Customized Nutrient Standards to Diagnose Nutrient Imbalance in Fertigated ‘Nanica’ Banana Groves

Abstract

Banana (Musa spp.) is an important fruit production in Brazil, but crop productivity is still too low. The ‘Nanica’ cultivar and fertigation have been introduced, but more accurate guidelines are needed to support fertilization decisions at the orchard scale. This study aimed to develop customized nutrient standards for fertigated ‘Nanica’. A commercial ‘Nanica’ orchard provided 129 observations on yield and foliar nutrient concentrations from 2010 to 2017 in eight groves of 3.26 ha each. Plant density averaged 1479 plants ha⁻¹. The diagnostic leaf was analyzed for 13 elements. Concentration values were transformed into centered log ratios (clr), weighted log ratios (wlr), and isometric log ratios (ilr) to account for nutrient interactions and normalize the data. Yield cutoff between low- and high yielders was set at 27 t ha⁻¹ semester⁻¹. The XGBoost classification models relating yield to tissue composition returned an area under curve averaging 0.715 for log ratio expressions. Nutrient standards were expressed as clr, wlr, and raw concentration means and standard deviations of performing specimens. The clr and wlr diagnoses of a low-yielding and imbalanced specimen against a benchmark specimen (Euclidean distance = 2.5) or the performing subpopulation (Mahalanobis distance = 37.6, p < 0.01) indicated Mn shortage and Na excess. Sufficiency concentration ranges may not agree with log ratio diagnoses, especially for Mn. The clr and wlr nutrient standards were site-specific, supporting precision farming. The concept developed in this paper is applicable to endogenous research conducted by stakeholders in orchards worldwide.

1. Introduction

Banana (Musa spp.) is a typical fruit of tropical regions. Brazil offers favorable climatic conditions for its establishment, development, and production [1]. Brazil is the sixth largest banana producer worldwide. The cultivated area is 456,000 hectares, the annual production is 6.9 million tons, but the average national productivity of 15 t ha⁻¹ is too low [2]. The ‘Nanica’ cultivar that belongs to the ‘Cavendish’ subgroup is the most widely cultivated and commercialized in the world [3,4]. Cultivars of the ‘Cavendish’ subgroup are widespread in Brazil because they present higher performance than cultivars of the other subgroups, good adaptability to most soil and climate conditions, and resistance to diseases such as ‘Panama disease’ caused by the soil-borne fungus Fusarium oxysporum f. sp. Cubense [3,5,6].

Soil acidity, aluminum toxicity, and the low natural fertility are primary factors for the low productivity of banana orchards in Brazil [7,8]. Intensive cropping of bananas using ammonia-based fertilizers decreases the soil acid-buffering capacity even more via ammonia-oxidizing bacteria [9]. Soil acidity is neutralized by lime in the upper soil layer, and aluminum toxicity is tackled by gypsum in lower soil layers [10,11,12]. Neutral to slightly alkaline pH that impacts nutrient bioavailability in the rhizosphere also decreases the severity of ‘Panama disease’ (Fusarium wilt) in banana roots [13].

The banana plant is nutrient demanding. Because Brazil is highly dependent on imported fertilizers, representing 97% of the N, 72% of the P₂O₅, and 97% of the K₂O [14], each kg of applied fertilizer must be optimized. However, national recommendations are based on a few trials conducted decades ago [15]. Present fertilizer guidelines based on those trials are thus obsolete for the fertigated banana. While fertilizer trials are expensive and time-consuming, observational data can be collected by stakeholders to support fertilization decisions for the intensive production of banana.

Fertigation of ‘Nanica’ can be adjusted locally to meet plants’ nutrient requirements, considering banana nutrient exports, the requirements being highest for K and N, and the amount of nutrients left on soil surface as plant residues following harvest [16,17]. After harvesting, the banana plant is replaced by other members of the family (daughter, granddaughter, etc.) grown concomitantly in the orchard. The biomass of pruned parents contributes to plant nutrition [16,18]. Adequate plant nutrition not only involves nutrient supply but also an adequate water supply and temperature to support nutrient transfers by ion diffusion and convection in the soil [19,20] and by short- and long-distance transport in the plant [21]. Local guidelines are thus needed to customize the fertigation of ‘Nanica’, a knowledge gap for precision farming.

Tissue tests are thought to integrate all factors that have interacted to affect plant growth, including nutrient availability [22]. However, nutrient interactions distort the interpretation of raw concentration values [23]. Indeed, components constrained to the closed space of a measurement unit cannot be interpreted without relating each of them to other components [24]. Nutrient interactions are interpreted in terms of the chemical stoichiometry required to maintain homeostasis with respect to the environment [25]. Nutrient interactions in plant tissues are thus expressed as dual ratios [26]. Because any nutrient interacts with more than one nutrient, several nutrient ratios should be considered simultaneously to identify the most limiting nutrient [26,27,28].

Dual nutrient ratios expressed as functions were first added up to indices by projecting them onto bisectors in the diagnosis and recommendation integrated system (DRIS) [29]. The log transformation of ratios needed to conduct statistical analyses [30] was used advantageously to run DRIS [31]. Log ratios have an additional advantage over untransformed dual ratios and ordinary log-transformed dual ratios of being additive as centered log ratios (clr) [32]. Compositional nutrient diagnosis (CND) integrated dual log ratios (dlr) into a clr expression to compute nutrient standards [33]. The clr is the average across dlrs. However, each dlr may bear a different importance for the target variable depending on the nature of database and the yield cutoff between high- and low-performing crops selected by stakeholders or derived statistically. A gain ratio coefficient was thus assigned to each dlr to generate the weighted log ratio (wlr) [27]. A subpopulation of high-yielding and nutritionally balanced specimens can be set apart to compute clr and wlr standards. Alternatively, a high-yielding and nutritionally balanced specimen of equal-length composition can also be selected as a reference for comparison with a defective specimen under the ceteris paribus assumption in a comparable agroecosystem [34].

