*Pierre Calka*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0005
- Subject:
- Mathematics, Geometry / Topology

Random tessellations and cellular structures occur in many domains of application, such as astrophysics, ecology, telecommunications, biochemistry and naturally cellular biology (see Stoyan, Kendall ...
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Random tessellations and cellular structures occur in many domains of application, such as astrophysics, ecology, telecommunications, biochemistry and naturally cellular biology (see Stoyan, Kendall and Mecke 1987 or Okabe, Boots, Sugihara and Chiu 2000 for complete surveys). The theoretical study of these objects was initiated in the second half of the twentieth century by D. G. Kendall, J. L. Meijering, E. N. Gilbert and R. E. Miles, notably. Two isotropic and stationary models have emerged as the most basic and useful: the Poisson hyperplane tessellation and the Poisson–Voronoi tessellation. Since then, a large majority of questions raised about random tessellations have concerned statistics of the population of cells (‘how many cells are triangles in the plane?’, ‘how many cells have a volume greater than one?’) or properties of a specific cell (typically the one containing the origin). Two types of results are presented below: exact distributional calculations and asymptotic estimations. In the first part, we describe the two basic constructions of random tessellations (i.e. by throwing random hyperplanes or by constructing Voronoi cells around random nuclei) and we introduce the fundamental notion of typical cell of a stationary tessellation. The second part is devoted to the presentation of exact distributional results on basic geometrical characteristics (number of hyperfaces, typical k‐face, etc.). The following part concerns asymptotic properties of the cells. It concentrates in particular on the well‐known D. G. Kendall conjecture which states that large planar cells in a Poisson line tessellation are close to the circular shape. In the last part, we present some recent models of iterated tessellations which appear naturally in applied fields (study of crack structures, telecommunications). Intentionally, this chapter does not contain an exhaustive presentation of all the models of random tessellations existing in the literature (in particular, dynamical constructions such as Johnson‐Mehl tessellations will be omitted). The aim of the text below is to provide a selective view of recent selected methods and results on a few specific models.Less

Random tessellations and cellular structures occur in many domains of application, such as astrophysics, ecology, telecommunications, biochemistry and naturally cellular biology (see Stoyan, Kendall and Mecke 1987 or Okabe, Boots, Sugihara and Chiu 2000 for complete surveys). The theoretical study of these objects was initiated in the second half of the twentieth century by D. G. Kendall, J. L. Meijering, E. N. Gilbert and R. E. Miles, notably. Two isotropic and stationary models have emerged as the most basic and useful: the Poisson hyperplane tessellation and the Poisson–Voronoi tessellation. Since then, a large majority of questions raised about random tessellations have concerned statistics of the population of cells (‘how many cells are triangles in the plane?’, ‘how many cells have a volume greater than one?’) or properties of a specific cell (typically the one containing the origin). Two types of results are presented below: exact distributional calculations and asymptotic estimations. In the first part, we describe the two basic constructions of random tessellations (i.e. by throwing random hyperplanes or by constructing Voronoi cells around random nuclei) and we introduce the fundamental notion of typical cell of a stationary tessellation. The second part is devoted to the presentation of exact distributional results on basic geometrical characteristics (number of hyperfaces, typical k‐face, etc.). The following part concerns asymptotic properties of the cells. It concentrates in particular on the well‐known D. G. Kendall conjecture which states that large planar cells in a Poisson line tessellation are close to the circular shape. In the last part, we present some recent models of iterated tessellations which appear naturally in applied fields (study of crack structures, telecommunications). Intentionally, this chapter does not contain an exhaustive presentation of all the models of random tessellations existing in the literature (in particular, dynamical constructions such as Johnson‐Mehl tessellations will be omitted). The aim of the text below is to provide a selective view of recent selected methods and results on a few specific models.

