A new class of [formula omitted]-transformations for the spatial analysis of Compositional Data

One of the most robust empirical findings in political science is that in multiparty democracies cabinet ministries are distributed in rough proportion to parties’ legislative seat shares, a pattern known as Gamson’s Law. Yet existing research often overlooks the fact that portfolio and seat shares are compositional—mutually dependent parts of a whole. Standard methods treat them as unconstrained, risking bias, misleading uncertainty estimates, and flawed inference. Unfortunately, the most common strategy for handling compositions—the additive log-ratio (ALR) transformation combined with seemingly unrelated regression (SUR)—fails when the number and identity of compositional elements (like parties) vary across cases. We propose the isometric log-ratio (ILR) transformation—new to political science—as an alternative that both respects compositional geometry and adapts to differing compositional structures. Monte Carlo simulations show that ILR sharply outperforms standard approaches, reducing bias, improving coverage probability, and increasing statistical power. While we apply it to portfolio allocation, ILR provides a general solution for modeling compositional outcomes with other potential uses, including in electoral competition, where ALR+SUR has required strong assumptions or ad hoc adjustments. Using this improved methodology, we find that seat–portfolio proportionality is weaker overall than conventionally reported and varies substantially across governments.

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.

This study introduces a novel groundwater pollution index (GPI) formulated through compositional data analysis (CoDa) and robust principal component analysis (RPCA) to enhance groundwater quality assessment. Using groundwater quality monitoring data from sites impacted by the 2010–2011 foot-and-mouth disease outbreak in South Korea, CoDa uncovers critical hydrochemical differences between leachate-influenced and background groundwater. The GPI was developed by selecting key subcompositional parts (NH4+-N, Cl-, and NO3--N) using RPCA, performing the isometric log-ratio (ILR) transformation, and normalizing the results to environmental standards, thereby providing a more precise and accurate assessment of pollution. Validated against government criteria, the GPI has shown its potential as an alternative assessment tool, with its reliability confirmed by receiver operating characteristic curve analysis. This study highlights the essential role of CoDa, especially the ILR -transformation, in overcoming the limitations of traditional statistical methods that often neglect the relative nature of hydrochemical data. Our results emphasize the utility of the GPI in significantly advancing groundwater quality monitoring and management by addressing a methodological gap in the quantitative assessment of groundwater pollution.

Loadings and scores of principal component analysis are popularly displayed together in a planar graph, called biplot, with an intuitive interpretation. In case of compositional data, multivariate observations that carry only relative information (represented usually in proportions or percentages), principal component analysis cannot be used for the original compositions. They first need to be transformed using the centered logratio (clr) transformation. If outlying observations occur in compositional data, even the clr (compositional) biplot can lead to useless conclusions. A robust alternative biplot can be computed by using the isometric logratio (ilr) transformation, and by robustly estimating location and covariance. The robust compositional biplot has a big potential in many applications, like in geology, analytical chemistry, or social sciences.

Digital soil mapping of soil particle-size fractions (PSFs) using log-ratio methods is a widely used technique. As a hybrid interpolator, regression kriging (RK) provides a way to improve prediction accuracy. However, there have been few comparisons with other techniques when RK is applied for compositional data, and it is not known if its performance based on different balances of isometric log-ratio (ILR) transformation is robust. Here, we compared the generalized linear model (GLM), random forest (RF), and their hybrid patterns (RK) using different transformed data based on three ILR balances, with 29 environmental covariables (ECs) for the prediction of soil PSFs in the upper reaches of the Heihe River Basin (HRB), China. The results showed that the RF performed best, with more accurate predictions, but the GLM produced a more unbiased prediction. As a hybrid interpolator, RK was recommended because it widened the data ranges of the prediction values, and modified the bias and accuracy of most models, especially the RF. The prediction maps generated from RK revealed more details of the soil sampling points than the other models. Different data distributions were produced for the three ILR balances. Using the most abundant component of the compositional data as the first component of the permutations was not considered to be the right choice because it produced the worst performance. Based on the relative abundance of the components, we recommend that the focus should be on data distribution. This study provides a reference for the mapping of soil PSFs combined with transformed data at the regional scale.

Plateau–orogenic belts host a substantial share of global gold resources, yet quantitative prospectivity mapping is challenged by complex mineralization and strongly heterogeneous, multi-scale datasets. Using the Mayoumu area (Tibet) as a representative orogenic gold district, we develop an integrated multi-source workflow that fuses remote-sensing alteration information with regional geochemical and structural constraints within an ensemble-learning framework. Alteration anomalies were mapped from GF-5 hyperspectral imagery using mixture-tuned matched filtering (MTMF) and from Sentinel-2 multispectral imagery using the iCrosta method to extend alteration signals across scales. Geochemical anomalies were extracted from 1:200,000 stream-sediment data through isometric log-ratio (ILR) transformation and robust principal component analysis (RPCA). At the same time, ore-controlling structures were quantified using Euclidean-distance-to-fault layers. Three Boosting-based ensemble models—gradient boosting decision tree (GBDT), extreme gradient boosting (XGBoost), and light gradient boosting machine (LightGBM)—were trained to predict mineral prospectivity. Performance was evaluated using confusion matrix metrics and ROC–AUC, and key predictors were interpreted using SHAP. All three models achieved AUC values > 0.90, with LightGBM performing best (AUC = 0.94) and delineating high-prospectivity zones that coincide with known occurrences and highlight additional targets. The proposed workflow provides a practical, transferable reference for gold prospectivity mapping in complex orogenic belts worldwide.