We hypothesized that observational data collected by stakeholders in a ‘Nanica’ orchard return a consistent nutrient diagnosis across nutrient expressions (concentrations, log ratios). The objectives of this study were to (1) elaborate clr, wlr, and raw concentration standards to manage ‘Nanica’ nutrition at a local scale and (2) diagnose the composition of a low-yielding and nutritionally imbalanced specimen across nutrient expressions as a check for consistency across nutrient expressions. We introduce Euclidean distance as measure of difference between two equal-length tissue compositions in an otherwise comparable environment, and the multivariate Mahalanobis distance to support the statistical significance of clr and wlr nutrient indices computed from local standards.

2. Materials and Methods

2.1. Database and Study Region

The research was conducted in partnership with Sítio Barreiras Fruticultura LTDA located in Missão Velha, Ceará state, Brazil (7°35′90″ south latitude and 39°21′17″ west longitude; altitude = 442 m). Soils were predominantly sandy, of low natural fertility, pedologically weakly developed, and classified as Typic Quartzipsamment [35]. The cultivar was ‘Nanica’ (AAA group, Cavendish subgroup). The age of banana orchards ranged from 6 to 19 years. Plant density averaged 1479 plants ha⁻¹ and remained the same throughout the experimental period. Data on fruit yield were collected in eight groves of 3.26 ha each.

According to the Köppen–Geiger classification, the tropical regional climate is Aw with a dry season in winter and rains concentrated in the summer. During the experimental period, maximum temperature ranged from 31 to 35 °C, and minimum temperature varied from 19 to 21 °C. Annual precipitation averaged 1006 mm [36]. Consecutive precipitation peaks occurred in 2010–2011 and 2013–2014. Meteorological data are presented in Figure 1. While minimum and maximum temperature varied little across years, irrigation was required to stabilize yield across seasons.

Management of clumps or families corresponding to mother and daughter plants is maintained across generations. On average, one bunch per plant is expected to be harvested each year. The production cycle can shorten or lengthen depending on climate, tree spacing, and nutrient management. Fruits are harvested every week, but the number of harvested bunches varies over time. In the northeast, the first semester (January to June) is the rainy season, and the second one (July to December) is the dry season (Figure 1). Even with irrigation, differences in semestral production levels are still noticeable.

Plants were fertigated weekly [37]. The nitrogen (N) fertilization, supplied mainly as urea (45% N) and complemented by monoammonium phosphate (11% N), ranged from 440 to 600 kg N ha⁻¹ year⁻¹. Potassium (K) was supplied as potassium chloride (60% K₂O) at rates ranging from 1100 to 1700 kg K₂O ha⁻¹ year⁻¹. Phosphate (P) was applied at planting as reactive natural phosphate (27% P₂O₅) followed by fertigation with monoammonium phosphate (48% P₂O₅) for total rate of 160 kg P₂O₅ ha⁻¹ year⁻¹. Micronutrients were supplied entirely via fertigation as follows: manganese sulfate fertilizer (26% Mn) applied at a rate of 60 kg ha⁻¹ year⁻¹, zinc sulfate fertilizer (20% Zn) applied at a rate of 96 kg ha⁻¹ year⁻¹, and boric acid fertilizer (17% B) applied at rates of 20 to 30 kg ha⁻¹ year⁻¹.

2.2. Soil and Tissue Analyses

Soil and leaves were collected twice a year from 2010 to 2017 in June and December, totaling 129 observations. The soil was sampled in the 0–20 cm layer, air-dried, ground, and analyzed [38]. The pH was measured in the volumetric soil/water ratio of 1:2.5. Soil P and K were extracted using the Mehlich-1 method. Soil Ca, Mg, and Al were extracted using 1 N KCl. Potential acidity (H + Al) was extracted using 0.5 M calcium acetate at pH 7.0. Elements were quantified by inductively coupled plasma optical emission spectroscopy. While soil test values varied widely in the orchard, precision fertigation offers the possibility to adjust nutrient supply spatially.

Cation exchange capacity (CEC) was calculated as the sum of the exchangeable cations (K, Ca, Mg) and potential acidity. Total carbon as determined by Walkley–Black oxidation was multiplied by 1.724 to obtain soil organic matter content. Soil test results are presented in

Table 1. Descriptive statistics of soil test results (0–20 cm) over the experimental period.

Soil Fertility Attributes	Minimum	Maximum	Mean	Standard Deviation
pHwater	6.5	7.8	7.3	0.3
	g kg−1
Organic matter content	5.0	36.0	18.0	6.5
	mg dm−3
P	37.0	220.0	115.9	44.3
K	50.8	449.6	155.1	98.3
	cmolc dm−3
Ca	1.9	15.3	4.7	3.1
Mg	0.3	3.0	1.0	0.7
Exchangeable acidity	0.4	1.8	1.0	26.1
Sum of the cations	2.5	18.9	6.2	4.0
Cation exchange capacity	3.3	19.8	7.2	4.0
	%
Base saturation	66.8	96.2	83.1	6.8

. The sum of mean values of cationic species differed from the mean sum of cations due to non-normal distributions of cationic species, impacting mean CEC and base saturation.

The 3rd leaf from the plant apex was sampled, removing 10 to 15 cm of the inner median part of the leaf blade and discarding the central vein [39]. Tissues were dried at 60 °C, ground to less than 1 mm and analyzed for nitrogen (N), phosphorus (P), potassium (K), calcium (Ca), magnesium (Mg), sulfur (S), boron (B), copper (Cu), iron (Fe), manganese (Mn), zinc (Zn), sodium (Na), and aluminum (Al) [38].

