Marked point process thesis
From: Martin M.
Category: island essay
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Point processes short introduction. We consider a compact set K. A point process X is defined as a mapping from a probability space to configurations of points of K. Since K is bounded, the conditions ensuring the measurability of such a mapping are verified see [ 24 ] for details : Figure d shows some realisations of a point process. Basically, it may be sawn as a random variable giving random configurations of points. The total number of points of a point process is a random variable.
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We start by providing canonical examples of MAPs. Again, we provide both pictorial explanation and more formal explanation. Figure 3. In a Poisson process, the intervals between consecutive events are independent and identically distributed exponential random variables. When there is a transition from a state to itself in the Markov chain, there is an event in the Poisson process. In a PH renewal process, the intervals between consecutive events are independent and identically distributed with a PH distribution.
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Poisson point process
In statistics and probability theory , a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. 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. 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.
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