Nutritional Diagnosis of Potato Crops Using the Multivariate Method

Abstract

The compositional nutrient diagnosis (CND) method considers the multiple relationships among nutrients and has been proposed to evaluate the nutritional status of plants in place of the univariate and bivariate methods. As it is mathematically based and considers the interactions among all nutrients at the same time, it avoids the errors and trends observed in the calculations of other methods estimating nutritional status, enabling a greater relationship with productivity. The objective of this study was to obtain the CND norms for high-yielding populations of potato crops. For this, 587 samples were used from 21 experimental areas in the state of São Paulo, Brazil to correlate the leaf nutrient contents and the yields of potato crops. Crops with yields higher than 48,993.24 kg ha⁻¹ were considered to have high yields, and the Mahalanobis distance separated the balanced samples from the nutritionally unbalanced ones. Thus, the CND-ilr method generated the norms and classified the 587 samples as nutritionally balanced with a high yield (5% of the total), nutritionally unbalanced with a low yield (92%), nutritionally unbalanced with a high yield (0.3%), or nutritionally balanced with a low yield (2.7%), with accuracy, sensitivity, specificity, NPV, and PPV scores of 96.9, 97.1, 93.6, 64.4, and 99.6%, respectively.

1. Introduction

Potato (Solanum tuberosum L.) is a key component of global food security, being grown on nearly 18 million hectares in more than 150 countries [1,2]. In Brazil, it is cultivated on 118.1 thousand hectares, with a production of 3.6 million tons in 2020 [3,4]. In the last ten years, there has been a 28.1% increase in the yield of potatoes grown in Brazil, driven by improvements in cultivation techniques and the adoption of more efficient varieties [2]. Such an increase ranks the country as the 21st largest producer in the world [5], with production concentrated in the south and southeast regions, mainly in the states of Minas Gerais, São Paulo, and Paraná [4].

There are several factors responsible for the yield and profitability of potato crops. One of the main ones is fertilization, since potato is a highly nutrient-demanding crop [6], which makes fertilizers responsible for 19% of the crop’s production cost [7]. Thus, nutritional monitoring is essential to evaluate and adjust fertilization management to ensure balanced nutrition, maximize the production potential of the crop, and avoid environmental damage [8].

Among the possibilities for monitoring nutritional status, the compositional nutrient diagnosis (CND) method has received increased attention from researchers. This method employs a multivariate approach that considers the interactions among all nutrients in order to evaluate the balance of the nutritional status of the plant [9,10]. When applied to foliar nutrient analysis, the method associates each nutrient of interest with the geometric mean of all nutrients in the leaf dry matter and allows researchers to infer the analytically determined balance of nutrients, as well as other elements not directly assayed [9,11].

The CND method makes it possible to group nutrients from the most limited, due to deficiency, to those at an excessive level. As a result, the generated norms are more independent of local conditions compared to the norms generated by calibration curves. The CND standards are a set of reference values for a plant’s nutrients that are used to assess its nutritional status. They allow one to more accurately identify the nutritional needs of each plant, avoiding nutrient deficiencies and excesses, as they take into account the interactions among nutrients [12]. The method also makes it possible to assign the same weights to deficiencies and excesses in the imbalance, which can be detected with the use of Mahalanobis distance [13]. This makes it possible to define the contribution of each nutrient to the nutritional composition of the dry matter and to assign a single standard deviation to each sample, with the possibility of its identification in the compositional space, and thus the exclusion of outliers, which increases the reliability in the interpretation of the results [13,14]. Compositional space refers to a coordinate system that represents the nutritional composition of a sample in relation to the content of all analyzed nutrients. Each nutrient occupies an axis in this space, and the position of the sample in this space is defined by the concentrations of each nutrient in its composition [9]. The identification of outliers in this compositional space is important to ensure the reliability of the interpretation of the results. By excluding these outliers from the compositional space, it is possible to obtain a more accurate view of the real distribution of the nutrient content in the samples, which facilitates the identification of patterns and trends in the nutritional composition of potato plants [15].

The methodology has been widely applied to annual crops, such as soybean [16,17], fruit trees [11,14], and maize [13,18]. However, few studies on the nutritional diagnosis of potato crop by CND have been conducted [19,20,21], among which none have been carried out under Brazilian conditions, which differ in terms of cultivars, climate, cultural management, and CND norms.

