Isometric Log-ratio Transformation Research Articles

Multivariate analysis of log-ratio transformed data and its priority in mining science, porphyry and polymetallic vein deposits case studies

Each mineralization style is characterized by typical signature associations between elements due to elemental interactions, therefore the coherence and closure effects problem must be overcome in geochemical processing. The coherence indicates the ratios between two components (rows or columns) remains the same whether they are considered in a subcomposition or in the full composition. The log-ratio transformation (LRT) has recognized as a standard procedure to support subcompositional coherence. The log-transformed data is applicable for geochemical data to unveil such associations, prior to applying the multivariate analysis like correspondence analysis (CA) and principal component analysis (PCA). At the present study, subcompositional coherence is overcome by inverse iso-metric log-ratio transformation for geochemical compositional data at two polymetallic and porphyry deposits. Based on Ilr-transformed data, Ag, Au, As, Pb, Te, Mo and rather S, W, Cu are enriched as polymetallic elements at Glojeh, while Au-Cu-(Mo) compositions indicate a porphyry deposit occurred in Dalli deposit. The ability to handle zero values in the data matrix and determining an elemental eccentricity from the center of each axis based on Euclidean distances are the advantages of CA method, with compression to LRT. Whereas, loading factors which spread in every direction and providing subcompositional coherence are the competitive advantages of PCA based on LRT, for both case studies. Results with these techniques show significant ability to draw an inference in such geochemical data, and in improving the performance of multivariate techniques using LRT.

Statistical evaluation of the geochemical variability in overbank sediments in Spain.

The overall objective of this study is to estimate, detect and specify the main sources of variance which affect the contents of the different elements in overbank sediments across Spain. These sources of variance were assessed and compared by means of a series of analyses of variance (ANOVAs), by regarding two parameters: their significance and their contribution to the total variance. Overbank sediments, sampled in erosion banks, were studied in several locations, in basins which drain different types of geological backgrounds and land uses (urban, mining, agricultural or pristine) across the Iberian Peninsula. Forty-eight elements (mostly in the < 63μm fraction) were analysed by ICP-OES, ICP-MS and INAA. After an isometric log ratio (ilr) transformation of the data, three ANOVA analyses were performed considering three perspectives: (1) local scale, (2) regional scale: within-profile perspective and (3) regional scale: inter-profile perspective. On a local scale, it was observed that the variability of rare earth elements (REE) depends mostly on the grain size and that heavy metals are also influenced by depth. In the analysis carried out on a regional scale, from a within-profile perspective, depth and duplicates do not influence significantly the variability of the element contents. Finally, from an inter-profile perspective, the selected sources of variance were land use and provenance, whose significance is the highest. While grain size and the selection of depth are of crucial importance in the final results, on local studies, land use and provenance are the ones that influence the most the composition of sediments in regional studies.

An XRPD and EPR spectroscopy study of microcrystalline calcite bioprecipitated by Bacillus subtilis

We report in this study the first XRPD and EPR spectroscopy characterisation of a biogenic calcite, obtained from the activity of the bacterium Bacillus subtilis. Microcrystalline calcite powders obtained from bacterial culture in a suitable precipitation liquid medium were analysed without further manipulation. Both techniques reveal unusual parameters, closely related to the biological source of the mineral, i.e., to the bioprecipitation process and in particular to the organic matrix observed inside calcite. In detail, XRPD analysis revealed that bacterial calcite has slightly higher c/a lattice parameters ratio than abiotic calcite. This correlation was already noticed in microcrystalline calcite samples grown by bio-mineralisation processes, but it had never been previously verified for bacterial biocalcites. EPR spectroscopy evidenced an anomalously large value of W6, a parameter that can be linked to occupation by different chemical species in the next nearest neighbouring sites. This parameter allows to clearly distinguish bacterial and abiotic calcite. This latter achievement was obtained after having reduced the parameters space into an unbiased Euclidean one, through an isometric log-ratio transformation. We conclude that this approach enables the coupled use of XRPD and EPR for identifying the traces of bacterial activity in fossil carbonate deposits.