2.3. CND Standards

The sample space ${\mathrm{S}}^d$ of tissue composition was defined as follows [33]:

$${\mathrm{S}}^d=\left[\left(\mathrm{N},\mathrm{P},\dots {\mathrm{R}}_d\right):\mathrm{N}>0,\mathrm{P}>0,\mathrm{K}>0,\dots x_D>0,\mathrm{N}+\mathrm{P}+\mathrm{K}+\cdots +x_D={10}^3\mathrm{g}{\mathrm{k}\mathrm{g}}^{-1}\right]$$

where $c_i$ is concentration of the i^th nutrient and $x_D$ is a filling value representing unquantified elements and computed by difference as follows:

$$x_D={10}^3-{\sum }_{i=1}^d{c}_i$$

The centered log ratio for component i (${clr}_i$) was computed as follows:

$${clr}_i=\mathrm{l}\mathrm{n}\left(x_i/G\right)$$

$$G={\left(N\times P\times K\times \dots \times x_D\right)}^{\frac{1}{D}}$$

where $G$ is the geometric mean across components and D is total number of components. The sum of the ${clr}_i$ values is zero due to symmetry about G. The ${clr}_i$ is the average across the D dual log ratios [40], as follows for N:

$$\begin{array}{l}ln\left(\frac{N}{G}\right)=\mathrm{l}\mathrm{n}{\left(\frac{N}{N}\times \frac{N}{P}\times \frac{N}{K}\times \frac{N}{Ca}\times \frac{N}{Mg}\times \frac{N}{S}\times \frac{N}{B}\times \frac{N}{Cu}\times \frac{N}{Zn}\times \frac{N}{Mn}\times \frac{N}{Fe}\times \dots \times \frac{N}{x_D}\right)}^{\frac{1}{D}}\\ =\frac{1}{D}\left[\mathrm{l}\mathrm{n}\left(\frac{N}{N}\right)+\mathrm{l}\mathrm{n}\left(\frac{N}{P}\right)+\dots +\mathrm{l}\mathrm{n}\left(\frac{N}{x_D}\right)\right]\end{array}$$

The clr, computed as an average across dual log ratios, does not account for the importance of each dual ratio regarding the target variable. Each dual log ratio was thus weighted by its gain ratio ${\phi }_j$ to compute the weighted log ratio (wlr) as follows:

$${wlr}_{x_i}=\frac{1}{D}{\sum }_{j=1}^D{\phi }_j\mathrm{l}\mathrm{n}\left(\frac{x_i}{x_j}\right),i\ne j$$

where $x_i$ and $x_j$ are components. The gain ratio in a machine learning classification is a metric indicating the efficiency of a dual log ratio in partitioning yields in the database about the selected cutoff value [27]. The gain ratio is thus agronomically meaningful for the database under study. If all ${\phi }_j=1$, ${wlr}_{x_i}={clr}_{x_i}$. Because $\frac{1}{D}$ is common to all ${wlr}_{x_i}$ variables, $\frac{1}{D}$ can be removed without affecting the results of statistical analysis. To compute wlr, nutrient $x_i$ is maintained as the numerator across dual log ratios. If $x_i$ is the denominator in the dual log ratio expression, it is moved to the numerator by multiplying the dlr by −1, i.e., $\mathrm{l}\mathrm{n}\left(\frac{x_j}{x_i}\right)=-\mathrm{l}\mathrm{n}\left(\frac{x_i}{x_j}\right)$.

Nutrient indices for clr and wlr were computed as follows for ${clr}_{xi}$ [33]:

$$Index{clr}_{xi}=\frac{\left({clr}_{xi}-{clr}_{xi}^{*}\right)}{s_{{clr}_{xi}}^{*}}$$

where ${clr}_i$ refers to the diagnosed specimen and the asterisk indicates the corresponding means and standard deviations for a high-yielding and nutritionally balanced subpopulation. The sum of clr and wlr variables is zero in conformity with the concept of nutrient balance.

The clr variables have Euclidian geometry [32] as does the wlr variable. The Euclidian distance ($\epsilon$) is computed between two equal-length D-part compositions [32], as follows for clr:

$$\epsilon =\sqrt{{\sum }_i^D{\left({clr}_i-{clr}_i^{\mathrm{*}}\right)}^2}$$

where the asterisk indicates the reference specimen. The Euclidian distance is useful to compare a deficient to a compositionally close-performing specimen located in an otherwise similar environment in the orchard. Nutrient indices computed as (${clr}_i-{clr}_i^{*}$) can be displayed in a histogram to illustrate relative nutrient shortage, sufficiency, or excess.

2.4. Mahalanobis Distance

Isometric log ratios are orthonormal variables arranged as ‘balances’ among groups of components. There are D-1 ilrs, i.e., the exact number of degrees of freedom available in a composition [41]. The ilr avoids computing singular matrices. The ${ilr}_i$ is a log ratio expression that contrasts pre-defined groups of components as follows [42]:

$${ilr}_i=\sqrt{{\displaystyle \frac{rs}{r+s}}}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{g_r}{g_s}}\right)$$

where r is the number of components in subset r, s is the number of components in subset s, $g_r$ is the geometric mean across components in subset r, and $g_s$ is the geometric mean across components in subset s. The ilr can be conceptualized in a simplified SBP contrasting any component with the remainder (

Table 2. Sequential binary partition shown by 13 contrasts between 14 tissue components (plus sign for component at numerator and minus sign for components at denominator).

ilr	N	P	K	Ca	Mg	S	B	Cu	Fe	Mn	Zn	Na	Al	xD	r	s
1	1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	1	13
2	0	1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	1	12
3	0	0	1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	1	11
4	0	0	0	1	−1	−1	−1	−1	−1	−1	−1	−1	−1	−1	1	10
5	0	0	0	0	1	−1	−1	−1	−1	−1	−1	−1	−1	−1	1	9
6	0	0	0	0	0	1	−1	−1	−1	−1	−1	−1	−1	−1	1	8
7	0	0	0	0	0	0	1	−1	−1	−1	−1	−1	−1	−1	1	7
8	0	0	0	0	0	0	0	1	−1	−1	−1	−1	−1	−1	1	6
9	0	0	0	0	0	0	0	0	1	−1	−1	−1	−1	−1	1	5
10	0	0	0	0	0	0	0	0	0	1	−1	−1	−1	−1	1	4
11	0	0	0	0	0	0	0	0	0	0	1	−1	−1	−1	1	3
12	0	0	0	0	0	0	0	0	0	0	0	1	−1	−1	1	2
13	0	0	0	0	0	0	0	0	0	0	0	0	1	−1	1	1

).