*Sergei Zuyev*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0016
- Subject:
- Mathematics, Geometry / Topology

Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and ...
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Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and analysis of modern telecommunications systems, in which spatial arrangement is typically a crucial consideration in their performance evaluation, optimization or future development. In this survey we aim to summarize the main stochastic geometry models and tools currently used in studying modern telecommunications. We outline specifics of wired, wireless fixed and ad hoc systems and show how stochastic geometry modelling helps in their analysis and optimization. Point and line processes, Palm theory, shot‐noise processes, random tessellations, Boolean models, percolation, random graphs and networks, spatial statistics and optimization: this is a far from exhaustive list of techniques used in studying contemporary telecommunications systems and which we shall briefly discuss.Less

Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and analysis of modern telecommunications systems, in which spatial arrangement is typically a crucial consideration in their performance evaluation, optimization or future development. In this survey we aim to summarize the main stochastic geometry models and tools currently used in studying modern telecommunications. We outline specifics of wired, wireless fixed and ad hoc systems and show how stochastic geometry modelling helps in their analysis and optimization. Point and line processes, Palm theory, shot‐noise processes, random tessellations, Boolean models, percolation, random graphs and networks, spatial statistics and optimization: this is a far from exhaustive list of techniques used in studying contemporary telecommunications systems and which we shall briefly discuss.

*David G. Hankin, Michael S. Mohr, and Ken B. Newman*

- Published in print:
- 2019
- Published Online:
- December 2019
- ISBN:
- 9780198815792
- eISBN:
- 9780191853463
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198815792.003.0012
- Subject:
- Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies, Ecology

In many ecological and natural resource settings, there may be a high degree of spatial structure or pattern to the distribution of target variable values across the landscape. For example, the ...
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In many ecological and natural resource settings, there may be a high degree of spatial structure or pattern to the distribution of target variable values across the landscape. For example, the number of trees per hectare killed by a bark beetle infestation may be exceptionally high in one region of a national forest and near zero elsewhere. In such circumstances it may be highly desirable or even required that a sample survey directed at estimation of total tree mortality across a forest be based on selection of random locations that have good spatial balance, i.e., locations are well spread over the landscape with relatively even distances between them. A simple random sample cannot guarantee good spatial balance. We present two methods that have been proposed for selection of spatially balanced samples: GRTS (Generalized Random Tessellation Stratified Sampling) and BAS (Balanced Acceptance Sampling). Selection of samples using the GRTS approach involves a complicated series of sequential steps that allows generation of spatially balanced samples selected from finite populations or from infinite study areas. Selection of samples using BAS relies on the Halton sequence, is conceptually simpler, and produces samples that generally have better spatial balance than those produced by GRTS. Both approaches rely on use of software that is available in the R statistical/programming environment. Estimation relies on the Horvitz–Thompson estimator. Illustrative examples of running the SPSURVEY software package (used for GRTS) and links to the SDraw package (used for BAS) are provided at http://global.oup.com/uk/companion/hankin.Less

In many ecological and natural resource settings, there may be a high degree of spatial structure or pattern to the distribution of target variable values across the landscape. For example, the number of trees per hectare killed by a bark beetle infestation may be exceptionally high in one region of a national forest and near zero elsewhere. In such circumstances it may be highly desirable or even required that a sample survey directed at estimation of total tree mortality across a forest be based on selection of random locations that have good *spatial balance*, i.e., locations are well spread over the landscape with relatively even distances between them. A simple random sample cannot guarantee good spatial balance. We present two methods that have been proposed for selection of spatially balanced samples: GRTS (Generalized Random Tessellation Stratified Sampling) and BAS (Balanced Acceptance Sampling). Selection of samples using the GRTS approach involves a complicated series of sequential steps that allows generation of spatially balanced samples selected from finite populations or from infinite study areas. Selection of samples using BAS relies on the Halton sequence, is conceptually simpler, and produces samples that generally have better spatial balance than those produced by GRTS. Both approaches rely on use of software that is available in the R statistical/programming environment. Estimation relies on the Horvitz–Thompson estimator. Illustrative examples of running the SPSURVEY software package (used for GRTS) and links to the SDraw package (used for BAS) are provided at http://global.oup.com/uk/companion/hankin.