmental and ecological studies, it is of interest to model compositional data. One approach is to consider positive random vectors that are subject to a unit-sum constraint. In landscape ecological studies, it is common that compositional data are also sampled in space with some elements of the composition absent at certain sampling sites. In this paper, we first propose a practical spatial multivariate ordered probit model for multivariate ordinal data, where the response variables can be viewed as the discretized non-negative compositions without the unit-sum constraint. We then propose a novel two-stage spatial mixture Dirichlet regression model. The first stage models the spatial dependence and the presence of exact zero values, and the second stage models all the non-zero compositional data. A maximum composite likelihood approach is developed for parameter estimation and inference in both the spatial multivariate ordered probit model and the two-stage spatial mixture Dirichlet regression model. The standard errors of the parameter estimates are computed by an estimate of the Godambe information matrix. A simulation study is conducted to evaluate the performance of the proposed models and methods. A land cover data example in landscape ecology further illustrates that accounting for spatial dependence can improve the accuracy in the prediction of presence/absence of different land covers as well as the magnitude of land cover compositions.

BEMMA: A Hierarchical Bayesian End-Member Modeling Analysis of Sediment Grain-Size Distributions

Sediment grain-size distributions provide rich information about sedimen- tary dynamics and potentially about environmental and climatic changes. However, entrainment, transport, and deposition, as a sequence of sorting process, modify original grain-size distributions of source materials, thereby resulting in complex distribution forms that are commonly multimodal and asymmetrical. However, nei- ther traditional descriptive statistics nor curving fitting methods are able to address this complexity fully. End-member modeling analysis, essentially based on polytope expansion, stands out as a flexible and robust method for the unmixing of sediment grain-size distributions. Yet there are still several key issues that remain unresolved. Here a hierarchical Bayesian end-member modeling analysis of grain-size distribu- tions, fully subject to the non-negative and unit-sum constraints on the distributions, is presented. Within the Bayesian framework, the number of end members, as well as the end-member spectra and fractions can be inferred sequentially using a reversible- jump Markov chain Monte Carlo algorithm in conjunction with Gibbs samplers. Test runs using both a synthetic and a real-world dataset from a small playa located on the southern margin of the Badain Jaran Desert, NW China, reveal that this model

A stochastic diffusion process for Lochner's generalized Dirichlet distribution

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner's generalized Dirichlet distribution [R. H. Lochner, “A generalized Dirichlet distribution in Bayesian life testing,” J. R. Stat. Soc. Ser. B (Methodol.) 37(1), 103–113 (1975)] as its asymptotic solution. Individual samples of a discrete ensemble, obtained from the system of stochastic differential equations, equivalent to the Fokker-Planck equation developed here, satisfy a unit-sum constraint at all times and ensure a bounded sample space, similarly to the process developed in [J. Bakosi and J. R. Ristorcelli, “A stochastic diffusion process for the Dirichlet distribution,” Int. J. Stoch. Anal. 2013, 7]. Consequently, the generalized Dirichlet diffusion process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Compared to the Dirichlet distribution and process, the additional parameters of the generlized Dirichlet distribution allow a more general class of physical processes to be modeled with a more general covariance matrix.

Spatial Regression Modeling for Compositional Data With Many Zeros

Compositional data analysis considers vectors of nonnegative-valued variables subject to a unit-sum constraint. Our interest lies in spatial compositional data, in particular, land use/land cover (LULC) data in the northeastern United States. Here, the observations are vectors providing the proportions of LULC types observed in each 3 km×3 km grid cell, yielding order 104 cells. On the same grid cells, we have an additional compositional dataset supplying forest fragmentation proportions. Potentially useful and available covariates include elevation range, road length, population, median household income, and housing levels. We propose a spatial regression model that is also able to capture flexible dependence among the components of the observation vectors at each location as well as spatial dependence across the locations of the simplex-restricted measurements. A key issue is the high incidence of observed zero proportions for the LULC dataset, requiring incorporation of local point masses at 0. We build a hierarchical model prescribing a power scaling first stage and using latent variables at the second stage with spatial structure for these variables supplied through a multivariate CAR specification. Analyses for the LULC and forest fragmentation data illustrate the interpretation of the regression coefficients and the benefit of incorporating spatial smoothing.

A Stochastic Diffusion Process for the Dirichlet Distribution

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.

Land Cover Type and Fire in Portugal: Do Fires Burn Land Cover Selectively?

The purpose of this study is to investigate if, or under what conditions, fires select given land cover types for burning. If fires burn unselectively then the land cover composition (the proportional area of various land cover types) of individual fires should approximate the land cover composition available in their neighborhood. In this study we test this hypothesis by performing statistical analyses of a data set consisting of paired vectors with the proportions of land cover types present in burned areas and in their respective surroundings. The statistical methods employed (a permutation technique and the Cmax statistic) are commonly used in resource selection studies where data is subject to a unit-sum constraint. The results of the analysis of 506 fires that burned in Portugal in 1991 indicate that fires are selective, with small fires exhibiting stronger land cover preferences than large fires. According to the results of a multiple comparison analysis performed for small fires, there is a marked preference for shrubland followed by other forest cover types, while agriculture is clearly avoided. A similar analysis is performed to test if fire selectivity is related to the ecological region where it occurs. The results obtained in this study contribute to the discussion on the relative importance of fuels as a drivers of fire spread.

