Agriculture Reference
In-Depth Information
Fig. 12.1
Cattell scree plot
12.9.4 Principal Component Analysis
3. Obtain the loading for the first principal com-
ponent
P
1
as
The principle followed in this method is to con-
struct a set of new variables called “principal
components” (
P
k
j¼
1
rxixj
P
i¼
1
P
k
j¼
P
i
's) as linear combination of the
given set of variables (mostly correlated) (
X
i
's)
a
1
j
¼
s
¼
1
j:
either
from the original
X
's or
from their
1
rxixj
standardized form, that is,
X
i
X
σ
x
;
; P
1
¼ a
11
x
1
þ a
12
x
2
þ a
13
x
3
þ L þ a
1
k
x
k
P
2
¼ a
21
x
1
þ a
22
x
2
þ a
23
x
3
þ L þ a
2
kx
k
: :
: :
: :
P
k
¼ ak
1
x
1
þ ak
2
x
2
þ ak
3
x
3
þ L þ a
kk
x
k
that is
4.
λ
1
¼
Latent root of
P
1
X
k
j¼
1
l
X
k
j¼
1
a
2
2
2
11
2
12
¼
j
¼
1
j
¼ a
þ a
2
2
þ a
13
þþa
1
k
:
's are called loading and are so chosen
that the principal components (
These
a
5. (
Percent variation absorbs and
accounted for by the
λ
1
/
k
)
100
¼
's) are orthogo-
nal, that is, uncorrelated, and the first principal
component
P
P
1
because
P
1
absorbs and accounts for the max-
imum possible proportion of the total variation of
the set of all
X
k
m¼
1
λ
m
¼ k:
P
2
, absorbs
and accounts for maximum of the remaining
variation in the
X
P
's, the second
, that is,
's, and so on.
The step-by-step procedure for PCA is as
follows:
1. Construct the correlation matrix of the explan-
atory variables.
2. Calculate the row total and column total.
X
Y
P
s
and substitute
X
s
P
s
6. Regress
on
in
to get
the relationship in original
X
-
Y
form.
Example
Using the data for yield
components in Example 12.2, let us demonstrate
the factor analysis and the PCA through SPSS.
12.4.
