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Brownian Bridge
A Brownian bridge is a continuous-time gaussian process ''B''(''t'') whose probability distribution is the conditional probability distribution of a standard Wiener process ''W''(''t'') (a mathematical model of Brownian motion) subject to the condition (when standardized) that ''W''(''T'') = 0, so that the process is pinned to the same value at both ''t'' = 0 and ''t'' = ''T''. More precisely: : B_t := (W_t\mid W_T=0),\;t \in ,T The expected value of the bridge at any t in the interval ,T/math> is zero, with variance \frac, implying that the most uncertainty is in the middle of the bridge, with zero uncertainty at the nodes. The covariance of ''B''(''s'') and ''B''(''t'') is \min(s,t)-\frac, or \frac if s . It is independent of W(T) Conversely, if B(t) is a Brownian bridge for t \in , 1/math> and Z is a standard normal random variable independent of B, then the process : W(t) = B(t) + tZ\, is a Wiener process for t \in , 1/math>. More generally, a Wiener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean, mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by Integral, integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Excursion
In probability theory a Brownian excursion process (BPE) is a stochastic process that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In particular, a Brownian excursion process is a Wiener process conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive. BEPs are important because, among other reasons, they naturally arise as the limit process of a number of conditional functional central limit theorems. Constructions A Brownian excursion process, e, is a Wiener process (or Brownian motion) conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive. Although note that, since the probability for an unrestricted Brownian bridge to be positive is zero, the conditioni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution Support (measure theory), supported on the real numbers, discrete or "mixed" as well as Continuous variable, continuous, is uniquely identified by a right-continuous Monotonic function, monotone increasing function (a càdlàg function) F \colon \mathbb R \rightarrow [0,1] satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from negative infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Inference
Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. In machine learning, the term ''inference'' is sometimes used instead to mean "make a prediction, by evaluating an already trained model"; in this context inferring properties of the model is referred to as ''training'' or ''learning'' (rather than ''inference''), and using a model for prediction is referred to as ''inference'' (instead of ''prediction''); se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov–Smirnov Test
In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric statistics, nonparametric test of the equality of continuous (or discontinuous, see #Discrete and mixed null distribution, Section 2.2), one-dimensional probability distributions. It can be used to test whether a random sample, sample came from a given reference probability distribution (one-sample K–S test), or to test whether two samples came from the same distribution (two-sample K–S test). Intuitively, it provides a method to qualitatively answer the question "How likely is it that we would see a collection of samples like this if they were drawn from that probability distribution?" or, in the second case, "How likely is it that we would see two sets of samples like this if they were drawn from the same (but unknown) probability distribution?". It is named after Andrey Kolmogorov and Nikolai Smirnov (mathematician), Nikolai Smirnov. The Kolmogorov–Smirnov statistic quantifies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Process
In probability theory, an empirical process is a stochastic process that characterizes the deviation of the empirical distribution function from its expectation. In mean field theory, limit theorems (as the number of objects becomes large) are considered and generalise the central limit theorem for empirical measures. Applications of the theory of empirical processes arise in non-parametric statistics. Definition For ''X''1, ''X''2, ... ''X''''n'' independent and identically-distributed random variables in R with common cumulative distribution function ''F''(''x''), the empirical distribution function is defined by :F_n(x)=\frac\sum_^n I_(X_i), where I''C'' is the indicator function of the set ''C''. For every (fixed) ''x'', ''F''''n''(''x'') is a sequence of random variables which converge to ''F''(''x'') almost surely by the strong law of large numbers. That is, ''F''''n'' converges to ''F'' pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donsker's Theorem
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem for empirical distribution functions. Specifically, the theorem states that an appropriately centered and scaled version of the empirical distribution function converges to a Gaussian process. Let X_1, X_2, X_3, \ldots be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let S_n:=\sum_^n X_i. The stochastic process S:=(S_n)_ is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by : W^(t) := \frac, \qquad t\in ,1 The central limit theorem asserts that W^(1) converges in distribution to a standard Gaussian random variable W(1) as n\to\infty. Donsker's invariance principle extends this convergence to the whole function W^:=(W^(t))_. More precisely, in its modern form, Do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Identically Distributed
Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist group Music Groups, labels, and genres * Independent music, a number of genres associated with independent labels * Independent record label, a record label not associated with a major label * Independent Albums, American albums chart Albums * ''Independent'' (Ai album), 2012 * ''Independent'' (Faze album), 2006 * ''Independent'' (Sacred Reich album), 1993 Songs * "Independent" (song), a 2007 song by Webbie * "Independent", a 2002 song by Ayumi Hamasaki from '' H'' News media organizations * Independent Media Center (also known as Indymedia or IMC), an open publishing network of journalist collectives that report on political and social issues, e.g., in ''The Indypendent'' newspaper of NYC * ITV (TV network) (Independent Television ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quanti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy Process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk. The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Further important examples include the Gamma process, the Pascal process, and the Meixner process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths. All Lévy processes are additive processes. Mathematical definition A Lévy process is a stochastic process X=\ that satisfi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