The arrangement of orthonormal balances with the use of the CND-ilr (isometric log ratio) methodology complements the CND-clr method because it employs an orthogonal balance between two groups of components (chemical elements evaluated from dry matter), not overlapped and arranged in the order of D−1 contrasts, where D is the number of components of the dry matter. Thus, the concept of CND-ilr was formulated, in which binary sequential partitions reflect the interactions of the nutrients. Binary sequential partitions can be defined according to the already known interactions of nutrients [11,14]. CND-clr uses the “centered log-ratio” (clr) transformation, which divides each nutrient content by the sum of all nutrient contents in the sample and then applies the natural logarithm of the result. This transformation seeks to centralize the data around zero and ensure that the scale of values is the same for all nutrients [22]. However, CND-clr has some limitations, such as sensitivity to the presence of outliers (extreme values) and heteroscedasticity (unequal variation in variance) in the data. To overcome these limitations, CND-ilr was proposed as an evolution of CND-clr [13]. CND-ilr uses the “isometric log-ratio” (ilr) transformation, which divides each pair of nutrient contents by the sum of the contents and applies the natural logarithm of the result. This transformation compares the nutrient contents in pairs, instead of comparing them with the sum of all contents, as in CND-clr [14]. Thus, the CND-ilr method represents a significant advance compared to CND-clr in several aspects, especially in the interpretation and precision of nutritional diagnosis, as it has a greater capacity to identify nutrient deficiencies and excesses, especially in cases of complex interactions among them, increasing the sensitivity of the method to detect subtle nutritional imbalances, which makes it more accurate and reliable [11,23].

The use of a meta-analysis database allows for a robust statistical evaluation, considering the variability between different studies and cultivation contexts. This approach is advantageous because it provides a broad database that can increase the generalizability of results and provide more robust insights into the usefulness of the CND-ilr method [24]. Thus, CND-ilr can effectively contribute to improving soil fertility management, potato nutrition, and increasing crop productivity.

In view of the advantages of the method, the proven feasibility of performing the nutritional diagnosis of crops, and the scarcity of studies on CND-ilr in potato crops, it is believed that the CND-ilr norms for the high-yielding populations of potato crops contribute to the evaluation of the nutritional status of potato crops, classifying the samples into nutritionally balanced and nutritionally unbalanced in high- and low-yield scenarios. Therefore, the objective of this study was to obtain the CND norms for high-yielding populations of potato crops.

2. Materials and Methods

2.1. Database

A database of potato tuber yields and the associated leaf nutrient contents for 752 experimental units was constructed from 21 experiments carried out between 2011 and 2019 in municipalities in the state of São Paulo, Brazil (Avaré, Bernardino de Campos, Botucatu, Cerqueira César, Itaí, Jaboticabal, São Manuel, and Taquarituba) (

Table 1. Characterization of the experiments that make up the database.

Id. 1	Municipality	Cultivar	Nutrient Studied	Growing Season
1	São Manuel	Agata	N	9 May to 20 August 2011
2	São Manuel	Agata	N	9 May to 20 August 2011
3	Avaré	Agata	N	9 May to 20 August 2011
4	São Manuel	Agata	N	9 May to 20 August 2011
5	São Manuel	Agata	N	9 May to 20 August 2011
6	Avaré	Agata	N	9 May to 20 August 2011
7	Botucatu	Agata	N	13 May to 20 August 2011
8	Botucatu	Agata	N	6 August to 6 November 2016
9	Botucatu	Agata	N	27 July to 4 November 2017
10	Botucatu	Electra	N	13 May to 20 August 2016
11	Botucatu	Electra	N	6 August to 6 November 2016
12	Botucatu	Electra	N	27 July to 4 November 2017
13	Avaré	Agata, Asterix, Atlantic, Markies, and Mondial	P	28 April to 18 August 2011
14	Itaí	Agata, Asterix, Atlantic, Markies, and Mondial	P	20 April to 23 July 2011
15	Cerqueira	Agata, Asterix, Atlantic, Markies, and Mondial	P	20 May to 25 August 2011
16	Taquarituba	Agata and Mondial	P	17 June to 24 September 2012
17	Taquarituba	Agata	K	15 June to 22 September 2012
18	Bernardino	Agata	K	18 April to 20 July 2013
19	Botucatu	Agata	K	12 July to 13 November 2013
20	Jaboticabal	Asterix	N	16 April to 11 July 2019
21	Jaboticabal	Agata	N	16 April to 4 July 2019

¹ Id: Identification of experiments: 1 to 6 [16]; 7 to 12 [25]; 13 to 16 [26]; 17 to 19 [27]; 20 to 21 [28].

