NormalDistribution

Description

The normal (or Gaussian) distribution object defines a normal distribution function with the provided mean and standard_deviation parameters. The probability density function (PDF) of the normal distribution is given by the Eq. (1).

$(1)var element = document.getElementById("moose-equation-32d0a8c2-61af-446d-ac20-3ccf4ab2e790");katex.render(" f(x \\; | \\; \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2\\pi\\sigma^2} } \\; e^{ -\\frac{(x-\\mu)^2}{2\\sigma^2} }", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where $var element = document.getElementById("moose-equation-5d2f5db3-2894-4344-ac8b-1e8f20154d3f");katex.render("\\mu", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the mean and $var element = document.getElementById("moose-equation-680f7a00-438b-4bcc-905d-3d34512cb369");katex.render("\\sigma", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the standard deviation ( $var element = document.getElementById("moose-equation-c53834ff-4603-4100-86d1-9f0899322faf");katex.render("\\sigma > 0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ) of the distribution.

This implementation of a normal distribution uses a numerical approximation described in Kennedy and Gentle (1980).

Example Input Syntax

The following input file defines a normal distribution with a mean of 0 and a standard deviation of 1.

[Distributions]
  [normal_test]
    type = NormalDistribution
    mean = 0
    standard_deviation = 1
  []
[]

/opt/civet/build_0/moose/modules/stochastic_tools/test/tests/distributions/normal.i

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 1
  ny = 1
[]

[Variables]
  [u]
  []
[]

[Distributions]
  [normal_test]
    type = NormalDistribution
    mean = 0
    standard_deviation = 1
  []
[]

[Postprocessors]
  [cdf]
    type = TestDistributionPostprocessor
    distribution = normal_test
    value = 0
    method = cdf
    execute_on = initial
  []
  [pdf]
    type = TestDistributionPostprocessor
    distribution = normal_test
    value = 0
    method = pdf
    execute_on = initial
  []
  [quantile]
    type = TestDistributionPostprocessor
    distribution = normal_test
    value = 0.5
    method = quantile
    execute_on = initial
  []
[]

[Executioner]
  type = Steady
[]

[Problem]
  solve = false
  kernel_coverage_check = false
[]

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

Input Parameters

meanMean (or expectation) of the distribution.
C++ Type:double
Options:
Description:Mean (or expectation) of the distribution.
standard_deviationStandard deviation of the distribution
C++ Type:double
Options:
Description:Standard deviation of the distribution

Required Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.

Advanced Parameters

Input Files

modules/stochastic_tools/test/tests/distributions/normal_direct.i
modules/stochastic_tools/test/tests/distributions/normal.i

modules/stochastic_tools/test/tests/distributions/normal_direct.i

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 1
  ny = 1
[]

[Variables]
  [u]
  []
[]

[Distributions]
  [normal_test]
    type = NormalDistribution
    mean = 0
    standard_deviation = 1
  []
[]

[Postprocessors]
  [cdf]
    type = TestDistributionDirectPostprocessor
    distribution = normal_test
    value = 0
    method = cdf
    execute_on = initial
  []
  [pdf]
    type = TestDistributionDirectPostprocessor
    distribution = normal_test
    value = 0
    method = pdf
    execute_on = initial
  []
  [quantile]
    type = TestDistributionDirectPostprocessor
    distribution = normal_test
    value = 0.5
    method = quantile
    execute_on = initial
  []
[]

[Executioner]
  type = Steady
[]

[Problem]
  solve = false
  kernel_coverage_check = false
[]

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

modules/stochastic_tools/test/tests/distributions/normal.i

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 1
  ny = 1
[]

[Variables]
  [u]
  []
[]

[Distributions]
  [normal_test]
    type = NormalDistribution
    mean = 0
    standard_deviation = 1
  []
[]

