Probability distributions (discrete and continuous) for use in the likelihood function. More...

Files
file	dnorm.hpp
	Univariate normal density.

Functions
template<class Type >
Type	dbeta (Type x, Type shape1, Type shape2, int give_log)
	Probability density function of the beta distribution. More...

template<class Type >
Type	dbinom (Type k, Type size, Type prob, int give_log=0)
	Probability mass function of the binomial distribution. More...

template<class Type >
Type	dbinom_robust (Type k, Type size, Type logit_p, int give_log=0)
	Density of binomial distribution parameterized via logit(prob) More...

template<class T1 , class T2 , class T3 >
T1	dcompois (T1 x, T2 mode, T3 nu, int give_log=0)
	Conway-Maxwell-Poisson. Calculate density. More...

template<class T1 , class T2 , class T3 >
T1	dcompois2 (T1 x, T2 mean, T3 nu, int give_log=0)
	Conway-Maxwell-Poisson. Calculate density parameterized via the mean. More...

template<class Type >
Type	df (Type x, Type df1, Type df2, int give_log)
	Probability density function of the Fisher distribution. More...

template<class Type >
Type	dgamma (Type y, Type shape, Type scale, int give_log=0)
	Density of X where X~gamma distributed.

template<class Type >
Type	dlgamma (Type y, Type shape, Type scale, int give_log=0)
	Density of log(X) where X~gamma distributed.

template<class Type >
Type	dlogis (Type x, Type location, Type scale, int give_log)
	Probability density function of the logistic distribution. More...

template<class Type >
Type	dmultinom (vector< Type > x, vector< Type > p, int give_log=0)
	Probability mass function of the multinomial distribution. More...

template<class Type >
Type	dnbinom (const Type &x, const Type &size, const Type &prob, int give_log=0)
	Negative binomial probability function.Parameterized through size and prob parameters, following R-convention.

template<class Type >
Type	dnbinom2 (const Type &x, const Type &mu, const Type &var, int give_log=0)
	Negative binomial probability function.Alternative parameterization through mean and variance parameters.

template<class Type >
Type	dnbinom_robust (const Type &x, const Type &log_mu, const Type &log_var_minus_mu, int give_log=0)
	Negative binomial probability function. More...

template<class Type >
Type	dnorm (Type x, Type mean, Type sd, int give_log=0)
	Probability density function of the normal distribution. More...

template<class Type >
Type	dpois (const Type &x, const Type &lambda, int give_log=0)
	Poisson probability function.

template<class Type >
Type	dsn (Type x, Type alpha, int give_log=0)
	Probability density function of the skew-normal distribution. More...

template<class Type >
Type	dt (Type x, Type df, int give_log)
	Probability density function of the Student t-distribution. More...

template<class Type >
Type	dtweedie (Type y, Type mu, Type phi, Type p, int give_log=0)
	dtweedie function (same as dtweedie.series from R package 'tweedie'). More...

template<class Type >
Type	dzinbinom (const Type &x, const Type &size, const Type &p, const Type &zip, int give_log=0)
	Zero-Inflated negative binomial probability function. More...

template<class Type >
Type	dzinbinom2 (const Type &x, const Type &mu, const Type &var, const Type &zip, int give_log=0)
	Zero-Inflated negative binomial probability function. More...

template<class Type >
Type	dzipois (const Type &x, const Type &lambda, const Type &zip, int give_log=0)
	Zero-Inflated Poisson probability function. More...

template<class Type >
Type	pbeta (Type q, Type shape1, Type shape2)
	Distribution function of the beta distribution (following R argument convention). More...

template<class Type >
Type	pgamma (Type q, Type shape, Type scale=1.)
	Distribution function of the gamma distribution (following R argument convention).

template<class Type >
Type	pnorm (Type q, Type mean=0., Type sd=1.)
	Distribution function of the normal distribution (following R argument convention).

template<class Type >
Type	ppois (Type q, Type lambda)
	Distribution function of the poisson distribution (following R argument convention).

template<class Type >
Type	qbeta (Type p, Type shape1, Type shape2)
	Quantile function of the beta distribution (following R argument convention). More...

template<class Type >
Type	qgamma (Type q, Type shape, Type scale=1.)
	Quantile function of the gamma distribution (following R argument convention).

template<class Type >
Type	qnorm (Type p, Type mean=0., Type sd=1.)
	Quantile function of the normal distribution (following R argument convention).