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

Park, N.-W. and Jang, D.-H., 2020. Geostatistical classification of intertidal surface sediments using log-ratio transformation and high-resolution remote sensing imagery. In: Jung, H.-S.; Lee, S.; Ryu, J.-H., and Cui, T. (eds.), Advances in Geospatial Research of Coastal Environments. Journal of Coastal Research, Special Issue No. 102, pp. 157-165. Coconut Creek (Florida), ISSN 0749-0208.This paper presents a multivariate geostatistical approach to classify intertidal surface sediments by combining compositional data analysis and high-resolution remote sensing imagery. An isometric log-ratio (ilr) transformation is first applied to the sediment composition data prior to employing geostatistical analysis to consider the compositional properties of the sediment compositions. To complement the information deficiency of sparse field measurements, high-resolution remote sensing imagery is considered as exhaustive soft information and integrated with the ilr transformed balances via simple kriging with varying local means (SKLM). An inverse ilr transformation is then applied to the SKLM results to obtain sediment compositions over the study area. Finally, Shepard's classification scheme is applied to the sediment compositions to classify the intertidal surface sediments. A case study was conducted on the Baramarae tidal flats in Korea with high-resolution KOMPSAT-2 imagery to demonstrate the effectiveness of the proposed geostatistical approach. The classification results produced by the integration of high-resolution remote sensing imagery via ilr transformation and SKLM outperformed those based on cokriging of sediment compositions, with an improvement of approximately 11.7 %p in overall accuracy. This improvement was attributed to the superior prediction capability of SKLM for sediment compositions. Further, detailed variations in the sediment distributions in the study area, which could not be observed when using only a limited number of sediment samples, could be revealed by integrating the high-resolution remote sensing imagery. Therefore, the geostatistical integration approach that properly accounts for both the property of sediment compositions and the exhaustive soft information from remote sensing imagery could be effectively applied for the classification of intertidal surface sediments.

Aggregating multiple types of complex data in stock market prediction: A model-independent framework

GIS is a geo-referenced data based system for manipulating digital spatial and thematic data. Because of its unique properties of spatial data (spatial dependence and spatial heterogeneity), there is an urgent need to investigate the mechanisms of spatial processes through exploring spatial data. The techniques of spatial analysis can be categorized into six groups: (1) generation of spatial data and its preliminary treatment; (2) spatialization of attribute data and scale transformation of spatial data; (3) visualization and exploration of spatial data; (4) geo-statistics; (5) analysis of lattice data; and (6) reversing information from complicated spatial data. To meet the real world solutions, a procedural diagram of spatial data analysis is suggested as below: (1) preliminary data treatment; (2) modeling; (3) solution; and (4) interpretation. Various techniques are being employed to investigate spatial processes, the identical study objective and some common characteristics of the study object may lead to the integration and convergence of the various approaches, and further the introduction of a general science: spatial mathematics. A same important issue is to develop robust spatial analysis computer packages for public uses.

Summary Treatment methods for produced water (PW) are significantly affected by a high concentration of total dissolved solids (TDS), a summation of dissolved organic and inorganic compositions. Understanding the constituents of TDS to eliminate or prevent chemical reactions is critical in the design optimization of the treatment processes. In this paper, two PW geochemical data sets generated from conventional and unconventional reservoirs in the Permian Basin were analyzed to correlate constituents with TDS. Compositional data sets from over 115,000 PW samples originally reported by the U.S. Geological Survey (USGS) and 45 oil and gas operations were analyzed. Data preprocessing, culling, systematized- and meta-analysis, and statistical techniques were adapted to associate the data. Subcompositional geochemical data were transformed into isometric log ratios and are presented in bivariate and multivariate plots. Results indicate that Na+ and Ca2+ were the dominant cations and Cl− was the dominant anion. No observable trend differences in the Na+, Cl−, Ca2+, Mg+, and SO42− concentrations between PW from conventional and unconventional wells were registered. Variations in the isometric log ratio of Na/Cl and Ca/SO4 with TDS revealed that Na/Cl was nearly constant over the range of TDS, suggesting mineral buffering or a lack of significant water/rock reactions involving Na and Cl, and that Ca/SO4 increased with TDS, indicating that low-salinity fluids may have dissolved anhydrite producing a value near zero, with Ca gain and/or SO4 loss with increasing salinity. In all 10 counties and 8 formations investigated in this work, the ln (Ca/SO4) denotes Ca gain/SO4 loss relative to their composition in anhydrite or Permian seawater. Likely mechanisms leading to elevated ln (Ca/SO4) include sulfate reduction, dolomitization of calcite, Na/Ca cation exchange, albitization, and anhydrite precipitation from Ca-rich fluids. Results from this work are important as it is revealed that Ca/SO4 and Na/Cl can potentially be predicted from PW TDS concentrations. This information was combined to create a reservoir or location-specific model to estimate Na, Cl, Ca, and SO4 concentrations in Permian Basin PW, a powerful tool to improve treatment and reuse options in areas with few direct data.

