Isometric Log-ratio Coordinates Research Articles

Comparing Two Geostatistical Simulation Algorithms for Modelling the Spatial Uncertainty of Texture in Forest Soils

Uncertainty assessment is an essential part of modeling and mapping the spatial variability of key soil properties, such as texture. The study aimed to compare sequential Gaussian simulation (SGS) and turning bands simulation (TBS) for assessing the uncertainty in unknown values of the textural fractions accounting for their compositional nature. The study area was a forest catchment (1.39 km2) with soils classified as Typic Xerumbrepts and Ultic Haploxeralf. Samples were collected at 135 locations (0.20 m depth) according to a design developed using a spatial simulated annealing algorithm. Isometric log-ratio (ilr) was used to transform the three textural fractions into a two-dimensional real vector of coordinates ilr.1 and ilr.2, then 100 realizations were simulated using SGS and TBS. The realizations obtained by SGS and TBS showed a strong similarity in reproducing the distribution of ilr.1 and ilr.2 with minimal differences in average conditional variances of all grid nodes. The variograms of ilr.1 and ilr.2 coordinates were better reproduced by the realizations obtained by TBS. Similar results in reproducing the texture data statistics by both algorithms of simulation were obtained. The maps of expected values and standard deviations of the three soil textural fractions obtained by SGS and TBS showed no notable visual differences or visual artifacts. The realizations obtained by SGS and TBS showed a strong similarity in reproducing the distribution of isometric log-ratio coordinates (ilr.1 and ilr.2). Overall, their variograms and data were better reproduced by the realizations obtained by TBS.

Spatial distribution of soil rare earth elements in Sicily (Italy)

Rare earth elements (REEs) are becoming increasingly interesting as indicators of soil processes, and modelling their spatial distribution has become a fundamental requirement for this purpose. This study aims to model and quantify the spatial distribution of soil REEs taking into account their compositional nature in their mapping, and to assess the associated spatial uncertainty. In particular, the cerium anomaly (Ce⁎) and the ratio of the sum (Σ) of light to heavy REEs (ΣLREEs/ΣHREEs) were calculated and analysed. The study was carried out in Sicily (Italy), one of the largest (25,832 km2) and most populous island in the Mediterranean Sea. Soil REE data were obtained from the soil database of Sicily region. Soil REEs were transformed into a vector of isometric log-ratio (ilr) coordinates and analysed using a geostatistical approach for predicting their values at unsampled locations and generating 100 REEs realizations using turning bands simulation. Each REE element and Ce⁎ were mapped as well as the ΣLREEs/ΣHREEs ratio. The joint variability of the investigated REEs and the ΣLREEs/ΣHREEs ratio provided insights into their abundance and distribution patterns. From the simulated realizations, standard deviation maps were calculated for the Ce⁎ and the ΣLREEs/ΣHREEs ratio, providing a measure of their spatial uncertainty. Although with some limitations, the study established a first baseline of the Ce⁎ and the ΣLREEs/ΣHREEs ratio. The maps of spatial uncertainty may be a useful tool for planning future soil sampling and optimise the choice of new sampling locations.

Geochemical anomaly recognition using Shapley values and cell-wise outlier detection: a case study in the Yuanbo Nang District, Gansu Province, China

Geochemical pattern recognition has long been of interest for geologists to reveal geochemical anomalies associated with mineralization. In regional-scale exploration, geochemical anomalies are derived conventionally from stream sediment samples and processed in the form of vectors, resulting in row-wise outliers. However, geochemical anomalies derived through various means of pattern recognition have shown their limits in depicting complex geochemical distributions. In this paper, we propose to utilize the Shapley value, linked to the Mahalanobis distance (MD), and cell-wise outlier detection to facilitate the recognition of anomalous geochemical indicator elements. First, by considering the compositional nature of geochemical data, multivariate outliers are detected based on the MD in isometric log-ratio coordinates. Secondly, to quantify the contributions of individual elements to the outlyingness of an outlier, Shapley values are used to express the MDs of data as outlyingness contributions of single elements. Finally, cell-wise outlier detection is introduced to examine and quantify the outlyingness of each cell in a geochemical data matrix. The outlying cells serve as criteria for further recognition of element associations. By analysing the Shapley values of individual elements and the outlying cells in a geochemical data matrix, more information contained in multivariate outliers can be recognized. Using this proposed methodology, the element associations that relate to regional mineralization in the study area were Au-only anomalies, Au–As–Sb anomalies, As–Sb–Hg anomalies and Ag-related anomalies.

