#include <STK_Law_Poisson.h>

Inheritance diagram for STK::Law::Poisson:

Public Types
typedef IUnivLaw< int >	Base

Public Member Functions
	Poisson (Real const &lambda=1.)
	constructor

virtual	~Poisson ()
	destructor

Real const &	lambda () const

void	setLambda (Real const &lambda)

virtual int	rand () const

virtual Real	pdf (int const &x) const
	compute the probability distribution function.

virtual Real	lpdf (int const &x) const
	compute the log probability distribution function.

virtual Real	cdf (Real const &t) const
	compute the cumulative distribution function Give the probability that a Poisson random variate is less or equal to t.

virtual int	icdf (Real const &p) const
	inverse cumulative distribution function The quantile is defined as the smallest value q such that F(q) >= p , where F is the cumulative distribution function.

Public Member Functions inherited from STK::Law::IUnivLaw< int >
virtual	~IUnivLaw ()
	Virtual destructor.

virtual Real	lcdf (Real const &t) const
	compute the lower tail log-cumulative distribution function Give the log-probability that a random variate is less or equal to t.

virtual Real	cdfc (Real const &t) const
	calculate the complement of cumulative distribution function, called in statistics the survival function.

virtual Real	lcdfc (Real const &t) const
	calculate the log-complement of cumulative distribution function Give the log-probability that a random variate is greater than t.

Public Member Functions inherited from STK::Law::ILawBase
String const &	name () const

Static Public Member Functions
static int	rand (Real const &lambda)

static Real	pdf (int const &x, Real const &lambda)
	compute the probability distribution function Give the value of the pdf at the point x.

static Real	lpdf (int const &x, Real const &lambda)
	compute the log probability distribution function Give the value of the log-pdf at the point x.

static Real	cdf (Real const &t, Real const &lambda)
	compute the cumulative distribution function Give the probability that a Poisson random variate is less or equal to t.

static int	icdf (Real const &p, Real const &lambda)
	inverse cumulative distribution function The quantile is defined as the smallest value x such that F(x) >= p , where F is the cumulative distribution function.

Protected Attributes
Real	lambda_
	mean of the Poisson distribution

Protected Attributes inherited from STK::Law::ILawBase
String	name_
	Name of the Law.

Additional Inherited Members
Protected Member Functions inherited from STK::Law::IUnivLaw< int >
	IUnivLaw (String const &name)
	Constructor.

	IUnivLaw (IUnivLaw const &law)
	copy Constructor.

Protected Member Functions inherited from STK::Law::ILawBase
	ILawBase (String const &name)
	Constructor.

	~ILawBase ()
	destructor.

Detailed Description

Poisson distribution law.

In probability theory and statistics, the Poisson distribution, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.

The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. How many such events will occur during a fixed time interval? Under the right circumstances, this is a random number with a Poisson distribution.

A discrete random variable X is said to have a Poisson distribution with parameter $\lambda >0$ (the mean) if the probability mass function of X is given by

$f(k; \lambda) = P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k=0,1,2,\ldots,$

The positive real number $\lambda$ is equal to the expected value of X and also to its variance.

Definition at line 69 of file STK_Law_Poisson.h.

Member Typedef Documentation

◆ Base

typedef IUnivLaw<int> STK::Law::Poisson::Base

Definition at line 72 of file STK_Law_Poisson.h.

Constructor & Destructor Documentation

◆ Poisson()

STK::Law::Poisson::Poisson ( Real const & lambda = 1. )

inline

constructor

Parameters

lambda mean of a Poisson distribution

Definition at line 76 of file STK_Law_Poisson.h.

77 : Base(_T("Poisson")), lambda_(lambda) {}

_T

#define _T(x)

Let x unmodified.

Definition STK_Typedefs.h:71

STK::Law::Poisson::lambda_

Real lambda_

mean of the Poisson distribution

Definition STK_Law_Poisson.h:165

STK::Law::Poisson::Base

IUnivLaw< int > Base

Definition STK_Law_Poisson.h:72

STK::Law::Poisson::lambda

Real const & lambda() const

Definition STK_Law_Poisson.h:81

◆ ~Poisson()

virtual STK::Law::Poisson::~Poisson ( )

inlinevirtual

destructor

Definition at line 79 of file STK_Law_Poisson.h.

79{}

Member Function Documentation

◆ cdf() [1/2]

Real STK::Law::Poisson::cdf ( Real const & t ) const

virtual

compute the cumulative distribution function Give the probability that a Poisson random variate is less or equal to t.

Parameters

t	a real value

Returns: the value of the cdf

Implements STK::Law::IUnivLaw< int >.

Definition at line 202 of file STK_Law_Poisson.cpp.

{
  // check trivial values
  if (t < 0)    return 0;
  if (lambda_ == 0.) return( 1. );
  // use gamma ratio function
  return Funct::gammaRatioQ(std::floor(t) + 1, lambda_);
}

References STK::Funct::gammaRatioQ(), and lambda_.

