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Gibbs particle processes
Petráková, Martina ; Beneš, Viktor (advisor) ; Pawlas, Zbyněk (referee)
We consider a recent result for the existence of infinite-volume marked Gibbs point processes and try to apply it to geometric models. At first, we reformulate a problematic assumption of the considered existence result and check that the theorem still holds. We use this result for the family of Gibbs facet processes (a special case of particle processes) and prove the existence for repulsive interactions. We find counterexamples for the process with attractive interactions and prove that the finite-volume Gibbs facet process in R2 does not exist in this case. We also study the class of Gibbs-Laguerre tessellations of R2 . We cannot use the mentioned existence result in general, but we are able to prove the existence of an infinite-volume Gibbs-Laguerre process with particular energy function, under the assumption that we almost surely see a point. 1
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Summary statistics of spatial point processes
Mirtes, Lukáš ; Lechnerová, Radka (advisor) ; Pawlas, Zbyněk (referee)
The paper presents introduction to spatial point processes and their characteristics. The reader is familiar with Poisson point process, which plays fundamental role in the theory of point processes. Basic properties and summary statistics are introruced with nonparametric estimations. The work is concluded by example of the point process including estimations of some charakteristics.
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Testy dobré shody v bodových procesech.
Perevorskyy, Taras ; Mrkvička, Tomáš (advisor) ; Pawlas, Zbyněk (referee)
The main objective of this thesis is to carry out a comparative analysis of the now available point process model tests. A script file has been written implementing the different point process test methods in the R program. With the help of this software, a simulation study has been conducted resulting in over 500 000 tests. A great attention has been paid to the only recently published envelopes test. Except for envelopes test other two simulation tests have been considered along with the J-test and the quadrant-count test. As test subjects 5 different process families were chosen. The obtained results have been carefully analyzed and the interesting patterns have been documented.
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Structural properties of graphs---probabilistic and deterministic point of view
Hladký, Jan ; Šámal, Robert (advisor) ; Pawlas, Zbyněk (referee)
We study bipartite subgraphs of a random cubic graph in the thesis. We show, that an edge-maximum bipartite subgraph of a random cubic graph on n vertices has asymptotically almost surely less then 3 2 · 0.9351n edges. We also show that the number of vertices of a vertex-maximum induced bipartite subgraph of a random cubic graph lies within interval [0.75n; 0.9082n]. To obtain the lower bound we design a randomized algorithm for finding a large induced bipartite subgraph of a random cubic graph. We discuss consequences of the results for graph homomorphisms, namely for Pentagon Conjecture posed by Nešetřil.
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Point processes on linear networks
Moravec, Jan ; Prokešová, Michaela (advisor) ; Pawlas, Zbyněk (referee)
The central theme of this thesis is the theory of point processes on linear net- works, in particular two kinds of the network K-function. The first part is devoted to the theory of stationary point processes in the plane, including the K-function and its estimator. The second part is concerned with the theory of point proces- ses on linear networks. There is defined the Okabe-Yamada network K -function and its estimator, the geometrically corrected network K-function, including its estimator, and there are explained their theoretical properties. In the third part we examine the ability of these two kinds of the network K-function to detect clustering or regularity in point processes on linear networks. There is explained the envelope test, the refined envelope test and the deviation tests. The software environment R with library spatstat is used for simulations.
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Segment point processes
Honzl, Ondřej ; Pawlas, Zbyněk (advisor) ; Beneš, Viktor (referee)
Naz-ev prace: Bodove proeesy usecek Aut.or: Ondrej Ilonzl Katedra: Katedra. pravdepodobnosti a matematicke statist,iky Vodonci bakalafske praee: RNDr. Zbynek Pawlas, Ph.D. e-mail vedouciho: zbynek.pawlas'O'mfl'.cmn.cz Abstrakt: Prace obsahnje strucny uvod do teoric bodovych procesii na nplnem se- parabilnim lokalne koutpakt iiiiu metrickem prost.oru. Hamcove je ziuinen si>ccii'ilni I>fipad st,acionariiilio ])roc:csu kouipaktiiich ninozin. Dale sc jiran1 vice1 /aincfujo na, Poissonuv prort\ usccek sc /nainyni ro/dcMrnim typickrho /rnu. V roviniirin pfi])aclo pak ukazujc n'i/nr odliady dolkovr int.i'ii/ily I'oissonova jiroccsu usccck, kU're jsou drfinovany na /aklade udaju /iskanych v okuc poxorovani. Hlavnim zajinoin prace st1 stava porovnavani tcclito odhadu die jujich rozpt.ylu. Cilem to- hoto sroviiavani ina byi stanovoni niezc \vlikoHti okna, klcra. fika, dokud jc Icpsi pou/it slozitojsi odhad a odkdy je naopak ro/ninno pouzit odliad. jclioz vvpocot jo snazsi, ale kl.cry pft'ilpoklada., zc inatiie vice iniormaci u po/orovanein JJI'OCCHU. Klicova slova: I'oissonuv i>roces. hodovy proces usccek, odhad delkove intonzit.y Title: Segment point Autlior: Ondrej Ilonzl Deijartuient: DepartiiieiiL ol'Prol)a.l>ilit.y and Mathcinalical Statistics Supervisor: KNDr. Zbyuek Pawlas, Ph.I). Supervisor's (^niail address:...
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