**2.2 An** *ad hoc* **arithmetic procedure based on retaining a prescribed proportion of the total variance**

Another *ad hoc* procedure is to retain a number of PCs sufficient to account for a prescribed proportion, say, 90% of the total variance, that total variance being trace **<sup>S</sup>** <sup>¼</sup> <sup>P</sup>*<sup>p</sup> <sup>j</sup>*¼<sup>1</sup> *<sup>λ</sup> <sup>j</sup>:* The Figure 90% is of course somewhat arbitrary, so it might be good to have some somewhat more objective criteria based on the pattern of the eigenvalues.

#### **2.3 Procedure based on the decrease of the eigenvalues**

Another procedure—a graphical procedure—is to plot *λ*1, *λ*2, … , *λ<sup>p</sup>* against 1, 2, … , *p:* The *λ*s are in decreasing order, so one then looks for a dropoff—an elbow in the curve and retains a number of PCs corresponding to the point before the leveling off of the curve, if it does indeed take an elbow shape. Such a plot, of the eigenvalues versus 1, 2, … , *p*, is called a *scree* plot, "scree" being the debris at the foot of a glacier (or, more generally, a collection of broken rock fragments at the base of crags, mountain cliffs, volcanoes, or valley shoulders).
