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Variable-order Markov Model
In the mathematical theory of stochastic processes, variable-order Markov (VOM) models are an important class of models that extend the well known Markov chain models. In contrast to the Markov chain models, where each random variable in a sequence with a Markov property depends on a fixed number of random variables, in VOM models this number of conditioning random variables may vary based on the specific observed realization. This realization sequence is often called the ''context''; therefore the VOM models are also called ''context trees''. VOM models are nicely rendered by colorized probabilistic suffix trees (PST). The flexibility in the number of conditioning random variables turns out to be of real advantage for many applications, such as statistical analysis, classification and prediction. Example Consider for example a sequence of random variables, each of which takes a value from the ternary alphabet . Specifically, consider the string ' constructed from infinite concate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Processes
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 Brownian motion pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Code
In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communication channel or storage in a storage medium. An early example is an invention of language, which enabled a person, through speech, to communicate what they thought, saw, heard, or felt to others. But speech limits the range of communication to the distance a voice can carry and limits the audience to those present when the speech is uttered. The invention of writing, which converted spoken language into visual symbols, extended the range of communication across space and time. The process of encoding converts information from a source into symbols for communication or storage. Decoding is the reverse process, converting code symbols back into a form that the recipient understands, such as English or/and Spanish. One reason for coding is to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-Markov Process
In probability and statistics, a Markov renewal process (MRP) is a random process that generalizes the notion of Markov jump processes. Other random processes like Markov chains, Poisson processes and renewal processes can be derived as special cases of MRP's. Definition Consider a state space \mathrm. Consider a set of random variables (X_n,T_n), where T_n are the jump times and X_n are the associated states in the Markov chain (see Figure). Let the inter-arrival time, \tau_n=T_n-T_. Then the sequence (X_n,T_n) is called a Markov renewal process if : \begin & \Pr(\tau_\le t, X_=j\mid(X_0, T_0), (X_1, T_1),\ldots, (X_n=i, T_n)) \\ pt= & \Pr(\tau_\le t, X_=j\mid X_n=i)\, \forall n \ge1,t\ge0, i,j \in \mathrm \end Relation to other stochastic processes # If we define a new stochastic process Y_t:=X_n for t \in _n,T_), then the process Y_t is called a semi-Markov process. Note the main difference between an MRP and a semi-Markov process is that the former is defined as a two- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis–Hastings algorithm. Application domains MCMC methods are primarily used for calculating numerical approximations of multi-dimensional integrals, for example in Bayesian statistics, computational physics, computational biology and computational linguistics. In Bayesian statistics, the recent development of MCMC methods has made it possible to compute large hierarchical models that require integrations over hundreds to thousands of unknown parameters. In rare even ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variable Order Bayesian Network
Variable-order Bayesian network (VOBN) models provide an important extension of both the Bayesian network models and the variable-order Markov models. VOBN models are used in machine learning in general and have shown great potential in bioinformatics applications. These models extend the widely used position weight matrix (PWM) models, Markov models, and Bayesian network (BN) models. In contrast to the BN models, where each random variable depends on a fixed subset of random variables, in VOBN models these subsets may vary based on the specific realization of observed variables. The observed realizations are often called the context and, hence, VOBN models are also known as context-specific Bayesian networks.{{cite conference , title=Context-specific independence in Bayesian networks , last1=Boutilier , first1=C. , last2=Friedman , first2=N. , author-link2=Nir Friedman , last3=Goldszmidt , first3=M. , last4=Koller , first4=D., author-link4=Daphne Koller , date=1996 , publisher= , b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Examples Of Markov Chains
This article contains examples of Markov chains and Markov processes in action. All examples are in the countable state space. For an overview of Markov chains in general state space, see Markov chains on a measurable state space. Discrete-time Board games played with dice A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. This is in contrast to card games such as blackjack, where the cards represent a 'memory' of the past moves. To see the difference, consider the probability for a certain event in the game. In the above-mentioned dice games, the only thing that matters is the current state of the board. The next state of the board depends on the current state, and the next roll of the dice. It doesn't depend on how things got to their current state. In a game such as blackjack, a player can gain an advantage by remembering which cards have already been shown (and hence which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Chains With Memory Of Variable Length