Robust CoDA balances and the role of the variance in complex riverine geochemical systems

This study introduces a robust method for analyzing the geochemical behavior of chemical species in river catchment water. It focuses on isometric log-ratio coordinates obtained from a sequential partition method that successively maximizes the explained variance in the data set. Robust orthonormal coordinates are created based on hierarchical clustering and robust estimation of the variation matrix. Applying this to the water chemistry of Italy's Arno and Tiber basins, the research reveals the associations of variables in data structure and processes across varying geological and climatic conditions. The method uncovers key contrasting geochemical processes and suggests that the behavior of simple balances characterized by lower variances (i.e., Ca2+/HCO3− and Na+/Cl−) are mainly influenced by random fluctuations with no differences between classical or robust methods. However, when balances describe more complex geochemical processes resulting in frequency distributions affected by the presence of bimodality or outliers, significant differences among the two approaches emerge, compromising the data interpretation. The proposed metodology offers more insights into the investigation of catchment geochemistry's resilience to hydroclimatic changes, marking a significant step in understanding large-scale environmental dynamics.

How to choose a proper representation of compositional data for mineral exploration?

Regional mineral exploration is based on geochemical data of which the nature is compositional and frequently involves a large number of components. Consequently, it mostly needs multivariate dimension reduction methods such as principal component analysis (PCA) and its various robust versions. The application of such methods, defined for real random variables, require the data to be represented in coordinates supported in the real space. However, a common problem in exploration geochemistry is to select the appropriate representation. Using centered (clr) and isometric (ilr) logratio coordinates to discriminate anomalous zones for orogeny gold exploration throughout Sweden revealed that there is, as expected, no difference between the two representation methods. The main difference affects the interpretation of the coordinates used. This is observed for regional scale exploration, while it is also needed to study different ways of representing geochemical data in local scale.

Association Between 24-Hour Movement Behaviors and Adiposity in Malaysian Schoolchildren: A Compositional Isotemporal Substitution Analysis.

Physical activity (PA), sedentary behavior (SB), and sleep are interrelated 24-hour movement behaviors that are important for the growth and well-being of children. This cross-sectional study examined the relationship between 24-hour movement behaviors and adiposity, and predicted changes in adiposity following compositional time reallocations in 7- to 12.9-year-old Malaysian children from the South East Asian Nutrition Surveys II Malaysia. A total of 381 children (mean age 9.7 [1.6]y, 57% girls) provided 24-hour wrist-worn GENEActiv accelerometry data which captured time spent for sleep, SB, light PA and moderate to vigorous PA (MVPA). Indicators of adiposity were derived from anthropometry and bioelectrical impedance analysis: body-mass-index-for-age, waist circumference, waist-to-height ratio, percent body fat, and body mass index. The composition of 4-part movement behaviors was expressed as isometric log-ratio coordinates which were entered into regression models. Isotemporal substitution analysis was used to assess changes in adiposity indicators when reallocating time between movement behaviors. Relative to other movement behaviors, time spent on MVPA was significantly associated with waist circumference, waist-to-height ratio, percent body fat, and fat mass index. A 15-minute one-to-one reallocation from other movement behaviors to MVPA predicted lower body-mass-index-for-age (-0.03 to -0.11), smaller waist circumference (-0.67to -1.28cm), lower waist-to-height ratio (-0.004 to -0.008), percent body fat (-0.87% to -1.47%), and fat mass index (-0.23 to -0.42). Replacing SB and light PA with sleep or MVPA was associated with lower adiposity. The overall composition of movement behavior was significantly associated with the adiposity of Malaysian schoolchildren. Promoting MVPA and sleep and reducing SB and light PA are important for prevention of childhood obesity.

