Package 'RTMBdist'

Title: Distributions Compatible with Automatic Differentiation by 'RTMB'
Description: Extends the functionality of the 'RTMB' <https://kaskr.r-universe.dev/RTMB> package by providing a collection of non-standard probability distributions compatible with automatic differentiation (AD). While 'RTMB' enables flexible and efficient modelling, including random effects, its built-in support is limited to standard distributions. The package adds additional AD-compatible distributions, broadening the range of models that can be implemented and estimated using 'RTMB'. Automatic differentiation and Laplace approximation are described in Kristensen et al. (2016) <doi:10.18637/jss.v070.i05>.
Authors: Jan-Ole Fischer [aut, cre] (ORCID: <https://orcid.org/0009-0004-1556-9053>)
Maintainer: Jan-Ole Fischer <[email protected]>
License: MIT + file LICENSE
Version: 1.0.4
Built: 2026-05-28 10:59:38 UTC
Source: https://github.com/janolefi/rtmbdist

Help Index

Smooth approximation to the absolute value function

Description

Smooth approximation to the absolute value function

Usage

abs_smooth(x, epsilon = 1e-06)

Arguments

x

vector of evaluation points
epsilon

smoothing constant

Details

We approximate the absolute value here as

∣x∣≈x2+ϵ\vert x \vert \approx \sqrt{x^2 + \epsilon}

Value

Smooth absolute value of x.

Examples

abs(0)
abs_smooth(0, 1e-4)

Box–Cox Cole and Green distribution (BCCG)

Description

Density, distribution function, quantile function, and random generation for the Box–Cox Cole and Green distribution.

Usage

dbccg(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)

pbccg(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)

qbccg(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)

rbccg(n, mu = 1, sigma = 0.1, nu = 1)

Arguments

x, q

vector of quantiles
mu

location parameter, must be positive.
sigma

scale parameter, must be positive.
nu

skewness parameter (real).
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dbccg and pbccg allows for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package. See gamlss.dist::BCCG for more details.

The density is

f(x;μ,σ,ν)=xν−1μνσ2πexp⁡ ⁣(−z22)[Φ ⁣(1σ∣ν∣)]−1,x>0,f(x; \mu, \sigma, \nu) = \frac{x^{\nu-1}}{\mu^{\nu} \sigma \sqrt{2\pi}} \exp\!\left(-\tfrac{z^2}{2}\right) \Bigl[\Phi\!\left(\tfrac{1}{\sigma|\nu|}\right)\Bigr]^{-1}, \quad x > 0,

where z=[(x/μ)ν−1]/(νσ)z = [(x/\mu)^\nu - 1]/(\nu\sigma) for ν≠0\nu \neq 0 and z=log⁡(x/μ)/σz = \log(x/\mu)/\sigma for ν=0\nu = 0, and Φ\Phi is the standard normal CDF.

Value

dbccg gives the density, pbccg gives the distribution function, qbccg gives the quantile function, and rbccg generates random deviates.

References

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

x <- rbccg(5, mu = 10, sigma = 0.2, nu = 0.5)
d <- dbccg(x, mu = 10, sigma = 0.2, nu = 0.5)
p <- pbccg(x, mu = 10, sigma = 0.2, nu = 0.5)
q <- qbccg(p, mu = 10, sigma = 0.2, nu = 0.5)

Box-Cox Power Exponential distribution (BCPE)

Description

Density, distribution function, quantile function, and random generation for the Box-Cox Power Exponential distribution.

Usage

dbcpe(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)

pbcpe(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

qbcpe(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

rbcpe(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)

Arguments

x, q

vector of quantiles
mu

location parameter, must be positive.
sigma

scale parameter, must be positive.
nu

vector of nu parameter values.
tau

vector of tau parameter values, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dbcpe and pbcpe allows for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package. See gamlss.dist::BCPE for more details.

The density is

f(x;μ,σ,ν,τ)=xν−1μνσfT(z;τ)FT ⁣(1/(σ∣ν∣);τ),x>0,f(x; \mu, \sigma, \nu, \tau) = \frac{x^{\nu-1}}{\mu^{\nu} \sigma} \frac{f_T(z;\tau)}{F_T\!\left(1/(\sigma|\nu|);\tau\right)}, \quad x > 0,

where z=[(x/μ)ν−1]/(νσ)z = [(x/\mu)^\nu - 1]/(\nu\sigma) for ν≠0\nu \neq 0 and z=log⁡(x/μ)/σz = \log(x/\mu)/\sigma for ν=0\nu = 0, and fT(⋅;τ)f_T(\cdot;\tau) and FT(⋅;τ)F_T(\cdot;\tau) are the PDF and CDF of the power exponential (PE) distribution with shape τ\tau.

