|
Phase-type Distribution
A phase-type distribution is a probability distribution constructed by a convolution or mixture of exponential distributions. It results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence in which each of the phases occurs may itself be a stochastic process. The distribution can be represented by a random variable describing the time until absorption of a Markov process with one absorbing state. Each of the states of the Markov process represents one of the phases. It has a discrete-time equivalent the discrete phase-type distribution. The set of phase-type distributions is dense in the field of all positive-valued distributions, that is, it can be used to approximate any positive-valued distribution. Definition Consider a continuous-time Markov process with ''m'' + 1 states, where ''m'' ≥ 1, such that the states 1,...,''m'' are transient states and state 0 is an absorbing state. Further, let the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts. The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions. Definitions Probability density function The probability density function (pdf) of an exponential distribution is : f(x;\lambda) = \begin \lambda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyper-exponential Distribution
In probability theory, a hyperexponential distribution is a continuous probability distribution whose probability density function of the random variable ''X'' is given by : f_X(x) = \sum_^n f_(x)\;p_i, where each ''Y''''i'' is an exponentially distributed random variable with rate parameter ''λ''''i'', and ''p''''i'' is the probability that ''X'' will take on the form of the exponential distribution with rate ''λ''''i''. It is named the ''hyper''exponential distribution since its coefficient of variation is greater than that of the exponential distribution, whose coefficient of variation is 1, and the hypoexponential distribution, which has a coefficient of variation smaller than one. While the exponential distribution is the continuous analogue of the geometric distribution, the hyperexponential distribution is not analogous to the hypergeometric distribution. The hyperexponential distribution is an example of a mixture density. An example of a hyperexponential random var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous-time Markov Process
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to the least value of a set of exponential random variables, one for each possible state it can move to, with the parameters determined by the current state. An example of a CTMC with three states \ is as follows: the process makes a transition after the amount of time specified by the holding time—an exponential random variable E_i, where ''i'' is its current state. Each random variable is independent and such that E_0\sim \text(6), E_1\sim \text(12) and E_2\sim \text(18). When a transition is to be made, the process moves according to the jump chain, a discrete-time Markov chain with stochastic matrix: :\begin 0 & \frac & \frac \\ \frac & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Phase-type Distribution
The discrete phase-type distribution is a probability distribution that results from a system of one or more inter-related geometric distributions occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random variable describing the time until absorption of an absorbing Markov chain with one absorbing state. Each of the states of the Markov chain represents one of the phases. It has continuous time equivalent in the phase-type distribution. Definition A terminating Markov chain is a Markov chain where all states are transient, except one which is absorbing. Reordering the states, the transition probability matrix of a terminating Markov chain with m transient states is : =\left begin&\mathbf^0\\\mathbf^\mathsf&1\end\right where is a m\times m matrix, \mathbf^0 and \mathbf are column vectors with m entries, and \mathbf^0+\mathbf=\mathbf. The transition matrix is characteri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expectation–maximization Algorithm
In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the ''E'' step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. History The EM algorithm was explained and given its name in a classic 1977 paper by Arthur Dempster, Nan Laird, and Donald Rubin. They pointed out that the method had been "proposed many times in special circumstances" by earlier authors. One of the earliest is the gene-counting method for estimating allele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heavy-tailed Distribution
In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distribution that is of interest, but a distribution may have a heavy left tail, or both tails may be heavy. There are three important subclasses of heavy-tailed distributions: the fat-tailed distributions, the long-tailed distributions and the subexponential distributions. In practice, all commonly used heavy-tailed distributions belong to the subexponential class. There is still some discrepancy over the use of the term heavy-tailed. There are two other definitions in use. Some authors use the term to refer to those distributions which do not have all their power moments finite; and some others to those distributions that do not have a finite variance. The definition given in this article is the most general in use, and includes all dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in ''Mathematica''. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. __TOC__ Notebook interface Wolfram Mathematica (called ''Mathematica'' by some of its users) is split into two parts: the kernel and the front end. The kernel interprets expressions (Wolfram Language code) and returns result expressions, which can then be displayed by the front end. The origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. As of 2020, MATLAB has more than 4 million users worldwide. They come from various backgrounds of engineering, science, and economics. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was based on his 1960s PhD thesis. Moler became a math professor at the University of New Mexico and starte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proceedings Of The Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, formerly titled ''Proceedings of the Cambridge Philosophical Society'', has been published since 1843. See also *Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ... External linksofficial website Academic journals associated with learned and professional societies Cambridge University Press academic journals Mathematics education in the United Kingdom Mathematics journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypoexponential Distribution
In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one. Overview The Erlang distribution is a series of ''k'' exponential distributions all with rate \lambda. The hypoexponential is a series of ''k'' exponential distributions each with their own rate \lambda_, the rate of the i^ exponential distribution. If we have ''k'' independently distributed exponential random variables \boldsymbol_, then the random variable, : \boldsymbol=\sum^_\boldsymbol_ is hypoexponentially distributed. The hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperexponential Distribution
In probability theory, a hyperexponential distribution is a continuous probability distribution whose probability density function of the random variable ''X'' is given by : f_X(x) = \sum_^n f_(x)\;p_i, where each ''Y''''i'' is an exponentially distributed random variable with rate parameter ''λ''''i'', and ''p''''i'' is the probability that ''X'' will take on the form of the exponential distribution with rate ''λ''''i''. It is named the ''hyper''exponential distribution since its coefficient of variation is greater than that of the exponential distribution, whose coefficient of variation is 1, and the hypoexponential distribution, which has a coefficient of variation smaller than one. While the exponential distribution is the continuous analogue of the geometric distribution, the hyperexponential distribution is not analogous to the hypergeometric distribution. The hyperexponential distribution is an example of a mixture density. An example of a hyperexponential random va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
