Isometric Log-ratio Transformation Research Articles

A comparison of isometric and amalgamation logratio balances in compositional data analysis

The isometric logratio transformation, in the form of what has been called a “balance”, has been promoted as a way to contrast two groups of parts in a compositional data set by forming ratios of their respective geometric means. This transformation has attractive theoretical properties and hence provides a useful reference, but geometric means are highly affected by parts with small relative values. When a comparison between two groups of parts is required in practical applications, such as the investigation and construction of models, while making use of substantive domain knowledge, it is demonstrated that the logratio of two amalgamations serves as an alternative, interpretable form of balance. A geochemical data set is considered, which has been analyzed previously by transforming to a set of isometric logratio balances. An alternative approach, using a reduced set of pairwise logratios of parts, optionally involving prescribed amalgamations, is very close to optimal in accounting for the variance in this compositional data set. These simpler transformations also have an exact back-transformation to the original parts. This approach highlights for this dataset which compositional parts are driving the data structure, using variables that are easy to interpret and that map well to research-driven objectives.

Compositional Associations of Sleep and Activities within the 24-h Cycle with Cardiometabolic Health Markers in Adults.

This study aimed to examine how compositions of 24-h time use and time reallocations between movement behaviors are associated with cardiometabolic health in a population-based sample of middle-age Finnish adults. Participants were 3443 adults 46 yr of age from the Northern Finland Birth Cohort 1966 study. Participants wore a hip-worn accelerometer for 14 d from which time spent in sedentary behavior (SB), light-intensity physical activity (LPA), and moderate- to vigorous-intensity physical activity (MVPA) were determined. These data were combined with self-reported sleep to obtain the 24-h time-use composition. Cardiometabolic outcomes included adiposity markers, blood lipid levels, and markers of glucose control and insulin sensitivity. Multivariable-adjusted regression analysis, using a compositional data analysis approach based on isometric log-ratio transformation, was used to examine associations between movement behaviors with cardiometabolic outcomes. More daily time in MVPA and LPA, relative to other movement behaviors, was consistently favorably associated with all cardiometabolic outcomes. For example, relative to time spent in other behaviors, 30 min·d-1 more MVPA and LPA were both associated with lower 2-h post-glucose load insulin level (-11.8% and -2.7%, respectively). Relative to other movement behaviors, more daily time in SB was adversely associated with adiposity measures, lipid levels, and markers of insulin sensitivity, and more daily time asleep was adversely associated with adiposity measures, blood lipid, fasting plasma glucose, and 2-h insulin. For example, 60 min·d-1 more SB and sleep relative to the remaining behaviors were both associated with higher 2-h insulin (3.5% and 5.7%, respectively). Altering daily movement behavior compositions to incorporate more MVPA at the expense of any other movement behavior, or more LPA at the expense of SB or sleep, could help to improve cardiometabolic health in midadulthood.

Regional-scale mapping for determining geochemical background values in soils of the Itacaiúnas River Basin, Brazil: The use of compositional data analysis (CoDA)