Value

dbcpe gives the density, pbcpe gives the distribution function, qbcpe gives the quantile function, and rbcpe generates random deviates.

References

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

x <- rbcpe(1, mu = 5, sigma = 0.1, nu = 1, tau = 1)
d <- dbcpe(x, mu = 5, sigma = 0.1, nu = 1, tau = 1)
p <- pbcpe(x, mu = 5, sigma = 0.1, nu = 1, tau = 1)
q <- qbcpe(p, mu = 5, sigma = 0.1, nu = 1, tau = 1)

Box–Cox t distribution (BCT)

Description

Density, distribution function, quantile function, and random generation for the Box–Cox t distribution.

Usage

dbct(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)

pbct(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

qbct(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

rbct(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)

Arguments

x, q

vector of quantiles
mu

location parameter, must be positive.
sigma

scale parameter, must be positive.
nu

skewness parameter (real).
tau

degrees of freedom, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dbct and pbct allows for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package. See gamlss.dist::BCT for more details.

The density is

f(x;μ,σ,ν,τ)=xν−1μνσft(z;τ)Ft ⁣(1/(σ∣ν∣);τ),x>0,f(x; \mu, \sigma, \nu, \tau) = \frac{x^{\nu-1}}{\mu^{\nu} \sigma} \frac{f_t(z;\tau)}{F_t\!\left(1/(\sigma|\nu|);\tau\right)}, \quad x > 0,

where z=[(x/μ)ν−1]/(νσ)z = [(x/\mu)^\nu - 1]/(\nu\sigma) for ν≠0\nu \neq 0 and z=log⁡(x/μ)/σz = \log(x/\mu)/\sigma for ν=0\nu = 0, and ft(⋅;τ)f_t(\cdot;\tau) and Ft(⋅;τ)F_t(\cdot;\tau) are the PDF and CDF of Student's tt distribution with τ\tau degrees of freedom.

Value

dbct gives the density, pbct gives the distribution function, qbct gives the quantile function, and rbct generates random deviates.

References

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

x <- rbct(1, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)
d <- dbct(x, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)
p <- pbct(x, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)
q <- qbct(p, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)

Reparameterised beta distribution

Description

Density, distribution function, quantile function, and random generation for the beta distribution reparameterised in terms of mean and concentration.

Usage

dbeta(x, shape1, shape2, log = FALSE, eps = 0)

dbeta2(x, mu, phi, log = FALSE, eps = 0)

pbeta2(q, mu, phi, lower.tail = TRUE, log.p = FALSE)

qbeta2(p, mu, phi, lower.tail = TRUE, log.p = FALSE)

rbeta2(n, mu, phi)

Arguments

x, q

vector of quantiles
shape1, shape2

non-negative parameters
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
eps

for internal use only, don't change.
mu

mean parameter, must be in the interval from 0 to 1.
phi

concentration parameter, must be positive.
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Currently, dbeta masks RTMB::dbeta because the latter has a numerically unstable gradient.

dbeta and dbeta2 compute the beta density. dbeta2 reparameterises by mean μ\mu and concentration ϕ\phi:

f(x;μ,ϕ)=xμϕ−1(1−x)(1−μ)ϕ−1B(μϕ, (1−μ)ϕ),x∈(0,1).f(x; \mu, \phi) = \frac{x^{\mu\phi - 1}(1-x)^{(1-\mu)\phi - 1}}{B(\mu\phi,\, (1-\mu)\phi)}, \quad x \in (0,1).

Value

dbeta2 gives the density, pbeta2 gives the distribution function, qbeta2 gives the quantile function, and rbeta2 generates random deviates.

Examples

set.seed(123)
x <- rbeta2(1, 0.5, 1)
d <- dbeta2(x, 0.5, 1)
p <- pbeta2(x, 0.5, 1)
q <- qbeta2(p, 0.5, 1)

Beta-binomial distribution

Description

Density and random generation for the beta-binomial distribution.

Usage

dbetabinom(x, size, shape1, shape2, log = FALSE)

rbetabinom(n, size, shape1, shape2)

Arguments

x

vector of non-negative counts.
size

vector of total counts (number of trials). Needs to be >= x.
shape1

positive shape parameter 1 of the Beta prior.
shape2

positive shape parameter 2 of the Beta prior.
log

logical; if TRUE, densities are returned on the log scale.
n

number of random values to return (for rbetabinom).

Details

This implementation of dbetabinom allows for automatic differentiation with RTMB.

P(X=k; n,a,b)=(nk)B(k+a, n−k+b)B(a, b),k=0,1,…,n.P(X = k;\, n, a, b) = \binom{n}{k} \frac{B(k+a,\, n-k+b)}{B(a,\, b)}, \quad k = 0, 1, \ldots, n.

