Since the dynamics of nutrient interaction (Serra et al.,

Since nutrient uptake anddistribution are affected by interactions within the plant, multivariateapproaches have been derived in a bid toovercome such difficulties of the univariate and bivariate approaches. Severalpieces of literature have shown that nutrient diagnosis using univariate andbivariate approaches presented conflicting diagnosis (Huang et al., 2012; Wairegi and van Asten,2012; Blanco-Macias et al., 2009; Silvaet al., 2004). As such, nutrientnorms from these approaches are numerically biased (Parent et al.

(2012a). In 1992, Parent and Dafir proposed a modified DRISusing the centered log-ratio technique – proposed by Aitchison (1986) for compositionaldata to conduct Compositional Nutrient Diagnosis (CND). CND and the univariateand bivariate indices have shown to be moderately to closely related to eachother (Parent et al., 1994a; Parent et al., 1994b; Wairegi and van Asten,2011; Parent, 2011; Wairegi and van Asten, 2012). With the multivariateapproach, other inferential techniques such as principal component analysis(PCA), canonical correlation, Chi-square test and others are often used toimprove the efficiency and provide an accurate diagnosis by recognizing thedynamics of nutrient interaction (Serra etal., 2016).

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CND is a multivariate approach that was developed toimprove nutrient diagnosis via univariate or bivariate approach.  It was proposed and developed by Parent andDafir (1992) and is based on the principles of Compositional Data Analysis(CDA) 166. The CND takes into consideration the interdependence of nutrientconcentrations in plants and that the sum of all dry matter concentrationsalways totals up to 100 %, or sums up to 1. CND method has anaccurately stated covariance matrix, allowing for the computation of ratiosoriginating from nutrient concentration that are mutually exclusive (Parent,2011) as opposed to DRIS approach which is empirical without clear and distinctoutline of the covariance matrix for conducting multivariate analysis (Barlog,2016).Nutrient indices are usedfor the interpretation of compositional nutrient data. It represents thedifference between a particular nutrient and its geometric mean relative to thedifference of the same nutrient to the geometric mean of the high-yieldingsubpopulation 292.5.3.

1 Establishing CND NormsCND norms are establishedfirst by forming a database of nutrient concentration and yield of the crop inquestion. According Serra et al.(2016), the nutrient concentration database must show normal distribution,thus, making it necessary to transform the nutrient concentration to correctnon-normal distribution. Several methods of nutrient concentrationtransformation have been proposed.

These include row centered log ratio,isometric log ratio e.t.c. Row centered log ratios (clr) have been the most widely used method of transformation.Recently, Parent etal (2013) demonstrated certain difficulties in CND-clr computationsuch as; the occurrence of a singular matrix in multivariate analysescomputations (due to closure of indices to a zero-sum) made clr aninappropriate transformation as the geometric mean of the whole unstructuredvector was affected by large variations in micronutrient concentrations due tofungicide applications. Because of these limitations, modified CND-clr was proposed by Parent et al.

(2013) and this approach uses theisometric log-ratio (ilr) transformation instead of row-centered logratios. This approach generates linearly independent variables computed asstructured balances of components or groups of components (Egozcue et al., 2003).

To date, CND-ilr hasbeen used to classify the nutrient composition of several crops (Parent, 2011;Parent et al., 2012b; Hernandes et al., 2012; Marchand et al., 2013; Parent et al., 2013). After transforming the nutrient concentration, thehigh yielding population of healthy leaves with no damage is selected.

Thedatabase might be divided into two subpopulations using the mean+0.5 standarddeviation as a criterion to separate the populations into a high yielding groupand low yielding group (Serra et al.,2010). Cumulative variance function fit to cubic (Khiari et al., 2001) and Boltzmann equation (Hernandez et al., 2008). Parent et al.

(1994) proposed Chi-squaredistribution function to define a CND threshold value for nutrientimbalance.  2.5.3.