This study aims at providing high quality soil geochemical data at a regional scale (1 sample per 25 km2) to understand the source and distribution of potentially toxic elements (PTEs: Al, As, Ba, Cd, Cu, Co, Cr, Fe, Hg, Mo, Mn, Ni, Pb, Se, U, V, and Zn) and to define their geochemical background threshold values (GBTVs) in the Itacaiúnas River Basin (IRB), Brazilian Amazon. A total of 2958 soil samples were collected at two depths (surface: 0–10 cm; bottom: 30–50 cm) and the fine fractions (<0.117 mm) were analyzed by ICP-MS and ICP-AES after aqua-regia digestion. The compositional nature of geochemical data was evaluated using the log-ratio approach and compared with traditional methods. Results show that compared to raw/log-transformation, centred log-ratio (clr) transformation improves spatial mapping and also better separated variables in Principal Component Analysis (PCA). Correlation analysis based on the isometric log-ratio (ilr) transformation in heat-maps provides reliable and better structure of the results. The spatial maps (clr) indicate that most of the elements in surface and bottom soils are mainly governed by underlying bedrock geology (parent lithologies and mineralization), while anthropogenic factors are insignificant. The resulting clr-based PCA and cluster analyses associated with geological information indicate that geochemical patterns of Fe-Ti-V-Cr-Sc can be linked to mafic to intermediate metavolcanic rocks and banded-iron formations, which are dominant in the Carajás basin, while Ni and Cr distribution pattern is attributed to mafic and ultramafic lithologies, and Cu anomalous values are mainly related to hydrothermal mineralized copper belts. Among the statistical methods applied for estimation of GBTVs, TIF produced the highest BTVs, followed by 98th and mMAD, with few exceptions, while mMAD was considered the more suitable for defining GBTVs. The number of potential anomalies of PTEs identified by these methods follow the order of MAD > 98th > TIF or MAD > TIF > 98th, and overall the anomalies are mostly related to the local lithology/mineralization with no indication of anthropogenic contamination. Compared with Brazilian guidelines (CONAMA Resolution No. 420/2009), the GBTVs determined for Cr, Co, Ni, and Cu by TIF, MAD, and 98th percentile mostly exceeded both IV and PV limits. However, these high values are due to the influence of local lithology. This reveals that the Brazilian CONAMA guidelines of these elements for the IRB are unrealistic because that suggests a large number of sites requiring investigation, which is not consistent with the geochemical evidence. Therefore, site-specific GBTVs must be considered for the accurate evaluation of anthropogenic contamination, as well as the ecological and human health risks of PTEs in this region.

Orthonormal balances as a means of characterizing dietary exposure

Evaluating the effect of nutritional components on health outcomes is hampered by technical difficulties that are closely related to the compositional nature of dietary data. Compositional data consist of parts of a whole and, on the basis of the assumption that any relevant information they provide is contained in the ratios between them, procedures based on log-ratio transformations have been proposed for their analysis. It has been previously suggested that nutrient compositions can be analyzed using isometric log-ratio transformations which allow the compositions to be represented by a system of orthogonal coordinates corresponding to the log-contrasts between the compositional parts. One particular expression of these contrasts is called balances, and the aim of this note is to provide a step-by-step description of how to construct orthonormal balances and show how they can be applied to dietary data by means of an example based on data coming from an Italian population-based study. Nutrient balances are new variables that can be included as covariates in regression models. After fitting a linear regression model using estimates of relative fat mass as a response variable, scores indicating the obesogenic potential of foods can be defined on the basis of their nutritional contents and the estimated effect of the balances on relative fat mass. This approach to evaluating the relationship between diet and health outcomes provides insights into the roles of specific nutrient ratios and foods within a comprehensive and integrated framework that complies with the essentially compositional nature of dietary data.

Preprocessing alternatives for compositional data related to water, sanitation and hygiene

The Sustainable Development Goals (SDGs) 6.1 and 6.2 measure the progress of urban and rural populations in their access to different levels of water, sanitation and hygiene (WASH) services, based on multiple sources of information. Service levels add up to 100%; therefore, they are compositional data (CoDa). Despite evidence of zero value, missing data and outliers in the sources of information, the treatment of these irregularities with different statistical techniques has not yet been analyzed for CoDa in the WASH sector. Thus, the results may present biased estimates, and the decisions based on these results will not necessarily be appropriate. In this article, we therefore: i) evaluate methodological imputation alternatives that address the problem of having either zero values or missing values, or both simultaneously; and ii) propose the need to complement the point-to-point identification of the WHO/UNICEF Joint Monitoring Program (JMP) with other robust alternatives, to deal with outliers depending on the number of data points. These suggestions have been considered here using statistics for CoDa with isometric log-ratio (ilr) transformation. A selection of illustrative cases is presented to compare performance of different alternatives.

Investigation of Adiposity Measures and Operational Taxonomic unit (OTU) Data Transformation Procedures in Stool Samples from a German Cohort Study Using Machine Learning Algorithms.