Referenced by icdf(), and icdf().

◆ cdf() [2/2]

Real STK::Law::Poisson::cdf	(	Real const &	t,
		Real const &	lambda
	)

static

compute the cumulative distribution function Give the probability that a Poisson random variate is less or equal to t.

Parameters

t	a real value
lambda	the mean

Returns: the value of the cdf at t

Definition at line 297 of file STK_Law_Poisson.cpp.

{
  // check trivial values
  if (t < 0) return 0;
  if (lambda == 0.) return( 1. );
  // use gamma ratio function
  return Funct::gammaRatioQ(std::floor(t) + 1, lambda);
}

References STK::Funct::gammaRatioQ(), and lambda().

◆ icdf() [1/2]

int STK::Law::Poisson::icdf ( Real const & p ) const

virtual

inverse cumulative distribution function The quantile is defined as the smallest value q such that F(q) >= p , where F is the cumulative distribution function.

Parameters

p	a probability number

Returns: the quantile for p

Implements STK::Law::IUnivLaw< int >.

Definition at line 215 of file STK_Law_Poisson.cpp.

{
  // check trivial values
  if (p == 0.) return 0;
  if (p == 1.) return Arithmetic<int>::max(); // infty does not exists for int
  // check values 0 and 1
  Real el = std::exp(-lambda_);
  if (p<el) return 0;
  if (p<(1+lambda_)*el) return 1;
  // othewise search
  Real q = gauss_icdf_fast(p), sqlambda = std::sqrt(lambda_);
  int k = std::max(0., round(lambda_ + q * sqlambda + (q-1.)*(q+1.)*(1.-q/(12.*sqlambda))/6));
  if (cdf(k) >= p)
  { /* decreasing search */
    while(1)
    {
      if ( cdf(k - 1) < p) return k;
      k--;
    }
  }
  else
  { /* increasing search */
    while(1)
    {
      k++;
      if( cdf(k) >= p) return k;
    }
  }
  return k; // avoid warning at compilation
}

References cdf(), and lambda_.

◆ icdf() [2/2]

int STK::Law::Poisson::icdf	(	Real const &	p,
		Real const &	lambda
	)

static

inverse cumulative distribution function The quantile is defined as the smallest value x such that F(x) >= p , where F is the cumulative distribution function.

Parameters

p	a probability number
lambda	the mean

Returns: the quantile for p

Definition at line 310 of file STK_Law_Poisson.cpp.

{
  // check trivial values
  if (p == 0.) return 0;
  if (p == 1.) return Arithmetic<int>::max();
  // check values 0 and 1
  Real el = std::exp(-lambda);
  if (p<el) return 0;
  if (p< (1+lambda)*el) return 1;
  // othewise search
  Real q = gauss_icdf_fast(p), sqlambda = std::sqrt(lambda);
  int k = round(lambda + q * sqlambda + (q-1.)*(q+1.)*(1.-q/(12.*sqlambda))/6);
  if (cdf(k, lambda) >= p)
  { /* decreasing search */
    while(1)
    {
      if ( cdf(k - 1,lambda) < p) return k;
      k--;
    }
  }
  else
  { /* increasing search */
    while(1)
    {
      k++;
      if( cdf(k, lambda) >= p) return k;
    }
  }
  return k; // avoid warning at compilation
}

References cdf(), and lambda().

◆ lambda()

Real const & STK::Law::Poisson::lambda ( ) const

inline

Returns: the mean

Definition at line 81 of file STK_Law_Poisson.h.

81{ return lambda_;}

References lambda_.

Referenced by cdf(), icdf(), lpdf(), pdf(), rand(), and setLambda().

◆ lpdf() [1/2]

Real STK::Law::Poisson::lpdf ( int const & x ) const

virtual

compute the log probability distribution function.

Give the value of the log-pdf at the point x.

Parameters

x	an integer value

Returns: the value of the log-pdf

Reimplemented from STK::Law::IUnivLaw< int >.

Definition at line 183 of file STK_Law_Poisson.cpp.

{
  // check trivial values
  if (x<0) return( -Arithmetic<Real>::infinity() );
  // if lambda is 0, we have P(X=0) = 1
  if (lambda_==0.) return( (x==0) ? 0 : -Arithmetic<Real>::infinity() );
  // special value
  if (x==0) return( -lambda_ );
  // stirling approximation and deviance
  return( -Funct::lgammaStirlingError(x)-Funct::dev0(x, lambda_)
          -Const::_LNSQRT2PI_-std::log((Real)x)/2.
        );
}

References STK::Funct::dev0(), lambda_, and STK::Funct::lgammaStirlingError().

Referenced by STK::PoissonBase< Derived >::lnComponentProbability().