Calibration Spiking of MIR-DRIFTS Soil Spectra for Carbon Predictions Using PLSR Extensions and Log-Ratio Transformations

There is a need to minimize the usage of traditional laboratory reference methods in favor of spectroscopy for routine soil carbon monitoring, with potential cost savings existing especially for labile pools. Mid-infrared spectroscopy has been associated with accurate soil carbon predictions, but the method has not been researched extensively in connection to C lability. More studies are also needed on reducing the numbers of samples and on how to account for the compositional nature of C pools. This study compares performance of two classes of partial least squares regression models to predict soil carbon in a global (models trained to data from a spectral library), local (models trained to data from a target area), and calibration-spiking (spectral library augmented with target-area spectra) scheme. Topsoil samples were+ scanned with a Fourier-transform infrared spectrometer, total and hot-water extractable carbon determined, and isometric log-ratio coordinates derived from the latter measurements. The best RMSEP was estimated as 0.38 and 0.23 percentage points TC for the district and field scale, respectively—values sufficiently low to make only qualitative predictions according to the RPD and RPIQ criteria. Models estimating soil carbon lability performed unsatisfactorily, presumably due to low labile pool concentration. Traditional weighing of spiking samples by including multiple copies thereof in training data yielded better results than canonical partial least squares regression modeling with embedded weighing. Although local modeling was associated with the most accurate predictions, calibration spiking addressed better the trade-off between data acquisition costs and model quality. Calibration spiking with compositional data analysis is, therefore, recommended for routine monitoring.

A new approach to assess the degree of contamination and determine sources and risks related to PTEs in an urban environment: the case study of Santiago (Chile)

In 2017, a geochemical survey was carried out across the Commune of Santiago, a local administrative unit located at the center of the namesake capital city of Chile, and the concentration of a number of major and trace elements (53 in total) was determined on 121 topsoil samples. Multifractal IDW (MIDW) interpolation method was applied to raw data to generate geochemical baseline maps of 15 potential toxic elements (PTEs); the concentration–area (C-A) plot was applied to MIDW grids to highlight the fractal distribution of geochemical data. Data of PTEs were elaborated to statistically determine local geochemical baselines and to assess the spatial variation of the degree of soil contamination by means of a new method taking into account both the severity of contamination and its complexity. Afterwards, to discriminate the sources of PTEs in soils, a robust Principal Component Analysis (PCA) was applied to data expressed in isometric log-ratio (ilr) coordinates. Based on PCA results, a Sequential Binary Partition (SBP) was also defined and balances were determined to generate contrasts among those elements considered as proxies of specific contamination sources (Urban traffic, productive settlements, etc.). A risk assessment was finally completed to potentially relate contamination sources to their potential effect on public health in the long term. A probabilistic approach, based on Monte Carlo method, was deemed more appropriate to include uncertainty due to spatial variation of geochemical data across the study area. Results showed how the integrated use of multivariate statistics and compositional data analysis gave the authors the chance to both discriminate between main contamination processes characterizing the soil of Santiago and to observe the existence of secondary phenomena that are normally difficult to constrain. Furthermore, it was demonstrated how a probabilistic approach in risk assessment could offer a more reliable view of the complexity of the process considering uncertainty as an integral part of the results.

Lithological correction of chemical weathering proxies based on K, Rb, and Mg contents for isolation of orbital signals in clastic sedimentary archives

