Introduction

In the beginning – in 1901 – there was Principal Component Analysis. On our journey through this book we have encountered many different methods for analysing multidimensional data, and many times on this journey, Principal Component Analysis reared its – some might say, ugly – head. About a hundred years since its birth, a renaissance of Principal Component Analysis (PCA) has led to new theoretical and practical advances for high-dimensional data and to SPCA, where S variously refers to simple, supervised and sparse. It seems appropriate, at the end of our journey, to return to where we started and take a fresh look at developments which have revitalised Principal Component Analysis. These include the availability of high-dimensional and functional data and the necessity for dimension reduction and feature selection and new and sparse ways of representing data.

Exciting developments in the analysis of high-dimensional data have been interacting with similar ones in Statistical Learning. It is not clear where analysis of data stops and learning from data starts. An essential part of both is the selection of ‘important’ and ‘relevant’ features or variables.