A semiparametric model for compositional data analysis in presence of covariates on the simplex

Compositional data occur as natural realizations of multivariate observations comprising element proportions of some whole quantity. Such observations predominate in disciplines like geology, biology, ecology, economics and chemistry. Due to unit sum constraint on compositional data, specialized statistical methods are required for analyzing these data. Dirichlet distributions were originally used to study compositional data even though this family of distribution is not appropriate (see Aitchison, 1986) because of their extreme independence properties. Aitchison (1982) endeavored to provide a viable alternative to existing methods by employing Logistic Normal distribution to analyze such constrained data. However this family does not include the Dirichlet class and is therefore unable to address the issue of extreme independence. In this paper generalized Liouville family is investigated to model compositional data which includes covariates. This class permits distributions that admit negative or mixed correlation and also contains non-Dirichlet distributions with non-positive correlation and overcomes deficits in the Dirichlet class. Semiparametric Bayesian methods are proposed to estimate the probability density. Predictive distributions are used to assess performance of the model. The methods are illustrated on a real data set.

A single-step method for identifying individual resources

Many methods of analysis have been used to test whether animals use resources in proportion to availability. Many of the most commonly used methods have been criticized because they ignore the induced correlation among cells in the multinomial classification table due to the unit-sum constraint. Some methods are also flawed because they assume that relocation data can be pooled across animals even though the animals may select habitats differently. We propose a method, based on the maximum of the joint chi-square, to compare habitat usage to availability for individual animals. This method is combined with a follow-up multiple comparison technique that takes account of the unit-sum constraint. We compare our proposed method to a method based on the sum of individual chi-squares and a method based on compositional analysis by analyzing data on gray partridge and ringnecked pheasant resource use. Simulation is also used to compare methods by evaluating their Type I error rates.

Quantitative analysis of detrital modes: statistically rigorous confidence regions in ternary diagrams and their use in sedimentary petrology

Detrital modes are the basic type of quantitative data used in sedimentary petrology. They represent estimates of the petrographic framework composition of sand or sandstone obtained by point counting of thin sections. At present, geological inferences from analysis of these numerical data are semi-quantitative only, because many of the data-processing methods employed in sedimentary petrology lack a firm theoretical basis. A review of existing methods reveals a number of fundamental statistical problems associated with the use of univariate statistics and the construction of so-called hexagonal fields of compositional variation. It is shown that these problems can be overcome by using multivariate methods that honour the non-negativity and unit-sum constraints on compositional data, and incorporate the covariance structure. Spread in compositional data can be modelled on two levels: (1) by regarding the grain as unit of observation, and the total point count as a sample from a multinomial distribution; (2) by regarding the total count as unit of observation, and a set of counts as a sample from an additive logistic normal distribution. Numerical methods are presented for constructing statistically rigorous confidence regions in ternary diagrams by inversion of significance tests for these two distributions. Statistical analyses of the framework composition and heavy-mineral assemblages of modern beach and river sands illustrate their use. The case histories are followed by a brief overview of popular multivariate methods for reconstructing sediment provenance.

Twenty-four-hour feeding and lying patterns of wild reindeer Rangifer tarandus tarandus in summer

We tested whether sunlight and insect harassment were important proximate factors (Zeitgebers) controlling feeding and lying patterns of wild reindeer (Rangifer tarandus tarandus) in summer. With no insect harassment, feeding and lying each exhibited a polycyclic rhythm with, on average, 4 bouts at equal intervals during each 24-h period (unit-sum constraints in a polycyclic rhythm). When insects were absent, the activity percentages were 50 ± 4% feeding (mean ± SE) and 33 ± 4% lying for daytime and 44 ± 3% feeding and 47 ± 4% lying for nighttime, i.e., the percentage of time spent feeding was equal for daytime and nighttime (nychthemeral), while for lying it was not. With insects present, percentages of time spent feeding and lying were 18 ± 3 and 13 ± 3% for daytime and 45 ± 8 and 41 ± 8% for nighttime, respectively, i.e., neither feeding nor lying exhibited a nychthemeral pattern. During periods of insect harassment, 24-h feeding patterns had no cyclic rhythm, whereas lying remained cyclic, but the percentages of time spent lying during peak bouts (amplitude) and at peak duration (period) were unequal between night and day. Reindeer did not attune their activity to the set points of sunrise and sunset (not crepuscular), demonstrating that daylight is most likely not a powerful Zeitgeber for reindeer during summer.