The analysis of the gut microbiome with respect to health care prevention and diagnostic purposes is increasingly the focus of current research. We analyzed around 2000 stool samples from the KORA (Cooperative Health Research in the Region of Augsburg) cohort using high-throughput 16S rRNA gene amplicon sequencing representing a total microbial diversity of 2089 operational taxonomic units (OTUs). We evaluated the combination of three different components to assess the reflection of obesity related to microbiota profiles: (i) four prediction methods (i.e., partial least squares (PLS), support vector machine regression (SVMReg), random forest (RF), and M5Rules); (ii) five OTU data transformation approaches (i.e., no transformation, relative abundance without and with log-transformation, as well as centered and isometric log-ratio transformations); and (iii) predictions from nine measurements of obesity (i.e., body mass index, three measures of body shape, and five measures of body composition). Our results showed a substantial impact of all three components. The applications of SVMReg and PLS in combination with logarithmic data transformations resulted in considerably predictive models for waist circumference-related endpoints. These combinations were at best able to explain almost 40% of the variance in obesity measurements based on stool microbiota data (i.e., OTUs) only. A reduced loss in predictive performance was seen after sex-stratification in waist–height ratio compared to other waist-related measurements. Moreover, our analysis showed that the contribution of OTUs less prevalent and abundant is minor concerning the predictive power of our models.

Comparison of additive and isometric log-ratio transformations combined with machine learning and regression kriging models for mapping soil particle size fractions

Digital soil mapping approaches relating to the soil particle size fractions (psf) face the challenge around how to establish the statistical or geostatistical models from large sets of environmental variables, especially in a situation with sparse soil profile data. Recently, many machine learning (ML) models have sprung up with advantages over statistical models. However, few studies focused on the comprehensive comparative analyses between ML and geostatistical models in the soil psf mapping. And the exploration of optimal combination of data transformation and model simulation was even less. Therefore, two transformed methods such as additive log-ratio (ALR) and isometric log-ratio (ILR) transformations combine with two ML models such as boosted regression tree (BRT), random forest (RF) and a classic geostatistical model of regression kriging (RK) were implemented to map soil psf in the Heihe River basin, China. A total of 640 samples and thirteen scorpan factors were collected and used for the comprehensive comparative analysis. Results showed that the scorpan factors such as temperature, precipitation, elevation, soil type, soil organic carbon, vegetation types and normalized difference vegetation index had important impacts on the soil psf mapping. ILR transformation was better than ALR transformation with advantage of improving stability of data distributions and ML models could also improve the mapping performance in comparison with RK models for better handling candidate factors. For these ML models, the RF models had better accuracy performance than the BRT models. In contrast, ILR transformation combined with RF model (ILR_RF) had the best performance, with the lowest root mean square error values (sand, 15.35%; silt, 14.20%; and clay, 6.66%), Aitchison distance value (0.86), standardized residual sum of squares value (0.60), and the highest concordance correlation coefficient value (0.73) and coefficient of determination value (56.69%) for clay content. In addition, ILR_RF had a relatively higher right ratio of soil texture type (68.44%) and better predict performance for most soil texture types. The predicted maps generated from ILR_RF presented more reasonable and smoother transitions. In the future, more ML models should be explored and more variables related to soil psf should be introduced into the models to improve the predictive performance.

A Novel Method to Visualize the Dietary Macronutrient Composition of Smaller Visceral Fat Accumulation.

The accumulation of visceral fat is considered a potential cause of a clustering of metabolic disorders including hypertension, hyperglycemia, and dyslipidemia. These disorders are some of the upstream determinants of serious diseases such as coronary heart disease, cerebrovascular disease, and dementia. In particular, the accumulation of visceral fat is considered to have a causal relationship with dietary habits. To clarify this relationship, we characterize dietary habits with dietary macronutrient composition and visceral fat accumulation with a measure of visceral fat area (VFA). We then employ a novel multiple regression model with VFA as the objective variable and macronutrient composition, gender, and age group as explanatory variables. The macronutrient composition is converted by the isometric log-ratio transformation since it is compositional data. The squared term of the transformed macronutrient composition is also included as an explanatory variable. To fit the data to the model, variable selection is performed based on Akaike's information criterion to exclude unnecessary interaction terms. The validity of the model is confirmed by a numerical simulation study. We then cross-sectionally analyze real-world data collected through community-wide health examinations of adults living in the Iwaki district in northern Japan. The macronutrient composition data is taken by the dietary history questionnaire and VFA is measured using a bioimpedance-type visceral fat meter. The main factors of macronutirent composition and their interactions with gender and age group are identified through analysis of variance and are significantly associated with VFA (p < 0.05). Moreover, the predicted VFA corresponding to the macronutrient composition stratified by gender and age group are obtained, and visualized seamlessly on a ternary plot. The results show that a diet with a high ratio of %protein to %fat generally corresponds to a lower VFA level. However, in middle-aged female subjects, higher VFA is found in lower %fat and higher %carbohydrate diets. In summary, the association between VFA and dietary macronutrient composition is significantly modulated depending on gender and age group in Iwaki district's adult population. The novel statistical analysis method in this study is useful in exploring favorable dietary macronutrient composition for lower level of visceral fat accumulation.

