Stochastic Geometry and its Applications

Chiu, Sung Nok; Stoyan, Dietrich; Kendall, Wilfrid S.; Mecke, Joseph

doi:10.1002/9781118658222

Cited by 1,907 publications

(57 citation statements)

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“…Therefore, although the martensitic transformation might not entirely fulfill the JMAK assumptions [20][21][22] , their equations will be used here as reasonable approximations. See Nok Chiu et al 24 for a more in depth mathematical discussion of usage of Boolean models in practical situations. Admittedly, the results therefrom derived here will need a careful evaluation.…”

Section: Formal Description Of Martensite Microstructural Pathmentioning

confidence: 99%

“…In the present work, we apply the "extended space" concept to characterize the martensite transformation in the absence of interaction with other units to relax the shape strain. Recently, the JMAK type problem has received rigorous mathematical treatment falling within the class of the so-called "birth-and-growth" processes 23,24 . The martensite product-units exhibit well defined morphologies and crystallographic habit, so that shape is preserved.…”

Section: Formal Description Of Martensite Microstructural Pathmentioning

confidence: 99%

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Microstructural Path Analysis of Martensite Dimensions in FeNiC and FeC Alloys

Guimarães

Rios

2015

Mat. Res.

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Section: Formal Description Of Martensite Microstructural Pathmentioning

confidence: 99%

Section: Formal Description Of Martensite Microstructural Pathmentioning

confidence: 99%

Microstructural Path Analysis of Martensite Dimensions in FeNiC and FeC Alloys

Guimarães

Rios

2015

Mat. Res.

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“…Since Boolean models in general have been paid attention in many respects there exist many results on the statistics of Boolean models in the literature, in particular several special methods of parameter estimation in Boolean models have been established and investigated (Heinrich, 1993;Molchanov, 1997;Chiu et al, 2013).…”

Section: D Boolean Model With Discsmentioning

confidence: 99%

“…It is, however, relatively easy -though still computationally demanding and afflicted with substantial random error -to approximately compute expected values of empirical characteristics through Monte Carlo simulations by doing a random simulation of the process and computing characteristics from the simulation result. Such empirical characteristics can be any type of descriptive statistic, for instance, for a random closed set, the densities of the Minkowski functionals like volume fraction or specific surface, the covariance and certain contact distribution functions (Schneider and Weil, 2008;Ohser and Schladitz, 2009;Chiu et al, 2013), or, for a random point pattern, the intensity (function), the pair correlation function and the nearest neighbour distance distribution function (Møller and Waagepetersen, 2003;Illian et al, 2008), or any other sort of summary statistic maybe tailored specifically for the given estimation problem. In particular, this includes those cases where a spatial random structure is practically observable only via planar sections or projections and, hence, only statistics based on sections or projections are available.…”

Section: Introductionmentioning

confidence: 99%

A Quasi-Likelihood Approach to Parameter Estimation for Simulatable Statistical Models

2014

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This paper introduces a parameter estimation method for a general class of statistical models. The method exclusively relies on the possibility to conduct simulations for the construction of interpolation-based metamodels of informative empirical characteristics and some subjectively chosen correlation structure of the underlying spatial random process. In the absence of likelihood functions for such statistical models, which is often the case in stochastic geometric modelling, the idea is to follow a quasi-likelihood (QL) approach to construct an optimal estimating function surrogate based on a set of interpolated summary statistics. Solving these estimating equations one can account for both the random errors due to simulations and the uncertainty about the meta-models. Thus, putting the QL approach to parameter estimation into a stochastic simulation setting the proposed method essentially consists of finding roots to a sequence of approximating quasiscore functions. As a simple demonstrating example, the proposed method is applied to a special parameter estimation problem of a planar Boolean model with discs. Here, the quasi-score function has a half-analytical, numerically tractable representation and allows for the comparison of the model parameter estimates found by the simulation-based method and obtained from solving the exact quasi-score equations.

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“…Its distribution Q is the mark distribution of Ψ on the space K of compact sets in R 2 (Stoyan et al, 1995, p. 216;Chiu et al, 2013).…”

Section: The Boolean Modelmentioning

confidence: 99%

Inclusion Ratio Based Estimator for the Mean Length of the Boolean Line Segment Model With an Application to Nanocrystalline Cellulose

Niilo-Rämä¹,

Kärkkäinen²,

Gasbarra³

et al. 2014

Image Anal Stereol

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A novel estimator for estimating the mean length of fibres is proposed for censored data observed in square shaped windows. Instead of observing the fibre lengths, we observe the ratio between the intensity estimates of minus-sampling and plus-sampling. It is well-known that both intensity estimators are biased. In the current work, we derive the ratio of these biases as a function of the mean length assuming a Boolean line segment model with exponentially distributed lengths and uniformly distributed directions. Having the observed ratio of the intensity estimators, the inverse of the derived function is suggested as a new estimator for the mean length. For this estimator, an approximation of its variance is derived. The accuracies of the approximations are evaluated by means of simulation experiments. The novel method is compared to other methods and applied to real-world industrial data from nanocellulose crystalline.

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Stochastic Geometry and its Applications

Cited by 1,907 publications

References 0 publications

Microstructural Path Analysis of Martensite Dimensions in FeNiC and FeC Alloys

Microstructural Path Analysis of Martensite Dimensions in FeNiC and FeC Alloys

A Quasi-Likelihood Approach to Parameter Estimation for Simulatable Statistical Models

Inclusion Ratio Based Estimator for the Mean Length of the Boolean Line Segment Model With an Application to Nanocrystalline Cellulose

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