Point process

In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on some underlying mathematical space such as the real line, the Cartesian plane, or more abstract spaces. Point processes can be used as mathematical models of phenomena or objects representable as points in some type of space. In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on some underlying mathematical space such as the real line, the Cartesian plane, or more abstract spaces. Point processes can be used as mathematical models of phenomena or objects representable as points in some type of space. There are different mathematical interpretations of a point process, such as a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. Others consider a point process as a stochastic process, where the process is indexed by sets of the underlying space on which it is defined, such as the real line or n {displaystyle n} -dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes. Sometimes the term 'point process' is not preferred, as historically the word 'process' denoted an evolution of some system in time, so point process is also called a random point field. Point processes are well studied objects in probability theory and the subject of powerful tools in statistics for modeling and analyzing spatial data, which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience, economics and others. Point processes on the real line form an important special case that is particularly amenable to study, because the points are ordered in a natural way, and the whole point process can be described completely by the (random) intervals between the points. These point processes are frequently used as models for random events in time, such as the arrival of customers in a queue (queueing theory), of impulses in a neuron (computational neuroscience), particles in a Geiger counter, location of radio stations in a telecommunication network or of searches on the world-wide web. In mathematics, a point process is a random element whose values are 'point patterns' on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure, it is sufficient for more applied purposes to think of a point pattern as a countable subset of S that has no limit points. Let S be a locally compact second countable Hausdorff space equipped with its Borel σ-algebra B(S). Write N {displaystyle {mathfrak {N}}} for the set of locally finite counting measures on S and N {displaystyle {mathcal {N}}} for the smallest σ-algebra on N {displaystyle {mathfrak {N}}} that renders all the point counts measurable for all relatively compact sets B in B(S).A point process on S is a measurable map from a probability space ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},P)} to the measurable space ( N , N ) {displaystyle ({mathfrak {N}},{mathcal {N}})} . By this definition, a point process is a special case of a random measure.