).

The potato cultivars used in the experiments are the main ones used in the state of São Paulo. Agata has a high yield and is the most planted cultivar in Brazil, occupying approximately 50% of the planted area [29]. Asterix is the most widely used cultivar in the pre-fried and frozen potato stick industry. Atlantic accounts for the largest demand in the domestic market for potato chips [30]. Electra is used for the fresh market. Markies is used for cooking and frying [31]. Mondial has a high yield and is suitable for the fresh market.

The climatic classification of experiments 1 to 19 is the Cfa type (a hot, humid, temperate (mesothermal) climate), and the average temperature of the hottest month was above 22 °C. Experiments 20 and 21 were carried out in a tropical climate, the Aw type, with drought in winter [32]. Irrigation was carried out using a center pivot or conventional sprinkler, and phytosanitary management was carried out according to the technical recommendations for the crop [33].

2.2. Foliar Diagnosis and Tuber Yield

In each experimental unit, the nutritional diagnostic leaves were sampled. The third leaf of the stem was collected approximately 30 days after the emergence of the apical tuft of fifteen plants per plot, with one complete leaf (leaf + petiole) per plant, as recommended by Lorenzi et al. [34]. The sampled leaves were washed, dried in an oven with forced ventilation at 65 °C until reaching a constant mass, and then ground in a Wiley mill with a 120-mesh sieve. Then, the dry leaves were digested, and the nutrient contents (N, P, K, Ca, Mg, S, B, Cu, Fe, Mn, and Zn) were read, according to methodologies proposed by Malavolta et al. [35]. In each experimental unit, in addition to the leaves for nutritional diagnosis, the tuber yield was evaluated, estimating the production for one hectare.

2.3. Obtaining the Norms Using the CND-ilr Method

The contents of all nutrients are expressed in mg kg⁻¹, and the yield is expressed in kg ha⁻¹. The samples in the database were sorted in descending order according to yield. For each nutrient, outliers were removed using interquartile variation (IQR) and a boxplot to display the distribution of the data. The boxplot was divided into quartiles, and the lower and upper limits were defined. Quartiles divide a series of ordered data into four equal parts: the first quartile (Q1) separates the bottom 25% from the top 75%; the median (Q2) divides the data into the bottom 50% and the top 50%; and the third quartile (Q3) separates the bottom 75% from the top 25%. This analysis is crucial to understanding the dispersion and central tendency of the data. The IQR was the difference between the Q1 and Q3 quartiles, so all values that exceeded the limits were considered outliers, as proposed by Rousseeuw and Croux [36], using the Mahalanobis distance [13]. This procedure is recommended for improving the evaluation parameters of the sample classifications and, consequently, the quality of the CND norms [10,11,14,23].

After excluding the outliers, the data normality was checked with the Shapiro–Wilk test, as indicated by Hair et al. [37], since data normality is a condition required for making valid inferences about population parameters.

Subsequently, for each sample, the filling value (Fv) was calculated by deducting the sum of the nutrient contents from 1,000,000 in the leaf dry matter. The filling value (Fv) represents the amount, in ppm, not accounted for by nutrients in the leaf analysis. Therefore, by subtracting the nutrient contents from 1,000,000 in the dry matter of the leaves, the Fv is obtained, which indicates the remaining concentration that is not composed of the analyzed nutrients [21]. The geometric mean (g) of the nutrient contents of each sample in the database [38] was calculated as follows:

$$\mathrm{g}=(\mathrm{N}\times \mathrm{P}\times \dots \times \mathrm{Z}\mathrm{n}\times {\mathrm{F}\mathrm{v})}^{\frac{1}{(\mathrm{d}+1)}}$$

(1)

where d is the number of nutrients evaluated.