The Mahalanobis distance ($\mathcal{M}$) is a metric measuring the distance between a composition and the centroid of a distribution. Using ilr variables, $\mathcal{M}$ is computed as follows:

$$\mathcal{M}=\sqrt{{\left({ilr}_i-{ilr}_i^{*}\right)}^T{COV}^{-1}\left({ilr}_i-{ilr}_i^{*}\right)}$$

where ${ilr}_i$ is the i^th ilr, ${ilr}_i^{*}$ is the centroid of the reference population (the ilr means of the high-yielding and nutritionally balanced subpopulation), ${COV}^{-1}$ is the inverse covariance matrix of the reference population, and $T$ indicates a transposed matrix. Because orthogonal variables may be correlated to each other, the covariance matrix is required to provide a statistically robust test [43]. The ${\mathcal{M}}^2$ is distributed like a ${\chi }_{D-1}^2$ variable and can thus be tested statistically. The significance level is selected by stakeholders, not necessarily the 0.05 or the 0.01 levels [44]. Nutrient indices are t-tests as Mahalanobis distances for single components weighted by their variance.

2.5. Statistical Analysis

Descriptive statistics were computed as mean, maximum and minimum values, standard deviation, and coefficient of variation. As a normality test, data distribution was classified as symmetrical where skewness was between −0.5 and +0.5, while skewness was high if less than −1 or greater than +1 [45]. Relationships among components were visualized as biplots using Codapack vs. 2.03.01. A two-dimensional loadings plot was drawn using D-1 clr variables (excluding the filling value) to illustrate the size and direction of the relationships among nutrients.

Non-parametric ML classification models set apart the subpopulation of nutritionally balanced and imbalanced specimens in a confusion matrix. Random forest is the classical ML model. Gradient boosting combines weak learners into a stronger learner by minimizing the loss function. The hyperparameters were those suggested by the software. For random forest, there were ten trees and at least five subsets per split. For XGBoost, hyperparameters were 100 trees, with a learning rate of 0.300, replicable training, six individual trees as limit depth, 1.00 as fractions of training instances, features for each tree, features for each level, and features for each split.

Cross-validation is a procedure that averages the splits of mutually exclusive subsets (folds). Classification ML models were run using the Orange Data Mining freeware (University of Ljubljana, Slovenia). The target variable was fruit yield. Yield was reported as the sum of harvests per plot from January to June (rainy season), and from July to December (dry season), respectively, and converted into kg ha⁻¹ semester⁻¹.

Yield cutoff can be selected by stakeholders or derived statistically using variance ratios [29], a cumulative variance ratio function [46] or a subpopulation of nutritionally balanced and high-yielding specimens set apart in a confusion matrix [47]. To derive yield cutoff, the cumulative variance ratio function of clr variables was computed [46] after removing outliers [48].

The data were stratified to avoid model overfitting by resampling the same category (year and semester) to avoid model overfitting. Soil and tissue compositional variables were analyzed statistically as raw, clr- or wlr-transformed data. The ML classification models classified specimens into four categories in a confusion matrix [47] as follows (Figure 2):

• True negative quadrant (TN): high-performing and nutritionally balanced specimens (upper left quadrant), likely no response to correcting measures.
• False negative quadrant (FN): low-performing but nutritionally balanced specimens (lower left quadrant), search needed to identify a limiting variable not documented in the database.
• False positive quadrant (FP): high-performing but nutritionally imbalanced specimens (upper right quadrant), requiring adjusting at least one variable out of target but not limiting crop performance.
• True positive quadrant (TP): low-performing and nutritionally imbalanced specimens (lower right quadrant), correct at least for one variable limiting crop performance.

Model accuracy is computed as follows:

$$Accuracy={\displaystyle \frac{TN+TP}{TN+TP+FN+FP}}$$

The area under curve (AUC) separates signals from noise in binary classification models. Separation is null if AUC = 0.5, little informative if AUC is between 0.5 and 0.7, moderately informative if AUC is between 0.7 and 0.9, very informative if AUC is >0.9, and perfect if AUC = 1 [49].

Proximate concentration ranges were computed as the means and standard deviations of nutrient concentrations in the high-yielding and nutritionally balanced subpopulation set apart in a confusion matrix by the machine learning (ML) classification model [50]. Five classes of nutrient status were suggested as a distance from the centroid of the high-yielding and nutritionally balanced subpopulation as follows [51]: shortage < centroid −4/3sd; trend toward shortage = centroid −4/3sd to centroid −2/3sd; sufficiency = centroid −2/3sd to centroid +2/3sd; trend toward excess = centroid +2/3sd to centroid +4/3sd; excess > centroid +4/3sd.

3. Results

3.1. Seasonal Variations in Banana Yields

The patterns of yield variations were grove- and season-specific (Figure 3). Yields were higher during the rainy season (Figure 1). Fertigation and irrigation did not totally offset the negative effects of the dry season, especially in 2017, which was preceded by a dry year in 2016. During the 2010 to 2016 period, yields were closer between dry and rainy seasons in the less productive grove compared to the more productive one, indicating site-specific constraints. Because nutrient standards are derived from a high-yield and nutritionally balanced subpopulation, nutrient diagnosis may not improve crop productivity substantially unless other stresses have been relieved.

3.2. Yield Cutoff

Yield averaged 52.0 ± 11.9 t ha⁻¹ year⁻¹, 29.4 ± 6.5 t ha⁻¹ year⁻¹ during the first semester (rainy season), and 22.8 ± 7.3 t ha⁻¹ year⁻¹ during the second semester (dry season). Yield cutoff was determined using a cumulative variance ratio function after excluding 11 outliers. Yield at the inflection point of the cubic curve was 27.1 t ha⁻¹ semester⁻¹ (Figure 4). There were 50 high-yielding specimens averaging 33.0 t ha⁻¹ semester⁻¹.

