Counting Process

	Counting Process A counting process is a stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ... \ with values that are non-negative, integer, and non-decreasing: # N(t)\geq0. # N(t) is an integer. # If s\leq t then N(s)\leq N(t). If s, then $N(t)-N(s)$ is the number of events occurred during the interval $(s,t].$ Examples of counting processes include Poisson processes and Renewal processes. Counting processes deal with the number of occurrences of something over time. An example of a counting process is the number of job arrivals to a queue over time. If a process h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Stochastic Process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ..., neuroscience, physics, image processing, signal processing, stochastic control, control theory, information theory, computer scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Poisson Process In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points randomly located on a Space (mathematics), mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science. This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Renewal Theory Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite expectation. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times. A renewal process has asymptotic properties analogous to the strong law of large numbers and central limit theorem. The renewal function m(t) (expected number of arrivals) and reward function g(t) (expected reward value) are of key importance in renewal theory. The renewal function satisfies a recursive integral equation, the renewal equation. The key renewal equation gives the limiting value of the convolution of m'(t) with a suitable non-negative function. The superposition of renewal processes can be studied as a special case of M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Markov Property In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process, which means that its future evolution is independent of its history. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the Markov property is known as a Markov chain. Introduction A stochastic process has the Markov property if the conditional probability distribution of future states of the process (cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Intensity Of Counting Processes The intensity \lambda of a counting process is a measure of the rate of change of its predictable part. If a stochastic process \ is a counting process, then it is a submartingale, and in particular its Doob-Meyer decomposition is :N(t) = M(t) + \Lambda(t) where M(t) is a martingale and \Lambda(t) is a predictable increasing process. \Lambda(t) is called the cumulative intensity of N(t) and it is related to \lambda by :\Lambda(t) = \int_^ \lambda(s)ds. Definition Given probability space (\Omega, \mathcal, \mathbb) and a counting process \ which is adapted to the filtration \, the intensity of N is the process \ defined by the following limit: : \lambda(t) = \lim_ \frac \mathbb \mathcal_t. The right-continuity property of counting processes allows us to take this limit from the right. Estimation In statistical learning, the variation between \lambda and its estimator In statistics, an estimator is a rule for calculating an estimate of a given quantity based on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Poisson Point Process In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science. This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]