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- Point_process_operation abstract "In probability and statistics, a point process operation or point process transformation is a type of mathematical operation performed on a random object known as a point process, which are often used as mathematical models of phenomena that can be represented as points randomly located in space. These operations can be purely random, deterministic or both, and are used to construct new point processes, which can be then also used as mathematical models. The operations may include removing or thinning points from a point process, combining or superimposing multiple point processes into one point process or transforming the underlying space of the point process into another space. Point process operations and the resulting point processes are used in the theory of point processes and related fields such as stochastic geometry and spatial statistics.One point process that gives particularly convenient results under random point process operations is the Poisson point process, The Poisson point process often exhibits a type of mathematical closure such that when a point process operation is applied to some Poisson point process, then provided some conditions on the point process operation, the resulting process will be often another Poisson point process operation, hence it is often used as a mathematical model.Point process operations have been studied in the mathematical limit as the number of random point process operations applied approaches infinity. This had led to convergence theorems of point process operations, which have their origins in the pioneering work of Conny Palm in 1940s and later Aleksandr Khinchin in the 1950s and 1960s who both studied point processes on the real line, in the context of studying the arrival of phone calls and queueing theory in general. Provided that the original point process and the point process operation meet certain mathematical conditions, then as point process operations are applied to the process, then often the resulting point process will behave stochastically more like a Poisson point process if it has a non-random mean measure, which gives the average number of points of the point process located in some region. In other words, in the limit as the number of operations applied approaches infinity, the point process will converge in distribution (or weakly) to a Poisson point process or, if its is measure measure is a random measure, to a Cox point process. Convergence results, such as the Palm-Khinchin theorem for renewal processes, are then also used to justify the use of the Poisson point process as a mathematical of various phenomena.".
- Point_process_operation wikiPageID "40766565".
- Point_process_operation wikiPageLength "12808".
- Point_process_operation wikiPageOutDegree "40".
- Point_process_operation wikiPageRevisionID "694160391".
- Point_process_operation wikiPageWikiLink Abstraction_(mathematics).
- Point_process_operation wikiPageWikiLink Aleksandr_Khinchin.
- Point_process_operation wikiPageWikiLink Borel_set.
- Point_process_operation wikiPageWikiLink Cartesian_coordinate_system.
- Point_process_operation wikiPageWikiLink Category:Point_processes.
- Point_process_operation wikiPageWikiLink Category:Spatial_processes.
- Point_process_operation wikiPageWikiLink Conny_Palm.
- Point_process_operation wikiPageWikiLink Convergence_theorems.
- Point_process_operation wikiPageWikiLink Countable_set.
- Point_process_operation wikiPageWikiLink Cox_process.
- Point_process_operation wikiPageWikiLink Determinism.
- Point_process_operation wikiPageWikiLink Independence_(probability_theory).
- Point_process_operation wikiPageWikiLink Independent_and_identically_distributed_random_variables.
- Point_process_operation wikiPageWikiLink Limit_(mathematics).
- Point_process_operation wikiPageWikiLink Mathematical_model.
- Point_process_operation wikiPageWikiLink Moment_measure.
- Point_process_operation wikiPageWikiLink Operation_(mathematics).
- Point_process_operation wikiPageWikiLink Point_(geometry).
- Point_process_operation wikiPageWikiLink Point_process.
- Point_process_operation wikiPageWikiLink Point_process_notation.
- Point_process_operation wikiPageWikiLink Poisson_point_process.
- Point_process_operation wikiPageWikiLink Polar_coordinate_system.
- Point_process_operation wikiPageWikiLink Probability.
- Point_process_operation wikiPageWikiLink Probability_distribution.
- Point_process_operation wikiPageWikiLink Queueing_theory.
- Point_process_operation wikiPageWikiLink Random_measure.
- Point_process_operation wikiPageWikiLink Randomness.
- Point_process_operation wikiPageWikiLink Set_(mathematics).
- Point_process_operation wikiPageWikiLink Space_(mathematics).
- Point_process_operation wikiPageWikiLink Spatial_analysis.
- Point_process_operation wikiPageWikiLink Statistics.
- Point_process_operation wikiPageWikiLink Stochastic_geometry.
- Point_process_operation wikiPageWikiLink Transformation_(function).
- Point_process_operation wikiPageWikiLink Union_(set_theory).
- Point_process_operation wikiPageWikiLinkText "Point process operation".
- Point_process_operation wikiPageWikiLinkText "point process operation".
- Point_process_operation wikiPageUsesTemplate Template:Main.
- Point_process_operation subject Category:Point_processes.
- Point_process_operation subject Category:Spatial_processes.
- Point_process_operation hypernym Operation.
- Point_process_operation type MilitaryConflict.
- Point_process_operation comment "In probability and statistics, a point process operation or point process transformation is a type of mathematical operation performed on a random object known as a point process, which are often used as mathematical models of phenomena that can be represented as points randomly located in space. These operations can be purely random, deterministic or both, and are used to construct new point processes, which can be then also used as mathematical models.".
- Point_process_operation label "Point process operation".
- Point_process_operation sameAs Q16155268.
- Point_process_operation sameAs m.0y6dlyj.
- Point_process_operation sameAs Q16155268.
- Point_process_operation wasDerivedFrom Point_process_operation?oldid=694160391.
- Point_process_operation isPrimaryTopicOf Point_process_operation.