Exponential distribution.
Functions relative to the exponential distribution.
template<class Type >
Type	pexp (Type x, Type rate)
	Cumulative distribution function of the exponential distribution. More...

template<class Type >
Type	dexp (Type x, Type rate, int give_log=0)
	Probability density function of the exponential distribution. More...

template<class Type >
Type	qexp (Type p, Type rate)
	Inverse cumulative distribution function of the exponential distribution. More...

Weibull distribution.
Functions relative to the Weibull distribution.
template<class Type >
Type	pweibull (Type x, Type shape, Type scale)
	Cumulative distribution function of the Weibull distribution. More...

template<class Type >
Type	dweibull (Type x, Type shape, Type scale, int give_log=0)
	Probability density function of the Weibull distribution. More...

template<class Type >
Type	qweibull (Type p, Type shape, Type scale)
	Inverse cumulative distribution function of the Weibull distribution. More...

Sinh-asinh distribution.
Functions relative to the sinh-asinh distribution.
template<class Type >
Type	dSHASHo (Type x, Type mu, Type sigma, Type nu, Type tau, int give_log=0)
	Probability density function of the sinh-asinh distribution. More...

template<class Type >
Type	pSHASHo (Type q, Type mu, Type sigma, Type nu, Type tau, int give_log=0)
	Cumulative distribution function of the sinh-asinh distribution. More...

template<class Type >
Type	qSHASHo (Type p, Type mu, Type sigma, Type nu, Type tau, int log_p=0)
	Quantile function of the sinh-asinh distribution. More...

template<class Type >
Type	norm2SHASHo (Type x, Type mu, Type sigma, Type nu, Type tau, int log_p=0)
	Transforms a normal variable into a sinh-asinh variable. More...

Detailed Description

Probability distributions (discrete and continuous) for use in the likelihood function.

Function Documentation

§ dbeta()

template<class Type >

Type dbeta	(	Type	x,
		Type	shape1,
		Type	shape2,
		int	give_log
	)

Probability density function of the beta distribution.

Parameters

shape1	First shape parameter. Must be strictly positive.
shape2	Second shape parameter. Must be strictly positive.
give_log	true if one wants the log-probability, false otherwise.

Definition at line 227 of file distributions_R.hpp.

§ dbinom()

template<class Type >

Type dbinom	(	Type	k,
		Type	size,
		Type	prob,
		int	give_log = `0`
	)

Probability mass function of the binomial distribution.

Parameters

k	Number of successes.
size	Number of trials.
prob	Probability of success.
give_log	true if one wants the log-probability, false otherwise.

Definition at line 183 of file distributions_R.hpp.

§ dbinom_robust()

template<class Type >

Type dbinom_robust	(	Type	k,
		Type	size,
		Type	logit_p,
		int	give_log = `0`
	)

Density of binomial distribution parameterized via logit(prob)

This version should be preferred when working on the logit scale as it is numerically stable for probabilities close to 0 or 1.

Definition at line 205 of file distributions_R.hpp.

§ dcompois()

template<class T1 , class T2 , class T3 >

T1 dcompois	(	T1	x,
		T2	mode,
		T3	nu,
		int	give_log = `0`
	)

Conway-Maxwell-Poisson. Calculate density.

\[ P(X=x) \propto \frac{\lambda^x}{(x!)^\nu}\:,x=0,1,\ldots \]

Silently returns NaN if not within the valid parameter range:

\[ (0 \leq x) \land (0 < \lambda) \land (0 < \nu) \]

.

