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Generalized Gamma Distribution
The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many distributions commonly used for parametric models in survival analysis (such as the exponential distribution, the Weibull distribution and the gamma distribution) are special cases of the generalized gamma, it is sometimes used to determine which parametric model is appropriate for a given set of data. Another example is the half-normal distribution. Characteristics The generalized gamma distribution has two shape parameters, d > 0 and p > 0, and a scale parameter, a > 0. For non-negative ''x'' from a generalized gamma distribution, the probability density function is : f(x; a, d, p) = \frac, where \Gamma(\cdot) denotes the gamma function. The cumulative distribution function is : F(x; a, d, p) = \frac , \text \, P\left( \frac, \left( \fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rayleigh Distribution
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribution is named after Lord Rayleigh (). A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions. Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, which is infrequent, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution. A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Integer Gamma Distribution
In probability and statistics, the generalized integer gamma distribution (GIG) is the distribution of the sum of independent gamma distributed random variables, all with integer shape parameters and different rate parameters. This is a special case of the generalized chi-squared distribution. A related concept is the generalized near-integer gamma distribution (GNIG). Definition The random variable X\! has a gamma distribution with shape parameter r and rate parameter \lambda if its probability density function is : f^_X(x)=\frac\,e^ x^~~~~~~(x>0;\,\lambda,r>0) and this fact is denoted by X\sim\Gamma(r,\lambda)\!. Let X_j\sim\Gamma(r_j,\lambda_j)\!, where (j=1,\dots,p), be p independent random variables, with all r_j being positive integers and all \lambda_j\! different. In other words, each variable has the Erlang distribution with different shape parameters. The uniqueness of each shape parameter comes without loss of generality, because any case where some of the \lamb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modified Half-normal Distribution
In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple families, including the half-normal distribution, truncated normal distribution, gamma distribution, and square root of the gamma distribution, all of which are special cases of the MHN distribution. Therefore, it is a flexible probability model for analyzing real-valued positive data. The name of the distribution is motivated by the similarities of its density function with that of the half-normal distribution. In addition to being used as a probability model, MHN distribution also appears in Markov chain Monte Carlo (MCMC)-based Bayesian procedures, including Bayesian modeling of the directional data, Bayesian binary regression, and Bayesian graphical modeling. In Bayesian analysis, new distributions often appear as a conditional po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified Gaussian Distribution
In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution (constant 0) and a continuous distribution (a truncated Gaussian distribution with interval (0,\infty)) as a result of censoring (statistics), censoring. Density function The probability density function of a rectified Gaussian distribution, for which random variables ''X'' having this distribution, derived from the normal distribution \mathcal(\mu,\sigma^2), are displayed as X \sim \mathcal^(\mu,\sigma^2) , is given by f(x;\mu,\sigma^2) =\Phi\delta(x)+ \frac\; e^\textrm(x). Here, \Phi(x) is the cumulative distribution function (cdf) of the standard normal distribution: \Phi(x) = \frac \int_^x e^ \, dt \quad x\in\mathbb, \delta(x) is the Dirac delta function \delta(x) = \begin +\infty, & x = 0 \\ 0, & x \ne 0 \end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Folded Normal Distribution
The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable ''X'' with mean ''μ'' and variance ''σ''2, the random variable ''Y'' = , ''X'', has a folded normal distribution. Such a case may be encountered if only the magnitude of some variable is recorded, but not its sign. The distribution is called "folded" because probability mass to the left of ''x'' = 0 is folded over by taking the absolute value. In the physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin. Definitions Density The probability density function (PDF) is given by :f_Y(x;\mu,\sigma^2)= \frac \, e^ + \frac \, e^ for ''x'' ≥ 0, and 0 everywhere else. An alternative formulation is given by : f\left(x \right)=\sqrte^\cosh, where cosh is the Hyperbolic cosine function. It fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Normal Distribution
In probability and statistics, the truncated normal distribution is the probability distribution derived from that of a normally distributed random variable by bounding the random variable from either below or above (or both). The truncated normal distribution has wide applications in statistics and econometrics. Definitions Suppose X has a normal distribution with mean \mu and variance \sigma^2 and lies within the interval (a,b), \text \; -\infty \leq a < b \leq \infty . Then conditional on has a truncated normal distribution. Its , , for , is given by and by otherwise. Here, |
Half-t Distribution
In statistics, the folded-''t'' and half-''t'' distributions are derived from Student's ''t''-distribution by taking the absolute values of variates. This is analogous to the folded-normal and the half-normal statistical distributions being derived from the normal distribution. Definitions The folded non-standardized ''t'' distribution is the distribution of the absolute value of the non-standardized ''t'' distribution with \nu degrees of freedom; its probability density function is given by: :g\left(x\right)\;=\;\frac\left\lbrace \left +\frac\frac\right+\left +\frac\frac\right \right\rbrace \qquad(\mbox\quad x \geq 0). The half-''t'' distribution results as the special case of \mu=0, and the standardized version as the special case of \sigma=1. If \mu=0, the folded-''t'' distribution reduces to the special case of the half-''t'' distribution. Its probability density function then simplifies to :g\left(x\right)\;=\;\frac \left(1+\frac\frac\right)^ \qquad(\mbox\quad x \geq 0). Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SciPy
SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing. SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, fast Fourier transform, signal and image processing, ordinary differential equation solvers and other tasks common in science and engineering. SciPy is also a family of conferences for users and developers of these tools: SciPy (in the United States), EuroSciPy (in Europe) and SciPy.in (in India). Enthought originated the SciPy conference in the United States and continues to sponsor many of the international conferences as well as host the SciPy website. The SciPy library is currently distributed under the BSD license, and its development is sponsored and supported by an open community of developers. It is also supported by NumFOCUS, a community foundation for supporting reproducible and accessible science. Components The SciPy package is at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python (programming Language)
Python is a high-level programming language, high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is type system#DYNAMIC, dynamically type-checked and garbage collection (computer science), garbage-collected. It supports multiple programming paradigms, including structured programming, structured (particularly procedural programming, procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC (programming language), ABC programming language, and he first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000. Python 3.0, released in 2008, was a major revision not completely backward-compatible with earlier versions. Python 2.7.18, released in 2020, was the last release of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
R (programming Language)
R is a programming language for statistical computing and Data and information visualization, data visualization. It has been widely adopted in the fields of data mining, bioinformatics, data analysis, and data science. The core R language is extended by a large number of R package, software packages, which contain Reusability, reusable code, documentation, and sample data. Some of the most popular R packages are in the tidyverse collection, which enhances functionality for visualizing, transforming, and modelling data, as well as improves the ease of programming (according to the authors and users). R is free and open-source software distributed under the GNU General Public License. The language is implemented primarily in C (programming language), C, Fortran, and Self-hosting (compilers), R itself. Preprocessor, Precompiled executables are available for the major operating systems (including Linux, MacOS, and Microsoft Windows). Its core is an interpreted language with a na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digamma Function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing and Concave function, strictly concave on (0,\infty), and it Asymptotic analysis, asymptotically behaves as :\psi(z) \sim \ln - \frac, for complex numbers with large modulus (, z, \rightarrow\infty) in the Circular sector, sector , \arg z, 0. The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning Gamma, double-gamma). Gamma. Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives: :\log \Gamma(z+1)=\log(z)+\log \Gamma(z), Differentiating both sides with respect to gives: :\psi(z+1)=\psi(z)+\frac Since the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
