Principal Components and Classification Analysis

In addition to reducing the dimensions of the original space of variables, Principal Components & Classification Analysis (PCCA) can also be used as a classification technique, to highlight the relations among variables and cases. To do this, the variables and the cases are plotted in the space generated by the factor axes. However, the relations among the variables and among the cases cannot be easily seen in the factor space, if its dimension is greater than two. In that case, to have a clearer picture, the projections of the points (variables or cases) must be studied in the two dimensional factor spaces (factor planes), formed by pairs of axes chosen from the set of factor axes.

Graphics associated with variables
The factor coordinates of the variables are the correlations of the variables and the factor axes (if the current analysis is based on the correlation matrix). In a 2D graphical representation, they fall in a circle, called the correlation circle, with the pair of factor axes as its axes. The variables, when projected onto this circle, reveal a lot about themselves. For instance, the further a point is from the origin of the circle, the greater the correlation of the corresponding variable with the factor axes. The variables that are correlated with a particular factor can thus be identified, thereby providing information as to which variables can explain the given factor.

Similarly, the position of the factor coordinate of a variable with respect to the factor axes classifies it into one or the other category. For example, starting with the first factor axis, the variables can be classified into two categories, depending upon which side of the factor axis the corresponding factor coordinates of the variables lie. In other words, the classification of variables is done according to the sign of the factor coordinates. More and more underlying classificatory information can be obtained from the plots of factor coordinates by repeating this exercise for other factor axes.

Graphics associated with cases
For the graphics associated with the cases, a pair of factor-axes is selected from amongst the set of factor axes. The points from the vector space generated by the cases are then projected onto the factor-plane generated by the selected pair of axes. These graphics help in classifying individuals (cases) into categories. The cases are classified according to the position of their corresponding coordinates with respect to the factor axis. By considering different pairs of axes from amongst the computed factors, more classes of cases can be derived.