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

Since nutrient uptake and
distribution are affected by interactions within the plant, multivariate
approaches have been derived in a bid to
overcome such difficulties of the univariate and bivariate approaches. Several
pieces of literature have shown that nutrient diagnosis using univariate and
bivariate approaches presented conflicting diagnosis (Huang et al., 2012; Wairegi and van Asten,
2012; Blanco-Macias et al., 2009; Silva
et al., 2004). As such, nutrient
norms from these approaches are numerically biased (Parent et al. (2012a). In 1992, Parent and Dafir proposed a modified DRIS
using the centered log-ratio technique – proposed by Aitchison (1986) for compositional
data to conduct Compositional Nutrient Diagnosis (CND). CND and the univariate
and bivariate indices have shown to be moderately to closely related to each
other (Parent et al., 1994a; Parent et al., 1994b; Wairegi and van Asten,
2011; Parent, 2011; Wairegi and van Asten, 2012). With the multivariate
approach, other inferential techniques such as principal component analysis
(PCA), canonical correlation, Chi-square test and others are often used to
improve the efficiency and provide an accurate diagnosis by recognizing the
dynamics of nutrient interaction (Serra et
al., 2016).

CND is a multivariate approach that was developed to
improve nutrient diagnosis via univariate or bivariate approach.  It was proposed and developed by Parent and
Dafir (1992) and is based on the principles of Compositional Data Analysis
(CDA) 166. The CND takes into consideration the interdependence of nutrient
concentrations in plants and that the sum of all dry matter concentrations
always totals up to 100 %, or sums up to 1.

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CND method has an
accurately stated covariance matrix, allowing for the computation of ratios
originating from nutrient concentration that are mutually exclusive (Parent,
2011) as opposed to DRIS approach which is empirical without clear and distinct
outline of the covariance matrix for conducting multivariate analysis (Barlog,
2016).

Nutrient indices are used
for the interpretation of compositional nutrient data. It represents the
difference between a particular nutrient and its geometric mean relative to the
difference of the same nutrient to the geometric mean of the high-yielding
subpopulation 29

2.5.3.1 Establishing CND Norms

CND norms are established
first by forming a database of nutrient concentration and yield of the crop in
question. According Serra et al.
(2016), the nutrient concentration database must show normal distribution,
thus, making it necessary to transform the nutrient concentration to correct
non-normal distribution. Several methods of nutrient concentration
transformation 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 et
al (2013) demonstrated certain difficulties in CND-clr computation
such as; the occurrence of a singular matrix in multivariate analyses
computations (due to closure of indices to a zero-sum) made clr an
inappropriate transformation as the geometric mean of the whole unstructured
vector was affected by large variations in micronutrient concentrations due to
fungicide applications.

Because of these limitations, modified CND-clr was proposed by Parent et al. (2013) and this approach uses the
isometric log-ratio (ilr) transformation instead of row-centered log
ratios. This approach generates linearly independent variables computed as
structured balances of components or groups of components (Egozcue et al., 2003). To date, CND-ilr has
been 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, the
high yielding population of healthy leaves with no damage is selected. The
database might be divided into two subpopulations using the mean+0.5 standard
deviation as a criterion to separate the populations into a high yielding group
and 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-square
distribution function to define a CND threshold value for nutrient
imbalance. 

2.5.3.2 Mathematical approach for establishing the CND norms

Parent and Dafir (1992) indicated that plant tissue
composition forms a d dimensional
nutrient arrangement i.e, simplex (Sd)
made of d+1 nutrient proportions
including a d nutrient and filling
value defined as follows:

 

                                             10

Where 100 is the dry matter concentration
(%)

N, P, K… = Nutrient proportion (%)

 = The filling value computed as

                                                       11

The nutrient proportions
become scale invariant after they have been divided by geometric mean (G) of the d+1 components including

 (Aitchinson, 1986) as follows;

                                                                 12

After calculation of the
geometric mean (G), the new
expression for the multi-nutrient is log-transformed to generate the
row-centered log ratios as follows:

                 13

The sum of the
row-centered log ratios must be equal to zero i.e;

                                                              14

From this, CND norms are
the mean and the standard deviations (SD) of row-centered log ratios of the
high yielding subpopulation from the yield and nutrient concentration database.
The additivity or independence among the compositional data is ascertained
using the clr transformation
(Aitchison, 1986).

