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Kushner Equation
In filtering theory the Kushner equation (after Harold Kushner) is an equation for the conditional probability density of the state of a stochastic non-linear dynamical system, given noisy measurements of the state. It therefore provides the solution of the nonlinear filtering problem in estimation theory. The equation is sometimes referred to as the Stratonovich–Kushner Stratonovich, R.L. (1960). ''Conditional Markov Processes''. Theory of Probability and Its Applications, 5, pp. 156–178. (or Kushner–Stratonovich) equation. Overview Assume the state of the system evolves according to :dx = f(x,t) \, dt + \sigma dw and a noisy measurement of the system state is available: :dz = h(x,t) \, dt + \eta dv where ''w'', ''v'' are independent Wiener processes. Then the conditional probability density ''p''(''x'', ''t'') of the state at time ''t'' is given by the Kushner equation: :dp(x,t) = L (x,t)dt + p(x,t) (x,t)-E_t h(x,t) \top \eta^\eta^ z-E_t h(x,t) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtering Problem (stochastic Processes)
In the theory of stochastic processes, filtering describes the problem of determining the state of a system from an incomplete and potentially noisy set of observations. While originally motivated by problems in engineering, filtering found applications in many fields from signal processing to finance. The problem of optimal non-linear filtering (even for the non-stationary case) was solved by Ruslan L. Stratonovich (1959, 1960), see also Harold J. Kushner's work and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of the filter known as Zakai equation. The solution, however, is infinite-dimensional in the general case. Certain approximations and special cases are well understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the Wiener filter and the Kalman-Bucy filter. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold J
Harold may refer to: People * Harold (given name), including a list of persons and fictional characters with the name * Harold (surname), surname in the English language * András Arató, known in meme culture as "Hide the Pain Harold" Arts and entertainment * ''Harold'' (film), a 2008 comedy film * ''Harold'', an 1876 poem by Alfred, Lord Tennyson * ''Harold, the Last of the Saxons'', an 1848 book by Edward Bulwer-Lytton, 1st Baron Lytton * '' Harold or the Norman Conquest'', an opera by Frederic Cowen * ''Harold'', an 1885 opera by Eduard Nápravník * Harold, a character from the cartoon ''The Grim Adventures of Billy & Mandy'' *Harold & Kumar, a US movie; Harold/Harry is the main actor in the show. Places ;In the United States * Alpine, Los Angeles County, California, an erstwhile settlement that was also known as Harold * Harold, Florida, an unincorporated community * Harold, Kentucky, an unincorporated community * Harold, Missouri, an unincorporated communi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Probability
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. If the event of interest is and the event is known or assumed to have occurred, "the conditional probability of given ", or "the probability of under the condition ", is usually written as or occasionally . This can also be understood as the fraction of probability B that intersects with A: P(A \mid B) = \frac. For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing. For example, the conditional probability that someone unwell (sick) is coughing might b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Filter
In signal processing, a nonlinear (or non-linear) filter is a filter whose output is not a linear function of its input. That is, if the filter outputs signals ''R'' and ''S'' for two input signals ''r'' and ''s'' separately, but does not always output ''αR'' + ''βS'' when the input is a linear combination ''αr'' + ''βs''. Both continuous-domain and discrete-domain filters may be nonlinear. A simple example of the former would be an electrical device whose output voltage ''R''(''t'') at any moment is the square of the input voltage ''r''(''t''); or which is the input clipped to a fixed range 'a'',''b'' namely ''R''(''t'') = max(''a'', min(''b'', ''r''(''t''))). An important example of the latter is the running-median filter, such that every output sample ''R''''i'' is the median of the last three input samples ''r''''i'', ''r''''i''−1, ''r''''i''−2. Like linear filters, nonlinear filters may be shift invariant or not. Non-linear filters hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimation Theory
Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An '' estimator'' attempts to approximate the unknown parameters using the measurements. In estimation theory, two approaches are generally considered: * The probabilistic approach (described in this article) assumes that the measured data is random with probability distribution dependent on the parameters of interest * The set-membership approach assumes that the measured data vector belongs to a set which depends on the parameter vector. Examples For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the parameter sought; the estimate is based on a small random sample of voters. Alternatively, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruslan L
Ruslan may refer to: * ''Ruslan'' (film), a 2009 film starring Steven Segal * Ruslan (given name), male name used mainly in Slavic countries, with list of people * Antonov An-124 ''Ruslan'', large Soviet cargo aircraft, later built in Ukraine and Russia * SS ''Ruslan'', a Russian cargo ship in the Third Aliyah in 1919 See also * Rusian (other) Rusian may refer to: * Old East Slavic, a language which some scholars refer to as ''Rusian'' * Ruthenian language, also known as ''Rusian'' * Rusian, a fictional character in '' And You Thought There Is Never a Girl Online?'' See also * Rus' ... * Ruslan and Ludmila (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Itō Calculus
Itō may refer to: *Itō (surname), a Japanese surname *Itō, Shizuoka, Shizuoka Prefecture, Japan *Ito District, Wakayama Prefecture, Japan See also * Itô's lemma, used in stochastic calculus *Itoh–Tsujii inversion algorithm, in field theory *Itô calculus, an extension of calculus to stochastic processes, named after Kiyoshi Itô *Ito (other) *ITO (other) Ito may refer to: Places * Ito Island, an island of Milne Bay Province, Papua New Guinea * Ito Airport, an airport in the Democratic Republic of the Congo * Ito District, Wakayama, a district located in Wakayama Prefecture, Japan * Itō, Shizuo ..., for the three-letter acronym {{DEFAULTSORT:Ito es:Ito fr:Ito nl:Ito ja:いとう pt:Ito ru:Ито ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruslan Stratonovich
Ruslan Leont'evich Stratonovich (russian: Русла́н Лео́нтьевич Страто́нович) was a Russian physicist, engineer, and probabilist and one of the founders of the theory of stochastic differential equations. Biography Ruslan Stratonovich was born May 31, 1930, in Moscow. He studied from 1947 at the Moscow State University, specializing in there under P. I. Kuznetsov on radio physics (a Soviet term for oscillation physics - including noise - in the broadest sense, but especially in the electromagnetic spectrum). In 1953 he graduated and came into contact with the mathematician Andrey Kolmogorov. In 1956 he received his doctorate (theory of correlated random points apply to the calculation of electronic noise). In 1969 he became professor of physics at the Moscow State University. Research Stratonovich invented a stochastic calculus which serves as an alternative to the Itō calculus; the Stratonovich calculus is most natural when physical laws are bei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown. It is one of the best known Lévy processes ( càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics. The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