For the 12 components (D) corresponding to the 11 nutrients and the Fv, the 11 orthonormal balances (D−1) and sequential binary partition were established [9,14]. The balances were defined using the interpretation of the data according to Seaborn, which is a data visualization library of the Python program, establishing high-level correlations among the nutrients. Using the clustermap function with predefined parameters (spearman parameters (method), Euclidean distance (metric), and a scale pattern equal to 1), a heatmap was constructed and is presented in the form of a dendrogram. The isometric logarithmic ratio (ilr) was calculated as proposed by Egozcue and Pawlowsky-Glahn [10]:

$${\mathrm{i}\mathrm{l}\mathrm{r}}_{\mathrm{j}}=\sqrt{\frac{{\mathrm{n}}_{\mathrm{j}}^{+}{\mathrm{n}}_{\mathrm{j}}^{-}}{{\mathrm{n}}_{\mathrm{j}}^{+}+{\mathrm{n}}_{\mathrm{j}}^{-}}\mathrm{l}\mathrm{n}\frac{\mathrm{g}\left({\mathrm{c}}_{\mathrm{j}}^{+}\right)}{\mathrm{g}\left({\mathrm{c}}_{\mathrm{j}}^{-}\right)}}$$

(2)

where ${\mathrm{n}}_{\mathrm{j}}^{+}$ is the number of components in the numerator (r); ${\mathrm{n}}_{\mathrm{j}}^{-}$ is the number of components in the denominator(s), determined from the sequential binary partition; $\mathrm{g}\left({\mathrm{c}}_{\mathrm{j}}^{+}\right)$ is the geometric mean between the components in the numerator; and $\mathrm{g}\left({\mathrm{c}}_{\mathrm{j}}^{-}\right)$ is the geometric mean between the components in the denominator.

For each ilr calculated, the mean of the high-yielding population was calculated, and then the covariance matrix of the high-yielding population was constructed. The Mahalanobis distance (D²) was calculated to separate the nutritionally balanced from the nutritionally unbalanced samples [13], based on the Euclidean space [14], as follows:

$${\mathrm{D}}^2=\sqrt{{\left({\mathrm{i}\mathrm{l}\mathrm{r}}_{\mathrm{j}}-{\mathrm{i}\mathrm{l}\mathrm{r}}_{\mathrm{j}}^{\mathrm{*}}\right)}^{\mathrm{T}}{\mathrm{C}\mathrm{O}\mathrm{V}}^{-1}({\mathrm{i}\mathrm{l}\mathrm{r}}_{\mathrm{j}}-{\mathrm{i}\mathrm{l}\mathrm{r}}_{\mathrm{j}}^{\mathrm{*}})}$$

(3)

where ${\mathrm{i}\mathrm{l}\mathrm{r}}_{\mathrm{j}}^{\mathrm{*}}$ is the barycenter of a reference population, COV is the covariance matrix of the reference population (TN), and ${\mathrm{i}\mathrm{l}\mathrm{r}}_{\mathrm{j}}$ is the sample being used in the calculation.

Also, using the Cate–Nelson method allows for maximizing the sum of squares between two partitions, with the following formula:

$${\mathrm{R}}^2=\left[\frac{{\left({{\sum }_{\mathrm{i}=1}^{\mathrm{k}}\mathrm{Y}}_{\mathrm{i}}\right)}^2}{\mathrm{k}}+\frac{{\left({{\sum }_{\mathrm{j}=\mathrm{k}}^{\mathrm{n}}\mathrm{Y}}_{\mathrm{j}}\right)}^2}{(\mathrm{n}-\mathrm{k})}\right]-\mathrm{C}\mathrm{F}$$

(4)

where Y is the yield; k is an elementary count that begins with the first observation ordered above $\mathrm{n}$; $\mathrm{n}$ is a variable that represents a certain value or cut-off point of the high-yielding population; and CF is a correction factor.

The D² value and its corresponding yield were observed in the sample that obtained the highest R² value (Figure 1).

The D² value and the yield represent the two cut-off points for determining the Cate–Nelson partition of the samples into four quadrants, which were named by Parent et al. [14] as follows: (1) True negative (TN) samples are nutritionally balanced and have a high yield and adequate nutritional status; they are classified as the “reference population”. (2) False positive (FP) samples indicate a Type I error; they are nutritionally unbalanced samples with a high yield. FP represents samples with a “luxury consumption” of nutrients or a high nutrient use efficiency. (3) False negative (FN) samples indicate a Type II error. FN corresponds to samples influenced by other production factors (climate, pests, diseases, and others), negatively affecting the production performance of the crop. (4) True positive (TP) samples have a low yield and are nutritionally unbalanced, where at least one nutrient is causing the imbalance.