A knowledge-driven way to interpret the isometric log-ratio transformation and mixture distributions of geochemical data

When extracting geo-information from geochemical data, it is essential to consider the compositional nature and geological signatures of the data. Isometric log-ratio transformation (ILR) produces an orthonormal basis of geochemical data and accounts for the compositional nature of the data. However, it still often remains difficult to interpret ILR-transformed variables because of the lack of a geological basis for a data-driven approach; therefore, it is necessary to find some geological knowledge-based criteria to help enable a more understandable interpretation following ILR transformation. Characterized by certain elements and elemental ratios, the chronological order of geological processes can be used to construct interpretable ILR transformation. This concept was applied to extract geo-information from stream sediment geochemical data in the Duolong mineral district, Tibet, China. Furthermore, the expectation-maximization (EM) algorithm modified by a minimum message length criterion (MML) was employed to investigate mixture distributions of the geochemical data. In the study area, mafic rocks with high concentrations of Cr and Ni were emplaced earlier than the porphyry Cu–Au deposits which have stable Cu/Au ratios. Based on these criteria and hierarchical cluster analysis, sequential binary partition was constructed among the Cu, Au, Cr, and Ni concentrations. The ILR-transformed variables follow either a bi-normal or tri-normal distribution, the subpopulations of which were interpreted as fingerprints inherited from mafic magmatic processes, Cu–Au hydrothermal systems, and later geological processes, respectively. The high-average subpopulation of ILR-transformed Cu corresponds to anomalies associated with the porphyry and epithermal Cu–Au systems, and two areas are predicted to have high Cu–Au mineralization potential. This study demonstrates that geological knowledge-driven ILR transformation is a promising way to efficiently extract geo-information from geochemical data.

Serum cholesterol elasticities in relation to macro- and micronutrient balances.

The application of the isometric log-ratio (ilr) transformation to dietary data leads to the definition of nutrient balances that represent new variables which can be included in regression models as covariates. However, their effects on the response variable are not easy to quantify. We then propose borrowing the concept of elasticity from econometrics as an appealing means of improving the interpretability of the ilr-related coefficients. Using data from an Italian population-based study, a multiple linear regression model of the serum total-/HDL-cholesterol ratio on nutrient balances was fitted and the estimated coefficients were used to derive elasticities. The elasticities measure the relative changes in the cholesterol ratio in response to the relative changes in specific nutrient ratios, while keeping unchanged the proportional relationships between the other dietary elements. In a comprehensive overview of the diet, this alternative approach to dietary data analysis allows isocaloric analysis and may provide interesting new insights.

ILRA: Novelty Detection in Face-Based Intervener Re-Identification

Transparency laws facilitate citizens to monitor the activities of political representatives. In this sense, automatic or manual diarization of parliamentary sessions is required, the latter being time consuming. In the present work, this problem is addressed as a person re-identification problem. Re-identification is defined as the process of matching individuals under different camera views. This paper, in particular, deals with open world person re-identification scenarios, where the captured probe in one camera is not always present in the gallery collected in another one, i.e., determining whether the probe belongs to a novel identity or not. This procedure is mandatory before matching the identity. In most cases, novelty detection is tackled applying a threshold founded in a linear separation of the identities. We propose a threshold-less approach to solve the novelty detection problem, which is based on a one-class classifier and therefore it does not need any user defined threshold. Unlike other approaches that combine audio-visual features, an Isometric LogRatio transformation of a posteriori (ILRA) probabilities is applied to local and deep computed descriptors extracted from the face, which exhibits symmetry and can be exploited in the re-identification process unlike audio streams. These features are used to train the one-class classifier to detect the novelty of the individual. The proposal is evaluated in real parliamentary session recordings that exhibit challenging variations in terms of pose and location of the interveners. The experimental evaluation explores different configuration sets where our system achieves significant improvement on the given scenario, obtaining an average F measure of 71.29% for online analyzed videos. In addition, ILRA performs better than face descriptors used in recent face-based closed world recognition approaches, achieving an average improvement of 1.6% with respect to a deep descriptor.