◆ lpdf() [2/2]

Real STK::Law::Poisson::lpdf	(	int const &	x,
		Real const &	lambda
	)

static

compute the log probability distribution function Give the value of the log-pdf at the point x.

Parameters

x	a binary value
lambda	the mean

Returns: the value of the log-pdf

Definition at line 277 of file STK_Law_Poisson.cpp.

{
  // check trivial values
  if (x<0) return( -Arithmetic<Real>::infinity() );
  // if lambda is 0, we have P(X=0) = 1
  if (lambda==0.) return( (x==0) ? 0 : -Arithmetic<Real>::infinity() );
  // special value
  if (x==0) return( -lambda );
  // stirling approximation and deviance
  return( -Funct::lgammaStirlingError((Real)x)-Funct::dev0((Real)x, lambda)
          -Const::_LNSQRT2PI_-std::log((Real)x)/2.
        );
}

References STK::Funct::dev0(), lambda(), and STK::Funct::lgammaStirlingError().

◆ pdf() [1/2]

Real STK::Law::Poisson::pdf ( int const & x ) const

virtual

compute the probability distribution function.

Give the value of the pdf at the point x.

Parameters

x	a binary value

Returns: the value of the pdf

Implements STK::Law::IUnivLaw< int >.

Definition at line 164 of file STK_Law_Poisson.cpp.

{
  // check trivial values
  if (x<0) return( 0. );
  // if lambda is 0, we have P(X=0) = 1
  if (lambda_==0.) return( (x==0) ? 1. : 0. );
  // special value
  if (x==0) return( Real(std::exp(-lambda_)) );
  // stirling approximation and deviance
  return( std::exp(-Funct::lgammaStirlingError((Real)x)-Funct::dev0((Real)x, lambda_))
          /(Const::_SQRT2PI_*std::sqrt(x))
        );
}

References STK::Funct::dev0(), lambda_, and STK::Funct::lgammaStirlingError().

◆ pdf() [2/2]

Real STK::Law::Poisson::pdf	(	int const &	x,
		Real const &	lambda
	)

static

compute the probability distribution function Give the value of the pdf at the point x.

Parameters

x	a binary value
lambda	the mean

Returns: the value of the pdf

Definition at line 257 of file STK_Law_Poisson.cpp.

{
  // check trivial values
  if (x<0) return( 0. );
  // if lambda is 0, we have P(X=0) = 1
  if (lambda==0.) return( (x==0) ? 1. : 0. );
  // special value
  if (x==0) return( Real(std::exp(-lambda)) );
  // stirling approximation and deviance
  return( std::exp(-Funct::lgammaStirlingError((Real)x)-Funct::dev0((Real)x, lambda))
          /(Const::_SQRT2PI_*std::sqrt(x))
        );
}

References STK::Funct::dev0(), lambda(), and STK::Funct::lgammaStirlingError().

◆ rand() [1/2]

int STK::Law::Poisson::rand ( ) const

virtual

Returns: a Poisson random variate .

Implements STK::Law::IUnivLaw< int >.

Definition at line 155 of file STK_Law_Poisson.cpp.

{
 return poisson_icd_raw(Law::generator.randUnif(), lambda_);
}

References lambda_.

Referenced by STK::PoissonBase< Derived >::rand().

◆ rand() [2/2]

int STK::Law::Poisson::rand ( Real const & lambda )

static

Parameters

lambda the mean

Returns: a int random variate.

Definition at line 249 of file STK_Law_Poisson.cpp.

250{ return poisson_icd_raw(Law::generator.randUnif(), lambda);}

References lambda().

◆ setLambda()

void STK::Law::Poisson::setLambda ( Real const & lambda )

inline

Parameters

lambda mean to set

Definition at line 83 of file STK_Law_Poisson.h.

    {
      if (lambda<0)
      { STKDOMAIN_ERROR_1ARG(Poisson::setLambda,lambda,lambda must be >= 0);}
      lambda_ = lambda;
    }

References lambda(), lambda_, setLambda(), and STKDOMAIN_ERROR_1ARG.

Referenced by setLambda().

Member Data Documentation

◆ lambda_

Real STK::Law::Poisson::lambda_

protected

mean of the Poisson distribution

Definition at line 165 of file STK_Law_Poisson.h.

Referenced by cdf(), icdf(), lambda(), lpdf(), pdf(), rand(), and setLambda().

The documentation for this class was generated from the following files:

Public Types

Public Member Functions

Static Public Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Member Typedef Documentation

◆ Base

Constructor & Destructor Documentation

◆ Poisson()

◆ ~Poisson()

Member Function Documentation

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ icdf() [1/2]

◆ icdf() [2/2]

◆ lambda()

◆ lpdf() [1/2]

◆ lpdf() [2/2]

◆ pdf() [1/2]

◆ pdf() [2/2]

◆ rand() [1/2]

◆ rand() [2/2]

◆ setLambda()

Member Data Documentation

◆ lambda_