The parameters to evaluate the classifications of the samples in the quadrants were calculated according to Parent et al. [14]:

Accuracy (ACC) is the probability of a sample being correctly identified as balanced or unbalanced, calculated by:

ACC = (TN + TP)/(TN + FN + TP + FP)

(5)

Sensitivity (SEN) is the probability that a low-yield sample is unbalanced, calculated by:

SEN = TP/(TP + FN)

(6)

A positive predicted value (PPV) is the probability that an unbalanced sample will return to poor performance, calculated by:

PPV = TP/(TP + FP)

(7)

Specificity (SPC) is the probability of a high-yield sample being balanced, calculated by:

SPC = TN/(TN + FP)

(8)

A negative predicted value (NPV) is the probability that a balanced sample will return to high performance, calculated by:

NPV = TN/(TN + FN)

(9)

Thus, the CND-ilr norms corresponded to the mean and the multiplication matrix, which were calculated from the CND-ilr coordinates of the reference population (TN).

3. Results

The normality test was performed on the 752 samples that made up the initial database, which showed normality (Shapiro–Wilk W = 0.8897, p = 0.999). However, 165 outliers were removed—140 using the IQR method and 25 using the two-tailed probability (p) of the chi-square distribution (p < 0.05). The test was performed again on the 587 remaining data, which showed normality, with W = 0.9206 and p = 0.999.

CND-ilr, Norms, Sequential Binary Partition, and Cate–Nelson Partition

In the heatmap, which shows the groups hierarchically, the correlations among the nutrients according to the study were observed using the Seaborn tool in Python (Figure 2), based on which 11 balances and sequential binary partitioning were constructed (

Table 2. Sequential binary partition (SBP) of n−1 (11) orthonormal balances.

ilr	Balances	N	K	P	S	Ca	Mg	B	Cu	Zn	Mn	Fe	Fv	r	s
1	[N|K]	−1	1	0	0	0	0	0	0	0	0	0	0	1	1
2	[N, K|Mg]	−1	−1	0	0	0	1	0	0	0	0	0	0	1	2
3	[Ca|Zn]	0	0	0	0	−1	0	0	0	1	0	0	0	1	1
4	[Fe|Mn]	0	0	0	0	0	0	0	0	0	1	−1	0	1	1
5	[Mg, N, K|Ca, Zn]	−1	−1	0	0	1	−1	0	0	1	0	0	0	2	3
6	[S|B]	0	0	0	−1	0	0	1	0	0	0	0	0	1	1
7	[P|Fe, Mn]	0	0	−1	0	0	0	0	0	0	1	1	0	2	1
8	[Cu|S, B]	0	0	0	1	0	0	1	−1	0	0	0	0	2	1
9	[Mg, N, K, Ca, Zn|Cu, S, B]	−1	−1	0	1	−1	−1	1	1	−1	0	0	0	3	5
10	[P, Fe, Mn|Mg, N, K, Ca, Zn, Cu, S, B]	1	1	−1	1	1	1	1	1	1	−1	−1	0	8	3
11	[Fv|P, Fe, Mn, Mg, N, K, Ca, Zn, Cu, S, B]	1	1	1	1	1	1	1	1	1	1	1	−1	11	1

Fv: filling value; r and s: number of positives (numerator) and negatives (denominator), respectively, of the formula for calculating multi-nutrient relationships (${\mathrm{i}\mathrm{l}\mathrm{r}}_{\mathrm{j}}$).

).

After optimizing the true negative (TN; n = 29), false positive (FP; n = 2), true positive (TP; n = 540), and false negative (FN; n = 16) quadrants (Figure 3), to achieve the highest accuracy (97%), cut-off points whose D² was 3.87 and yield was 48,993 kg ha⁻¹ were obtained with n = 587, establishing a high-yielding population with an increase of 63.3% compared to the Brazilian average of 30,000 kg ha⁻¹, as indicated by IBGE [4].