2 Mathematical approach for establishing the CND normsParent and Dafir (1992) indicated that plant tissuecomposition forms a d dimensionalnutrient arrangement i.e, simplex (Sd)made of d+1 nutrient proportionsincluding a d nutrient and fillingvalue defined as follows:                                               10Where 100 is the dry matter concentration(%)N, P, K… = Nutrient proportion (%)  = The filling value computed as                                                        11The nutrient proportionsbecome scale invariant after they have been divided by geometric mean (G) of the d+1 components including  (Aitchinson, 1986) as follows;                                                                  12After calculation of thegeometric mean (G), the newexpression for the multi-nutrient is log-transformed to generate therow-centered log ratios as follows:                  13The sum of therow-centered log ratios must be equal to zero i.e;                                                               14From this, CND norms arethe mean and the standard deviations (SD) of row-centered log ratios of thehigh yielding subpopulation from the yield and nutrient concentration database.The additivity or independence among the compositional data is ascertainedusing the clr transformation(Aitchison, 1986). After obtaining the clr, there is need to iterate apartition of the database into two subpopulations using the Cate-Nelsonprocedure after the arranging the yield data in decreasing order as describedby Khiari et al.

(2001). After thisstage, it is necessary to iterate a partition of the database between twosubpopulations using the Cate-Nelson procedure once the observations have beenranked in a decreasing yield order (Khiari etal, 2001). In the first partition,the two highest yield values form one group, and the remainder of yield valuesforms another group; thereafter, the three highest yield values form the other.This process is repeated until the two lowest yield values form one group, andthe remainder of yield values forms the other.

At each iteration, the firstsubpopulation comprises n1 observations, and the secondcomprises n2 observations for a total of n observations(n = n1 + n1) in the whole database.For the two subpopulations obtained at each iteration, one must compute thevariance of CND VX values. The variance ratio forcomponent X can be estimated as follows:                                         15The cumulative varianceratio function (  isthen computed as the sum of variance ratios at the ith iteration from the top.

The cumulated varianceratios for a given iteration is computed as a proportion of the total sum ofvariance ratios across all iterations to compare the discrimination power ofthe VX between low-yield and high-yield subpopulations on a commonscale. It is computed as:                                                                    16Where n1-1 ispartition number and n is the total number of observations (n1+ n2). The denominator is the sum of variance ratios acrossall iterations, and thus, is a constant for nutrient X. The cumulativefunction  relatedto yield (Y) shows a cubic pattern:                                                       17Whereh = intercepta,b and c= parameter coefficients.The optimum partitionbetween the two subpopulations is defined as the inflection portion (IP) and is computed by as the pointwhere the model shows changes in concavity and is obtained by equating thederivative of the equation above to zero as:                                                                 18And then the secondderivative as:                                                                  19The yield cutoff value isobtained as  and the highest yield cutoffvalue across nutrient expressions can be selected to ascertain that minimumyield target for a high-yield subpopulation will be classified as high yieldwhatever the nutrition expression.2.

5.3.3 Establishing the CND IndexAfter establishing theCND norms as means and standard deviation of the clr of the nutrient concentration in the maize ear leaf tissuedenoted as  +  and  + .The CND index (I) denotedas  … , were calculated from the clr as follows: , , , ,…     20the index is defined as the distance of a givennutrient Xi from its geometric mean12, which is relative to the distance of the same nutrient from the geometricmean of the target population (reference population with high yield).From this point of view above, it is expected that whenCND index is closer to zero, Xinutrient is less imbalanced than others in the analysis. Serra et al. (2010a, b) observed CND indexclose to zero showed higher nutritional balance.

The CND indices arestandardized and linearized variables as dimensions of a circle (d+1=2), a sphere (d+1=3), or hypersphere (d+1>3)in a d-dimensional space. CND nutrient imbalance index (NII)The NII is the CND r2as recommended by Parent and Dafir (1992) and is given as                                       21Its radius, r,computed from the CND nutrient indices, thus characterizes each specimen.

Thesum of d + 1 squared independent, unit-normal variables produces a newvariable having a ?2 distribution with d + 1 degrees of freedom(Ross, 1987). Because CND indices are independent, unit-normal variables, theCND r2 values must have a ?2 distributionfunction. This is why it is recommended that the highest yield cutoff value(highest discrimination power) among d + 1 nutrient computations beretained to calculate the proportion of the low-yield subpopulation below yieldcutoff used as the critical value for the cumulative distribution function. Asdefined by equation 19 and 20, the closer to zero that CND indices are, andthus the CND r2 or ?2 values are, the higher theprobability to obtain a high yield.