Testing for mediation effect with application to human microbiome data.

Mediation analysis has been commonly used to study the effect of an exposure on an outcome through a mediator. In this paper, we are interested in exploring the mediation mechanism of microbiome, whose special features make the analysis challenging. We consider the isometric logratio transformation of the relative abundance as the mediator variable. Then, we present a de-biased Lasso estimate for the mediator of interest and derive its standard error estimator, which can be used to develop a test procedure for the interested mediation effect. Extensive simulation studies are conducted to assess the performance of our method. We apply the proposed approach to test the mediation effect of human gut microbiome between the dietary fiber intake and body mass index.

A Bayesian analysis for pseudo-compositional data with spatial structure.

We proposed a Bayesian analysis of pseudo-compositional data in presence of a latent factor, assuming a spatial structure. This development was motivated by a dataset containing information on the number of newborns of primiparous mothers living in each of the microregions of the state of Sao Paulo, Brazil, in the year of 2015, stratified by the age of the mothers (15-18, 19-29 and 30 years or more). Considering that data on newborns are not stochastically distributed among the three age groups, but they are explained in relation to women's population structure, we adopted the expression "pseudo-compositional data" to refer to this data structure. The hypothesis of interest establishes that the age of the first pregnancy is associated with the economic conditions of the geographic area where the mother lives. The incidence of poverty was included as an independent variable. Additive log-ratio (alr) and isometric log-ratio (ilr) transformations were considered, as is usually done in the analysis of compositional data. The model included a random effect related to the spatial effect assumed to have a conditional autoregressive structure. A Bayesian Markov Chain Monte Carlo (MCMC) simulation procedure was used to get the posterior summaries of interest. The model based on the (ilr) transformation was well fitted to the data, showing that in the microregions with the highest incidence of poverty, there are higher proportions of women who have their first child in adolescence, while in the microregions with the lowest incidence of poverty, there are higher proportions of women who have their first child after the age of 30 years. From these results it is possible to conclude that this Bayesian approach was very useful in the estimation of the parameters of the proposed model. The proposed method should have a broad application to other problems involving pseudo-compositional data.

Ionome balance analysis of mango fruit from orchard with and without resin canal discolouration

Resin canal discolouration (RCD) NDASH a quality defect that appears during fruit ripening, reduces the market price and consumer appeal of mango fruits. Abiotic and biotic factors such as fruit maturity, time of harvest after rainfall event and opportunistic bacteria are implicated in RCD incidence. No study been has been conducted elucidating the possible contribution of nutrients on RCD. In this study, we analysed the nutrient contents of developing mango fruits from an orchard with high RCD incidence (RCD+) and fruits from an orchard without RCD incidence (RCD-). Fruit samples were collected from upper, lower, sunny (west) and more shaded (east) side of the tree, oven-dried, ground and analysed for macronutrient (N, P, K, Ca, Mg, S) and micronutrient (B, Cu, Fe, Mn, Zn) content. Nutrient balance was determined based on compositional data analysis using isometric log ratio (ilr) transformation. Results showed that fruit from RCD+ orchard had higher variations in Ca/K and Ca/Fe ratios while RCD- fruit had higher variations in Ca/Cu and Ca/Zn ratios. Principal component analysis indicated a distinct separation of nutrient content between mango fruit from RCD+ and RCD- orchards. Moreover, dendrogram of fruit ionome showed an imbalance of Ca/Cu, B/Mn and Fe/Zn ratios. These variations in nutrient ratios may indicate a possible role of nutrient imbalances to RCD. Further analysis is needed to confirm the usefulness of the balance approach to diagnose susceptibility of mango to RCD and develop a remediation program.

