*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.

*Cozman Fabio and Cohen Ira*

- Published in print:
- 2006
- Published Online:
- August 2013
- ISBN:
- 9780262033589
- eISBN:
- 9780262255899
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262033589.003.0004
- Subject:
- Computer Science, Machine Learning

This chapter presents a number of conclusions. Firstly, labeled and unlabeled data contribute to a reduction in variance in semi-supervised learning under maximum-likelihood estimation. Secondly, ...
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This chapter presents a number of conclusions. Firstly, labeled and unlabeled data contribute to a reduction in variance in semi-supervised learning under maximum-likelihood estimation. Secondly, when the model is “correct,” maximum-likelihood methods are asymptotically unbiased both with labeled and unlabeled data. Thirdly, when the model is “incorrect,” there may be different asymptotic biases for different values of λ. Asymptotic classification error may also vary with λ—an increase in the number of unlabeled samples may lead to a larger estimation asymptotic bias and to a larger classification error. If the performance obtained from a given set of labeled data is better than the performance with infinitely many unlabeled samples, then at some point the addition of unlabeled data must decrease performance.Less

This chapter presents a number of conclusions. Firstly, labeled and unlabeled data contribute to a reduction in variance in semi-supervised learning under maximum-likelihood estimation. Secondly, when the model is “correct,” maximum-likelihood methods are asymptotically unbiased both with labeled and unlabeled data. Thirdly, when the model is “incorrect,” there may be different asymptotic biases for different values of λ. Asymptotic classification error may also vary with λ—an increase in the number of unlabeled samples may lead to a larger estimation asymptotic bias and to a larger classification error. If the performance obtained from a given set of labeled data is better than the performance with infinitely many unlabeled samples, then at some point the addition of unlabeled data must decrease performance.