20.500.12556/RUL-107495 Hawkesovi procesi magistrsko delo Hawkes processes Predstavimo analitično definicijo Hawkesovega procesa in pripadajoče pojme: pogojna intenzivnost, jedrna funkcija in količnik razvejitve. Izpeljani so pogoji za asimptotsko stacionarnost in limitno intenzivnost ter odsotnost eksplozije in izumrtje. Pogled na Hawkesove procese kot na rojstne procese z imigracijami omogoča izpeljavo porazdelitve velikosti družin, števila družin, porazdelitve časa obstoja družine ter rodovnega funkcionala procesa. S pomočjo kompenzatorja procesa lahko transformiramo Hawkesov proces v Poissonovega, kar je uporabno pri analizi slučajnega vzorca. Implementiramo različne algoritme za simulacijo: Ogatov algoritem redčenja, paralelna simulacija generacij ter popolna simulacija šibko stacionarnega Hawkesovega procesa. Definiramo in simuliramo tudi večrazsežni Hawkesov proces ter predstavimo nekaj primerov uporabe Hawkesovih procesov. We study the Hawkes process and its main components: conditional intensity, kernel function and branching ratio, deriving the conditions for asymptotic stationarity, extinction and explosion. The interpretation of Hawkes processes as cluster processes or immigration-birth cluster processes is explained. We present some results about the number and lenght of clusters, using the concept of the probability generating functional. By using the compensator and the random time change theorem, a Hawkes process can be transformed into a homogeneous Poisson process, which can be used for goodness-of-fit test. The various algorithms for generating Hawkes process are presented: the thinning algorithm by Ogata, the clustering algorithm and perfect simulation of weakly stationary Hawkes process. The multivariate Hawkes processes are introduced and its corresponding simulation algorithm is implemented. In conclusion, we present some applications of Hawkes processes in research. Hawkesov proces slučajni procesi točkasti procesi simulacija Galton-Watsonov proces Poissonov proces kompenzator pogojna intenzivnost Hawkes process stochastic processes point processes simulation Galton-Watson process Poisson process compensator conditional intensity true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2019-04-20 07:45:02 2019-04-20 07:45:10 2024-05-30 12:13:28 0000-00-00 00:00:00 2019 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.2 95951 18627929 651.pdf 651.pdf 1 9470D311738AE51B4F4310EBB15ECC9B a2ce2f0a46c9ad0e2c2e15cccd8be445da85927cf8bfc948f12c9923c4cc6225 71ac30be-a1b6-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=118324 Fakulteta za matematiko in fiziko 0 0 0