Parameters

x	Observation
mode	Approximate mode \( \lambda^\frac{1}{\nu} \)
nu	\( \nu \)

Examples:: compois.cpp.

Definition at line 627 of file distributions_R.hpp.

§ dcompois2()

template<class T1 , class T2 , class T3 >

T1 dcompois2	(	T1	x,
		T2	mean,
		T3	nu,
		int	give_log = `0`
	)

Conway-Maxwell-Poisson. Calculate density parameterized via the mean.

Silently returns NaN if not within the valid parameter range:

\[ (0 \leq x) \land (0 < E[X]) \land (0 < \nu) \]

.

Parameters

x	Observation
mean	\( E[X] \)
nu	\( \nu \)

Examples:: compois.cpp.

Definition at line 647 of file distributions_R.hpp.

§ dexp()

template<class Type >

Type dexp	(	Type	x,
		Type	rate,
		int	give_log = `0`
	)

Probability density function of the exponential distribution.

Parameters

rate	Rate parameter. Must be strictly positive.
give_log	true if one wants the log-probability, false otherwise.

Definition at line 94 of file distributions_R.hpp.

§ df()

template<class Type >

Type df	(	Type	x,
		Type	df1,
		Type	df2,
		int	give_log
	)

Probability density function of the Fisher distribution.

Parameters

df1	Degrees of freedom 1.
df2	Degrees of freedom 2.
give_log	true if one wants the log-probability, false otherwise.

Definition at line 246 of file distributions_R.hpp.

§ dlogis()

template<class Type >

Type dlogis	(	Type	x,
		Type	location,
		Type	scale,
		int	give_log
	)

Probability density function of the logistic distribution.

Parameters

location	Location parameter.
scale	Scale parameter. Must be strictly positive.
give_log	true if one wants the log-probability, false otherwise.

Definition at line 263 of file distributions_R.hpp.

§ dmultinom()

template<class Type >

Type dmultinom	(	vector< Type >	x,
		vector< Type >	p,
		int	give_log = `0`
	)

Probability mass function of the multinomial distribution.

Parameters

x	Vector of length K of integers.
p	Vector of length K, specifying the probability for the K classes (note, unlike in R these must sum to 1).
give_log	true if one wants the log-probability, false otherwise.

Definition at line 312 of file distributions_R.hpp.

§ dnbinom_robust()

template<class Type >

Type dnbinom_robust	(	const Type &	x,
		const Type &	log_mu,
		const Type &	log_var_minus_mu,
		int	give_log = `0`
	)

inline

Negative binomial probability function.

More robust parameterization through \(log(\mu)\) and \(log(\sigma^2-\mu)\) parameters.

Definition at line 103 of file lgamma.hpp.

§ dnorm()

template<class Type >

Type dnorm	(	Type	x,
		Type	mean,
		Type	sd,
		int	give_log = `0`
	)

Probability density function of the normal distribution.

Parameters

x	vector of observations
mean	mean of the normal distribution
sd	standard deviation of the normal distribution. must be strictly positive.
give_log	true if one wants the log-probability, false otherwise.

Examples:: adaptive_integration.cpp, fft.cpp, linreg.cpp, linreg_parallel.cpp, longlinreg.cpp, lr_test.cpp, mvrw.cpp, mvrw_sparse.cpp, register_atomic.cpp, register_atomic_parallel.cpp, sam.cpp, sde_linear.cpp, sdv_multi.cpp, socatt.cpp, thetalog.cpp, TMBad/sam.cpp, TMBad/thetalog.cpp, transform.cpp, transform2.cpp, transform_parallel.cpp, validation/MVRandomWalkValidation.cpp, validation/randomwalkvalidation.cpp, and validation/rickervalidation.cpp.

Definition at line 17 of file dnorm.hpp.

Referenced by density::ARk_t< scalartype_ >::operator()().

§ dSHASHo()

template<class Type >

Type dSHASHo	(	Type	x,
		Type	mu,
		Type	sigma,
		Type	nu,
		Type	tau,
		int	give_log = `0`
	)

Probability density function of the sinh-asinh distribution.