The extraction of palaeoenvironmental (palaeoclimatic) signals from the chemical composition of siliciclastic sediments is valuable for the reconstruction of past environments, particularly in continental basins. Here we test novel weathering proxies, which are less sensitive to lithological control than the previously used raw element ratios K/Al, K/Ti, and K/Rb: (1) local enrichment factors of K/Al, Mg/Al, and K/Rb, i.e., the element ratios corrected for grain size- and matrix composition using local background functions (Al/Si, Fe, and Ca as explanatory variables) and ordinary regression and (2) robust regression residuals of those element ratios based on isometric log-ratio coordinates of the most relevant “lithogenic” elements (Ca, Fe, Rb, Si, Zr) in the chemical composition. Chemical weathering proxies can be obtained from departures of chemical composition of sedimentary profiles from relationships with other chemical elements, in particular those with grain-size control. The resulting weathering proxies were examined for the Miocene deposits from the Most Basin, the Czech Republic, which recorded one of the major warm episodes of the Cenozoic time – the Miocene Climatic Optimum. The performance of weathering proxies has been checked by (1) comparison of individual proposed proxies in one drill core HK930, (2) detailed analysis of orbital signals in the relevant compositional functions in HK930; and (3) lateral correlation of three cores HK930, DU7, and DO565 of the same basin. The novel proxies show lateral stability and orbital signatures of short eccentricity, obliquity, and precession, confirming their usefulness in palaeoenvironmental studies. Corrections for grain-size and carbonate contents should help to isolate climatic content from the weathering proxies, although in the studied sediments it weakened the precession component in the orbital signal, as grain-size proxies and other compositional data also carried orbital signals. We propose to consider these proxy ideas in palaeoclimatic reconstructions based on chemical weathering proxies.

Associations of physical activity and sedentary behavior with cardiometabolic biomarkers in prediabetes and type 2 diabetes: a compositional data analysis

ABSTRACT Objectives To investigate the associations between objectively measured sedentary behavior (SB), light-intensity physical activity (LIPA) and moderate-to-vigorous physical activity (MVPA) and cardiometabolic and endocrine biomarkers, and to estimate the associations of reallocating time from one behavior to another with cardiometabolic and endocrine biomarkers. Methods Baseline data from participants diagnosed with prediabetes or type 2 diabetes, n = 175, 58% men, mean (SD) age = 64.4 (7.7), recruited to a physical activity intervention was used. Time spent in SB, LIPA and MVPA was measured by accelerometer and transformed into isometric log-ratio coordinates. The associations between time spent in SB, LIPA and MVPA and biomarkers were examined by linear regression models. The change in each outcome of reallocating time between the three behaviors was estimated. Results The findings show strong positive associations of time spent in MVPA and negative associations of time spent in SB relative to time spent in the other behaviors with sagittal abdominal diameter (SAD) and homeostasis model assessment for insulin resistance (HOMA-IR) and negative associations of time spent in SB with high-density lipoprotein (HDL) cholesterol. Theoretically, reallocation of 19 minutes MVPA to SB or to LIPA was associated with a 17% and 17% larger SAD, 39% and 36% larger HOMA-IR values and 3.3% and 2.3% lower levels of HDL, respectively. Conclusion In conclusion, our analysis from a time-use perspective supports the current evidence that sedentary time is devastating for the cardiometabolic health. While LIPA probably requires more time, maintaining or increasing time in MVPA are the most important features of the time use behaviors when promoting a favorable cardiometabolic risk profile in adults with prediabetes and type 2 diabetes. Trial registration ClinicalTrials.gov, NCT02374788. Registered 2 March 2015 – Retrospectively registered, https://clinicaltrials.gov/ct2/show/NCT02374788

Associations of School Day Sedentary Behavior and Physical Activity With Gross Motor Skills: Use of Compositional Data Analysis

To examine the relationships among school day sedentary times (SED), light physical activity (LPA), and moderate to vigorous physical activity (MVPA) with gross motor skills in children using Compositional Data Analysis. Participants were 409 children (mean age = 8.4 [1.8] y) recruited across 5 low-income schools. Gross motor skills were assessed using the test for gross motor development—third edition (TGMD-3), and physical activity was assessed using accelerometers. Isometric log-ratio coordinates were calculated by quantifying the relative proportion of percentage of the school day spent in SED, LPA, and MVPA. The associations of the isometric log-ratio coordinates with the TGMD-3 scores were estimated using general linear mixed-effects models adjusted for age, body mass index, estimated aerobic capacity, and school affiliation. A higher proportion of the school day spent in %MVPA relative to %SED and %LPA was significantly associated with higher TGMD-3 total scores (γMVPA = 14.44, P = .01). This relationship was also observed for the ball skills subtest scores (γMVPA = 16.12, P = .003). Replacing %SED and %LPA with %MVPA during school hours may be an effective strategy for improving gross motor skills, specifically ball skills, in low-income elementary school-aged children.