3.3. Gain Ratios

The Na/Mn dual log ratio showed the highest gain ratio (0.050) at a yield cutoff of 27 t ha⁻¹ semester⁻¹ (Appendix A 

Table A1. Gain ratios associated with the 91 nutrient log ratios in banana ‘Nanica’ foliar tissues at yield cutoff set at 27 t ha⁻¹ semester⁻¹.

Log Ratio	Gain Ratio	Log Ratio	Gain Ratio	Log Ratio	Gain Ratio	Log Ratio	Gain Ratio
N/P	0.012	P/Al	0.012	Ca/Mg	0.022	S/xD	0.003
N/K	0.011	P/xD	0.016	Ca/S	0.012	B/Cu	0.023
N/Na	0.025	K/Na	0.010	Ca/B	0.014	B/Fe	0.006
N/Ca	0.004	K/Ca	0.007	Ca/Cu	0.012	B/Mn	0.041
N/Mg	0.001	K/Mg	0.003	Ca/Fe	0.024	B/Zn	0.006
N/S	0.016	K/S	0.000	Ca/Mn	0.029	B/Al	0.013
N/B	0.015	K/B	0.014	Ca/Zn	0.020	B/xD	0.009
N/Cu	0.005	K/Cu	0.007	Ca/Al	0.020	Cu/Fe	0.046
N/Fe	0.005	K/Fe	0.003	Ca/xD	0.010	Cu/Mn	0.022
N/Mn	0.028	K/Mn	0.035	Mg/S	0.008	Cu/Zn	0.004
N/Zn	0.007	K/Zn	0.005	Mg/B	0.010	Cu/Al	0.008
N/Al	0.025	K/Al	0.014	Mg/Cu	0.003	Cu/xD	0.010
N/xD	0.012	K/xD	0.004	Mg/Fe	0.001	Fe/Mn	0.035
P/K	0.010	Na/Ca	0.013	Mg/Mn	0.022	Fe/Zn	0.001
P/Na	0.020	Na/Mg	0.019	Mg/Zn	0.003	Fe/Al	0.019
P/Ca	0.018	Na/S	0.020	Mg/Al	0.015	Fe/xD	0.004
P/Mg	0.009	Na/B	0.005	Mg/xD	0.022	Mn/Zn	0.024
P/S	0.008	Na/Cu	0.014	S/B	0.013	Mn/Al	0.025
P/B	0.029	Na/Fe	0.019	S/Cu	0.011	Mn/xD	0.034
P/Cu	0.012	Na/Mn	0.050	S/Fe	0.010	Zn/Al	0.033
P/Fe	0.010	Na/Zn	0.017	S/Mn	0.034	Zn/xD	0.004
P/Mn	0.028	Na/Al	0.008	S/Zn	0.001	Al/xD	0.010
P/Zn	0.003	Na/xD	0.022	S/Al	0.010	-	-

). Gain ratios for Cu/Fe, K/Mn, Fe/Mn, B/Mn, Mn/xD, S/Mn, Ca/Mn, N/Mn, N/Na, Ca/Fe, Mg/Mn, Mn/Al, Mn/Zn, B/Cu, Mg/xD, Cu/Mn, Ca/S, Ca/Zn, Ca/Al, Na/xD, and P/Na varied between 0.035 and 0.020, while others were lower down to 0.000 for K/S, indicating the agronomically variable importance of dual ratios for the banana database at the selected yield cutoff. The gain ratios were intermediate for N/K and N/P, and small for K/Ca and K/Mg.

3.4. Machine Learning Models

Machine learning classification models set apart the nutritionally balanced and high-yielding subpopulation concerning yield cutoff. Gradient boosting (XGBoost) was more accurate than random forest. Results for XGBoost are presented in

Table 3. Area under curve (AUC), accuracy, and numbers of true negative (TN), true positive (TP), false negative (FN), and false positive (FP) specimens totaling 129 specimens in the confusion matrix.

AUC	Accuracy	TN	TP	FN	FP
Raw concentration
0.684	0.597	31	49	23	26
Centered log ratio (clr)
0.717	0.682	35	53	19	22
Weighted log ratio (wlr)
0.713	0.674	34	53	19	23

. Raw concentrations showed the lowest AUC and accuracy. Models showed areas under curve averaging 0.715 and were moderately informative. There were 34–35 TN specimens across nutrient expressions to compute nutrient standards, excluding 22–23 high-yielding but nutritionally imbalanced FP specimens.

The means and standard deviations of nutrient expressions for the high-yielding and nutritionally balanced subpopulation are presented in

Table 4. Nutrient clr and wlr standards (mean and standard deviation as sd) of the high-yielding and nutritionally balanced specimens at yield cutoff of 27.1 t ha⁻¹ semester⁻¹. Statistical differences with the TP means indicated following the sd values.

Component	clr		wlr		Raw Concentration
	Mean	sd	Mean	sd	Mean	sd
					----g kg−1----
N	3.664	0.113 ns	0.737	0.027 ns	21.8	1.9 ns
P	1.105	0.138 ns	0.275	0.034 ns	1.7	0.3 *
K	4.174	0.207 ns	0.644	0.032 *	37.0	7.5 ns
Ca	2.696	0.167 ns	0.742	$0.039†$	8.4	1.7 ns
Mg	1.548	0.150 ns	0.156	0.025 ns	2.6	0.4 ns
S	1.033	0.124 ns	0.279	0.028 *	1.6	0.2 ns
					----mg kg−1----
B	−4.227	0.475 **	−0.865	0.095 **	9.50	4.31 ns
Cu	−4.559	0.238 *	−0.689	0.048 ns	6.01	2.02 ns
Fe	−2.114	0.110 ns	−0.137	0.031 *	67.71	10.35 ns
Mn	−1.513	0.650 **	−0.638	0.249 **	155.86	96.02 **
Zn	−3.547	0.178 ns	−0.366	0.031 ns	16.26	3.40 ns
Na	−2.610	0.473 **	−0.759	0.125 **	41.9	19.6 **
Al	−3.077	0.469 ns	−0.669	0.114 ns	25.22	12.29 *
xD	7.426	0.118 ns	1.289	0.026 *	926.51	9.29 ns

ns, †, *, **: nonsignificant and significant at the 0.10, 0.05, and 0.01 levels, respectively.

