The next two chapters address classification issues from two varying perspectives. When considering groups of objects in a multivariate data set, two situations can arise. Given a data set containing measurements on individuals, in some cases we want to see if some natural groups or classes of individuals exist, and in other cases, we want to classify the individuals according to a set of existing groups. Cluster analysis develops tools and methods concerning the former case, that is, given a data matrix containing multivariate measurements on a large number of individuals (or objects), the objective is to build some natural sub-groups or clusters of individuals. This is done by grouping individuals that are “similar” according to some appropriate criterion.

In statistics, data sets mostly come in matrix form and the characteristics of the data can be written in terms of matrix operations. Understanding matrix algebra is crucial for investigating the properties of the observed data, see, e.g., Searle (1982), Lütkepohl (1996), Harville (1997, 2001), Seber (2008), or Puntanen, Styan, & Isotalo (2011).

This book is the fourth volume of the Handbook of Computational Statistics. As with the other handbooks in the series, it is a collection of articles on specific aspects of the broad field, written by experts in those areas. The purpose is to provide a survey and summary on each topic, ranging from basic background material through the current frontiers of research. The development of the field of computational statistics has been rather fragmented. We hope that the articles in this handbook series can provide a more unified framework for the field.

For a large range of options such as the American options the boundary conditions of the Black-Scholes differential equation are too complex to solve analytically. Therefore, one relies on numerical price computation. The best known method is to approximate the stock price process by a discrete time stochastic process, or, as in the approach followed by Cox, Ross, Rubinstein, to model the stock price process as a discrete time process from the start. The binomial model is a convenient tool for pricing European options.

We have seen in the previous chapters how very simple graphical devices can help in understanding the structure and dependency of data. The graphical tools were based on either univariate (bivariate) data representations or on “slick” transformations of multivariate information perceivable by the human eye. Most of the tools are extremely useful in a modelling step, but unfortunately, do not give the full picture of the data set. One reason for this is that the graphical tools presented capture only certain dimensions of the data and do not necessarily concentrate on those dimensions or sub-parts of the data under analysis that carry the maximum structural information. In Part III of this book, powerful tools for reducing the dimension of a data set will be presented. In this chapter, as a starting point, simple and basic tools are used to describe dependency. They are constructed from elementary facts of probability theory and introductory statistics (e.g. the covariance and correlation between two variables).

Empirical studies involving economic variables such as price level, real output and nominal interest rates have been shown to exhibit some degree of persistence. Moreover, findings across several asset markets have revealed a high persistence of volatility shocks and that over sufficiently long periods of time the volatility is typically stationary with “mean reverting” behaviour. Such series are reported to be characterised by distinct, but non periodic, cyclical patterns and their behaviour is such that current values are not only influenced by immediate past values but values from previous time periods. The terminology associated with a series with such characteristics is “long memory” or “long range dependence”. If financial time series exhibit persistence or long-memory, then their unconditional probability distribution may not be normal. This has important implications for many areas in finance, especially asset pricing, option pricing, portfolio allocation and risk management.

One of the major tasks of statistics is to obtain information about populations. The set of all elements that are of interest for a statistical analysis is called a population. The population must be defined precisely and comprehensively so that one can immediately determine whether an element belongs to it or not.