In recent years, the need for a temporal dimension in traditional spatial information systems and for high-level models useful for the conceptual design of the resulting spatiotemporal systems has become clear. Although having in common a need to manage spatial data and their changes over time, various spatiotemporal applications may manage different types of spatiotemporal data and may be based on very different models of space, time, and change. For example, the term spatiotemporal data is used to refer both to temporal changes in spatial extents, such as redrawing the boundaries of a voting precinct or land deed, and to changes in the value of thematic (i.e., alphanumeric) data across time or space, such as variation in soil acidity measurements depending on the measurement location and date. A spatiotemporal application may be concerned with either or both types of data. This, in turn, is likely to influence the underlying model of space employed, e.g., the two types of spatiotemporal data generally correspond to an object- versus a field-based spatial model. For either type of spatiotemporal data, change may occur in discrete steps, e.g., changes in land deed boundaries, or in a continuous process, e.g., changes in the position of a moving object such as a car. Another type of spatiotemporal data is composite data whose components vary depending on time or location. An example is the minimum combination of equipment and wards required in a certain category of hospital (e.g., general, maternity, psychiatric), where the relevant regulations determining the applicable base standards vary by locality and time period.Request access from your librarian to read this chapter's full text.

In recent years, the need for a temporal dimension in traditional spatial information systems and for high-level models useful for the conceptual design of the resulting spatiotemporal systems has become clear. Although having in common a need to manage spatial data and their changes over time, various spatiotemporal applications may manage different types of spatiotemporal data and may be based on very different models of space, time, and change. For example, the term spatiotemporal data is used to refer both to temporal changes in spatial extents, such as redrawing the boundaries of a voting precinct or land deed, and to changes in the value of thematic (i.e., alphanumeric) data across time or space, such as variation in soil acidity measurements depending on the measurement location and date. A spatiotemporal application may be concerned with either or both types of data. This, in turn, is likely to influence the underlying model of space employed, e.g., the two types of spatiotemporal data generally correspond to an object- versus a field-based spatial model. For either type of spatiotemporal data, change may occur in discrete steps, e.g., changes in land deed boundaries, or in a continuous process, e.g., changes in the position of a moving object such as a car. Another type of spatiotemporal data is composite data whose components vary depending on time or location. An example is the minimum combination of equipment and wards required in a certain category of hospital (e.g., general, maternity, psychiatric), where the relevant regulations determining the applicable base standards vary by locality and time period.Request access from your librarian to read this chapter's full text.

In a number of recent articles Riani, Cerioli, Atkinson and others advocate the technique of monitoring robust estimates computed over a range of key parameter values. Through this approach the diagnostic tools of choice can be tuned in such a way that highly robust estimators which are as efficient as possible are obtained. This approach is applicable to various robust multivariate estimates like S- and MM-estimates, MVE and MCD as well as to the Forward Search in whichmonitoring is part of the robust method. Key tool for detection of multivariate outliers and for monitoring of robust estimates is the Mahalanobis distances and statistics related to these distances. However, the results obtained with thistool in case of compositional data might be unrealistic since compositional data contain relative rather than absolute information and need to be transformed to the usual Euclidean geometry before the standard statistical tools can be applied. Various data transformations of compositional data have been introduced in the literature and theoretical results on the equivalence of the additive, the centered, and the isometric logratio transformation in the context of outlier identification exist. To illustrate the problem of monitoring compositional data and to demonstrate the usefulness of monitoring in this case we start with a simple example and then analyze a real life data set presenting the technologicalstructure of manufactured exports. The analysis is conducted with the R package fsdaR, which makes the analytical and graphical tools provided in the MATLAB FSDA library available for R users.

Meal timing is suggested to influence the obesity risk in children. Our aim was to analyse the effect of energy and nutrient distributions at eating occasions (EO), including breakfast, lunch, supper, and snacks, on the BMI z-score (zBMI) during childhood in 729 healthy children. BMI and three-day dietary protocols were obtained at 3, 4, 5, 6, and 8 years of age, and dietary data were analysed as the percentage of the mean total energy intake (TEI; %E). Intakes at EOs were transformed via an isometric log–ratio transformation and added as exposure variables to linear mixed-effects models. Stratified analyses by country and recategorization of EOs by adding intake from snacks to respective meals for further analyses were performed. The exclusion of subjects with less than three observations and the exclusion of subjects who skipped one EO or consumed 5% energy or less at one EO were examined in sensitivity analyses. Around 23% of the children were overweight at a given time point. Overweight and normal-weight children showed different distributions of dietary intakes over the day; overweight children consumed higher intakes at lunch and lower intakes of snacks. However, no significant effects of timing of EOs on zBMI were found in regression analyses.