. As shown by foliar concentrations, K and N were the most required. There was evidence of distortion in the statistical analysis of raw concentrations. While the measurement unit was 1000 g kg⁻¹, the sum of the means of raw components was 1280 g kg⁻¹. In contrast, the unbiased clr and wlr means added up to zero due to symmetry about geometric means.

The TN and TP subpopulations were compared statistically across nutrient expressions (Table 4). There were significant differences in raw concentration values between the means of TN and TP subpopulations: concentrations were too low for P and too high for Na and Al in TP specimens. The clr values that considered interactions among elements showed that Cu and Mn levels were significantly too low and B and Na levels were significantly too high in TP specimens. The wlr values impacted even more the difference between TN and TP specimens: the K, Ca, S, B, Fe, and Na levels were too high, while the Mn level was too low in TP specimens. Differences were highly significant for Mn and Na across nutrient expressions.

3.5. Nutrient Diagnosis of a TP Specimen

A low-yielding grove (24.2 t ha⁻¹) was compared to a productive and nutritionally balanced and compositionally close grove (43.9 t ha⁻¹). The Euclidean distance was 2.5. Nutrient indices were displayed on a histogram (Figure 5). The clr and wlr diagnoses showed a similar distribution of nutrient indices. Mn and Na were the most imbalanced nutrients. Tissue Mn should apparently be increased and tissue Na abated. A statistical check was also conducted using the Mahalanobis distance and clr and wlr indices.

At the population level, the TP specimen was compared to the distribution of nutritionally balanced and high-yielding subpopulation. The means and covariance matrix of ilr-transformed composition of the nutritionally balanced and high-yielding specimen (TN) and the ilr-transformed composition of the nutritionally imbalanced and tissue composition of the low-yielding specimen (TP) are presented in

Table 5. Statistics to compute the Mahalanobis distance between the ilr-transformed TN tissue compositional data and that of a TP specimen. Arrangements of ilr 1 to 13 isometric log ratios (ilr) are shown in the sequential binary partition in Table 2.

ilr1	ilr2	ilr3	ilr4	ilr5	ilr6	ilr7	ilr8	ilr9	ilr10	ilr11	ilr12	ilr13
Mean of the ilr-transformed composition of the nutritionally balanced and high-yielding specimens (TN)
1.099616	0.446392	1.518919	1.202533	1.014105	1.003872	−1.195872	−0.341005	−1.199372	−0.465335	−2.749738	−3.395556	−10.574180
ilr-transformed composition of a nutritionally imbalanced and low-yielding specimen (TP)
1.073531	0.422587	1.551604	1.269577	1.024079	1.040595	−1.246025	−0.290038	−1.173491	−1.045245	−2.613914	−2.799243	−10.714404
(TP-TN) ilr difference
−0.026085	−0.023805	0.032684	0.067044	0.009974	0.036723	−0.050153	0.050967	0.025881	−0.579910	0.135823	0.596313	−0.140224
Inverse of the covariance matrix of the ilr-transformed composition of the nutritionally balanced and high-yielding specimens
−1171.485630	−425.140449	470.097385			23.976591	119.754920	88.297282	142.716638	−24.266344	4.576614	3.931754	−13.423271
10.379220	75.051196	−72.335434	−14.774672	25.925068	76.471097	7.408502	−22.216753	−37.638756	3.643241	0.999374	−0.554455	0.191546
488.476936	171.485995	−231.181443	−58.346272	−88.163768	4.497523	−34.265432	−50.809648	−92.318766	13.767308	−0.044168	−1.776809	5.087421
−92.406487	−671.181843	248.492663	33.739819	−46.969246	8.160745	−9.953451	62.124006	116.660360	−13.309471	−4.973716	2.223179	−0.126761
−4055.300828	−3027.987740	2369.438311	−222.508045	−422.583366	84.495000	109.855424	430.454426	892.243446	−101.177562	−15.660031	13.866356	−31.924611
−2373.579880	−1864.799041	1393.270266	−160.475097	−264.757685	43.771185	91.575737	380.303606	469.797523	−57.212075	−5.251788	7.925890	−20.912940
2154.059723	1106.106202	−1036.671592	91.393120	172.793767	−35.965676	−80.473550	−185.097747	−345.701395	53.992235	−4.261817	−8.903370	23.254971
692.344223	1151.770760	−546.069010	83.740391	97.789662	−52.781080	−20.531540	−146.482121	−225.143546	15.722678	3.408736	2.893182	6.487391
360.043159	361.591845	−287.759560	22.610770	50.916168	−10.185511	−9.604703	−44.869764	−82.480538	9.236108	8.278468	−1.497122	2.534541
−352.584983	−461.623083	233.436104	−37.770377	−39.627325	20.962002	10.912831	58.239313	94.192434	−2.846327	−1.006582	1.416291	−3.055273
−997.250893	−757.105915	621.461183	−53.751257	−111.064373	23.978571	32.405248	104.199722	191.497415	−26.779499	−1.520314	4.595450	−7.918040
−480.275563	−940.641288	433.303316	−62.506876	−78.333462	40.678449	12.820739	108.310374	175.094179	−16.669098	−3.837867	3.678854	−4.197692
−844.533126	−539.219695	427.946520	−44.310979	−70.559867	19.768576	30.468202	86.615010	146.779649	−20.773302	−0.912366	3.320854	−6.286117

. The ${\mathcal{M}}^2$ was 37.6 (p < 0.01). Nutrient indices were thus statistically reliable. The clr and wlr diagnoses (Figure 6) were like the nutrient indices presented in Figure 5. Compared to clr, the wlr diagnosis showed differences in Mn and Na status significant at the 0.10 level, indicating a more sensitive diagnosis after considering the gain ratios associated with each dlr. The Mn and Na were the most imbalanced elements. The Mn was in relative shortage and Na in relative excess. Other elements except N, P, Cu, and Al showed some relative excess compared to the centroid of the high-yielding and nutritionally balanced subpopulation.

