**1.1 Introduction**

Sometimes, researchers know how many principal components (PCs) they need. For example, to construct an optimal scatterplot, the scores of the sample on the first two principal components will be used to obtain an optimal plot. For an optimal threedimensional scatterplot, the scores on the first three principal components will be used. In many applications, however, the researchers will question how many principal components they need. This chapter discusses the application of various methods to the problem of reduction of dimensionality, in the sense of choosing an adequate

number of principal components to retain to represent a dataset. The methods discussed include *ad hoc* methods, likelihood-based methods, and model selection criteria (MSCs), especially Akaike's information criterion (AIC) and Bayesian information criterion (BIC). This chapter applies the concepts of [1, 2] to this particular problem.