Robust variogram estimation combined with isometric log-ratio transformation for improved accuracy of soil particle-size fraction mapping

Mapping soil particle-size fractions (psf) plays an important role in regional hydrological, ecological, geological, agricultural and environmental studies. To map soil compositional data like soil psf, interpolators such as compositional kriging and the combination of log-ratio transformations with ordinary kriging or cokriging were developed. In addition, robust estimators were proposed for these interpolators to improve the variogram models. However, few studies have focused on how to choose log-ratio transformation, kriging, cokriging, or robust variogram estimation methods based on data characteristics to achieve optimal performance when mapping soil psf by comprehensive comparative analysis. Here, we selected different compositional kriging, log-ratio kriging, log-ratio cokriging and log-ratio cokriging methods combined with a robust variogram estimator to improve the accuracy of spatial predictions of soil psf when using 262 soil samples from the upper reaches of the Heihe River in China. In this study, a comprehensive comparative analysis of soil psf maps generated by using different interpolators is presented, and appropriate methods for mapping psf based on the characteristics of the available data are explored. The results show that using isometric log-ratio (ILR) transformation with different interpolators can achieve relatively better performance than the other log-ratio transformation methods. In addition, combining the interpolators with robust variogram estimators significantly improve the prediction accuracy compared with using standard estimators, which presented reasonable and smooth transitions when mapping soil psf. Combining ILR cokriging with a robust variogram estimator had the best accuracy, with the lowest root mean squared error (sand, 10.50%; silt, 11.24%; clay, 7.32%), an Aitchison's distance of 0.76, a standardized residual sum of squares of 0.70 and a relatively higher rate of correctly predicting soil texture types 90.04%. In the future, guideline for using log-ratio transformation methods with linear regression, a generalized linear model or random forest should be developed and combined with ancillary variables to improve the interpolators.

Mapping Soil Particle-Size Fractions Using Additive Log-Ratio (ALR) and Isometric Log-Ratio (ILR) Transformations and Proximally Sensed Ancillary Data

AbstractTogether, the three particle size fractions (PSFs) of clay, silt, and sand are the most fundamental soil properties because the relative abundance influences the physical, chemical, and biological activities in soil. Unfortunately, determining PSFs requires a laboratory method which is time-consuming. One way to add value is to use digital soil mapping, which relies on empirical models, such as multiple linear regression (MLR), to couple ancillary data to PSFs. This approach does not account for the special requirements of compositional data. Here, ancillary data were coupled, via MLR modelling, to additive log-ratio (ALR) or isometric log-ratio (ILR) transformations of the PSFs to meet these requirements. These three approaches (MLR vs. ALR-MLR and ILR-MLR) were evaluated along with the use of different ancillary data that included proximally sensed gamma-ray spectrometry, electromagnetic induction, and elevation data. In addition, how the prediction might be improved was examined using ancillary data that was measured on transects and was compared to data interpolated from transects spaced far apart. Although the ALR-MLR approach did not produce significantly better results, it predicted soil PSFs that summed to 100 and had the advantage of interpreting the ancillary data relative to the original coordinates (i.e. clay, silt, and sand). For the prediction of PSFs at various depths, all ancillary data were useful. Elevation and gamma-ray data were slightly better for topsoil and elevation and electromagnetic (EM) data were better for subsoil prediction. In addition, a smaller transect spacing (26 m) and number of samples (9–16) might be adopted for mapping soil PSFs and soil texture across the study field. The ALR-MLR approach can be applied elsewhere to map the spatial distribution of clay minerals.