Some practical aspects of predicting texture data in digital soil mapping

Soil texture is the most well-known composition in soil science. When separate components of the texture (sand, silt, and clay) are predicted independently in digital soil mapping (DSM), there is no guarantee that the separate estimates will sum to 100%. Log-ratio transformations before DSM modelling are alternatives to guarantee a constant sum of the estimates. Little is known about the effect of non-summing to 100% and transformations of particle-size fractions (PSFs) when DSM products were used to predict soil functional properties using pedotransfer functions (PTFs). Therefore, this study was conducted to investigate the effect of different soil texture modelling methods on the estimation of available soil water capacity (AWC) and the total amount of irrigation water (TIW) required for wheat production on a 4600 ha area in Khuzestan province, southwestern Iran. Specifically, this study aimed i) to assess the performance of random forest models (RF) to predict untransformed (UT) and transformed PSFs using environmental covariates; ii) to study the effects of three widely used log-ratio transformations including additive, centroid and isometric log-ratio transformations (alr, clr, and ilr respectively) on the estimations of AWC and TIW. A total of 150 soil samples were collected from the surface layers (0–30 cm) based on the conditioned Latin hypercube sampling (cLHS) procedure. Results indicated that, in terms of root mean square error (RMSE), RF provided similar accuracies in predicting PSFs for both untransformed and transformed data. However, transformation resulted in biased estimates. In addition, RF prediction based on untransformed data resulted in more correctly soil texture classes allocation when compared to transformed data. The spatial distribution of the sum of the predicted untransformed fractions indicated only small parts of the area conformed to the 100% sum. Almost the same accuracies for estimates of AWC were obtained when both untransformed and transformed predicted texture components were used as the inputs to PTFs. Data transformation can result in biased estimates of AWC. The findings indicated no significant difference between transformation methods in predicting AWC and TIW. The general patterns of the spatial distribution of the predicted AWCs across the whole area were the same for transformed and untransformed data (except for clr transformation).

Temporal density extrapolation using a dynamic basis approach

Density estimation is a versatile technique underlying many data mining tasks and techniques, ranging from exploration and presentation of static data, to probabilistic classification, or identifying changes or irregularities in streaming data. With the pervasiveness of embedded systems and digitisation, this latter type of streaming and evolving data becomes more important. Nevertheless, research in density estimation has so far focused on stationary data, leaving the task of of extrapolating and predicting density at time points outside a training window an open problem. For this task, temporal density extrapolation (TDX) is proposed. This novel method models and predicts gradual monotonous changes in a distribution. It is based on the expansion of basis functions, whose weights are modelled as functions of compositional data over time by using an isometric log-ratio transformation. Extrapolated density estimates are then obtained by extrapolating the weights to the requested time point, and querying the density from the basis functions with back-transformed weights. Our approach aims for broad applicability by neither being restricted to a specific parametric distribution, nor relying on cluster structure in the data. It requires only two additional extrapolation-specific parameters, for which reasonable defaults exist. Experimental evaluation on various data streams, synthetic as well as from the real-world domains of credit scoring and environmental health, shows that the model manages to capture monotonous drift patterns accurately and better than existing methods. Thereby, it requires not more than 1.5 times the run time of a corresponding static density estimation approach.

A flexible spatial autoregressive modelling framework for mixed covariates of multiple data types

Mixed spatial autoregressive (SAR) models with numerical covariates have been well studied. However, as non-numerical data, such as functional data and compositional data, receive substantial amounts of attention and are applied to economics, medicine and meteorology, it becomes necessary to develop flexible SAR models with multiple data types. In this article, we integrate three types of covariates, functional, compositional and numerical, in an SAR model. The new model has the merits of classical functional linear models and compositional linear models with scalar responses. Moreover, we develop an estimation method for the proposed model, which is based on functional principal component analysis (FPCA), the isometric logratio (ilr) transformation and the maximum likelihood estimation (MLE) method. Monte Carlo experiments demonstrate the effectiveness of the estimators. A real dataset is also used to illustrate the utility of the proposed model.