Parameters

mu	Location.
sigma	Scale.
nu	Skewness.
tau	Kurtosis.
give_log	true if one wants the log-probability, false otherwise.

Notation adopted from R package "gamlss.dist".

Probability density given in (2) in Jones and Pewsey (2009) Biometrika (2009) 96 (4): 761-780.

It is not possible to call this function with nu a vector or tau a vector.

Definition at line 339 of file distributions_R.hpp.

§ dsn()

template<class Type >

Type dsn	(	Type	x,
		Type	alpha,
		int	give_log = `0`
	)

Probability density function of the skew-normal distribution.

Parameters

alpha	Slant parameter.
give_log	true if one wants the log-probability, false otherwise.

Definition at line 279 of file distributions_R.hpp.

§ dt()

template<class Type >

Type dt	(	Type	x,
		Type	df,
		int	give_log
	)

Probability density function of the Student t-distribution.

Parameters

df	Degree of freedom.
give_log	true if one wants the log-probability, false otherwise.

Examples:: hmm.cpp, and sde_linear.cpp.

Definition at line 295 of file distributions_R.hpp.

§ dtweedie()

template<class Type >

Type dtweedie	(	Type	y,
		Type	mu,
		Type	phi,
		Type	p,
		int	give_log = `0`
	)

dtweedie function (same as dtweedie.series from R package 'tweedie').

Silently returns NaN if not within the valid parameter range:

\[ (0 \leq y) \land (0 < \mu) \land (0 < \phi) \land (1 < p) \land (p < 2) \]

.

This implementation can be used for both constant and variable y. However, note that the derivative wrt. y is hardcoded to return zero (!).

Note: Parameter order differs from the R version.

Warning: The derivative wrt. the y argument is disabled (zero). Hence the tweedie distribution can only be used for data (not random effects).

Examples:: tweedie.cpp.

Definition at line 554 of file distributions_R.hpp.

§ dweibull()

template<class Type >

Type dweibull	(	Type	x,
		Type	shape,
		Type	scale,
		int	give_log = `0`
	)

Probability density function of the Weibull distribution.

Parameters

shape	Shape parameter. Must be strictly positive.
scale	Scale parameter. Must be strictly positive.
give_log	true if one wants the log-probability, false otherwise.

Definition at line 145 of file distributions_R.hpp.

§ dzinbinom()

template<class Type >

Type dzinbinom	(	const Type &	x,
		const Type &	size,
		const Type &	p,
		const Type &	zip,
		int	give_log = `0`
	)

inline

Zero-Inflated negative binomial probability function.

Parameterized through size and prob parameters, following R-convention. No vectorized version is currently available.

Parameters

zip	is the probaility of having extra zeros

Definition at line 175 of file lgamma.hpp.

§ dzinbinom2()

template<class Type >

Type dzinbinom2	(	const Type &	x,
		const Type &	mu,
		const Type &	var,
		const Type &	zip,
		int	give_log = `0`
	)

inline

Zero-Inflated negative binomial probability function.

Alternative parameterization through mean and variance parameters (conditional on not being an extra zero). No vectorized version is currently available.

Parameters

zip	is the probaility of having extra zeros

Definition at line 192 of file lgamma.hpp.

§ dzipois()

template<class Type >

Type dzipois	(	const Type &	x,
		const Type &	lambda,
		const Type &	zip,
		int	give_log = `0`
	)

inline

Zero-Inflated Poisson probability function.

Parameters

zip	is the probaility of having extra zeros

Definition at line 158 of file lgamma.hpp.

§ norm2SHASHo()

template<class Type >

Type norm2SHASHo	(	Type	x,
		Type	mu,
		Type	sigma,
		Type	nu,
		Type	tau,
		int	log_p = `0`
	)

Transforms a normal variable into a sinh-asinh variable.

Parameters

mu	Location parameter of the result sinh-asinh distribution.
sigma	Scale parameter of the result sinh-asinh distribution.
nu	Skewness parameter of the result sinh-asinh distribution.
tau	Kurtosis parameter of the result sinh-asinh distribution.
log_p	true if p is log-probability, false otherwise.

