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###### Ancestral reconstruction
For discrete-valued traits (such as "pollinator type"), this process is typically taken to be a Markov chain; for continuous- ... Huelsenbeck and Bollback first proposed a hierarchical Bayes method to ancestral reconstruction by using Markov chain Monte ... The typical means of modelling evolution of this trait is via a continuous-time Markov chain, which may be briefly described as ... Asymmetrical Markov k {\displaystyle k} -state 2 parameter model (Figure 4): in this model, the state space is ordered (so that ...
###### Markovchain
Hidden Markov model Markov blanket Markov chain geostatistics Markov chain mixing time Markov chain Monte Carlo Markov decision ... process Markov information source Markov random field Quantum Markov chain Telescoping Markov chain Variable-order Markov model ... A beautiful visual explanation of Markov Chains Chapter 5: Markov Chain Models Making Sense and Nonsense of Markov Chains ... A Markov chain is a stochastic process with the Markov property. The term "Markov chain" refers to the sequence of random ...
###### Additive Markovchain
In probability theory, an additive Markov chain is a Markov chain with an additive conditional probability function. Here the ... A binary additive Markov chain is where the state space of the chain consists on two values only, Xn ∈ { x1, x2 }. For example ... Examples of Markov chains S.S. Melnyk, O.V. Usatenko, and V.A. Yampol'skii. (2006) "Memory functions of the additive Markov ... An additive Markov chain of order m is a sequence of random variables X1, X2, X3, ..., possessing the following property: the ...
###### Absorbing Markovchain
In an absorbing Markov chain, a state that is not absorbing is called transient. Let an absorbing Markov chain with transition ... 3: Absorbing Markov Chains". In Gehring, F. W.; Halmos, P. R. Finite Markov Chains (Second ed.). New York Berlin Heidelberg ... Like general Markov chains, there can be continuous-time absorbing Markov chains with an infinite state space. However, this ... In the mathematical theory of probability, an absorbing Markov chain is a Markov chain in which every state can reach an ...
###### Markovchain geostatistics
A Markov chain random field is still a single spatial Markov chain. The spatial Markov chain moves or jumps in a space and ... Markov chain geostatistics uses Markov chain spatial models, simulation algorithms and associated spatial correlation measures ... e.g., transiogram) based on the Markov chain random field theory, which extends a single Markov chain into a multi-dimensional ... is proposed as the accompanying spatial measure of Markov chain random fields. Li, W. 2007. Markov chain random fields for ...
###### Markovchain mixing time
In probability theory, the mixing time of a Markov chain is the time until the Markov chain is "close" to its steady state ... More precisely, a fundamental result about Markov chains is that a finite state irreducible aperiodic chain has a unique ... Such problems can, for sufficiently large number of colors, be answered using the Markov chain Monte Carlo method and showing ... Mixing (mathematics) for a formal definition of mixing Aldous, David; Fill, Jim, Reversible Markov Chains and Random Walks on ...
###### Markovchain approximation method
In case of need, one must as well approximate the cost function for one that matches up the Markov chain chosen to approximate ... F. B. Hanson, "Markov Chain Approximation", in C. T. Leondes, ed., Stochastic Digital Control System Techniques, Academic Press ... In numerical methods for stochastic differential equations, the Markov chain approximation method (MCAM) belongs to the several ... The basic idea of the MCAM is to approximate the original controlled process by a chosen controlled markov process on a finite ...
###### Markovchain Monte Carlo
These interacting Markov chain Monte Carlo samplers can be interpreted as a way to run in parallel a sequence of Markov chain ... In principle, any Markov chain Monte Carlo sampler can be turned into an interacting Markov chain Monte Carlo sampler. ... Random walk Monte Carlo methods make up a large subclass of Markov chain Monte Carlo methods. Markov chain Monte Carlo methods ... In contrast to traditional Markov chain Monte Carlo methods, the precision parameter of this class of interacting Markov chain ...
###### Markovchains on a measurable state space
A Markov chain on a measurable state space is a discrete-time-homogenous Markov chain with a measurable space as state space. ... The definition of Markov chains has evolved during the 20th century. In 1953 the term Markov chain was used for stochastic ... Sean Meyn and Richard L. Tweedie: Markov Chains and Stochastic Stability. 2nd edition, 2009. Daniel Revuz: Markov Chains. 2nd ... denotes the Markov chain according to a Markov kernel p {\displaystyle p} with stationary measure μ {\displaystyle \mu } , then ...
###### Time reversibility
Kolmogorov's criterion defines the condition for a Markov chain or continuous-time Markov chain to be time-reversible. Time ... Markov chains, and piecewise deterministic Markov processes. Time reversal method works based on the linear reciprocity of the ... Norris, J. R. (1998). Markov Chains. Cambridge University Press. ISBN 0521633966. Löpker, A.; Palmowski, Z. (2013). "On time ... Markov processes can only be reversible if their stationary distributions have the property of detailed balance: p ( x t = i , ...
###### Balance equation
... of a Markov chain, when such a distribution exists. For a continuous time Markov chain with state space S, transition rate from ... For a continuous time Markov chain (CTMC) with transition rate matrix Q, if π i {\displaystyle \pi _{i}} can be found such that ... In probability theory, a balance equation is an equation that describes the probability flux associated with a Markov chain in ... For a discrete time Markov chain with transition matrix Q and equilibrium distribution π {\displaystyle \pi } , the global ...
###### Foster's theorem
It uses the fact that positive recurrent Markov chains exhibit a notion of "Lyapunov stability" in terms of returning to any ... Consider an irreducible discrete-time Markov chain on a countable state space S having a transition probability matrix P with ... Brémaud, P. (1999). "Lyapunov Functions and Martingales". Markov Chains. p. 167. doi:10.1007/978-1-4757-3124-8_5. ISBN 978-1- ... Foster's theorem states that the Markov chain is positive recurrent if and only if there exists a Lyapunov function V : S → R ...
###### Kemeny's constant
... required for a Markov chain to transition from a starting state i to a random destination state sampled from the Markov chain's ... It is in that sense a constant, although it is different for different Markov chains. When first published by John Kemeny in ... For a finite ergodic Markov chain with transition matrix P and invariant distribution π, write mij for the mean first passage ... Kemeny, J. G.; Snell, J. L. (1960). Finite Markov Chains. Princeton, NJ: D. Van Nostrand. (Corollary 4.3.6) Catral, M.; ...
###### Hydrological modelling
Markov Chains are a mathematical technique for determine the probability of a state or event based on a previous state or event ... Markov Chains were first used to model rainfall event length in days in 1976, and continues to be used for flood risk ... "Markov Chains explained visually". Explained Visually. Retrieved 2017-04-21. Haan, C. T.; Allen, D. M.; Street, J. O. (1976-06- ... "A Markov Chain Model of daily rainfall". Water Resources Research. 12 (3): 443-449. Bibcode:1976WRR....12..443H. doi:10.1029/ ...
###### Entropy (information theory)
See Markov chain. Entropy is one of several ways to measure diversity. Specifically, Shannon entropy is the logarithm of 1D, ... For a second order Markov source, the entropy rate is H ( S ) = − ∑ i p i ∑ j p i ( j ) ∑ k p i , j ( k ) log 2 ⁡ p i , j ( k ... A common way to define entropy for text is based on the Markov model of text. For an order-0 source (each character is selected ... For a first-order Markov source (one in which the probability of selecting a character is dependent only on the immediately ...