Isometric Log Ratio Research Articles

Nutrient Balances of New Zealand Kiwifruit (Actinidia deliciosa cv. Hayward) at High Yield Level

The productivity of kiwifruit (Actinidia deliciosa) orchards varies widely in New Zealand. We hypothesized that such variation was attributable in part to nutrient imbalance. However, nutrient imbalance is currently diagnosed using critical nutrient ranges that are noisy and biased. Unbiased diagnosis can be performed using the isometric log ratio (ilr) technique for ad hoc balances. Our objective was to present, calibrate, and validate ilr balance standards to diagnose nutrient problems in New Zealand kiwifruit orchards. We collected leaf analytical data [nitrogen (N), sulfur (S), chloride (Cl), phosphorus (P), potassium (K), calcium (Ca), magnesium (Mg), boron (B), copper (Cu), zinc (Zn), manganese (Mn), and iron (Fe)] in 433 kiwifruit orchards grown under conventional or organic farming in various agroecological zones of the North Island of New Zealand. Nutrients were arranged into sound balances illustrated by a mobile-fulcrums-buckets metaphor with balance sliders at fulcrums (domain of balances where statistics are computed) and concentrations in buckets (domain of concentrations where nutrient levels are examined relatively to each other). We developed optimum ranges of ilr balances to reach high yield levels and computed a nutrient imbalance predictor as the Mahalanobis distance. The critical Mahalanobis distance was 4.45 ± 0.13 at high yield level (>44 049 ± 334 kg ha−1), averaged from a five-fold cross-validation test; the test performance was 85 percent using the receiver operating characteristic curve, indicating that the test was informative for diagnostic purposes. After assessing relative nutrient levels in the concentrations domain, the most limiting nutrients appeared to be Cl, Mg, and Fe in poor-yield imbalanced specimens and Cl and S in organic orchards. Indeed, kiwifruit has high demand for Cl. Balance standards should be further tested in field trials.

The isometric log-ratio transform for probabilistic multi-label anatomical shape representation.

Sources of uncertainty in the boundaries of structures in medical images have motivated the use of probabilistic labels in segmentation applications. An important component in many medical image segmentation tasks is the use of a shape model, often generated by applying statistical techniques to training data. Standard statistical techniques (e.g., principal component analysis) often assume data lies in an unconstrained vector space, but probabilistic labels are constrained to the unit simplex. If these statistical techniques are used directly on probabilistic labels, relative uncertainty information can be sacrificed. A standard method for facilitating analysis of probabilistic labels is to map them to a vector space using the LogOdds transform. However, the LogOdds transform is asymmetric in one of the labels, which skews results in some applications. The isometric log-ratio (ILR) transform is a symmetrized version of the LogOdds transform, and is so named as it is an isometry between the Aitchison geometry, the inherent geometry of the simplex, and standard Euclidean geometry. We explore how to interpret the Aitchison geometry when applied to probabilistic labels in medical image segmentation applications. We demonstrate the differences when applying the LogOdds transform or the ILR transform to probabilistic labels prior to statistical analysis. Specifically, we show that statistical analysis of ILR transformed data better captures the variability of anatomical shapes in cases where multiple different foreground regions share boundaries (as opposed to foreground-background boundaries).

Digital Mapping of Soil Texture Using RADARSAT-2 Polarimetric Synthetic Aperture Radar Data