It is not possible to call this function with nu a vector or tau a vector.

Definition at line 417 of file distributions_R.hpp.

§ pbeta()

template<class Type >

Type pbeta	(	Type	q,
		Type	shape1,
		Type	shape2
	)

Distribution function of the beta distribution (following R argument convention).

Note: Non-centrality parameter (ncp) not implemented.

Definition at line 433 of file distributions_R.hpp.

§ pexp()

template<class Type >

Type pexp	(	Type	x,
		Type	rate
	)

Cumulative distribution function of the exponential distribution.

Parameters

rate	Rate parameter. Must be strictly positive.

Definition at line 80 of file distributions_R.hpp.

§ pSHASHo()

template<class Type >

Type pSHASHo	(	Type	q,
		Type	mu,
		Type	sigma,
		Type	nu,
		Type	tau,
		int	give_log = `0`
	)

Cumulative distribution function of the sinh-asinh distribution.

Parameters

mu	Location.
sigma	Scale.
nu	Skewness.
tau	Kurtosis.
give_log	true if one wants the log-probability, false otherwise.

Notation adopted from R package "gamlss.dist".

It is not possible to call this function with nu a vector or tau a vector.

Definition at line 368 of file distributions_R.hpp.

§ pweibull()

template<class Type >

Type pweibull	(	Type	x,
		Type	shape,
		Type	scale
	)

Cumulative distribution function of the Weibull distribution.

Parameters

shape	Shape parameter. Must be strictly positive.
scale	Scale parameter. Must be strictly positive.

Definition at line 130 of file distributions_R.hpp.

§ qbeta()

template<class Type >

Type qbeta	(	Type	p,
		Type	shape1,
		Type	shape2
	)

Quantile function of the beta distribution (following R argument convention).

Note: Non-centrality parameter (ncp) not implemented.

Examples:: transform2.cpp.

Definition at line 450 of file distributions_R.hpp.

§ qexp()

template<class Type >

Type qexp	(	Type	p,
		Type	rate
	)

Inverse cumulative distribution function of the exponential distribution.

Parameters

rate	Rate parameter. Must be strictly positive.

Definition at line 110 of file distributions_R.hpp.

§ qSHASHo()

template<class Type >

Type qSHASHo	(	Type	p,
		Type	mu,
		Type	sigma,
		Type	nu,
		Type	tau,
		int	log_p = `0`
	)

Quantile function of the sinh-asinh distribution.

Parameters

mu	Location.
sigma	Scale.
nu	Skewness.
tau	Kurtosis.
log_p	true if p is log-probability, false otherwise.

Notation adopted from R package "gamlss.dist".

It is not possible to call this function with nu a vector or tau a vector.

Definition at line 396 of file distributions_R.hpp.

§ qweibull()

template<class Type >

Type qweibull	(	Type	p,
		Type	shape,
		Type	scale
	)

Inverse cumulative distribution function of the Weibull distribution.

Parameters

p	Probability ; must be between 0 and 1.
shape	Shape parameter. Must be strictly positive.
scale	Scale parameter. Must be strictly positive.

Definition at line 163 of file distributions_R.hpp.

Files

Functions

Exponential distribution.

Weibull distribution.

Sinh-asinh distribution.

Detailed Description

Function Documentation

§ dbeta()

§ dbinom()

§ dbinom_robust()

§ dcompois()

§ dcompois2()

§ dexp()

§ df()

§ dlogis()

§ dmultinom()

§ dnbinom_robust()

§ dnorm()

§ dSHASHo()

§ dsn()

§ dt()

§ dtweedie()

§ dweibull()

§ dzinbinom()

§ dzinbinom2()

§ dzipois()

§ norm2SHASHo()

§ pbeta()

§ pexp()

§ pSHASHo()

§ pweibull()

§ qbeta()

§ qexp()

§ qSHASHo()

§ qweibull()