Gamma Distribution

Gamma distribution

Overview

This object defines a gamma distribution with "shape" parameter ( $var element = document.getElementById("moose-equation-367428b6-5929-446c-a9f1-149a15ad8d37");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ or $var element = document.getElementById("moose-equation-82251bff-1949-45da-bfb0-1b09fb342680");katex.render("\\alpha", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ) and "scale" parameter ( $var element = document.getElementById("moose-equation-71e4214b-2530-4aef-8aa2-9ceb22f08f48");katex.render("\\theta", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ or $var element = document.getElementById("moose-equation-efd05e76-3006-437d-b78a-e27bea8cf29c");katex.render("1/\\beta", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ). The probability density function, cumulative distribution function, and quantile are defined in Eq. (1), Eq. (2), and Eq. (3), respectively.

(1)var element = document.getElementById("moose-equation-e13add14-6539-482d-ae39-00c3e7d9981c");katex.render("f(x; \\alpha, \\beta) = \\frac{\\beta^\\alpha x^{\\alpha-1} e^{-\\beta x}}{\\Gamma(\\alpha)},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

(2)var element = document.getElementById("moose-equation-03d63104-f8d7-4755-b3f1-3e6a3ece8f79");katex.render("F(x; \\alpha, \\beta) = \\Gamma(\\alpha, \\beta x),", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

(3)var element = document.getElementById("moose-equation-e73d02a4-9fe5-45d2-a4ed-154138594c35");katex.render("F^{-1}(p; \\alpha, \\beta) = \\frac{\\Gamma^{-1}(\\alpha, p)}{\\beta},", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-56dcc62b-d5c9-45a4-b7a0-b9d557080a53");katex.render("x,\\alpha,\\beta > 0", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-735cbeda-c2cf-44b2-a3e5-07132850e19a");katex.render("0\\leq p \\leq 1", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . $var element = document.getElementById("moose-equation-ea0e39c3-39c0-4e9c-882b-a61a411444fa");katex.render("\\Gamma(a)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the gamma function defined by Eq. (4), $var element = document.getElementById("moose-equation-1decbefd-337d-4f01-8f98-53462d9e0e21");katex.render("\\Gamma(a, x)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the lower incomplete gamma function defined by Eq. (5), and $var element = document.getElementById("moose-equation-518544e0-5d7c-4ec0-b0b9-10116d9780cf");katex.render("\\Gamma^{-1}(a, p)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the inverse of the incomplete gamma function. $var element = document.getElementById("moose-equation-94d3d70e-60f2-4607-9b65-31daead33d7a");katex.render("\\Gamma", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-d04615dc-62da-438c-a7e0-5a22815f7233");katex.render("\\Gamma^{-1}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are computed using a custom iterative procedure, which obtains an accuracy of about $var element = document.getElementById("moose-equation-56694e06-8d14-4a5d-b743-ab05b53b304a");katex.render("10^{-14}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ .

(4)var element = document.getElementById("moose-equation-e40c6b06-e845-450c-93e4-dfddd9600a47");katex.render("\\Gamma(a) = \\int_0^\\infty x^{a-1}e^{-x}dx", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

(5)var element = document.getElementById("moose-equation-37cdd738-1282-4bea-b9ec-730ffb51f9b9");katex.render("\\Gamma(a, x) = \\frac{1}{\\Gamma(a)}\\int_0^x t^{a-1}e^{-t}dt", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Input Parameters

shapeShape parameter (k or alpha).
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Shape parameter (k or alpha).

Required Parameters

scale1Scale parameter (theta or 1/beta).
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scale parameter (theta or 1/beta).

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:No
Description:Set the enabled status of the MooseObject.

Advanced Parameters

Input Files

(modules/stochastic_tools/test/tests/distributions/gamma.i)

(modules/stochastic_tools/test/tests/distributions/gamma.i)

[StochasticTools]
[]

[Distributions]
  [gamma1]
    type = Gamma
    shape = 7.7229537456741735
    scale = 1
  []
  [gamma2]
    type = Gamma
    shape = 0.606859544986121
    scale = 0.4477587035791184
  []
[]

[Postprocessors]
  [cdf1]
    type = TestDistributionPostprocessor
    distribution = gamma1
    value = 8.565580942828962
    method = cdf
    execute_on = initial
  []
  [pdf1]
    type = TestDistributionPostprocessor
    distribution = gamma1
    value = 5.25180066219824
    method = pdf
    execute_on = initial
  []
  [quantile1]
    type = TestDistributionPostprocessor
    distribution = gamma1
    value = 0.25545064031407105
    method = quantile
    execute_on = initial
  []
  [cdf2]
    type = TestDistributionPostprocessor
    distribution = gamma2
    value = 1
    method = cdf
    execute_on = initial
  []
  [pdf2]
    type = TestDistributionPostprocessor
    distribution = gamma2
    value = 1
    method = pdf
    execute_on = initial
  []
  [quantile2]
    type = TestDistributionPostprocessor
    distribution = gamma2
    value = 0.5
    method = quantile
    execute_on = initial
  []
[]

[Outputs]
  execute_on = 'INITIAL'
  csv = true
[]

Overview
Input Parameters
Input Files