Compositional data analysis as an alternative paradigm for nutritional studies

Although the compositional nature of dietary data is well recognized, little attention has been given to the methods specifically developed for the statistical analysis of compositional data. The use of standard statistical procedures that ignore the relative nature of compositional elements can lead to spurious results when applied to crude data. This note proposes using a compositional data approach for the statistical analysis of nutritional data. The analyses of data from an Italian population study of 673 non-diabetic women aged 40-74 years were based on isometric log-ratio (ilr) transformation applied to three dietary compositions: seven-part macronutrients, nine-part vitamins and six-part minerals. The ilr transformation produces new variables that represent specific contrasts (balances) between the compositional parts. Different sequential binary partitions have been described and used as a means of flexibly defining balances on the basis of a researcher's interest, and the new variables (that are suitable for undergoing standard statistical procedures) have been included as covariates in linear regression models in order to examine the isocaloric associations between specific dietary balances and waist circumference (WC). Regardless of the dietary compositions, total energy was positively and fibre intake was negatively associated with WC. Net of these effects and keeping constant the proportional relationships between the other dietary components, WC was negatively associated with the relative increase in starches, and positively associated with the relative niacin content. Vitamin C was negatively associated with WC, whereas folate content was positively related. The proposed approach allows a clear interpretation of the relative roles of different dietary components within a holistic overview of a diet. The analyses involving WC provide some useful insights.

Integration of auto-encoder network with density-based spatial clustering for geochemical anomaly detection for mineral exploration

Auto-encoder network can be used for dimensionality reduction of data and for re-construction of sample population with unknown, complex multivariate probability distribution, where small-probability samples have little contribution to the auto-encoder network, leading to high re-construction error. In this paper, the trained auto-encoder networks were used to detect geochemical anomalies. Compared with deep auto-encoder network, the density-based spatial clustering application with noise (DBSCAN) regards noise samples (e.g., geochemically anomalous samples) that differ from core samples (e.g., geochemically background samples) as anomalies. Therefore, the learned representations from the code layer in the auto-encoder network are clustered by DBSCAN to detect noise samples representing geochemical anomalies. As benchmark for evaluating the performance of auto-encoder network and DBSCAN, and in consideration of the compositional nature of geochemical data, the compositional multivariate outlier detection was also applied. We applied these methods to two forms of the geochemical data, namely (1) without any transformation and (2) with isometric log ratio transformation. The similarities of the resulting anomaly maps in terms of data forms indicate that the auto-encoder network is effective for detecting multivariate geochemical anomalies. Differences between the anomaly maps indicate, however, that the compositional nature of geochemical data affects the performance of multivariate geochemical anomaly detection. Nevertheless, the assessment, by receiver operating characteristics analysis, of the geochemical anomalies derived using the different methodologies described implies that the detected geochemical anomalies are related to Au mineralization. Finally, the Youden index, which measures the relationship between binary anomalies and known deposits, was used for optimal threshold selection to create an optimal mineral potential map from the derived continuous geochemical anomaly data. The spatial distribution of geochemical anomalies at/around faults and magmatic rocks provides insights to where further detailed exploration is warranted in the study area.

Joint simulation of compositional and categorical data via direct sampling technique – Application to improve mineral resource confidence

Ore deposits usually consist of ore materials with different discrete (e.g. rock and alteration types) and continuous (e.g. geochemical and mineral composition) features. Financial feasibility studies are highly dependent on the modelling of these features and their associated joint uncertainties. Few geostatistical techniques have been developed for the joint modelling of high-dimensional mixed data (continuous and categorical) or constrained data, such as compositional data. The compositional nature of the mineral and geochemical data induces several challenges for multivariate geostatistical techniques, because such data carry relative information and are known for spurious statistical and spatial correlation effects. This paper investigates the application of the direct sampling algorithm for joint modelling of compositional and categorical data. In some mining projects the amount of available data may be enormous in some parts of the deposit and if the density of measurements is sufficient, multivariate geospatial patterns can be derived from that data and be simulated (without model inference) at other undersampled areas of the deposit with similar characteristics. In this context, the direct sampling multiple-point simulation method can be implemented for this reconstruction process. The compositional nature of the data is addressed via implementing an isometric log-ratio transformation. The approach is illustrated through two case studies, one synthetic and one real. The accuracy of the results is checked against a set of validation data, revealing the potential of the proposed methodology for joint modelling of compositional and categorical information. The direct sampling technique can be considered as a smart move to assess the future risk and uncertainty of a resource by making use of all the information hidden within the early data.