[Postprocessors]
  [cdf]
    type = TestDistributionPostprocessor
    distribution = normal_test
    value = 0
    method = cdf
    execute_on = initial
  []
  [pdf]
    type = TestDistributionPostprocessor
    distribution = normal_test
    value = 0
    method = pdf
    execute_on = initial
  []
  [quantile]
    type = TestDistributionPostprocessor
    distribution = normal_test
    value = 0.5
    method = quantile
    execute_on = initial
  []
[]

[Executioner]
  type = Steady
[]

[Problem]
  solve = false
  kernel_coverage_check = false
[]

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

Child Objects

modules/stochastic_tools/include/distributions/TruncatedNormalDistribution.h
modules/stochastic_tools/include/distributions/JohnsonSBDistribution.h

modules/stochastic_tools/include/distributions/TruncatedNormalDistribution.h

// This file is part of the MOOSE framework
// https://www.mooseframework.org
//
// All rights reserved, see COPYRIGHT for full restrictions
// https://github.com/idaholab/moose/blob/master/COPYRIGHT
//
// Licensed under LGPL 2.1, please see LICENSE for details
// https://www.gnu.org/licenses/lgpl-2.1.html

#pragma once

#include "NormalDistribution.h"

class TruncatedNormalDistribution;

template <>
InputParameters validParams<TruncatedNormalDistribution>();

/**
 * A class used to generate a truncated normal distribution
 */
class TruncatedNormalDistribution : public NormalDistribution
{
public:
  static InputParameters validParams();

  TruncatedNormalDistribution(const InputParameters & parameters);

  virtual Real pdf(const Real & x) const override;
  virtual Real cdf(const Real & x) const override;
  virtual Real quantile(const Real & p) const override;

  static Real pdf(const Real & x,
                  const Real & mean,
                  const Real & std_dev,
                  const Real & lower_bound,
                  const Real & upper_bound);
  static Real cdf(const Real & x,
                  const Real & mean,
                  const Real & std_dev,
                  const Real & lower_bound,
                  const Real & upper_bound);
  static Real quantile(const Real & p,
                       const Real & mean,
                       const Real & std_dev,
                       const Real & lower_bound,
                       const Real & upper_bound);

protected:
  /// The lower bound for the distribution
  const Real & _lower_bound;

  /// The upper bound for the distribution
  const Real & _upper_bound;
};

modules/stochastic_tools/include/distributions/JohnsonSBDistribution.h

// This file is part of the MOOSE framework
// https://www.mooseframework.org
//
// All rights reserved, see COPYRIGHT for full restrictions
// https://github.com/idaholab/moose/blob/master/COPYRIGHT
//
// Licensed under LGPL 2.1, please see LICENSE for details
// https://www.gnu.org/licenses/lgpl-2.1.html

#pragma once

#include "NormalDistribution.h"

class JohnsonSBDistribution;

template <>
InputParameters validParams<JohnsonSBDistribution>();

/**
 * A class used to generate a Johnson SB distribution
 */
class JohnsonSBDistribution : public NormalDistribution
{
public:
  static InputParameters validParams();

  JohnsonSBDistribution(const InputParameters & parameters);

  virtual Real pdf(const Real & x) const override;
  virtual Real cdf(const Real & x) const override;
  virtual Real quantile(const Real & p) const override;

  static Real
  pdf(const Real & x, const Real & a, const Real & b, const Real & alpha_1, const Real & alpha_2);
  static Real
  cdf(const Real & x, const Real & a, const Real & b, const Real & alpha_1, const Real & alpha_2);
  static Real quantile(
      const Real & p, const Real & a, const Real & b, const Real & alpha_1, const Real & alpha_2);

protected:
  /// The lower location parameter, a
  const Real & _lower;

  /// The upper location parameter, b
  const Real & _upper;

  /// The first shape parameter, alpha_1
  const Real & _alpha_1;

  /// The second shape parameter, alpha_2
  const Real & _alpha_2;
};

References

Kennedy and Gentle. Statistical Computing. Marcel Dekker, 1980.

Description
Example Input Syntax
Input Parameters
Input Files
Child Objects
References

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