Application of cluster analysis to geochemical compositional data for identifying ore-related geochemical anomalies

Cluster analysis is a well-known technique that is used to analyze various types of data. In this study, cluster analysis is applied to geochemical data that describe 1444 stream sediment samples collected in northwestern Xinjiang with a sample spacing of approximately 2 km. Three algorithms (the hierarchical, k-means, and fuzzy c-means algorithms) and six data transformation methods (the z-score standardization, ZST; the logarithmic transformation, LT; the additive log-ratio transformation, ALT; the centered log-ratio transformation, CLT; the isometric log-ratio transformation, ILT; and no transformation, NT) are compared in terms of their effects on the cluster analysis of the geochemical compositional data. The study shows that, on the one hand, the ZST does not affect the results of column- or variable-based (R-type) cluster analysis, whereas the other methods, including the LT, the ALT, and the CLT, have substantial effects on the results. On the other hand, the results of the row- or observation-based (Q-type) cluster analysis obtained from the geochemical data after applying NT and the ZST are relatively poor. However, we derive some improved results from the geochemical data after applying the CLT, the ILT, the LT, and the ALT. Moreover, the k-means and fuzzy c-means clustering algorithms are more reliable than the hierarchical algorithm when they are used to cluster the geochemical data. We apply cluster analysis to the geochemical data to explore for Au deposits within the study area, and we obtain a good correlation between the results retrieved by combining the CLT or the ILT with the k-means or fuzzy c-means algorithms and the potential zones of Au mineralization. Therefore, we suggest that the combination of the CLT or the ILT with the k-means or fuzzy c-means algorithms is an effective tool to identify potential zones of mineralization from geochemical data.

Mapping of compositional properties of coal using isometric log-ratio transformation and sequential Gaussian simulation – A comparative study for spatial ultimate analyses data

Chemical properties of coal largely determine coal handling, processing, beneficiation methods, and design of coal-fired power plants. Furthermore, these properties impact coal strength, coal blending during mining, as well as coal's gas content, which is important for mining safety. In order for these processes and quantitative predictions to be successful, safer, and economically feasible, it is important to determine and map chemical properties of coals accurately in order to infer these properties prior to mining.Ultimate analysis quantifies principal chemical elements in coal. These elements are C, H, N, S, O, and, depending on the basis, ash, and/or moisture. The basis for the data is determined by the condition of the sample at the time of analysis, with an “as-received” basis being the closest to sampling conditions and thus to the in-situ conditions of the coal. The parts determined or calculated as the result of ultimate analyses are compositions, reported in weight percent, and pose the challenges of statistical analyses of compositional data. The treatment of parts using proper compositional methods may be even more important in mapping them, as most mapping methods carry uncertainty due to partial sampling as well.In this work, we map the ultimate analyses parts of the Springfield coal from an Indiana section of the Illinois basin, USA, using sequential Gaussian simulation of isometric log-ratio transformed compositions. We compare the results with those of direct simulations of compositional parts. We also compare the implications of these approaches in calculating other properties using correlations to identify the differences and consequences. Although the study here is for coal, the methods described in the paper are applicable to any situation involving compositional data and its mapping.

A new method for correlation analysis of compositional (environmental) data – a worked example

Most data in environmental sciences and geochemistry are compositional. Already the unit used to report the data (e.g., μg/l, mg/kg, wt%) implies that the analytical results for each element are not free to vary independently of the other measured variables. This is often neglected in statistical analysis, where a simple log-transformation of the single variables is insufficient to put the data into an acceptable geometry. This is also important for bivariate data analysis and for correlation analysis, for which the data need to be appropriately log-ratio transformed. A new approach based on the isometric log-ratio (ilr) transformation, leading to so-called symmetric coordinates, is presented here. Summarizing the correlations in a heat-map gives a powerful tool for bivariate data analysis. Here an application of the new method using a data set from a regional geochemical mapping project based on soil O and C horizon samples is demonstrated. Differences to ‘classical’ correlation analysis based on log-transformed data are highlighted. The fact that some expected strong positive correlations appear and remain unchanged even following a log-ratio transformation has probably led to the misconception that the special nature of compositional data can be ignored when working with trace elements. The example dataset is employed to demonstrate that using ‘classical’ correlation analysis and plotting XY diagrams, scatterplots, based on the original or simply log-transformed data can easily lead to severe misinterpretations of the relationships between elements.