Value

dbetabinom gives the density and rbetabinom generates random samples.

Examples

set.seed(123)
x <- rbetabinom(1, 10, 2, 5)
d <- dbetabinom(x, 10, 2, 5)

Beta prime distribution

Description

Density, distribution function, quantile function, and random generation for the Beta prime distribution.

Usage

dbetaprime(x, shape1, shape2, log = FALSE)

pbetaprime(q, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

qbetaprime(p, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

rbetaprime(n, shape1, shape2)

Arguments

x, q

vector of quantiles
shape1, shape2

non-negative shape parameters of the corresponding Beta distribution
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

If X∼Beta(α,β)X \sim \text{Beta}(\alpha, \beta), then X1−X∼Betaprime(α,β)\frac{X}{1-X} \sim \text{Betaprime}(\alpha, \beta)

f(x; a,b)=xa−1(1+x)a+b B(a,b),x>0.f(x;\, a, b) = \frac{x^{a-1}}{(1+x)^{a+b}\, B(a,b)}, \quad x > 0.

Value

dbetaprime gives the density, pbetaprime gives the distribution function, qbetaprime gives the quantile function, and rbetaprime generates random deviates.

Examples

x <- rbetaprime(1, 2, 1)
d <- dbetaprime(x, 2, 1)
p <- pbetaprime(x, 2, 1)
q <- qbetaprime(p, 2, 1)

Clayton copula constructors

Description

Construct a function that computes the log density or CDF of the bivariate Clayton copula, intended to be used with dcopula.

Usage

cclayton(theta)

Cclayton(theta)

Arguments

theta

Positive dependence parameter (θ>0\theta > 0).

Details

The Clayton copula density is

c(u,v;θ)=(1+θ)(uv)−(1+θ)(u−θ+v−θ−1)−(2θ+1)/θ,θ>0.c(u,v;\theta) = (1+\theta) (uv)^{-(1+\theta)} \left( u^{-\theta} + v^{-\theta} - 1 \right)^{-(2\theta+1)/\theta}, \quad \theta > 0.

Value

A function of two arguments (u,v) returning log copula density (cclayton) or copula CDF (Cclayton).

See Also

cgaussian(), cgumbel(), cfrank()

Examples

x <- c(0.5, 1); y <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cclayton(2), log = TRUE)

# CDF version (for discrete copulas)
Cclayton(1.5)(0.5, 0.4)

Frank copula constructor

Description

Returns a function computing the log density of the bivariate Frank copula, intended to be used with dcopula.

Usage

cfrank(theta)

Cfrank(theta)

Arguments

theta

Dependence parameter (θ≠0\theta \neq 0).

Details

The Frank copula density is

c(u,v;θ)=θ(1−e−θ)e−θ(u+v)[(e−θu−1)(e−θv−1)+(1−e−θ)]2,θ≠0.c(u,v;\theta) = \frac{\theta (1-e^{-\theta}) e^{-\theta(u+v)}} {\left[(e^{-\theta u}-1)(e^{-\theta v}-1) + (1 - e^{-\theta}) \right]^2}, \quad \theta \ne 0.

Value

A function of two arguments (u, v) returning either the log copula density (cfrank) or the copula CDF (Cfrank).

See Also

cgaussian(), cclayton(), cgumbel()

Examples

x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cfrank(2), log = TRUE)

Gaussian copula constructor

Description

Returns a function computing the log density of the bivariate Gaussian copula, intended to be used with dcopula.

Usage

cgaussian(rho = 0)

Arguments

rho

Correlation parameter (−1<rho<1-1 < rho < 1).

Value

Function of two arguments (u,v) returning log copula density.

The Gaussian copula density is

c(u,v;ρ)=11−ρ2exp⁡{−12(1−ρ2)(z12−2ρz1z2+z22)+12(z12+z22)},c(u,v;\rho) = \frac{1}{\sqrt{1-\rho^2}} \exp\left\{-\frac{1}{2(1-\rho^2)} (z_1^2 - 2 \rho z_1 z_2 + z_2^2) + \frac{1}{2}(z_1^2 + z_2^2) \right\},

where z1=Φ−1(u)z_1 = \Phi^{-1}(u), z2=Φ−1(v)z_2 = \Phi^{-1}(v), and −1<ρ<1-1 < \rho < 1.

See Also

cclayton(), cgumbel(), cfrank()

Examples

x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cgaussian(0.5), log = TRUE)

Multivariate Gaussian copula constructor parameterised by inverse correlation matrix

Description

Returns a function computing the log density of the multivariate Gaussian copula, parameterised by the inverse correlation matrix.