This study aimed to assess the contribution of RADARSAT-2 polarimetric synthetic aperture radar (SAR) data to digital mapping of soil surface texture of Rouville County, near Montreal, Canada. First, compositional data transformation using isometric log ratio (ilr) was applied to soil texture data to transpose the simplex into multidimensional real space, which is better suited to multivariate analysis. Thereafter, two assumptions on the relationships between RADARSAT-2 data and the ilr components were evaluated: (i) as linearly dependent by applying cokriging (CK) and regression kriging (RK); and (ii) as nonlinearly dependent by applying the e-insensitive and nonlinear support vector regression (SVR). The results were compared with ordinary kriging (OK). The environmental variables used to define the covariates were monopolarization SAR channels, the parameters extracted from entropy/anisotropy/ mean α polarimetric, Freeman and Durden, and Touzi decompositions. Using 283 soil samples for training and 89 for validation, the results showed that the root mean square error (RMSE) of prediction obtained by OK was 8% for silt, 13% for clay, and 13% for sand. The SVR produced the best prediction accuracy compared with the geostatistical interpolation techniques. Compared with OK, the improvement of the digital mapping accuracies (in terms of RMSE reduction) with SVR was significant: RMSE was reduced by 18% for sand, 17% for silt, and 35% for clay. It was followed by RK, with RMSE reduction ranging from 6 to 13%, and then CK (3–5%). Using SAR polarimetric data extracted from a RADARSAT-2 image as covariates was found to be very useful for digital mapping of soil surface texture.

Mapping of Fe mineralization-associated geochemical signatures using logratio transformed stream sediment geochemical data in eastern Tianshan, China

Geochemical data used in this paper to identify geochemical signatures associated with Fe mineralization in eastern Tianshan mineral district, China are compositional data. In order to eliminate the spurious relationships between compositions, isometric logratio (ilr) transformation is currently employed to deal with the closure effects. The opened geochemical data are further analyzed by principal component analysis. By back-transforming to the clr space, PCA results of ilr transformed data with geological meanings of their counterparts assist in recognition of spatial distributions of both intermediate-felsic igneous rocks and fault systems in the study area. In comparison with log-transformed geochemical data, the ilr transformed ones are theoretically reasonable to apply standard statistics. In addition, multivariate outliers are identified to investigate their associations with the formation of these two geological features. At the end, this paper suggests that the ilr transformation is necessary to be applied routinely before applying statistical treatments to raw geochemical data.

Identification of geochemical anomalies associated with mineralization in the Fanshan district, Fujian, China

The Fanshan district is a prospective area to explore for epithermal-type Cu–Au mineralization that is similar to the Zijinshan Cu–Au deposit in the neighboring region which is the largest Cu–Au deposit in southern China. In this study, the robust principal component analysis (RPCA) method and spectrum-area (S-A) fractal model were applied to multi-element data from 2042 stream sediment samples collected in this region. The isometric logratio (ilr) transformation was used to open the geochemical data for reducing the effects of data closure problem. The resulting PC1 obtained using RPCA displays two different compositional assemblages: (I) Pb, Zn, Sn, W, Mo, Bi, Hg, and Ag, perhaps representing hydrothermal-type Zn–Pb polymetallic mineralization, and (II) As, Au, Cu, Sb and Mn, likely representing epithermal-type Cu–Au mineralization. The decomposed anomaly map from the PC1 obtained from the S-A model illustrates that highly anomalous areas occurring around the intrusions are correlated with hydrothermal-type Zn–Pb polymetallic mineralization, and areas with lower anomalies such as Dafanshan and Xiaofanshan are perhaps associated with Cu–Au polymetallic mineralization, and should be further studied to explore for epithermal-type Cu–Au mineralization.

Whole-coal versus ash basis in coal geochemistry: A mathematical approach to consistent interpretations

Several standard methods require coal to be ashed prior to geochemical analysis. Researchers, however, are commonly interested in the compositional nature of the whole-coal, not its ash. Coal geochemical data for any given sample can, therefore, be reported in the ash basis on which it is analyzed or the whole-coal basis to which the ash basis data are back calculated. Basic univariate (mean, variance, distribution, etc.) and bivariate (correlation coefficients, etc.) measures of the same suite of samples can be very different depending which reporting basis the researcher uses. These differences are not real, but an artifact resulting from the compositional nature of most geochemical data. The technical term for this artifact is subcompositional incoherence. Since compositional data are forced to a constant sum, such as 100% or 1,000,000ppm, they possess curvilinear properties which make the Euclidean principles on which most statistical tests rely inappropriate, leading to erroneous results. Applying the isometric logratio (ilr) transformation to compositional data allows them to be represented in Euclidean space and evaluated using traditional tests without fear of producing mathematically inconsistent results. When applied to coal geochemical data, the issues related to differences between the two reporting bases are resolved as demonstrated in this paper using major oxide and trace metal data from the Pennsylvanian-age Pond Creek coal of eastern Kentucky, USA. Following ilr transformation, univariate statistics, such as mean and variance, still differ between the ash basis and whole-coal basis, but in predictable and calculated manners. Further, the stability between two different components, a bivariate measure, is identical, regardless of the reporting basis. The application of ilr transformations addresses both the erroneous results of Euclidean-based measurements on compositional data as well as the inconsistencies observed on coal geochemical data reported on different bases.

