*Wilfrid S. Kendall and Ilya Molchanov (eds)*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.001.0001
- Subject:
- Mathematics, Geometry / Topology

Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, ...
More

Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds of applied science, from image analysis through to materials science. Recently, still more stimulus has arisen from exciting new links with rapidly developing areas of mathematics, from fractals through percolation theory to randomized allocation schemes. Coupled with many ongoing developments arising from all sorts of applications, the area is changing and developing rapidly. This book is intended to lay foundations for future research directions by collecting together seventeen chapters contributed by leading researchers in the field, both theoreticians and people involved in applications, surveying these new developments both in theory and in applications. It will introduce and lay foundations for appreciating the fresh perspectives, new ideas, and interdisciplinary connections now arising from stochastic geometry and from other areas of mathematics now connecting to this area.Less

Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, their probability theory, and the challenging problems raised by their statistical analysis. It has grown rapidly in response to challenges in all kinds of applied science, from image analysis through to materials science. Recently, still more stimulus has arisen from exciting new links with rapidly developing areas of mathematics, from fractals through percolation theory to randomized allocation schemes. Coupled with many ongoing developments arising from all sorts of applications, the area is changing and developing rapidly. This book is intended to lay foundations for future research directions by collecting together seventeen chapters contributed by leading researchers in the field, both theoreticians and people involved in applications, surveying these new developments both in theory and in applications. It will introduce and lay foundations for appreciating the fresh perspectives, new ideas, and interdisciplinary connections now arising from stochastic geometry and from other areas of mathematics now connecting to this area.

*Cascos Ignacio*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0012
- Subject:
- Mathematics, Geometry / Topology

This chapter presents several ways to measure the degree of centrality of a point with respect to a multivariate probability distribution or a data cloud. Such degree of centrality is called depth, ...
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This chapter presents several ways to measure the degree of centrality of a point with respect to a multivariate probability distribution or a data cloud. Such degree of centrality is called depth, and it can be used to extend a wide range of univariate techniques that are based on the natural order on the real line to the multivariate setting.Less

This chapter presents several ways to measure the degree of centrality of a point with respect to a multivariate probability distribution or a data cloud. Such degree of centrality is called *depth*, and it can be used to extend a wide range of univariate techniques that are based on the natural order on the real line to the multivariate setting.

*Rolf Schneider and Wolfgang Weil*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0001
- Subject:
- Mathematics, Geometry / Topology

The aim of this chapter is to introduce the basic tools and structures of stochastic geometry and thus to lay the foundations for much of the book. Before this, a brief historic account will reflect ...
More

The aim of this chapter is to introduce the basic tools and structures of stochastic geometry and thus to lay the foundations for much of the book. Before this, a brief historic account will reflect the development from elementary geometric probabilities over heuristic principles in applications to the advanced models employed in modern stochastic geometry. After the basic geometric and stochastic concepts have been presented, their interplay will be demonstrated by typical examples.Less

The aim of this chapter is to introduce the basic tools and structures of stochastic geometry and thus to lay the foundations for much of the book. Before this, a brief historic account will reflect the development from elementary geometric probabilities over heuristic principles in applications to the advanced models employed in modern stochastic geometry. After the basic geometric and stochastic concepts have been presented, their interplay will be demonstrated by typical examples.