Agriculture Reference
In-Depth Information
discriminant analysis comes into play in taking
decisions. The existence of two or more groups
with respect to several characters/variables
measured at interval or ratio level is presumed.
The groups are mutually exclusive. Given any
object, it can be placed in any one of the groups
presumed. As because the functional form used
to combine the group characteristics towards
classifying an object into a particular class or
group is linear, it is also known as linear discrim-
inant analysis (LDA). The basic assumption on
which the entire LDA is based on is that each and
every group or class belongs to multivariate nor-
mal population. This assumption, though not
necessary, warrants for precise estimation of
probabilities and subsequent test of significance.
Let
12.10 Discriminant Analysis
One of the most important multivariate statisti-
cal tools to distinguish among the individual
elements of a population is the discriminant
analysis. The essence of this analysis lies in
providing specific group to each and every
element of a population based on its difference
from other elements of the population consid-
ering multiple characters measured on all
elements. Discriminant analysis has got various
uses in agricultural and allied sectors, finance,
business, etc. To a breeder in the field of agri-
culture, it is important to know which of the
genetic stocks are prospective with respect to
certain characters of interest in a particular
crop, so that these can be selected and
exploited in future breeding improvement pro-
gram. Thus, the object of such selection is to
place a particular genotype into homogeneous
group or otherwise. On the basis of information
on multiple parameters, a bank manager may
wish to know whether a loan applicant is good
or otherwise. Before launching any product into
the market, a manufacturer may wish to know
the areas where the company should concen-
trate on the basis of certain common
characteristics of the consumers of the product.
The objectives of the entire discriminant analy-
sis are (1) to test the existence or otherwise of
any significant differences among the groups
presumed, (2) to test whether the variables
under consideration are contributing towards
intergroup discrimination, (3) to work out a
linear combination of a variables which maxi-
mize the ratio of the squared differences
between the group means and variance within
the group, and (4) given any object or individ-
ual to assign a particular group or class. It is
essentially a statistical technique to discrimi-
nate the individuals or objects belonging to
two or more groups based on multiple
characters/variables simultaneously.
The idea of discriminant analysis was first
proposed by Fisher in 1936. Discriminant analy-
sis has got various uses in agricultural and allied
sectors, finance, business, etc. In all these cases,
X
k
,
measured on all the individual elements. One
writes linear combination of all these variables
to make
Ψ ¼ α
1
X
1
þ α
2
X
2
þ α
3
X
3
þ α
4
X
4
þþ
α
k
1
X
k
1
þ α
k
X
k
, where there are
k
number of
variables combined linearly,
there be
k
variables,
X
1
,
X
2
,
...
,
α
i
's are called dis-
criminant coefficients, and
is the value of the
discriminant function of a particular individual/
object. The
Ψ
α
i
's are estimated in such a way that
based on
values the ratio of variance between
population to that of the within population is
maximized. The technique is to frame
Ψ
k
sets of
simultaneous equations so that solutions of these
equations provide the estimates of
α
i
's. Without
losing generality, one can measure the variables
from their respective means.
If we consider
the simplest case of
two
populations with
variables, then the discrimi-
nant function can be written in the form
k
Ψ ¼ α
1
X
1
þ α
2
X
2
þ α
3
X
3
þ α
4
X
4
þ
þ α
k
1
X
k
1
þ α
k
X
k
:
Thus, for group 1
Ψ
1
¼ α
1
X
þ α
2
X
þ α
3
X
þ α
4
X
1
1
1
2
1
3
1
4
þ
þ α
k
1
X
þ α
k
X
1
k
1
1
k
;
and for group 2
