18. Highly Interactive, Computationally Intensive Techniques

It is generally accepted that training in statistics must include some exposure to the mechanics of computational statistics. This exposure to computational methods is of an essential nature when we consider extremely high dimensional data. Computer aided techniques can help us discover dependencies in high dimensions without complicated mathematical tools. A draftman's plot (i.e., a matrix of pairwise scatterplots like in Figure 1.14) may lead us immediately to a theoretical hypothesis (on a lower dimensional space) about the relationship of the variables. Computer aided techniques are therefore at the heart of multivariate statistical analysis.

In this chapter we first present the concept of Simplicial Depth--a multivariate extension of the data depth concept of Section 1.1. We then present Projection Pursuit--a semiparametric technique which is based on a one-dimensional, flexible regression or on the idea of density smoothing applied to PCA type projections. A similar model is underlying the Sliced Inverse Regression (SIR) technique which we discuss in Section 18.3.