Associations Among School Day Sedentary Behavior, Physical Activity, and Motor Skills: A Compositional Data Analysis

PURPOSE: A novel analytic approach, Compositional Data Analysis (CoDa), has recently been used to analyze physical activity data. CoDa assumes co-dependence among physical activity compositional parts within a time-constrained data analytic framework, which makes it appropriate for assessing and analyzing physical activity behavior during school hours. The purpose of this study was to examine the relationships among school day sedentary times (SED), light physical activity (LPA), and moderate-to-vigorous physical activity (MVPA) with gross motor skills in children using Compositional Data Analysis (CoDa). METHODS: Participants were 409 children (Mean age = 8.4±1.8 years) recruited across five low-income schools. Gross Motor Skills were assessed using the Test of Gross Motor Development - 3rdEdition (TGMD-3) and physical activity was assessed using accelerometers. Isometric Log Ratio coordinates (ILRs) were calculated quantifying the relative proportion of percent of the school day (%) spent in SED, LPA, and MVPA. The associations of the ILRs with the TGMD-3 scores were estimated using general linear mixed effects models adjusted for age, BMI, and estimated VO2Peak. RESULTS: A higher proportion of the school day spent in %MVPA relative to %SED and %LPA significantly associated with higher TGMD-3 total scores (γMVPA= 14.44, P= 0.012). This relationship was also observed for the ball skills subtest scores (γMVPA= 16.12, P= 0.003). A 5% reallocation in % of time spent in SED and LPA to MVPA was independently associated with a 0.48 and 1.74-point increase in ball skills scores, respectively, after controlling for the potential confounders of age, BMI, and estimated VO2Peak. CONCLUSIONS: Replacing %SED and %LPA with %MVPA during school hours may an effective strategy for improving gross motor skills, specifically ball skills, in low-income elementary school-aged children.

Application of Data Analytics Techniques to Establish Geometallurgical Relationships to Bond Work Index at the Paracutu Mine, Minas Gerais, Brazil

Analysis of geometallurgical data is essential to building geometallurgical models that capture physical variability in the orebody and can be used for the optimization of mine planning and the prediction of milling circuit performance. However, multivariate complexity and compositional data constraints can make this analysis challenging. This study applies unsupervised and supervised learning to establish relationships between the Bond ball mill work index (BWI) and geomechanical, geophysical and geochemical variables for the Paracatu gold orebody. The regolith and fresh rock geometallurgical domains are established from two cluster sets resulting from K-means clustering of the first three principal component (PC) scores of isometric log-ratio (ilr) coordinates of geochemical data and standardized BWI, geomechanical and geophysical data. The first PC is attributed to weathering and reveals a strong relationship between BWI and rock strength and fracture intensity in the regolith. Random forest (RF) classification of BWI in the fresh rock identifies the greater importance of geochemical ilr balances relative to geomechanical and geophysical variables.

A kernel density approach for replacing rounded zeros in compositional data sets

The logratio methodology widely used in compositional data analysis is not applicable when some components have rounded zeros. There are many univariate and multivariate methods that have been used to deal with rounded zeros. However, both of them have restrictions: the univariate methods replaced the rounded zeros only using the information of the corresponding component; the multivariate methods need to assume the distribution of transformed data. When the form of the distribution function is unknown, a multivariate nonparametric replacement approach is proposed in this paper. The proposed method uses conditional expected value based on isometric logratio coordinates to replace rounded zeros, in which the conditional density is estimated through multivariate Gauss kernel function. The permutation invariance and invariance under change of orthonormal basis are also presented. Simulation studies show that the proposed method has better performance than previous methods as the percentage of rounded zeros increases. The proposed method is also applied on the moss data from the Kola project.