Results presented in Figure 5 and Figure 6 are supported by relationships shown in the biplot (Figure 7). The Mn, Na, and Al loaded the most. Al and Na showed opposite directions on the ilr.2 axis. Na is an indicator of soil alkalinity, while Al reflects some acidification in the rhizosphere. Mn loaded in a different direction along axis ilr.1 where it surpassed all other nutrients. However, the nutrient imbalance specific to the TP specimen involved primarily Mn and Na.

3.6. Proximate Sufficiency Ranges

Sufficiency ranges (

Table 6. Sufficiency ranges of elements in ‘Nanica’ foliar tissues excluding outliers.

Element	Shortage	Trend to Shortage	Sufficiency	Trend to Excess	Excess
	g kg−1
N	<19.3	19.3–20.5	20.5–23.1	23.1–24.3	>24.3
P	<1.3	1.3–1.5	1.5–1.9	1.9–2.1	>2.1
K	<27.0	27.0–32.0	32.0–42.0	42.0–47.0	>47.0
Ca	<6.1	6.1 –7.3	7.3–9.5	9.5–10.7	>10.7
Mg	<2.1	2.1–2.3	2.3–2.9	2.9–3.1	>3.1
S	<1.3	1.3–1.5	1.5–1.7	1.7–1.9	>1.9
	mg kg−1
B	<3.8	3.8–6.6	6.6–12.4	12.4–15.2	>15.2
Cu	<3.3	3.3–4.7	4.7–7.4	7.4–8.7	>8.7
Fe	<53.9	53.9–60.8	60.8–74.6	74.6–81.5	>81.5
Mn	<27.8	27.8–91.8	91.8–219.9	219.9–283.9	>283.9
Zn	<11.7	11.7–14.0	14.0–18.5	18.5–20.8	>20.8
Na	<15.8	15.8–28.8	28.8–55.0	55.0–68.0	>68.0
Al	<8.8	8.8–17.0	17.0–33.4	33.4–41.6	>41.6

) were computed as raw concentration means and standard deviations of the nutritionally balanced and high-yielding specimens (Table 4). Sufficiency ranges were wide for Mn (39.4–158.0 mg kg⁻¹ for Mn) and narrower for Na (28.4–57.1 mg kg⁻¹ for Na).

Compared to clr and wlr, which diagnosed Mn shortage and Na excess in the TP specimen, the sufficiency range diagnosis (SRD) indicated Mn sufficiency and Na excess. Indeed, the distribution of tissue Mn concentrations in the TN subpopulation was highly skewed (1.27), indicating non-normal distribution of tissue Mn concentrations compared to clr Mn (−0.39). In contrast, the distribution of Na concentrations in the TN subpopulation was moderately skewed (0.92) compared to clr Na that showed symmetrical distribution (0.49). While SRD disagreed with the clr and wlr for Mn diagnosis, it agreed for Na.

4. Discussion

4.1. Nutrient Interactions

Ninety-one dual log ratios were computed to derive wlr variables. In comparison, 117 interactions have been reported between macro- and micronutrients for different agricultural crops [52]. In the database under study, Mn and Na were associated with the highest gain ratios. Plant-specific interactions have been documented physiologically between Mn and P, B, Ca, Mg, Zn, Fe, and Al [53,54,55,56] and between Na and K [57]. The importance of each dual log ratio can vary widely depending on the nutrient network or ‘ionome’ [58] in the database documented by stakeholders and the selected yield cutoff.

4.2. Mn Shortage and Na Excess

Soil organic matter and available nutrients are soil-quality attributes [59]. Soil organic matter content was highly variable in the orchard, indicating a wide variation in soil quality. Soil pH varied widely between 6.5 and 7.8. Neutral to slightly alkaline pH aims to decrease the severity of the root Panama disease in banana roots but impacts nutrient availability in the soil [13]. Mn availability is limited in soils subject to over-liming, especially in weakly buffered coarse-textured Quartzipsamments [60]. Exchangeable forms of Mn occur at pH less than 5.0 but are redistributed among less available organic and oxide fractions as pH increases above 5.8 [61]. On the other hand, the intensive cropping of bananas can decrease the soil acid-buffering capacity in the rhizosphere via ammonia-oxidizing bacteria [9]. The negative relationship between Al and Na in the biplot (Figure 7) apparently indicated an opposition between acidification and alkalinization. Urea hydrolyzed into ammonia likely acidified the rhizosphere in the application zone, impacting Al availability in microsites.

Excess Na is generally attributed to salinity buildup in the soil. Salts can accumulate unequally in the orchard due to natural phenomena and the even fertilization of groves despite spatially uneven yields. The cause of groundwater salinization in the semiarid regions of Bahia state, south of Ceará and at comparable altitude and distance from the seashore, was attributed to local geology and salt concentration due to evaporation rather than to sea sprays [62]. In Juazeiro, Bahia, average rainfall was 800 mm during seven to nine months, including long periods of drought, compared to 1006 mm for Missão Velha, Ceará. Na salts can be partly leached during heavy rainfall. As anticipated by the balance concept, correcting the Mn shortage could increase yield and dilute tissue Na. Mn shortage can be corrected by combining foliar fungicides and fertilizer sprays [63].

4.3. Customized Nutrient Standards

Nutrient variability measured by the coefficient of variation (CV) was greater for micro- than macro-nutrients in the orchard under study, as also reported in India [64] and Equator [65]. In Brazil, the main banana production areas showed wide variations of nutrient concentrations in soils and foliar tissues due in part to heavy or variable applications of fertilizers [7,8,66,67]. New nutrient standards should thus be developed as cultivar and orchard management change and yield potential increases [7,68]. Moreover, nutrient standards developed at the orchard scale are unprecedented because present state fertilization guidelines do not provide state-wide tissue standards to interpret banana tissue tests [15].