Usage

cgmrf(Q)

Arguments

Q

Inverse of a positive definite correlation matrix with unit diagonal. Can either be sparse or dense matrix.

Details

Caution: Parameterising the inverse correlation directly is difficult, as inverting it needs to yield a positive definite matrix with unit diagonal. Hence we still advise parameterising the correaltion matrix R and computing its inverse. This function is useful when you need access to the precision (i.e. inverse correlation) in your likelihood function.

Value

Function with matrix argument U returning log copula density.

See Also

cmvgauss()

Examples

x <- c(0.5, 1); y <- c(1, 2); z <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE); d3 <- dbeta(z, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2); p3 <- pbeta(z, 2, 1)
R <- matrix(c(1,0.5,0.3,0.5,1,0.4,0.3,0.4,1), nrow = 3)

## Based on correlation matrix
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cmvgauss(R), log = TRUE)

## Based on precision matrix
Q <- solve(R)
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cgmrf(Q), log = TRUE)

## Parameterisation inside a model
# using RTMB::unstructured to get a valid correlation matrix
library(RTMB)
d <- 5 # dimension
cor_func <- unstructured(d)
npar <- length(cor_func$parms())
R <- cor_func$corr(rep(0.1, npar))

Gumbel copula constructors

Description

Construct functions that compute either the log density or the CDF of the bivariate Gumbel copula, intended for use with dcopula.

Usage

cgumbel(theta)

Cgumbel(theta)

Arguments

theta

Dependence parameter (θ>=1\theta >= 1).

Details

The Gumbel copula density

c(u,v;θ)=exp⁡[−((−log⁡u)θ+(−log⁡v)θ)1/θ]⋅h(u,v;θ),c(u,v;\theta) = \exp\Big[-\big((-\log u)^\theta + (-\log v)^\theta\big)^{1/\theta}\Big] \cdot h(u,v;\theta),

where h(u,v;θ)h(u,v;\theta) contains the derivative terms ensuring the function is a density.

Value

A function of two arguments (u, v) returning either the log copula density (cgumbel) or the copula CDF (Cgumbel).

See Also

cgaussian(), cclayton(), cfrank()

Examples

x <- c(0.5, 1); y <- c(0.2, 0.4)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cgumbel(1.5), log = TRUE)

# CDF version (for discrete copulas)
Cgumbel(1.5)(0.5, 0.4)

Multivariate Gaussian copula constructor

Description

Returns a function computing the log density of the multivariate Gaussian copula, intended to be used with dmvcopula.

Usage

cmvgauss(R)

Arguments

R

Positive definite correlation matrix (unit diagonal)

Value

Function with matrix argument U returning log copula density.

See Also

cgmrf()

Examples

x <- c(0.5, 1); y <- c(1, 2); z <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE); d3 <- dbeta(z, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2); p3 <- pbeta(z, 2, 1)
R <- matrix(c(1,0.5,0.3,0.5,1,0.4,0.3,0.4,1), nrow = 3)

## Based on correlation matrix
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cmvgauss(R), log = TRUE)

## Based on precision matrix
Q <- solve(R)
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cgmrf(Q), log = TRUE)

## Parameterisation inside a model
# using RTMB::unstructured to get a valid correlation matrix
library(RTMB)
d <- 5 # dimension
cor_func <- unstructured(d)
npar <- length(cor_func$parms())
R <- cor_func$corr(rep(0.1, npar))

Joint density under a bivariate copula

Description

Computes the joint density (or log-density) of a bivariate distribution constructed from two arbitrary margins combined with a specified copula.

Usage

dcopula(d1, d2, p1, p2, copula = cgaussian(0), log = FALSE)

Arguments

d1, d2

Marginal density values. If log = TRUE, supply the log-density. If log = FALSE, supply the raw density.
p1, p2

Marginal CDF values. Need not be supplied on log scale.
copula

A function of two arguments (u, v) returning the log copula density log⁡c(u,v)\log c(u,v). You can either construct this yourself or use the copula constructors available (see details)
log

Logical; if TRUE, return the log joint density. In this case, d1 and d2 must be on the log scale.

Details

The joint density is

f(x,y)=c(F1(x),F2(y)) f1(x)f2(y),f(x,y) = c(F_1(x), F_2(y)) \, f_1(x) f_2(y),

where FiF_i are the marginal CDFs, fif_i are the marginal densities, and cc is the copula density.

The marginal densities d1, d2 and CDFs p1, p2 must be differentiable for automatic differentiation (AD) to work.

Available copula constructors are:

Value

Joint density (or log-density) under the bivariate copula.

