**2. Some** *ad hoc* **arithmetic procedures for determining an appropriate number of PCs**

#### **2.1 Procedure based on the average of the eigenvalues**

The mean *λ* of the eigenvalues is the sum over the number

$$
\overline{\lambda} = \sum\_{j=1}^{p} \lambda\_j / p. \tag{28}
$$

The sum of the eigenvalues turns out to be equal to the *trace* of the covariance matrix; therefore, the mean eigenvalue is equal to the trace divided by *p:*

One procedure for deciding on the number of PCs to retain is to retain those for which the eigenvalues are greater than average, that is, greater than *λ:* When working in terms of the correlation matrix, this average value is 1. To see this, recall that the correlation matrix is a special case of the covariance matrix, namely, the correlation matrix is the covariance matrix of the standardized variables. It is often preferable to work in terms of the correlation matrix rather than the covariance matrix, to control the effects of different units of measurement and different variances. If a variable has high variance relative to the other variables, the PC will be pulled in the direction of the variable with large variance.

When **S** is taken to be the sample *correlation* matrix, the trace of the matrix is simply *p*, and therefore, the mean *λ* of the eigenvalues is 1.