Innovative monitoring tools for the complex spatial dynamics of river chemistry: case study for the Alpine region

Chemical reactions in aqueous geochemical systems are driven by nonequilibrium conditions, and their dynamics can be deduced through the distributional analysis (identification of probability laws) of complex compositional indices. In this perspective, compositional data analysis offers the possibility to investigate the behavior of the composition as a whole instead of isolated chemical species, with the awareness that multispecies systems are characterized by the simultaneous interactions among all their parts. We addressed this problem using D − 1 isometric log-ratio coordinates describing the D compositional dataset of the river chemistry of the Alpine region (D number of variables), thus working in the $${{\mathbb{R}}^{D - 1}}$$ statistical sample space. The D − 1 coordinates were chosen using the decreasing variance criterion so that each one could provide information about different space–time properties for the investigated geochemical system. Coordinates dominated by heterogeneity appear to be able to capture regime shifts only on a long-time period and monitor processes on a very wide scale. On the other hand, coordinates characterized by lower variability present multimodality, thus capturing the presence of alternative states in the analyzed spatial domain also for the current time. Further developments are needed to determine the ranges of conditions for which variability and other statistics can be useful indicators of regime shifts on different time–space scales in geochemical systems.

The isometric log-ratio (ilr)-ion plot: A proposed alternative to the Piper diagram

The Piper diagram has been a staple for the analysis of water chemistry data since its introduction in 1944. It was conceived to be a method for water classification, determination of potential water mixing between end-members, and to aid in the identification of chemical reactions controlling a sample set. This study uses the information gleaned over the years since the release of the Piper diagram and proposes an alternative to it, capturing the strengths of the original diagram while adding new ideas to increase its robustness. The new method uses compositional data analysis to create 4 isometric log-ratio coordinates for the 6 major chemical species analyzed in the Piper diagram and transforms the data to a 4-field bi-plot, the ilr-ion plot. This ilr-ion plot conveys all of the information in the Piper diagram (water mixing, water types, and chemical reactions) while also visualizing additional data, the ability to examine Ca2+/Mg2+ versus Cl-/SO42−. The Piper and the ilr-ion plot were also compared using multiple synthetic and real datasets in order to illustrate the caveats and the advantages of using either diagram to analyze water chemistry data. Although there are challenges with using the ilr-ion plot (e.g., missing or zero values zeros in the dataset must be imputed by positive real numbers), it appears that the use of compositional data analysis coupled with the ilr-ion plot provides a more in-depth and complete analysis of water quality data compared to the original Piper diagram.

Regression imputation with Q-mode clustering for rounded zero replacement in high-dimensional compositional data

ABSTRACTThe logratio methodology is not applicable when rounded zeros occur in compositional data. There are many methods to deal with rounded zeros. However, some methods are not suitable for analyzing data sets with high dimensionality. Recently, related methods have been developed, but they cannot balance the calculation time and accuracy. For further improvement, we propose a method based on regression imputation with Q-mode clustering. This method forms the groups of parts and builds partial least squares regression with these groups using centered logratio coordinates. We also prove that using centered logratio coordinates or isometric logratio coordinates in the response of partial least squares regression have the equivalent results for the replacement of rounded zeros. Simulation study and real example are conducted to analyze the performance of the proposed method. The results show that the proposed method can reduce the calculation time in higher dimensions and improve the quality of results.

The compositional isotemporal substitution model: A method for estimating changes in a health outcome for reallocation of time between sleep, physical activity and sedentary behaviour.

How people use their time has been linked with their health. For example, spending more time being physically active is known to be beneficial for health, whereas long durations of sitting have been associated with unfavourable health outcomes. Accordingly, public health messages have advocated swapping strategies to promote the reallocation of time between parts of the time-use composition, such as "Move More, Sit Less", with the aim of achieving optimal distribution of time for health. However, the majority of research underpinning these public health messages has not considered daily time use as a composition, and has ignored the relative nature of time-use data. We present a way of applying compositional data analysis to estimate change in a health outcome when fixed durations of time are reallocated from one part of a particular time-use composition to another, while the remaining parts are kept constant, based on a multiple linear regression model on isometric log ratio coordinates. In an example, we examine the expected differences in Body Mass Index z-scores for reallocations of time between sleep, physical activity and sedentary behaviour.

