Tissue analysis is commonly used in ecology and agronomy to portray plant nutrient signatures. Nutrient concentration data, or ionomes, belong to the compositional data class, i.e., multivariate data that are proportions of some whole, hence carrying important numerical properties. Statistics computed across raw or ordinary log-transformed nutrient data are intrinsically biased, hence possibly leading to wrong inferences. Our objective was to present a sound and robust approach based on a novel nutrient balance concept to classify plant ionomes. We analyzed leaf N, P, K, Ca, and Mg of two wild and six domesticated fruit species from Canada, Brazil, and New Zealand sampled during reproductive stages. Nutrient concentrations were (1) analyzed without transformation, (2) ordinary log-transformed as commonly but incorrectly applied in practice, (3) additive log-ratio (alr) transformed as surrogate to stoichiometric rules, and (4) converted to isometric log-ratios (ilr) arranged as sound nutrient balance variables. Raw concentration and ordinary log transformation both led to biased multivariate analysis due to redundancy between interacting nutrients. The alr- and ilr-transformed data provided unbiased discriminant analyses of plant ionomes, where wild and domesticated species formed distinct groups and the ionomes of species and cultivars were differentiated without numerical bias. The ilr nutrient balance concept is preferable to alr, because the ilr technique projects the most important interactions between nutrients into a convenient Euclidean space. This novel numerical approach allows rectifying historical biases and supervising phenotypic plasticity in plant nutrition studies.

Plant ionomes and soil nutrients are commonly diagnosed in agronomy using concentration and nutrient ratio ranges. However, both diagnoses are biased by redundancy of information, subcompositional incoherence and non-normal distribution inherent to compositional data, potentially leading to conflicting results and wrong inferences. Our objective was to present an unbiased statistical approach of plant nutrient diagnosis using a balance concept and mango (Mangifera indica) as test crop. We collected foliar samples at flowering stage in 175 mango orchards. The ionomes comprised 11 nutrients (S, N, P, K, Ca, Mg, B, Cu, Zn, Mn, Fe). Traditional multivariate methods were found to be biased. Ionomes were thus represented by unbiased balances computed as isometric log ratios (ilr). Soil fertility attributes (pH and bioavailable nutrients) were transformed into balances to conduct discriminant analysis. The orchards differed more from genotype than soil nutrient signatures. A customized receiver operating characteristic (ROC) iterative procedure was developed to classify tissue ionomes between balanced/misbalanced and high/low-yielders. The ROC partitioning procedure showed that the critical Mahalanobis distance of 4.08 separating balanced from imbalanced specimens about yield cut-off of 128.5 kg fruit tree⁻¹ proved to be a fairly informative test (area under curve = 0.84–0.92). The [P | N,S] and [Mn | Cu,Zn] balances were found to be potential sources of misbalance in the less productive orchards, and should thus be further investigated in field experiments. We propose using a coherent pan balance diagnostic method with median ilr values of top yielders centered at fulcrums of a mobile and the critical Mahalanobis distance as a guide for global nutrient balance. Nutrient concentrations in weighing pans assisted appreciating nutrients as relative shortage, adequacy or excess in balances.

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.

Statistical modeling is commonly used to relate the performance of potato (Solanum tuberosum L.) to fertilizer requirements. Prescribing optimal nutrient doses is challenging because of the involvement of many variables including weather, soils, land management, genotypes, and severity of pests and diseases. Where sufficient data are available, machine learning algorithms can be used to predict crop performance. The objective of this study was to determine an optimal model predicting nitrogen, phosphorus and potassium requirements for high tuber yield and quality (size and specific gravity) as impacted by weather, soils and land management variables. We exploited a data set of 273 field experiments conducted from 1979 to 2017 in Quebec (Canada). We developed, evaluated and compared predictions from a hierarchical Mitscherlich model, k-nearest neighbors, random forest, neural networks and Gaussian processes. Machine learning models returned R² values of 0.49–0.59 for tuber marketable yield prediction, which were higher than the Mitscherlich model R² (0.37). The models were more likely to predict medium-size tubers (R² = 0.60–0.69) and tuber specific gravity (R² = 0.58–0.67) than large-size tubers (R² = 0.55–0.64) and marketable yield. Response surfaces from the Mitscherlich model, neural networks and Gaussian processes returned smooth responses that agreed more with actual evidence than discontinuous curves derived from k-nearest neighbors and random forest models. When conditioned to obtain optimal dosages from dose-response surfaces given constant weather, soil and land management conditions, some disagreements occurred between models. Due to their built-in ability to develop recommendations within a probabilistic risk-assessment framework, Gaussian processes stood out as the most promising algorithm to support decisions that minimize economic or agronomic risks.

Over the past 20 years, the use of center-pivot irrigation has increased tomato (Solanum lycopersicum L.) yields in Brazil from 42 Mg ha-1 to more than 80 Mg ha-1. In the absence of field trials to support fertilizer recommendations, substantial amounts of phosphorus (P) have been applied to crops. Additional P dosing has been based on an equilibrated nutrient P budget adjusted for low-P fertilizer-use efficiency in high-P fixing tropical soils. To document nutrient requirements and prevent over-fertilization, tissue samples and crop yield data can be acquired through crop surveys and fertilizer trials. Nevertheless, most tissue diagnostic methods pose numerical difficulties that can be avoided by using the nutrient balance concept. The objectives of this study were to model the response of irrigated tomato crops to P fertilization in low- and high-P soils and to provide tissue diagnostic models for high crop yield. Three P trials, arranged in a randomized block design with six P treatments (0–437 kg P ha-1) and three or four replications, were established on a low-P soil in 2013 and high-P soils in 2013 and 2014, totaling 66 plots in all. Together with crop yield data, 65 tissue samples were collected from tomato farms. We found no significant yield response to P fertilization, despite large differences in soil-test P (coefficient of variation, 24%). High- and low-yield classes (cutoff: 91 Mg fruits ha-1) were classified by balance models with 78% to 81% accuracy using logit and Cate–Nelson partitioning models. The critical Mahalanobis distance for the partition was 5.31. Tomato yields were apparently not limited by P but were limited by calcium. There was no evidence that P fertilization should differ between center-pivot-irrigated and rain-fed crops. Use of the P budget method to arrive at the P requirement for tomato crops proved to be fallacious, as several nutrients should be rebalanced in Brazilian tomato cropping systems.