See Also

ddcopula(), dmvcopula()

Examples

# Normal + Exponential margins with Gaussian copula
x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cgaussian(0.5), log = TRUE)

# Normal + Beta margins with Clayton copula
x <- c(0.5, 1); y <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cclayton(2), log = TRUE)

# Normal + Beta margins with Gumbel copula
x <- c(0.5, 1); y <- c(0.2, 0.4)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dbeta(y, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pbeta(y, 2, 1)
dcopula(d1, d2, p1, p2, copula = cgumbel(1.5), log = TRUE)

# Normal + Exponential margins with Frank copula
x <- c(0.5, 1); y <- c(1, 2)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2)
dcopula(d1, d2, p1, p2, copula = cfrank(2), log = TRUE)

Joint probability under a discrete bivariate copula

Description

Computes the joint probability mass function of two discrete margins combined with a copula CDF.

Usage

ddcopula(d1, d2, p1, p2, Copula, log = FALSE)

Arguments

d1, d2

Marginal p.m.f. values at the observed points, not on log-scale.
p1, p2

Marginal CDF values at the observed points.
Copula

A function of two arguments returning the copula CDF.
log

Logical; if TRUE, return the log joint density. In this case, d1 and d2 must be on the log scale.

Details

The joint probability mass function for two discrete margins is

Pr⁡(Y1=y1,Y2=y2)=C(F1(y1),F2(y2))−C(F1(y1−1),F2(y2))−C(F1(y1),F2(y2−1))+C(F1(y1−1),F2(y2−1)),\Pr(Y_1 = y_1, Y_2 = y_2) = C(F_1(y_1), F_2(y_2)) - C(F_1(y_1-1), F_2(y_2)) - C(F_1(y_1), F_2(y_2-1)) + C(F_1(y_1-1), F_2(y_2-1)),

where FiF_i are the marginal CDFs, and CC is the copula CDF.

Available copula CDF constructors are:

Value

Joint probability (or log-probability) under chosen copula

See Also

dcopula(), dmvcopula()

Examples

x <- c(3,5); y <- c(2,4)
d1 <- dpois(x, 4); d2 <- dpois(y, 3)
p1 <- ppois(x, 4); p2 <- ppois(y, 3)
ddcopula(d1, d2, p1, p2, Copula = Cclayton(2), log = FALSE)

Dirichlet distribution

Description

Density and and random generation for the Dirichlet distribution.

Usage

ddirichlet(x, alpha, log = FALSE)

rdirichlet(n, alpha)

Arguments

x

vector or matrix of quantiles. If x is a vector, it needs to sum to one. If x is a matrix, each row should sum to one.
alpha

vector or matrix of positive shape parameters
log

logical; if TRUE, densities pp are returned as log⁡(p)\log(p).
n

number of random values to return.

Details

This implementation of ddirichlet allows for automatic differentiation with RTMB.

f(x; α)=Γ ⁣(∑iαi)∏iΓ(αi)∏ixiαi−1,f(\mathbf{x};\,\boldsymbol{\alpha}) = \frac{\Gamma\!\left(\sum_i \alpha_i\right)}{\prod_i \Gamma(\alpha_i)} \prod_i x_i^{\alpha_i - 1},

where x\mathbf{x} lies on the unit simplex and αi>0\alpha_i > 0.

Value

ddirichlet gives the density, rdirichlet generates random deviates.

Examples

# single alpha
alpha <- c(1,2,3)
x <- rdirichlet(1, alpha)
d <- ddirichlet(x, alpha)
# vectorised over alpha
alpha <- rbind(alpha, 2*alpha)
x <- rdirichlet(2, alpha)

Dirichlet-multinomial distribution

Description

Density and and random generation for the Dirichlet-multinomial distribution.

Usage

ddirmult(x, size, alpha, log = FALSE)

rdirmult(n, size, alpha)

Arguments

x

vector or matrix of non-negative counts, where rows are observations and columns are categories.
size

vector of total counts for each observation. Needs to match the row sums of x.
alpha

vector or matrix of positive shape parameters
log

logical; if TRUE, densities pp are returned as log⁡(p)\log(p).
n

number of random values to return.

Details

This implementation of ddirmult allows for automatic differentiation with RTMB.

P(x; α,n)=Γ(α0) Γ(n+1)Γ(n+α0)∏iΓ(xi+αi)Γ(αi) Γ(xi+1),P(\mathbf{x};\,\boldsymbol{\alpha}, n) = \frac{\Gamma(\alpha_0)\,\Gamma(n+1)}{\Gamma(n+\alpha_0)} \prod_i \frac{\Gamma(x_i + \alpha_i)}{\Gamma(\alpha_i)\,\Gamma(x_i+1)},

where α0=∑iαi\alpha_0 = \sum_i \alpha_i and n=∑ixin = \sum_i x_i.