Compositional data for global monitoring: The case of drinking water and sanitation

IntroductionAt a global level, access to safe drinking water and sanitation has been monitored by the Joint Monitoring Programme (JMP) of WHO and UNICEF. The methods employed are based on analysis of data from household surveys and linear regression modelling of these results over time. However, there is evidence of non-linearity in the JMP data. In addition, the compositional nature of these data is not taken into consideration. This article seeks to address these two previous shortcomings in order to produce more accurate estimates. MethodsWe employed an isometric log-ratio transformation designed for compositional data. We applied linear and non-linear time regressions to both the original and the transformed data. Specifically, different modelling alternatives for non-linear trajectories were analysed, all of which are based on a generalized additive model (GAM). Results and discussionNon-linear methods, such as GAM, may be used for modelling non-linear trajectories in the JMP data. This projection method is particularly suited for data-rich countries. Moreover, the ilr transformation of compositional data is conceptually sound and fairly simple to implement. It helps improve the performance of both linear and non-linear regression models, specifically in the occurrence of extreme data points, i.e. when coverage rates are near either 0% or 100%.

Multidimensional classification of magma types for altered igneous rocks and application to their tectonomagmatic discrimination and igneous provenance of siliciclastic sediments

A new multidimensional scheme consistent with the International Union of Geological Sciences (IUGS) is proposed for the classification of igneous rocks in terms of four magma types: ultrabasic, basic, intermediate, and acid. Our procedure is based on an extensive database of major element composition of a total of 33,868 relatively fresh rock samples having a multinormal distribution (initial database with 37,215 samples). Multinormally distributed database in terms of log-ratios of samples was ascertained by a new computer program DOMuDaF, in which the discordancy test was applied at the 99.9% confidence level. Isometric log-ratio (ilr) transformation was used to provide overall percent correct classification of 88.7%, 75.8%, 88.0%, and 80.9% for ultrabasic, basic, intermediate, and acid rocks, respectively. Given the known mathematical and uncertainty propagation properties, this transformation could be adopted for routine applications. The incorrect classification was mainly for the “neighbour” magma types, e.g., basic for ultrabasic and vice versa. Some of these misclassifications do not have any effect on multidimensional tectonic discrimination. For an efficient application of this multidimensional scheme, a new computer program MagClaMSys_ilr (MagClaMSys—Magma Classification Major-element based System) was written, which is available for on-line processing on http://tlaloc.ier.unam.mx/index.html. This classification scheme was tested from newly compiled data for relatively fresh Neogene igneous rocks and was found to be consistent with the conventional IUGS procedure. The new scheme was successfully applied to inter-laboratory data for three geochemical reference materials (basalts JB-1 and JB-1a, and andesite JA-3) from Japan and showed that the inferred magma types are consistent with the rock name (basic for basalts JB-1 and JB-1a and intermediate for andesite JA-3). The scheme was also successfully applied to five case studies of older Archaean to Mesozoic igneous rocks. Similar or more reliable results were obtained from existing tectonomagmatic discrimination diagrams when used in conjunction with the new computer program as compared to the IUGS scheme. The application to three case studies of igneous provenance of sedimentary rocks was demonstrated as a novel approach. Finally, we show that the new scheme is more robust for post-emplacement compositional changes than the conventional IUGS procedure.

Applying compositional data methodology to nutritional epidemiology

The purpose of epidemiological studies of nutrition and disease is to investigate the effects of specific dietary components regardless of total energy intake, but this is sometimes hampered by the compositional nature of dietary data. Compositional data are those that measure parts of a whole, such as percentages or proportions, and particular methodologies have been developed to allow their statistical analysis and theoretical and practical applications in various sciences. This paper describes the use of a compositional data perspective for statistical analyses in the field of nutritional epidemiology. The approach is based on isometric log-ratio transformation and has been previously proposed for the construction of regression models using compositional explanatory variables. The new isometric log-ratio variables allow full inferences about each element of dietary composition and adjustment by total energy intake. Using data from an Italian population-based study, logistic regression models were fitted to evaluate the effects of the intake of macronutrients (proteins, fats and carbohydrates) on the odds of having metabolic syndrome in middle-aged subjects.

