st: RE: principal component analysis-creating linear combinations

[Nick Cox <n.j.cox@durham.ac.uk>]

The easiest and best way to create the principal components themselves is use -predict- after -pca-. There is no need for you to do the calculation by typing out coefficients in a linear equation. That is even at best problematic in terms of keeping precision.

The default of -pca- is to use the correlation matrix; that is entirely equivalent to using standardised variables, so that there is absolutely no need to standardise yourself, except possibly as an exercise. 

I wouldn't call the eigenvectors the PCs myself, although there are varying habits on this. 

Nick 
n.j.cox@durham.ac.uk 

James Wu

Suppose we ran pca on four variables, x1, x2, x3, x4 as follows:

. pca  x1 x2 x3 x4, components (3)

Principal components/correlation                  Number of obs    =       659
Number of comp.  =         3
Trace            =         4
Rotation: (unrotated = principal)             Rho              =    0.9550
--------------------------------------------------------------------------
Component    Eigenvalue   Difference         Proportion   Cumulative
-------------+------------------------------------------------------------
Comp1       2.42894      1.67142             0.6072       0.6072
Comp2       .757515      .124084             0.1894       0.7966
Comp3       .633431      .453314             0.1584       0.9550
Comp4       .180117            .             0.0450       1.0000
--------------------------------------------------------------------------
Principal components (eigenvectors)
----------------------------------------------------------
Variable     Comp1     Comp2     Comp3  Unexplained
-------------+------------------------------+-------------
x1    0.3894    0.8726   -0.2945    .00004265
x2    0.4517    0.0966    0.8858     .0003491
x3    0.5733   -0.3179   -0.2218       .09384
x4    0.5619   -0.3580   -0.2817       .08588
----------------------------------------------------------


Now, suppose that you decide to retain the firs two principal
components, and then you want to create two variables that are linear
combinations of the original four variables.

Question1:  Would it be simply to create by multiply the Principal
Components (eigenvectors, columns)  with the orginal variables, say,
Y1=0.3894*x1+0.4517*x2+0.5733*x3+0.5619*x4 and
Y2=0.8726*x1+0.0966*x2-0.3179*x3-0.3580*x4?

Question 2: Assuming that I am correct in creating new variables by
simply multiplying the Principal components (eigenvectors) with the
orginal variables (Question 1),
if these four original variables are in different units of
measurement, then should we standardize the original four variables
(so that each of standardized original variable has mean 0 and std of
1) before computing the multiproducts as in my Question 1?

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