Value

ddirmult gives the density and rdirmult generates random samples.

Examples

# single alpha
alpha <- c(1,2,3)
size <- 10
x <- rdirmult(1, size, alpha)
d <- ddirmult(x, size, alpha)
# vectorised over alpha and size
alpha <- rbind(alpha, 2*alpha)
size <- c(size, 3*size)
x <- rdirmult(2, size, alpha)

Joint density under a multivariate copula

Description

Computes the joint density (or log-density) of a distribution constructed from any number of arbitrary margins combined with a specified copula.

Usage

dmvcopula(D, P, copula = cmvgauss(diag(ncol(D))), log = FALSE)

Arguments

D

Matrix of marginal density values of with rows corresponding to observations and columns corresponding to dimensions. If log = TRUE, supply the log-densities. If log = FALSE, supply the raw densities
P

Matrix of marginal CDF values of the same dimension as D. Need not be supplied on log scale.
copula

A function of a matrix argument U returning the log copula density log⁡c(u1,...ud)\log c(u_1, ... u_d). The columns of U correspond to dimensions. You can either construct this yourself or use the copula constructors available (see details)
log

Logical; if TRUE, return the log joint density. In this case, D must be on the log scale.

Details

The joint density is

f(x1,…,xd)=c(F1(x1),…,Fd(xd)) f1(x1)…fd(xd),f(x_1, \dots, x_d) = c(F_1(x_1), \dots, F_d(x_d)) \, f_1(x_1) \dots f_d(x_d),

where FiF_i are the marginal CDFs, fif_i are the marginal densities, and cc is the copula density.

The marginal densities d_1, ..., d_d and CDFs p_1, ..., p_d must be differentiable for automatic differentiation (AD) to work.

Available multivariate copula constructors are:

  • cmvgauss (Multivariate Gaussian copula)

  • cgmrf (Multivariate Gaussian copula parameterised by precision (inverse correlation) matrix)

Value

Joint density (or log-density) under the chosen copula.

See Also

dcopula(), ddcopula()

Examples

x <- c(0.5, 1); y <- c(1, 2); z <- c(0.2, 0.8)
d1 <- dnorm(x, 1, log = TRUE); d2 <- dexp(y, 2, log = TRUE); d3 <- dbeta(z, 2, 1, log = TRUE)
p1 <- pnorm(x, 1); p2 <- pexp(y, 2); p3 <- pbeta(z, 2, 1)
R <- matrix(c(1,0.5,0.3,0.5,1,0.4,0.3,0.4,1), nrow = 3)

### Multivariate Gaussian copula
## Based on correlation matrix
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cmvgauss(R), log = TRUE)

## Based on precision matrix
Q <- solve(R)
dmvcopula(cbind(d1, d2, d3), cbind(p1, p2, p3), copula = cgmrf(Q), log = TRUE)

## Parameterisation inside a model
# using RTMB::unstructured to get a valid correlation matrix
library(RTMB)
d <- 5 # dimension
cor_func <- unstructured(d)
npar <- length(cor_func$parms())
R <- cor_func$corr(rep(0.1, npar))

AD-compatible error function and complementary error function

Description

AD-compatible error function and complementary error function

Usage

erf(x)

erfc(x)

Arguments

x

vector of evaluation points

Value

erf(x) returns the error function and erfc(x) returns the complementary error function.

Examples

erf(1)
erfc(1)

Exponentially modified Gaussian distribution

Description

Density, distribution function, quantile function, and random generation for the exponentially modified Gaussian distribution.

Usage

dexgauss(x, mu = 0, sigma = 1, lambda = 1, log = FALSE)

pexgauss(q, mu = 0, sigma = 1, lambda = 1, lower.tail = TRUE, log.p = FALSE)

qexgauss(p, mu = 0, sigma = 1, lambda = 1, lower.tail = TRUE, log.p = FALSE)

rexgauss(n, mu = 0, sigma = 1, lambda = 1)

Arguments

x, q

vector of quantiles
mu

mean parameter of the Gaussian part
sigma

standard deviation parameter of the Gaussian part, must be positive.
lambda

rate parameter of the exponential part, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dexgauss and pexgauss allows for automatic differentiation with RTMB. qexgauss and rexgauss are reparameterised imports from gamlss.dist::exGAUS.

If X∼N(μ,σ2)X \sim N(\mu, \sigma^2) and Y∼Exp(λ)Y \sim \text{Exp}(\lambda), then Z=X+YZ = X + Y follows the exponentially modified Gaussian distribution with parameters μ\mu, σ\sigma, and λ\lambda.

