|
First Passage Time
In statistics, first-hitting-time models are simplified models that estimate the amount of time that passes before some random or stochastic process crosses a barrier, boundary or reaches a specified state, termed the first hitting time, or the first passage time. Accurate models give insight into the physical system under observation, and have been the topic of research in very diverse fields, from economics to ecology. The idea that a first hitting time of a stochastic process might describe the time to occurrence of an event has a long history, starting with an interest in the first passage time of Wiener diffusion processes in economics and then in physics in the early 1900s. Modeling the probability of financial ruin as a first passage time was an early application in the field of insurance. An interest in the mathematical properties of first-hitting-times and statistical models and methods for analysis of survival data appeared steadily between the middle and end of the 20t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion Equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation when bulk velocity is zero. It is equivalent to the heat equation under some circumstances. Statement The equation is usually written as: \frac = \nabla \cdot \big D(\phi,\mathbf) \ \nabla\phi(\mathbf,t) \big where is the density of the diffusing material at location and time and is the collective diffusion coefficient for density at location ; and represents the vector differential operator del. If the diffusion coefficient depends on the density then the equatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proportional Hazards Models
Proportional hazards models are a class of survival models in statistics. Survival models relate the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. In a proportional hazards model, the unique effect of a unit increase in a covariate is multiplicative with respect to the hazard rate. The hazard rate at time t is the probability per short time d''t'' that an event will occur between t and t + dt given that up to time t no event has occurred yet. For example, taking a drug may halve one's hazard rate for a stroke occurring, or, changing the material from which a manufactured component is constructed, may double its hazard rate for failure. Other types of survival models such as accelerated failure time models do not exhibit proportional hazards. The accelerated failure time model describes a situation where the biological or mechanical life history of an event is accelerated (or decelerated). Background ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Survival Analysis
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory, reliability analysis or reliability engineering in engineering, duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer certain questions, such as what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival? To answer such questions, it is necessary to define "lifetime". In the case of biological survival, death is unambiguous, but for mechanical reliability, failure may not be well-defined, for there may well be mechanical systems in which failure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Narrow Escape Problem
The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology. The mathematical formulation is the following: a Brownian particle ( ion, molecule, or protein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. When escape is even more stringent due to severe geometrical restrictions at the place of escape, the narrow escape problem becomes the dire strait problem. The narrow escape problem was proposed in the context of biology and biophysics by D. Holcman and Z. Schuss, and later on with A.Singer and led to the narrow escape theory in applied mathematics and computational biology. Formulation The motion of a particle is described by the Smoluchowski limit of the Langevin equation: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
In probability theory and statistics, a Markov chain or Markov process is a stochastic process 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). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Process
A gamma process, also called the ''Moran-Gamma subordinator'', is a two-parameter stochastic process which models the accumulation of ''effort'' or ''wear'' over time. The gamma process has independent and stationary increments which follow the gamma distribution, hence the name. The gamma process is studied in mathematics, statistics, probability theory, and stochastics, with particular applications in deterioration modeling and mathematical finance. Notation The gamma process is often abbreviated as X(t)\equiv\Gamma(t;\gamma, \lambda) where t represents the time from 0. The shape parameter \gamma (inversely) controls the jump size, and the rate parameter \lambda controls the rate of jump arrivals, analogously with the gamma distribution. Both \gamma and \lambda must be greater than 0. We use the gamma function and gamma distribution in this article, so the reader should distinguish between \Gamma(\cdot) (the gamma function), \Gamma(\gamma, \lambda) (the gamma distribution), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Process
In mathematics, the Wiener process (or Brownian motion, due to its historical connection with Brownian motion, the physical process of the same name) is a real-valued continuous-time stochastic process discovered by Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary increments, stationary independent increments). It occurs frequently in pure and applied mathematics, economy, 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 martingale (probability theory), 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 processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Lévy Distribution
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile."van der Waals profile" appears with lowercase "van" in almost all sources, such as: ''Statistical mechanics of the liquid surface'' by Clive Anthony Croxton, 1980, A Wiley-Interscience publication, , and in ''Journal of technical physics'', Volume 36, by Instytut Podstawowych Problemów Techniki (Polska Akademia Nauk), publisher: Państwowe Wydawn. Naukowe., 1995/ref> It is a special case of the inverse-gamma distribution. It is a stable distribution. Definition The probability density function of the Lévy distribution over the domain x \ge \mu is : f(x; \mu, c) = \sqrt \, \frac, where \mu is the location parameter, and c is the scale parameter. The cumulative distribution function is : F(x; \mu, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integration, contour integral which is path-independent because \exp(-t^2) is Holomorphic function, holomorphic on the whole complex plane \mathbb. In many applications, the function argument is a real number, in which case the function value is also real. In some old texts, the error function is defined without the factor of \frac. This nonelementary integral is a sigmoid function, sigmoid function that occurs often in probability, statistics, and partial differential equations. In statistics, for non-negative real values of , the error function has the following interpretation: for a real random variable that is normal distribution, normally distributed with mean 0 and standard deviation \frac, is the probability that falls in the range . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