Application of classical statistics and multifractals to delineate Au mineralization-related geochemical anomalies from stream sediment data: a case study in Xinghai-Zeku, Qinghai, China

The Xinghai-Zeku area of western China is one of the most important Au polymetallic metallogenic belts in Qinghai Province, China. To guide the mineral exploration in this area, log and isometric log-ratio (ilr) transformations of stream sediment data are evaluated using principal component analysis (PCA) combined with geological data to study the relationship among different elements. In addition, Mean + 2 standard deviations (Mean + 2STD), Median + 2 median absolute deviations (Median + 2MAD), and concentration-area (C-A) models are applied to identify pathfinder threshold values and geochemical patterns are decomposed using a spectrum-area (S-A) model. The results show that: (1) PCA for the ilr-transformed data can accurately describe three different geochemical assemblages, Au-As-Sb, that represent Au-As-Sb mineralization in fracture zones; (2) anomalies of Au + As + Sb are more suitable for targeting Au-As deposits than those of the single element Au; (3) the C-A model is useful for indentifying geochemical anomalies associated with mineralization because results obtained by the C–A fractal model and the geological characteristics are well correlated; (4) background and anomaly maps for Au + As + Sb from the S-A model in conjunction with the Mean + 2STD successfully identify weaker anomalies by reducing anomalous areas; and (5) using Au + As + Sb anomalies identified by C-A in conjunction with the S-A model and Mean + 2STD method are effective in exploration for Au deposits in the area.

Contrasting Nutritional Acclimation of Sugar Maple (Acer saccharum Marsh.) and Red Maple (Acer rubrum L.) to Increasing Conifers and Soil Acidity as Demonstrated by Foliar Nutrient Balances

Sugar maple (Acer saccharum Marshall, SM) is believed to be more sensitive to acidic and nutrient-poor soils associated with conifer-dominated stands than red maple (Acer rubrum L., RM). Greater foliar nutrient use efficiency (FNUE) of RM is likely the cause for this difference. In the context of climate change, this greater FNUE could be key in favouring northward migration of RM over SM. We used the concept of foliar nutrient balances to study the nutrition of SM and RM seedlings along an increasing gradient in forest floor acidity conditioned by increasing proportions of conifers (pH values ranging from 4.39 under hardwoods, to 4.29 under mixed hardwood-conifer stands and 4.05 under conifer-dominated stands). Nutrients were subjected to isometric log-ratio (ilr) transformation, which views the leaf as one closed system and considers interactions between nutrients. The ilr method eliminates numerical biases and weak statistical inferences based on raw or “operationally’’ log-transformed data. We analyzed foliar nutrients of SM and RM seedlings and found that the [Ca,Mg,K|P,N] and [Ca,Mg|K] balances of SM seedlings were significantly different among soil acidity levels, whereas they did not vary for RM seedlings. For SM seedlings, these differences among soil acidity levels were due to a significant decrease in foliar Ca and Mg concentrations with increasing forest floor acidity. Similar differences in foliar balances were also found between healthy and declining SM stands estimated from literature values. Conversely, foliar balances of RM seedlings did not differ among soil acidity levels, even though untransformed foliar nutrient concentrations were significantly different. This result highlights the importance of using ilr transformation, since it provides more sensitive results than standard testing of untransformed nutrient concentrations. The lower nutrient requirements of RM and its greater capacity to maintain nutrient equilibrium are factors that could explain its competitive success and recent northward expansion. This study underscores the importance of using nutrient balances to study the redistribution of plant species in natural ecosystems under climate change.