The density is

f(x; μ,σ,λ)=λexp⁡ ⁣(λμ+λ2σ22−λx) Φ ⁣(x−μ−λσ2σ),f(x;\,\mu,\sigma,\lambda) = \lambda \exp\!\Bigl(\lambda\mu + \tfrac{\lambda^2\sigma^2}{2} - \lambda x\Bigr)\, \Phi\!\left(\frac{x - \mu - \lambda\sigma^2}{\sigma}\right),

where Φ\Phi is the standard normal CDF.

Value

dexgauss gives the density, pexgauss gives the distribution function, qexgauss gives the quantile function, and rexgauss generates random deviates.

References

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

x <- rexgauss(1, 1, 2, 2)
d <- dexgauss(x, 1, 2, 2)
p <- pexgauss(x, 1, 2, 2)
q <- qexgauss(p, 1, 2, 2)

Folded normal distribution

Description

Density, distribution function, and random generation for the folded normal distribution.

Usage

dfoldnorm(x, mu = 0, sigma = 1, log = FALSE)

pfoldnorm(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rfoldnorm(n, mu = 0, sigma = 1)

Arguments

x, q

vector of quantiles
mu

location parameter
sigma

scale parameter, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return
p

vector of probabilities

Details

This implementation of dfoldnorm allows for automatic differentiation with RTMB.

f(x; μ,σ)=1σ2π[exp⁡ ⁣(−(x−μ)22σ2)+exp⁡ ⁣(−(x+μ)22σ2)],x≥0.f(x;\,\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}\left[\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) + \exp\!\left(-\frac{(x+\mu)^2}{2\sigma^2}\right)\right], \quad x \geq 0.

Value

dfoldnorm gives the density, pfoldnorm gives the distribution function, and rfoldnorm generates random deviates.

Examples

x <- rfoldnorm(1, 1, 2)
d <- dfoldnorm(x, 1, 2)
p <- pfoldnorm(x, 1, 2)

Reparameterised gamma distribution

Description

Density, distribution function, quantile function, and random generation for the gamma distribution reparameterised in terms of mean and standard deviation.

Usage

dgamma2(x, mean = 1, sd = 1, log = FALSE)

pgamma2(q, mean = 1, sd = 1, lower.tail = TRUE, log.p = FALSE)

qgamma2(p, mean = 1, sd = 1, lower.tail = TRUE, log.p = FALSE)

rgamma2(n, mean = 1, sd = 1)

Arguments

x, q

vector of quantiles
mean

mean parameter, must be positive.
sd

standard deviation parameter, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Reparameterises the gamma distribution via shape=μ2/s2\text{shape} = \mu^2/s^2 and scale=s2/μ\text{scale} = s^2/\mu:

f(x; μ,s)=xμ2/s2−1exp⁡(−xs2/μ2)(s2/μ)μ2/s2 Γ(μ2/s2),x>0.f(x;\,\mu, s) = \frac{x^{\mu^2/s^2 - 1} \exp(-x s^2/\mu^2)}{(s^2/\mu)^{\mu^2/s^2}\,\Gamma(\mu^2/s^2)}, \quad x > 0.

Value

dgamma2 gives the density, pgamma2 gives the distribution function, qgamma2 gives the quantile function, and rgamma2 generates random deviates.

Examples

x <- rgamma2(1)
d <- dgamma2(x)
p <- pgamma2(x)
q <- qgamma2(p)

Generalised Gamma distribution (GG)

Description

Density, distribution function, quantile function, and random generation for the generalised Gamma distribution.

Usage

dgengamma(x, mu = 1, sigma = 0.5, nu = 1, log = FALSE)

pgengamma(q, mu = 1, sigma = 0.5, nu = 1, lower.tail = TRUE, log.p = FALSE)

qgengamma(p, mu = 1, sigma = 0.5, nu = 1, lower.tail = TRUE, log.p = FALSE)

rgengamma(n, mu = 1, sigma = 0.5, nu = 1)

Arguments

x, q

vector of quantiles
mu

location parameter, must be positive.
sigma

scale parameter, must be positive.
nu

skewness parameter (real).
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dgengamma, pgengamma, and qgengamma allows for automatic differentiation with RTMB.

For ν≠0\nu \neq 0 the density is that of a generalised gamma: setting z=(x/μ)νz = (x/\mu)^\nu,

f(x; μ,σ,ν)=∣ν∣x (σ∣ν∣)2/(σν)2 Γ ⁣(1/(σν)2) z1/(σν)2 exp⁡ ⁣(−z(σ∣ν∣)2),x>0.f(x;\,\mu,\sigma,\nu) = \frac{|\nu|}{x\,({\sigma|\nu|})^{2/(\sigma\nu)^2}\,\Gamma\!\left(1/(\sigma\nu)^2\right)}\, z^{1/(\sigma\nu)^2}\, \exp\!\left(-\frac{z}{(\sigma|\nu|)^2}\right), \quad x > 0.