After obtaining the clr, there is need to iterate a
partition of the database into two subpopulations using the Cate-Nelson
procedure after the arranging the yield data in decreasing order as described
by Khiari et al. (2001). After this
stage, it is necessary to iterate a partition of the database between two
subpopulations using the Cate-Nelson procedure once the observations have been
ranked in a decreasing yield order (Khiari et
al, 2001).

In the first partition,
the two highest yield values form one group, and the remainder of yield values
forms another group; thereafter, the three highest yield values form the other.
This process is repeated until the two lowest yield values form one group, and
the remainder of yield values forms the other. At each iteration, the first
subpopulation comprises n1 observations, and the second
comprises 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 the
variance of CND VX values.

The variance ratio for
component X can be estimated as follows:

                                        15

The cumulative variance
ratio function (

 is
then computed as the sum of variance ratios at the ith iteration from the top. The cumulated variance
ratios for a given iteration is computed as a proportion of the total sum of
variance ratios across all iterations to compare the discrimination power of
the VX between low-yield and high-yield subpopulations on a common
scale. It is computed as:

                                                                   16

Where n1-1 is
partition number and n is the total number of observations (n1
+ n2). The denominator is the sum of variance ratios across
all iterations, and thus, is a constant for nutrient X. The cumulative
function

 related
to yield (Y) shows a cubic pattern:

                                                      17

Where
h = intercept

a,
b and c= parameter coefficients.

The optimum partition
between the two subpopulations is defined as the inflection portion (IP) and is computed by as the point
where the model shows changes in concavity and is obtained by equating the
derivative of the equation above to zero as:

                                                                18

And then the second
derivative as:

                                                                 19

The yield cutoff value is
obtained as

 and the highest yield cutoff
value across nutrient expressions can be selected to ascertain that minimum
yield target for a high-yield subpopulation will be classified as high yield
whatever the nutrition expression.

2.5.3.3 Establishing the CND Index

After establishing the
CND norms as means and standard deviation of the clr of the nutrient concentration in the maize ear leaf tissue
denoted as

 +

 and

 +

.

The CND index (I) denoted
as

 …

, were calculated from the clr as follows:

,

,

,

,…

    20

the index is defined as the distance of a given
nutrient Xi from its geometric mean
12, which is relative to the distance of the same nutrient from the geometric
mean of the target population (reference population with high yield).

From this point of view above, it is expected that when
CND index is closer to zero, Xi
nutrient is less imbalanced than others in the analysis. Serra et al. (2010a, b) observed CND index
close to zero showed higher nutritional balance.

The CND indices are
standardized 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.

2.5.3.4 CND nutrient imbalance index (NII)

The NII is the CND r2
as recommended by Parent and Dafir (1992) and is given as

                                      21

Its radius, r,
computed from the CND nutrient indices, thus characterizes each specimen. The
sum of d + 1 squared independent, unit-normal variables produces a new
variable having a ?2 distribution with d + 1 degrees of freedom
(Ross, 1987). Because CND indices are independent, unit-normal variables, the
CND r2 values must have a ?2 distribution
function. This is why it is recommended that the highest yield cutoff value
(highest discrimination power) among d + 1 nutrient computations be
retained to calculate the proportion of the low-yield subpopulation below yield
cutoff used as the critical value for the cumulative distribution function. As
defined by equation 19 and 20, the closer to zero that CND indices are, and
thus the CND r2 or ?2 values are, the higher the
probability to obtain a high yield.