Graham R. Wood

The bisection method in higher dimensions

Is the familiar bisection method part of some larger scheme? The aim of this paper is to present a natural and useful generalisation of the bisection method to higher dimensions. The algorithm preserves the salient features of the bisection method: it is simple, convergence is assured and linear, and it proceeds via a sequence of brackets whose infinite intersection is the set of points desired. Brackets are unions of similar simplexes. An immediate application is to the global minimisation of a Lipschitz continuous function defined on a compact subset of Euclidean space.

Multidimensional bisection: The performance and the context

Two aspects of the multidimensional bisection algorithms for the global optimisation of Lipschitz continuous functions are investigated. Firstly, for several test functions we examine the numerical performance of the deepest point algorithm and two acceleration procedures. Secondly, we phrase the branch and bound framework of Horst and Tuy in terms of covers, and show the algorithms to be included in this framework. A result of Basso on the convergence of localisations is extended to higher dimensions.

Hoshin Kanri: a systematic approach to breakthrough improvement

A critical ingredient in modern Japanese management is ‘Hoshin Kanri’ or ‘target and means control’. This paper describes the Hoshin Kanri cycle and how it is managed within an organization. Underlying principles are crystallized and two graphics used to sum up the key ideas.

Randomized Block Design

In preceding chapters our measurements have largely arisen through applying experimental treatments to experimental units. For example, we measured the “bulk” of two and three day conditioned samples of wool. When the experimental units are entirely uniform, the only major influence on the measured value is the treatment. In practice, however the experimental units are seldom entirely uniform; for example, the nature of the wool may vary from sample to sample. This variation in the experimental units will in turn influence our measurements, and disguise the variation between treatments, the issue of central interest. Intelligent design, which recognizes the variation between experimental units, can help overcome this problem.

Latin Square Design

In this chapter we introduce the latin square design. This design is used to take account of two sources of variation between the experimental units, in an orthogonal fashion. In practice, the design is not widely used. However, we include it in this book since it illuminates the ideas of orthogonal blocking in a very elegant manner.

Single Population Questions

What is the mean of the single population which we are studying? Is it reasonable that the mean is zero? These are the types of question we deal with in this chapter.

The Statistical Tool Kit

We complement the previous chapter by now summarizing thestatisticalconcepts and techniques which will be used in succeeding chapters. We begin in §1 with the notion of a population, a sample, a random variable and a probability distribution. Properties of combinations of random variables are dealt with in §2. We discuss estimation in §3, and finish in §4 with the definitions of theFand t distributions.

The Geometric Tool Kit

This book is aboutvisualizingthe classical methods of statistical analysis. The critical link between the numbers of a data set and the picture we analyze is the notion of a vector, a visual representation of that set of numbers. This chapter introduces you to the basic ideas of the geometry of vectors. Vectors are defined in §1, then in §2 we demonstrate how to combine them. Angles between vectors are defined in §3, projections of vectors are discussed in §4, and we conclude in §5 with a discussion of Pythagoras’ Theorem. Rest assured that we discuss only those ideas which are necessary for the statistics we do later!

Questions About Two Populations

In this chapter we deal with questions concerning two normal populations. The central question to be answered is “Are the two means equal?” If they are deemed unequal then typically we wish to estimate with some precision the difference between them. Situations of this type are common For example, in designing office furniture it would be of importance to know to what extent the female back is in general shorter than the male back. Furniture designed for each sex should then take into account the difference in back lengths.

Tool Kits At Work

Here we set the scene for the remainder of the book. We begin in §1 with an overview of the scientific method, or question — research — answer process, in which statistical data analysis plays an integral part. In §2 we home in upon the novel contribution of this text, namely a consistent geometric approach to data analysis. While this is a feature which sets this book apart from other texts, we emphasize that throughout we shall view data analysis as just a part of the design and analysis of each research study.