Stepwise regression for recognition of geochemical anomalies: Case study in Takab area, NW Iran

Stream sediment geochemical data represent compositional materials derived from various sources, including single or multiple lithologic units, soil types, rocks types, etc. In order to delineate geochemical anomalies, stream sediment geochemical data are usually subjected to suitable multivariate analysis, and not simply using univariate threshold values because these are not reliable for delineation of geochemical anomalies in areas with complex geological units. Relationships among multiple major/trace elements and rock types are more important than single major/trace elements for delineation of geochemical anomalies. In this study we present an approach based on robust stepwise multiple regression using values major oxides (SiO2, Al2O3, Fe2O3, MnO, and MgO) in stream sediments to predict elemental content related to rock types and to recognize geochemical anomalies. The major/trace element data were subjected to isometric logratio transformation to address the compositional data closure problem. For further examination of the stepwise regression method, its performance was compared to robust principal components analysis (RPCA), median+2MAD and concentration-area (C-A) fractal methods. The results show that multi-element anomalies obtained by the stepwise regression method, compared to those obtained by the other methods, have stronger spatial association with the known deposits, such as Chichaklo and Ay-Ghale-Si in the Takab 1:25,000 scale geological map (NW) Iran, and the anomalies have stronger spatial correlation with structural features and prospects, and thus can be used as guides to new exploration targets.

Associations between sleep duration, sedentary time, physical activity, and health indicators among Canadian children and youth using compositional analyses.

The purpose of this study was to examine the relationships between movement behaviours (sleep duration, sedentary time, physical activity) and health indicators in a representative sample of children and youth using compositional analyses. Cross-sectional findings are based on 4169 children and youth (aged 6-17 years) from cycles 1 to 3 of the Canadian Health Measures Survey. Sedentary time (SB), light-intensity physical activity (LPA), and moderate- to vigorous-intensity physical activity (MVPA) were accelerometer-derived. Sleep duration was subjectively measured. Body mass index z scores, waist circumference, blood pressure, behavioural strengths and difficulties, and aerobic fitness were measured in the full sample. Triglycerides, high-density lipoprotein-cholesterol, C-reactive protein, and insulin were measured in a fasting subsample. The composition of movement behaviours was entered into linear regression models via an isometric log ratio transformation and was found to be associated with all health indicators (p < 0.01). Relative to other movement behaviours, time spent in SB or LPA was positively associated (p < 0.04) and time spent in MVPA or sleep was negatively associated (p < 0.02) with obesity risk markers. Similarly, LPA was positively associated (p < 0.005) and sleep was negatively associated (p < 0.03) with unfavourable behavioural strengths and difficulties scores and systolic blood pressure. Relative to other movement behaviours, time spent in SB was negatively associated (p < 0.001) and time spent in MVPA (p < 0.001) was positively associated with aerobic fitness. Likewise, MVPA was also negatively associated with several cardiometabolic risk markers (p < 0.008). Compositional data analyses provide novel insights into collective health implications of 24-h movement behaviours and can facilitate interesting avenues for future investigations.

Calorific value and compositional ultimate analysis with a case study of a Texas lignite

Measurements to determine coal quality as fuel include proximate analysis, ultimate analysis and calorific value. The latter is an attribute taking non-negative real values, so a simple transformation is sufficient for its spatial modeling applying geostatistics. The analyses, however, involve proportions that follow the properties of compositional data, thus requiring special preprocessing for an adequate modeling already described in a previous publication for the case of proximate analysis data.11Olea, R.A., Luppens, J.A., 2015. Mapping of coal quality using stochastic simulation and isometric logratio transformation with an application to a Texas lignite. International Journal of Coal Geology, 152, 80–93. Here we model the results of calorific value and ultimate analysis. We propose to use two different binary partitions, one per analysis, map the corresponding isometric logratio transformations, and backtransform the results. The methodology is illustrated using the same coal bed in the previous paper modeling proximate analysis data. Results are summarized using probability maps that, in the case of this deposit, show a prominent channel crossing the deposit and separating the best quality coal from that of lower quality.

Compositional biplots including external non-compositional variables

Biplots represent a widely used statistical tool for visualizing the resulting loadings and scores of a dimension reduction technique applied to multivariate data. If the underlying data carry only relative information (i.e. compositional data expressed in proportions, mg/kg, etc.) they have to be pre-processed with a logratio transformation before the dimension reduction is carried out. In the context of principal component analysis, the resulting biplot is called compositional biplot. We introduce an alternative, the ilr biplot, which is based on a special choice of orthonormal coordinates resulting from an isometric logratio (ilr) transformation. This allows to incorporate also external non-compositional variables, and to study the relations to the compositional variables. The methodology is demonstrated on real data sets.