The present study indicates that the high-yielding population has a production variation between 48,993 and 61,328 kg ha⁻¹; that is, the samples classified in the TN group have a yield above the national average. The population classified as nutritionally unbalanced with a low yield (TP) had minimum and maximum yields of 2778 and 48,638 kg ha⁻¹, respectively, which indicates great potential for an increased yield with adequate nutritional balances of the production areas. Despite being nutritionally unbalanced, there were 92 samples in the TP quadrant that had a yield above the Brazilian average [4]. This indicates that the Brazilian average potato yield could be improved if the crop generally responds to enhanced nutrient balance.

Of the total samples, 29 samples (5%) were classified as TN, i.e., as a reference population. Of these, 20 and 9 samples were from the N and P experiments, respectively. Parent et al. [14], Modesto et al. [18], and Marchand et al. [39], in similar studies with mango, sweet corn, and cranberry crops, found that 13, 10, and 15% of the database samples were classified as TN, respectively. Regarding the TP group, it was represented by 540 (92%) of the samples, with 148, 318, and 74 samples from the N, P, and K experiments, respectively (

Table 3. Partitioning the database samples into true negative (TN), false positive (FP), false negative (FN), and true positive (TP) quadrants.

Quadrants	Number of Experimental Samples			Total of Samples	%
	N	P	K
TN	20	9	0	29	5.0
TP	148	318	74	540	92.0
FP	2	0	0	2	0.3
FN	6	10	0	16	2.7
Total	176	337	74	587	100.0

TN = reference population, nutritionally balanced with high yield; TP = nutritionally unbalanced population with low yield; FP = nutritionally unbalanced population with high yield; and FN = nutritionally balanced population with low yield.

).

The results of the accuracy, sensitivity, specificity, NPV, and PPV parameters were 96.9, 97.1, 93.6, 64.4%, and 99.6%, respectively. The high values of the parameters can be attributed to the fact that the database consists of samples obtained from experiments with nutrient doses that allowed for greater discrepancies among the yields as a function of the leaf nutrient contents.

For the initial database, without the elimination of outliers, the values obtained for accuracy, sensitivity, specificity, NPV, and PPV were 70.6, 70.7, 50.0, 0.45, and 99.8%, respectively. Therefore, with the removal of outliers, the parameters of accuracy, sensitivity, specificity, and NPV improved by 37, 37, 87, and 14%, while the PPV parameter decreased by 0.2%. When comparing the nutrient balances between the TN and the TP, FN, and FP classes, the t-test detected significant differences (

Table 4. Results of the t-test comparing the means of the samples classified as true negative (TN) and the TP, FN, and FP classes of orthonormal balances.

ilr	Balances	TN	TP	FN	FP
1	[N|K]	−0.10	0.03 **	−0.01 ns	−0.27 ns
2	[N, K|Mg]	−1.91	−1.80 **	−1.88 ns	−1.86 ns
3	[Ca|Zn]	−3.54	−3.66 ns	−3.30 ns	−3.77 ns
4	[Fe|Mn]	−0.71	−0.66 ns	−0.76 ns	−0.68 ns
5	[Mg, N, K|Ca, Zn]	−3.78	−3.46 **	−3.44 *	−3.95 ns
6	[S|B]	−3.29	−3.28 ns	−3.38 *	−3.03 ns
7	[P|Fe, Mn]	−1.99	−2.25 *	−2.33 ns	−1.64 ns
8	[Cu|S, B]	0.56	1.46 **	0.96 ns	0.63 ns
9	[Mg, N, K, Ca, Zn|Cu, S, B]	−4.49	−5.30 **	−4.96 *	−4.75 ns
10	[P, Fe, Mn|Mg, N, K, Ca, Zn, Cu, S, B]	1.62	1.37 **	1.84 ns	0.89 *
11	[Fv|P, Fe, Mn, Mg, N, K, Ca, Zn, Cu, S, B]	−6.28	−6.27 ns	−6.22 *	−6.33 ns

^ns = not significant; * and ** = significant to 5 and 1% of probability according to F test, respectively; TN = reference population, nutritionally balanced with high yield; TP = nutritionally unbalanced population with low yield; FP = nutritionally unbalanced population with high yield; and FN = nutritionally balanced population with low yield.

).