Predictors and outcomes of social network compositions: A compositional structural equation modeling approach

Proportions of a total, including social network compositions (proportions of partner, family, friends, etc.) lie in a restricted space, which challenges statistical analysis. Network compositions can be both dependent and explanatory variables and are usually measured with error by survey instruments. Structural equation models make it possible to correct measurement error bias. Coenders et al. (2011) fitted a factor analysis model to transformed network compositions. In this article, we use another transformation called an isometric log-ratio and we extend the model to include predictors and outcomes. The findings and hypotheses in the literature can be reformulated with isometric log-ratios in a more interpretable manner. For instance, we find relationships of gender with partner support, of education and extraversion with friend support, and of family support with tie multiplexity and closeness.

Linear regression with compositional explanatory variables

Compositional explanatory variables should not be directly used in a linear regression model because any inference statistic can become misleading. While various approaches for this problem were proposed, here an approach based on the isometric logratio (ilr) transformation is used. It turns out that the resulting model is easy to handle, and that parameter estimation can be done in like in usual linear regression. Moreover, it is possible to use the ilr variables for inference statistics in order to obtain an appropriate interpretation of the model.

Interpretation of multivariate outliers for compositional data

Compositional data—and most data in geochemistry are of this type—carry relative rather than absolute information. For multivariate outlier detection methods this implies that not the given data but appropriately transformed data need to be used. We use the isometric logratio (ilr) transformation, which seems to be generally the most proper one for theoretical and practical reasons. In this space it is difficult to interpret the outliers, because the reason for outlyingness can be complex. Therefore we introduce tools that support the interpretation of outliers by representing multivariate information in biplots, maps, and univariate scatterplots.

Diagnosis of the nutrient compositional space of fruit crops

Tissue analysis is a useful tool for the nutrient management of fruit orchards. The mineral composition of diagnostic tissues expressed as nutrient concentration on a dry weight basis has long been used to assess the status of 'pure' nutrients. When nutrients are mixed and interact in plant tissues, their proportions or concentrations change relatively to each other as a result of synergism, antagonism, or neutrality, hence producing resonance within the closed space of tissue composition. Ternary diagrams and nutrient ratios are early representations of interacting nutrients in the compositional space. Dual and multiple interactions were integrated by the Diagnosis and Recommendation Integrated System (DRIS) into nutrient indexes and by Compositional Nutrient Diagnosis into centered log ratios (CND-clr). DRIS has some computational flaws such as using a dry matter index that is not a part as well as nutrient products (e.g. NxCa) instead of ratios. DRIS and CND-clr integrate all possible nutrient interactions without defining an ad hoc interactive model. They diagnose D components while D-1 could be diagnosed in the D-compositional Hilbert space. The isometric log ratio (ilr) coordinates overcome these problems using orthonormal binary nutrient partitions instead of dual ratios. In this study, it is presented a nutrient interactive model as well as computation methods for DRIS and CND-clr and CND-ilr coordinates (CND-ilr) using leaf analytical data from an experimental apple orchard in Southwestern Quebec, Canada. It was computed the Aitchison and Mahalanobis distances across ilr coordinates as measures of nutrient imbalance. The effect of changing nutrient concentrations on ilr coordinates are simulated to identify the ones contributing the most to nutrient imbalance.