I.S. Molchanov. Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley Series in Probability and Statistics. John Wiley & Sons, 1997. Home/Search Document Not in Database Summary Related Articles Check |
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Random Sets and Histograms - Javier Nunez-Garcia And (Correct)
....of the random set. The capacityf5flHflH nal is definedf or sets (insteadf or f milies of sets) This makes its use more convenient than the distribution of the random setitself . Posterior developed random set theories and applications, are based on capacityf unctionals(f or example in [10] [12]) The single point coveragef unctionof a random set X is defined as af33flj: n c X : R d # [0, 1] such that c X (x) PX (x # X) # x # R d . 2) Def (2) defines af35: restriction and is also called a possibility measure [7] Note that the coveragef5BB:fl n is equal to the expectation of ....
I.S. Molchanov. Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley Series in Probability and Statistics. John Wiley & Sons, 1997.
Likelihood Devices in Spatial Statistics - van Zwet (1999) (Correct)
....fraction and the set covariance, which can be easily estimated by their observed image counterparts. The resulting estimators are unbiased, and expressions for the variance can be obtained from Robbins theorem (e.g. Stoyan et al. 1987) Under mild ergodicity assumptions they are strongly consistent (Molchanov, 1997) as the observation window expands to the entire plane. Aggregate functionals such as the contact distribution and pair correlation function are of interest when fitting the Boolean model to a data image. Usually, estimation is hampered by edge e#ects, but minus sampling ideas (Ripley, 1988, ....
I.S. Molchanov (1997). Statistics of the Boolean model for Practitioners and Mathematicians. John Wiley & Sons.
Perfect Simulation in Stochastic Geometry - Kendall, Thönnes (27 citations) (Correct)
.... and mining engineering motivate the modelling of populations of (random) objects using Boolean random sets: a point pattern of germs is generated according to a Poisson point process and at each germ an object or grain is located which is an independent realisation of a random compact set (see [23, 24, 25]) Often experimental data are available which provide further information on the location of grains. One may know that certain locations are covered by grains, one may know certain locations are not covered by grains, one may know the number of grains covering a certain location or one may know ....
Molchanov, I. (1997) Statistics of the Boolean Model for Practitioners and Mathematicians J. Wiley & Sons, Chichester.
Steepest Descent Algorithms in Space of Measures - Molchanov, Zuyev (2000) Self-citation (Molchanov) (Correct)
....Zuyev, 2000b) 3.4. Maximisation of the covered volume in a Boolean model Let again Z = fz 1 ; z 2 ; g be a Poisson point process in X R d with a nite intensity measure . If B r (x) is a ball of radius r centred at x, then = z i 2Z B r (z i ) is called a Boolean model, see (Molchanov, 1997). The set B r (0) is called the typical grain. It is easy to see that Pfx = 2 g = expf (B r (x) g : algo sub.tex; 1 11 2000; 15:52; p.8 Steepest descent for measures 9 By Fubini s theorem, the expected area left uncovered by can be written as f( Z X Pfx = 2 gdx = Z X expf (B r ....
Molchanov, I. S.: 1997, Statistics of the Boolean Model for Practitioners and Mathematicians. Chichester: Wiley.
Design of Inhomogeneous Materials With Given Global Properties - Molchanov, Chiu, Zuyev (2000) Self-citation (Molchanov) (Correct)
....which are not necessarily isotropic. A random set 0 having the same distribution as all i , i 1, is called the typical grain. The ensemble of the fully penetrable and uncorrelated grains = x i 2 (x i i ) where x i i = fx i y : y 2 i g, is called a Boolean model [11, 12]. If the underlying Poisson process is homogeneous, then we also say that is homogeneous. The total mass of , D) determines the mean total number of grains. Consider a Boolean model inside a region D. The mean d dimensional phase 2 volume (that corresponds to length if d = 1 or to area if ....
I. S. Molchanov, Statistics of the Boolean Model for Practitioners and Mathematicians (Wiley, Chichester, 1997).
Steepest Descent Algorithms in Space of Measures - Molchanov, Zuyev (2000) Self-citation (Molchanov) (Correct)
....0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 . Figure 2: A sample of points superimposed with contour plots of the corresponding P mean for the total mass a; a) a = 1; b) a = 5; c) a = 25; d) a = 100. 14 is called a Boolean model, see [14]. The set B r (0) is called the typical grain. Note that all arguments below can be easily generalised for random typical grain with a rather general shape and random size. It is easy to see that Pfx = 2 g = expf (B r (x) g : By Fubini s theorem, the expected area left uncovered by can be ....
I. S. Molchanov. Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester, 1997.
Design of Inhomogeneous Materials With Given Global Properties - Molchanov, Chiu, Zuyev (2000) Self-citation (Molchanov) (Correct)
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I. S. Molchanov. Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester, 1997.
Maximum Likelihood Estimation for the Bombing Model - van Lieshout, van Zwet (Correct)
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Molchanov, I.S. (1997), Statistics of the Boolean Model for Practitioners and Mathematicians , Chichester: Wiley. 21
Maximum Likelihood Estimation for the Bombing Model - van Lieshout, van Zwet (2000) (Correct)
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Molchanov, I.S. (1997), Statistics of the Boolean Model for Practitioners and Mathematicians, Chichester: Wiley.
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