When comparing the TN and TP classes, it is possible to observe differences in balances 1, 2, 5, 7, 8, 9, and 10, indicating that there were differences between the nutritional balance classes and the main sources of potential nutritional imbalances that impaired the crop yield in the TP group. Thus, there were four differences in the balances between the TN and FN classes (Table 4), although the samples of both classes were nutritionally balanced.

The CND-ilr norms generated corresponded to the multiplication matrix (multi-nutrient) and to the mean calculated from the CND-ilr coordinates of the reference population (TN) (

Table 5. CND-ilr norms (multi-nutrient and average matrix) calculated from the CND-ilr coordinates of the reference population (TN).

	ilr1	ilr2	ilr3	ilr4	ilr5	ilr6	ilr7	ilr8	ilr9	ilr10	ilr11
ilr1	0.0307
ilr2	0.0097	0.0209
ilr3	0.0533	0.0371	0.1758
ilr4	−0.0121	−0.0026	−0.0280	0.0391
ilr5	0.0545	0.0362	0.1537	−0.0126	0.1659
ilr6	−0.0120	−0.0085	−0.0425	−0.0010	−0.0373	0.0332
ilr7	−0.0782	−0.0524	−0.2000	0.0363	−0.1844	0.0421	0.3075
ilr8	0.1078	0.0629	0.2691	−0.0479	0.2572	−0.0707	−0.3835	0.6397
ilr9	−0.1058	−0.0714	−0.2590	0.0425	−0.2568	0.0484	0.3757	−0.5579	0.5553
ilr10	0.0651	0.0411	0.1493	−0.0122	0.1562	−0.0442	−0.2097	0.2538	−0.2549	0.2095
ilr11	0.0084	0.0041	0.0249	−0.0062	0.0252	−0.0032	−0.0258	0.0318	−0.0345	0.0221	0.0059
Average *	−0.0974	−1.9180	−3.5445	−0.7164	−3.7686	−3.2896	−2.0332	0.6681	−4.5701	1.5875	−6.2811

* Average reference population (TN) of the CND-ilr.

).

Figure 4 presents a balance dendrogram derived using an ad hoc sequential binary partition (SBP) that obtained the linkages among groups of components. SBP is a (D−1) × D matrix, where the parts labeled “+1” (group numerator) are balanced with parts labeled “−1” (group denominator) in each row, and the part labeled “0” is excluded (Table 2). The composition was partitioned sequentially in each ordered row into two contrasts until the subcompositions (+1) and (−1) each contained a single role, with the overall mean values of ilr in the fulcrums and univariate confidence intervals of 0.05 for the TN and TP classes.

4. Discussion

Variations in nutrient contents can establish significant differences in a given balance, such as in the [Mg, N, K|Ca, Zn] and [S|B] balances. However, when these are part of another, more comprehensive balance considering more nutrients, for example, [P, Fe, Mn|Mg, N, K, Ca, Zn, Cu, S, B], these contents are understood as being in balance (Table 4), as also observed by Parent et al. [14].

In the comparisons between the TN and FP classes, there was a significant difference only in balance 10 (Table 4), which shows nutritional imbalance despite having a high yield. This was probably due to the nutrient content being within the range considered “luxury consumption”, to the contamination of the leaf sample with the nutrient not belonging to the leaf content, or, exceptionally, to high nutrient use efficiency [14]. Balances can be represented metaphorically, such as by a moving autonomous diagram with fulcrums or the weighing pans of a scale, where the nutrient contents directly impact the nutritional balances in the fulcrums after a change [11,14].

The balances that were significant (Figure 4) corroborate the results found in Table 4, whose ilrs were 1, 2, 5, 7, 8, 9, and 10. Between the TN and TP classes, there was a significant difference in all balances in which N was present, except for the balance that compared all nutrients with the Fv, which was not significant (Figure 4). Also, the balances of [N|K], [N, K|Mg], [Mg, N, K,|Ca, Zn], and [Mg, N, K, Ca, Zn|Cu, S, B] had negative means in the samples in the TP quadrant, which were −0.54, −2.69, −4.46, and −6.61, respectively.