Stochastic chains with memory of variable length are a family of stochastic chains of finite order in a finite alphabet, such as, for every time pass, only one finite suffix of the past, called context, is necessary to predict the next symbol. These models were introduced in the information theory literature by Jorma Rissanen in 1983, as a universal tool to data compression, but recently have been used to model data in different areas such as biology, linguistics and music. Definition A stochastic chain with memory of variable length is a stochastic chain (X_n)_, taking values in a finite alphabet A, and characterized by a probabilistic context tree (\tau,p), so that *\tau is the group of all contexts. A context X_,\ldots,X_, being l the size of the context, is a finite portion of the past X_,\ldots,X_, which is relevant to predict the next symbol X_; *p is a family of transition probabilities associated with each context. History The class of stochastic chains with memory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence Analysis In Social Sciences
In social sciences, sequence analysis (SA) is concerned with the analysis of sets of categorical sequences that typically describe longitudinal data. Analyzed sequences are encoded representations of, for example, individual life trajectories such as family formation, school to work transitions, working careers, but they may also describe daily or weekly time use or represent the evolution of observed or self-reported health, of political behaviors, or the development stages of organizations. Such sequences are chronologically ordered unlike words or DNA sequences for example. SA is a longitudinal analysis approach that is holistic in the sense that it considers each sequence as a whole. SA is essentially exploratory. Broadly, SA provides a comprehensible overall picture of sets of sequences with the objective of characterizing the structure of the set of sequences, finding the salient characteristics of groups, identifying typical paths, comparing groups, and more generally studyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sharon R
Sharon ( he, שָׁרוֹן ''Šārôn'' "plain") is a given name as well as an Israeli surname. In English-speaking areas, Sharon is now predominantly a feminine given name. However, historically it was also used as a masculine given name. In Israel, it is used both as a masculine and a feminine given name. Etymology The Hebrew word simply means "plain", but in the Hebrew Bible, is the name specifically given to the fertile plain between the Samarian Hills and the coast, known (tautologically) as Sharon plain in English. The phrase "rose of Sharon" (חבצלת השרון ''ḥăḇaṣṣeleṯ ha-sharon'') occurs in the KJV translation of the Song of Solomon ("I am the rose of Sharon, the lily of the valley"), and has since been used in reference to a number of flowering plants. Unlike other unisex names that have come to be used almost exclusively as feminine (e.g. Evelyn), ''Sharon'' was never predominantly a masculine name. Usage before 1925 is very rare and was apparentl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haplotyping
A haplotype ( haploid genotype) is a group of alleles in an organism that are inherited together from a single parent. Many organisms contain genetic material ( DNA) which is inherited from two parents. Normally these organisms have their DNA organized in two sets of pairwise similar chromosomes. The offspring gets one chromosome in each pair from each parent. A set of pairs of chromosomes is called diploid and a set of only one half of each pair is called haploid. The haploid genotype (haplotype) is a genotype that considers the singular chromosomes rather than the pairs of chromosomes. It can be all the chromosomes from one of the parents or a minor part of a chromosome, for example a sequence of 9000 base pairs. However, there are other uses of this term. First, it is used to mean a collection of specific alleles (that is, specific DNA sequences) in a cluster of tightly linked genes on a chromosome that are likely to be inherited together—that is, they are likely to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spam Filtering
Various anti-spam techniques are used to prevent email spam (unsolicited bulk email). No technique is a complete solution to the spam problem, and each has trade-offs between incorrectly rejecting legitimate email (false positives) as opposed to not rejecting all spam email (false negatives) – and the associated costs in time, effort, and cost of wrongfully obstructing good mail. Anti-spam techniques can be broken into four broad categories: those that require actions by individuals, those that can be automated by email administrators, those that can be automated by email senders and those employed by researchers and law enforcement officials. End-user techniques There are a number of techniques that individuals can use to restrict the availability of their email addresses, with the goal of reducing their chance of receiving spam. Discretion Sharing an email address only among a limited group of correspondents is one way to limit the chance that the address will be "harvest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