The semestral orchard productivity averaging 33 t ha⁻¹ semester⁻¹ across high-yielders was much higher than the Brazilian average of 15 t ha⁻¹ year⁻¹ [2]. K deficiency is generally the most limiting nutrient in Brazil [66,67]. However, K and N did not appear to limit banana yield in the orchard under study. Using customized nutrient standards to test nutrient balance in a low-yielding grove, nutrient expressions reached different conclusions. The SRD diagnosis is interpreted as shortage, sufficiency, and excess rather than physiologically verified deficiency, transition zone, adequacy, luxury consumption, excess, or toxicity [69,70]. Log ratio diagnoses are interpreted in relative terms. The log ratio diagnoses disagreed with SRD on Mn status.

The Mn paradox is due to the noisy and non-normally distributed raw concentration values compared to the Mn clr values. The high coefficient of variation of tissue Mn concentration is impacted both by nutrient interactions and the skewed distribution of Mn concentration values. Indeed, Mn is one of the most variable nutrients in the foliar tissues of fruit trees, making Mn concentration very difficult to interpret [71]. Mn availability is related to soil pH [70] and to a plant’s own ability to accumulate large amounts of Mn [64]. Adding Mn may rebalance tissue composition through interactions with other nutrients, and the dilution of nutrients present in relative excess as biomass production increases. Nevertheless, agronomic Mn fertilization trials should be conducted to validate the effect of Mn additions on tissue Mn status, nutrient rebalancing, and crop yield.

Stakeholders did not anticipate that Na could be a potential yield-limiting element in the soil of the orchard under study. Tissue Na may reach excessive levels in banana tissues where the electrical conductivity (EC) of irrigation water exceeds 1.1 dS m⁻¹ [37]. Tissue Na diagnosis should be cross-checked for the quality of irrigation water and by soil salinity tests especially in low landscape positions that promote salt accumulation. Soil lixiviation may be needed as a local correcting measure to abate soil Na.

4.4. Universality of Nutrient Standards

Nutrient standards specific to the orchard under study differed from CND standards for the Cavendish subgroup in Santa Catarina state, southern Brazil [72], and for the cultivars Robusta and Ney Poovan in India [64], using the same CND methodology. There were also differences between nutrient sufficiency ranges for ‘Nanica’ in northeastern Brazil, measured in the present study, and for ‘Nanica’ using the DRIS methodology in São Paulo state [73] and for ‘Cavendish’ using the CND methodology in Santa Catarina state [72]. Nutrient standards thus depend on region, management, and cultivar and are not universal. Nutrient standards should be regionalized [51,74,75] because site-specific nutrient standards depend on unique combinations of cultivar, technological level, management, soil, and climate [73,76], avoiding the heroic ceteris paribus assumption behind ‘universal’ nutrient standards. The clr and wlr nutrient standards were site-specific, supporting precision farming.

4.5. Multivariate Distances

Plants are nutritionally more balanced and high-yielding as the multivariate distance between diagnosed and reference compositions approaches zero. Although higher multivariate distances are generally associated with nutritionally imbalanced and low-yielding crops as shown for banana [77] and other fruit crops [51], the relationship may also be weak due to yield-limiting factors other than nutrients [65,73]. The FN subpopulation indicates that factors other than nutrients must limit yield. The FP subpopulation indicates luxury consumption of nutrients or suboptimal concentrations. Tissue composition of a TP specimen could be compared one to one to a compositionally close TN specimen growing in otherwise similar conditions. The Euclidean distance between two compositions and nutrient indices showed the pattern of nutrient imbalance. The significance of the difference between two equal-length compositions can be further tested statistically using the Mahalanobis distance across ilr-transformed data and clr or wlr indices.

4.6. Nutrient Sufficiency Ranges Are Intrinsically Distorted

Sufficiency ranges are risky to elaborate using a small dataset and statistically distorted raw compositions. The SRD and CND could disagree due to the noise injected by nutrient interactions and the non-normal data distribution in SRD. This is why log ratio transformations were found to be more appropriate than raw concentration values to process compositional data [32] like tissue data.

4.7. Decision-Making Flowchart

A flowchart for a fertilization decision is summarized in Figure 8. Path A requires that a well-documented database allows computing nutrient standards. When Path B is followed, two equal-length compositions are compared using the Euclidean distance between a defective composition and a benchmark composition under otherwise similar environmental conditions. The ML model integrates the explanatory variables assembled in a database by stakeholders to predict yield.

The diagnosis identifies the most limiting nutrients. The recommendation selects limiting nutrients that must be addressed as a priority by fertilization and other means, given that a change in any nutrient level must impact the level of others. Adding nutrients in relative shortage to stimulate plant growth may correct nutrients in relative excess through dilution. The decision to fertilize includes nutrient source, dosage, placement, timing and associated risks, and the practices required to correct the problem identified by stakeholders. Actions are taken using the appropriate technologies.

5. Conclusions

Current guidelines to adjust banana fertilization at the orchard scale in northeastern Brazil may not be appropriate at the orchard scale. The compositional nutrient diagnosis (CND) method made it possible to elaborate nutrient standards for ‘Nanica’ banana down to the grove scale using a small dataset collected by stakeholders. Log ratio diagnoses such as clr and wlr showed relative Mn shortage and Na excess in a low-yielding grove significantly imbalanced in the orchard under study. Concentration ranges for Mn were large due to skewed data distribution and noisy nutrient interactions that inflated the variance of concentration values. Nutrient diagnosis using clr or wlr transformations should thus guide any diagnosis using concentration ranges familiar to stakeholders to avoid misinterpreting SRD.

A multivariate distance can be computed as the difference between two equal-length compositions because log ratios are additive. Compared to the Euclidean distance computed across clr variables, the Mahalanobis distance computed across ilr variables has the additional advantage of providing a statistic (${\chi }_{D-1}^2$) supporting fertilization decisions based on nutrient indices. The correction of any one nutrient through fertilization or another profitable measure may impact other nutrients by rebalancing the whole tissue system.