N was mainly responsible for causing imbalance. The N deficiency of the samples in TP, represented by values of −0.54 in the [N|K] balance and −2.69 in the [N, K|Mg] balance (Figure 4), is directly related to the yield due to its high demand by potato crops, being the second most accumulated nutrient [40], with a strong association with leaf growth, photosynthesis, and tuber production [40,41]. In the [Mg, N, K|Ca, Zn] balance, there are also competitive interactions among the K, Ca, and Mg cations, as reported by Hawkesford [42].

In the [Fe|Mn] balance, there was no difference between the TN and TP samples. Fe and Mn are directly related to the genesis of the Latossolo Vermelho (Oxisol) in the experimental areas, being supplied in sufficient quantities to the plant. However, in the [P|Fe, Mn] balance, there was an imbalance caused by the excess of P, which can be justified by the very high doses evaluated in the experiments, even for soils with high contents of the nutrient. In well-conducted fertilizer calibration experiments, it was indicated that the high doses optimized the fitting of the response curves; however, it was not optimized in the partition for the best nutritional balances.

Difficulties in fitting deterministic models to facts and data, controllable or not, in order to derive indications of the adequate doses of nutrients, highlight the historical difficulties [43] of this technique employing a limited number of calibration experiments. Rozane et al. [23] and Lima Neto [44] showed that the number of experiments with fertilizers could be reduced by observing large amounts of data from current production bases, as well as through the compositional analysis of numerous calibration experiments [11,39], as performed in the present study.

For the [S|B] balance, the indices of the TP samples did not differ from the indices of the TN samples. The [S|B] balance in the TP quadrant is probably similar to that observed for a population with a high yield (nutritionally balanced); i.e., this does not seem to be the problem of the TP samples. However, the [Cu|S, B] balance was nutritionally unbalanced, which indicates that the limiting factor was Cu deficiency and/or that the samples in the TP quadrant did not receive fungicide applications, which generally contain high concentrations of Cu, sufficiently equal to TN, as also reported by Rozane et al. [23] and Modesto et al. [18], and that the imbalance caused by Cu is not effectively an imbalance, but rather an increase in the Cu content in TP due to the deposition of the element on the leaves, generating contamination. Also, excess P in the soil can reduce the availability of Cu to plants, which explains the low Cu content in leaf tissue, since Cu deficiency is not common in clay soils. Thus, the imbalance in the [Mg, N, K, Ca, Zn|Cu, S, B] balance was caused by N and Cu, both due to deficiency. In the [P, Fe, Mn|Mg, N, K, Ca, Zn, Cu, S, B] balance, the main nutrients responsible for nutritional imbalances were P on the left side, due to excess, and N and Cu on the right side, due to deficiency (Figure 4).

There was no significant difference between the TN and TP groups for the following balances: [Ca|Zn], [Fe|Mn], [S|B] and [Fv|P, Fe, Mn, Mg, N, K, Ca, Zn, Cu, S, B], which probably did not negatively affect the production of potato tubers. All nutrients evaluated were compared with the filling value (Fv), as nutrient accumulation in the leaf tissues or as a dilution measure [45].

From the observed balances (Figure 4), it was found that the nutritional balances were more important than the absolute values themselves, since there was no significant difference between the TN and TP groups for [Fv|P, Fe, Mn, Mg, N, K, Ca, Zn, Cu, S, B]. Thus, nutritional interactions should be considered in the concept of nutritional balance, where nutrient groups need to be optimally balanced, rather than just considering the individual nutrient contents [10,11,14].

5. Conclusions

Considering the Mahalanobis distance of 3.87 and yield of 48,993.24 kg ha⁻¹, the CND-Ilr method generated the following norms—Ilr1 (−0.0974), Ilr2 (−1.9180), Ilr3 (−3.5445), Ilr4 (−0.7164), Ilr5 (−3.7686), Ilr6 (−3.2896), Ilr7 (−2.0332), Ilr8 (0.6681), Ilr9 (−4.5701), Ilr10 (1.5875), and Ilr11 (−6.2811)—and classified 5% of the 587 samples as nutritionally balanced with a high yield, 92% as nutritionally unbalanced with a low yield, 0.3% as nutritionally unbalanced with a high yield, and 2.7% as nutritionally balanced with a low yield, with accuracy, sensitivity, specificity, NPV, and PPV scores of 96.9, 97.1, 93.6, 64.4, and 99.6%, respectively.