Lithogeochemistry and chemostratigraphy of the Rosemont Cu-Mo-Ag skarn deposit, SE Tucson Arizona: A simplicial geometry approach

The Rosemont skarn deposit contains resources exceeding 1 billion tons of Cu-Mo-Ag mineralization. The deposit comprises three major structural-stratigraphic domains informally named Lower Plate, Upper Plate, and West Block. The Lower Plate is composed dominantly of Paleozoic chemical sedimentary rocks forming an east dipping homoclinal sequence. The Upper Plate overlies the Lower Plate and is composed dominantly of Mesozoic and Cenozoic siliciclastic sedimentary rocks. The West Block is dominated by Precambrian granitoids structurally interleaved with panels of Paleozoic chemical and siliciclastic sedimentary rocks. Most of the economic mineralization is hosted in the Lower Plate.Lithological variability, structural complications of the stratigraphy, and calc-silicate metasomatism complicates the visual identification of lithologies resulting in uncertain stratigraphic domains based on drill core observations. To circumvent this problem, a deposit-scale lithogeochemical and chemostratigraphic model was developed using multivariate statistics following a compositional data analysis approach to respect the relative scale and multivariate nature of geochemical compositions. Accordingly, hierarchical cluster analysis of variables using the variation matrix as a measure of similarity, and principal component analysis on centered logratio coefficients (clr) were used to explore relationships between variables and to reduce dimensionality. Simplicial projections including centered tetrahedral and ternary diagrams were used to develop a lithogeochemical classification for the sedimentary rocks of the Rosemont deposit. A ternary diagram, or 3-part simplex with vertices Ca, Mg, and a composite variable given by the geometric mean of Cr, Ni, Co, V, P, Hf, Zr, Th, Ti, Al, Nb, Sc, Ta, Y, Ce, and La contains a rich data structure for lithogeochemical classification.The lithological attributes of the Rosemont deposit can be subdivided into 7 lithogeochemical classes evident on the ternary diagram. Limestone, dolostone, and siliciclastic-crystalline classes are key end members. The remaining 4 classes represent mixed chemical-siliciclastic lithogeochemical attributes discriminating complex lithological variations induced by the incorporation of siliciclastic component in chemical sedimentary rocks. The lithogeochemical classification derived from the ternary diagram is supported by K-means clustering applied to two balances representing isometric logratio (ilr) coordinates of the 3-part simplex.The geospatial distribution of the lithogeochemical classes allows the simplification of the stratigraphy of the Lower Plate into three relevant chemostratigraphic units, namely a Lower Limestone Unit and an Upper Dolostone Unit separated by a Mixed Unit of chemical-siliciclastic sedimentary rocks. It is noteworthy that ore grades are controlled by the relative proportion of chemical to siliciclastic component of the mineralized rocks. Higher metal grades are characteristic of relatively pure chemical sedimentary rocks in the Lower Limestone and Upper Dolostone Units; whereas low metal grades characterize the Mixed Unit with a large proportion of siliciclastic component. As a corollary, the lithogeochemical classification can be used as a skarn fertility tool to predict the economic potential of chemical-siliciclastic sedimentary sequences in geological environments permissive of skarn mineralization.

Compositional data analysis for physical activity, sedentary time and sleep research.

The health effects of daily activity behaviours (physical activity, sedentary time and sleep) are widely studied. While previous research has largely examined activity behaviours in isolation, recent studies have adjusted for multiple behaviours. However, the inclusion of all activity behaviours in traditional multivariate analyses has not been possible due to the perfect multicollinearity of 24-h time budget data. The ensuing lack of adjustment for known effects on the outcome undermines the validity of study findings. We describe a statistical approach that enables the inclusion of all daily activity behaviours, based on the principles of compositional data analysis. Using data from the International Study of Childhood Obesity, Lifestyle and the Environment, we demonstrate the application of compositional multiple linear regression to estimate adiposity from children's daily activity behaviours expressed as isometric log-ratio coordinates. We present a novel method for predicting change in a continuous outcome based on relative changes within a composition, and for calculating associated confidence intervals to allow for statistical inference. The compositional data analysis presented overcomes the lack of adjustment that has plagued traditional statistical methods in the field, and provides robust and reliable insights into the health effects of daily activity behaviours.