njit-etd2003-081 - New Jersey Institute of Technology

109
3.14.1 Basic Concepts of Principal Components Analysis
Suppose that there is a random sample of N observations on X 1 and X2 . For ease of
interpretation the sample mean is subtracted from each observation, thus obtaining
Note that this technique makes the means of x 1 and x 2 equal to zero but does not
alter the sample variances Si and S 2 or the correlation r. The basic idea is to create two
new variables, C 1 and C2 called the principal components. These new variables are
linear functions of x 1 and x 2 and can therefore be written as
It should be noted that for any set of values of the coefficients a11 , a12 , a 21 , a 22 ,
one can introduce the N observed .,j and x 2 and obtain N values of CI and C 2 . The
means and variances of the N values of C 1 and C2 are
where S,2 =VarXi. Equation 3.98 is true because the means of A and x 2 are zero.
The coefficients are chosen to satisfy three requirements:
1. The VarC1 is as large as possible.
2. The N values of C 1 and C2 are uncorrelated.