Distribution of potentially toxic metals (As, Cu, Hg, Pb and Zn) in the topsoil of the Pannonian Basin System and associated parts of the surrounding orogens

ABSTRACTThe Pannonian Basin System (PBS) and associated parts of the surrounding orogens (SOs) were selected for a regional geochemical mapping of concentrations of potentially toxic metals, with the sample media the topsoil horizon. Geochemical data form a sample space that is different from Euclidean, and therefore standard statistical methods are not applicable for such data. To apply standard statistical methods, data were transformed to Euclidean space by using a special case of an isometric log-ratio transformation for a composition consisting of two parts (the first part is the metal concentration of interest; the second part is the total concentration of all other elements in the composition). Estimations were performed using deterministic (Ordinary Kriging) and stochastic (sequential Gaussian simulations (SGS)) methods. The resulting maps provide insight into the distribution of certain metals in the PBS and associated parts of the SOs, and are intended to serve as base maps for further studies. The maps obtained with SGS provide additional information on the variability (uncertainty) of estimation.

Mapping of coal quality using stochastic simulation and isometric logratio transformation with an application to a Texas lignite

Coal is a chemically complex commodity that often contains most of the natural elements in the periodic table. Coal constituents are conventionally grouped into four components (proximate analysis): fixed carbon, ash, inherent moisture, and volatile matter. These four parts, customarily measured as weight losses and expressed as percentages, share all properties and statistical challenges of compositional data. Consequently, adequate modeling should be done in terms of a logratio transformation, a requirement that is commonly overlooked by modelers. The transformation of choice is the isometric logratio transformation because of its geometrical and statistical advantages. The modeling is done through a series of realizations prepared by applying sequential simulation for the purpose of displaying the parts in maps incorporating uncertainty. The approach makes realistic assumptions and the results honor the data and basic considerations, such as percentages between 0 and 100, all four parts adding to 100% at any location in the study area, and a style of spatial fluctuation in the realizations equal to that of the data. The realizations are used to prepare different results, including probability distributions across a deposit, E-type maps displaying average properties, and probability maps summarizing joint fluctuations of several parts. Application of these maps to a lignite bed clearly delineates the deposit boundary, reveals a channel cutting across, and shows that the most favorable coal quality is to the north and deteriorates toward the southeast.

Meta-analysis in the Selection of Groups in Varieties of Citrus

Brazil is the largest producer of oranges (Citrus sinensis) in the world. The nutrient management of tree orchards is designed from experiments with a limited number of varieties. This knowledge is transferred to other varieties by diagnosing tissue nutrient composition and tree demand. Compositional data analysis has been first applied to tissue analysis of agricultural crops using centered log ratios with compositional nutrient diagnosis (CND-clr). The isometric log ratio (ilr) transformation is a new approach based on binary nutrient ratios and the principle of orthogonality (CND-ilr). We analyzed eleven nutrients: nitrogen (N), sulfur (S), phosphorus (P), potassium (K), calcium (Ca), magnesium (Mg), boron (B), copper (Cu), zinc (Zn), manganese (Mn), and iron (Fe) in leaf tissue samples across 108 commercial plots (thirty-one grow Valencia, twenty-two Hamlim, twenty Pera, and thirty-five Natal). Nutrients were partitioned between macro- and micronutrients as well as anionic and cationic species. The effect size of varieties over Valencia was quantified by the mean and standard deviation of ilr values across ilr coordinates. Specific varietal nutrient profiles and ilr norms were defined. The nutrient profile of orange varieties could be classified into homogeneous groups to take advantage of fertilizer trials conducted on varieties of the same group. The Aitchison distance and a perturbation vector could be instrumental for diagnostic purposes and nutrient management.