Research Methodology - Dr. Krishan K. Pandey

Multivariate Analysis Techniques 331
Correlation Matrix, R
Variables
X 1 X 2 X 3 …. X k
X 1 r 11 r 12 r 13 …. r 1k
X 2 r 21 r 22 r 23 …. r 3k
Variables X 3 r 31 r 32 r 33 …. r 3k
. . . .
. . . .
X k r k1 r k2 r k3 …. r kk
The main diagonal spaces include unities since such elements are self-correlations. The
correlation matrix happens to be a symmetrical matrix.
(ii) Presuming the correlation matrix to be positive manifold (if this is not so, then reflections as
mentioned in case of centroid method must be made), the first step is to obtain the sum of
coefficients in each column, including the diagonal element. The vector of column sums is
referred to as U a1 and when U a1 is normalized, we call it V a1 . This is done by squaring and
summing the column sums in U a1 and then dividing each element in U a1 by the square root
of the sum of squares (which may be termed as normalizing factor). Then elements in V a1
are accumulatively multiplied by the first row of R to obtain the first element in a new
vector U a2 . For instance, in multiplying V a1 by the first row of R, the first element in V a1
would be multiplied by the r 11 value and this would be added to the product of the second
element in V a1 multiplied by the r 12 value, which would be added to the product of third
element in V a1 multiplied by the r 13 value, and so on for all the corresponding elements in V a1
and the first row of R. To obtain the second element of U a2 , the same process would be
repeated i.e., the elements in V a1 are accumulatively multiplied by the 2nd row of R. The
same process would be repeated for each row of R and the result would be a new vector
U a2 . Then U a2 would be normalized to obtain V a2 . One would then compare V a1 and V a2 . If
they are nearly identical, then convergence is said to have occurred (If convergence does
not occur, one should go on using these trial vectors again and again till convergence
occurs). Suppose the convergence occurs when we work out V a8 in which case V a7 will be
taken as V a (the characteristic vector) which can be converted into loadings on the first
principal component when we multiply the said vector (i.e., each element of V a ) by the
square root of the number we obtain for normalizing U a8 .
(iii) To obtain factor B, one seeks solutions for V b , and the actual factor loadings for second
component factor, B. The same procedures are used as we had adopted for finding the first
factor, except that one operates off the first residual matrix, R 1 rather than the original
correlation matrix R (We operate on R 1 in just the same way as we did in case of centroid
method stated earlier).
(iv) This very procedure is repeated over and over again to obtain the successive PC factors
(viz. C, D, etc.).