Particle-size distribution model wherein random samples are drawn from a mixture of a finite set of doubly-truncated univariate normal probability density functions: More...

#include <multiNormal.H>

Inheritance diagram for multiNormal:

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Collaboration diagram for multiNormal:

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Public Member Functions
	TypeName ("multiNormal")
	Runtime type information.
	multiNormal (const dictionary &dict, Random &rndGen)
	Construct from components.
	multiNormal (const multiNormal &p)
	Copy construct.
virtual autoPtr< distributionModel >	clone () const
	Construct and return a clone.
void	operator= (const multiNormal &)=delete
	No copy assignment.
virtual	~multiNormal ()=default
	Destructor.
virtual scalar	sample () const
	Sample the distribution.
scalar	sample (const scalar mu, const scalar sigma) const
	Sample the normal distribution.
virtual scalar	meanValue () const
	Return the theoretical mean of the distribution.
Public Member Functions inherited from distributionModel
	TypeName ("distributionModel")
	Runtime type information.
	declareRunTimeSelectionTable (autoPtr, distributionModel, dictionary,(const dictionary &dict, Random &rndGen),(dict, rndGen))
	Declare runtime constructor selection table.
	distributionModel (const word &name, const dictionary &dict, Random &rndGen)
	Construct from dictionary.
	distributionModel (const distributionModel &p)
	Copy construct.
virtual	~distributionModel ()=default
	Destructor.
virtual scalar	minValue () const
	Return the minimum of the distribution.
virtual scalar	maxValue () const
	Return the maximum of the distribution.

Additional Inherited Members
Static Public Member Functions inherited from distributionModel
static autoPtr< distributionModel >	New (const dictionary &dict, Random &rndGen)
	Selector.
Protected Member Functions inherited from distributionModel
virtual void	check () const
	Check that the distribution model is valid.
Protected Attributes inherited from distributionModel
const dictionary	distributionModelDict_
	Coefficients dictionary.
Random &	rndGen_
	Reference to the random number generator.
scalar	minValue_
	Minimum of the distribution.
scalar	maxValue_
	Maximum of the distribution.

Detailed Description

Particle-size distribution model wherein random samples are drawn from a mixture of a finite set of doubly-truncated univariate normal probability density functions:

$\‍[ g (\mathbf{x}; \mathbf{\mu}, \mathbf{\sigma}, A, B) = \sum_i w_i f(x_i; \mu_i, \sigma_i, A, B) \‍]$

with for any distribution:

$\‍[ f(x; \mu, \sigma, A, B) = \frac{1}{\sigma} \frac{ \phi \left( \frac{x - \mu}{\sigma} \right) }{ \Phi \left( \frac{B - \mu}{\sigma} \right) - \Phi \left( \frac{A - \mu}{\sigma} \right) } \‍]$

where

$f(x; \mu, \sigma, A, B)$	=	Doubly-truncated univariate normal distribution
$\mu$	=	Mean of the parent general normal distribution
$\sigma$	=	Standard deviation of the parent general normal distribution
$\phi(\cdot)$	=	General normal probability density function
$\Phi(\cdot)$	=	General normal cumulative distribution function
	=	Sample
	=	Minimum of the distribution (the same for each distribution)
	=	Maximum of the distribution (the same for each distribution)
	=	Weighting factor

Constraints:

$\infty > B > A > 0$
$x \in [B,A]$
$\sigma^2 > 0$

Random samples are generated by a combination of the inverse transform sampling technique and categorical sampling in three steps:

Draw a sample from the uniform probability density function on the unit interval
Find the interval among normalised cumulative weight intervals wherein resides
Draw a sample from the distribution corresponding to the interval by using the quantile function of the doubly-truncated univariate normal probability density function by the following expressions (similar to distributionModels::normal):

$\‍[ x = \mu + \sigma \sqrt{2} \, {erf}^{-1} \left( 2 p - 1 \right) \‍]$

with

$\‍[ p = u \, \left( \Phi\left( \frac{B - \mu}{\sigma} \right) - \Phi\left( \frac{A - \mu}{\sigma} \right) \right) + \Phi\left( \frac{A - \mu}{\sigma} \right) \‍]$

$\‍[ \Phi(\xi) = \frac{1}{2} \left( 1 + {erf}\left(\frac{\xi - \mu}{\sigma \sqrt{2} }\right) \right) \‍]$

where $ u $ is another sample drawn from the uniform probability density function on the unit interval $ (0, 1) $ .

Usage

Minimal example by using constant/<CloudProperties>:

subModels
{
    injectionModels
    {
        <name>
        {
            ...

            sizeDistribution
            {
                type        multiNormal;
                multiNormalDistribution
                {
                    minValue  <min>;
                    maxValue  <max>;
                    mu
                    (
                        <mean1>
                        <mean2>
                        ...
                    );
                    sigma
                    (
                        <standard deviation1>
                        <standard deviation2>
                        ...
                    );
                    weight
                    (
                        <weight1>
                        <weight2>
                        ...
                    );
                }
            }
        }
    }
}

where the entries mean:

Property	Description	Type	Reqd	Deflt
`type`	Type name: multiNormal	word	yes	-
`multiNormalDistribution`	Distribution settings	dict	yes	-
`minValue`	Minimum of the distribution	scalar	yes	-
`maxValue`	Maximum of the distribution	scalar	yes	-
`mu`	List of means of parent general normal distributions	scalarList	yes	-
`sigma`	List of standard deviations of parent general normal distributions	scalarList	yes	-
`weight`	List of weights of a given distribution in the distribution mixture	scalarList	yes	-

Notes

The sum of normal distributions (i.e. a mixture distribution) should not be confused with the sum of normally-distributed random variables.
minValue and maxValue are the same for all distributions in the distribution mixture.
weight should always be input in a non-decreasing (i.e. monotonic) order.

Source files

Definition at line 281 of file multiNormal.H.