For ν=0\nu = 0 the distribution reduces to a log-normal with log-mean log⁡μ\log\mu and log-standard-deviation σ\sigma.

Value

dgengamma gives the density, pgengamma gives the distribution function, qgengamma gives the quantile function, and rgengamma generates random deviates.

References

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

x <- rgengamma(5, mu = 4, sigma = 0.5, nu = 0.5)
d <- dgengamma(x, mu = 4, sigma = 0.5, nu = 0.5)
p <- pgengamma(x, mu = 4, sigma = 0.5, nu = 0.5)
q <- qgengamma(p, mu = 4, sigma = 0.5, nu = 0.5)

Generalised Poisson distribution

Description

Probability mass function, distribution function, and random generation for the generalised Poisson distribution.

Usage

dgenpois(x, lambda = 1, phi = 1, log = FALSE)

pgenpois(q, lambda = 1, phi = 1, lower.tail = TRUE, log.p = FALSE)

qgenpois(p, lambda = 1, phi = 1,
         lower.tail = TRUE, log.p = FALSE, max.value = 10000)

rgenpois(n, lambda = 1, phi = 1, max.value = 10000)

Arguments

x, q

integer vector of counts
lambda

vector of positive means
phi

vector of non-negative dispersion parameters
log, log.p

logical; return log-density if TRUE
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
max.value

a constant, set to the default value of 10000 for how far the algorithm should look for q.
n

number of random values to return.

Details

This implementation of dgenpois allows for automatic differentiation with RTMB. The other functions are imported from gamlss.dist::GPO.

The distribution has mean λ\lambda and variance λ(1+ϕλ)2\lambda(1 + \phi \lambda)^2. For ϕ=0\phi = 0 it reduces to the Poisson distribution, however ϕ\phi must be strictly positive here.

P(X=x; λ,ϕ)=λ (1+ϕx)x−1 e−λ(1+ϕx)/(1+ϕλ)(1+ϕλ)x x!,x=0,1,2,…P(X = x;\,\lambda,\phi) = \frac{\lambda\,(1+\phi x)^{x-1}\,e^{-\lambda(1+\phi x)/(1+\phi\lambda)}}{(1+\phi\lambda)^x\, x!}, \quad x = 0, 1, 2, \ldots

Value

dgenpois gives the probability mass function, pgenpois gives the distribution function, qgenpois gives the quantile function, and rgenpois generates random deviates.

Examples

set.seed(123)
x <- rgenpois(1, 2, 3)
d <- dgenpois(x, 2, 3)
p <- pgenpois(x, 2, 3)
q <- qgenpois(p, 2, 3)

Gompertz distribution

Description

Density, distribution function, quantile function, and random generation for the Gompertz distribution.

Usage

dgompertz(x, eta = 1, b = 1, log = FALSE)

pgompertz(q, eta = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

qgompertz(p, eta = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

rgompertz(n, eta = 1, b = 1)

Arguments

x, q

vector of quantiles (non-negative)
eta

shape parameter, must be positive
b

rate parameter, must be positive
log, log.p

logical; if TRUE, probabilities/densities are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

The Gompertz distribution with shape η>0\eta > 0 and rate b>0b > 0 has density

f(x; η,b)=bη ebxexp⁡ ⁣(−η(ebx−1)),x≥0,f(x;\,\eta,b) = b\eta\,e^{bx}\exp\!\bigl(-\eta(e^{bx}-1)\bigr), \quad x \geq 0,

with CDF F(x)=1−exp⁡(−η(ebx−1))F(x) = 1 - \exp(-\eta(e^{bx}-1)) and quantile function Q(p)=log⁡(1−log⁡(1−p)/η) / bQ(p) = \log(1 - \log(1-p)/\eta)\,/\,b.

Value

dgompertz gives the density, pgompertz gives the distribution function, qgompertz gives the quantile function, and rgompertz generates random deviates.

References

https://en.wikipedia.org/wiki/Gompertz_distribution

Examples

x <- rgompertz(1, eta = 1, b = 1)
d <- dgompertz(x, eta = 1, b = 1)
p <- pgompertz(x, eta = 1, b = 1)
q <- qgompertz(p, eta = 1, b = 1)

Gumbel distribution

Description

Density, distribution function, quantile function, and random generation for the Gumbel distribution.

Usage

dgumbel(x, location = 0, scale = 1, log = FALSE)

pgumbel(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)

qgumbel(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)

rgumbel(n, location = 0, scale = 1)

Arguments

x, q

vector of quantiles
location

location parameter
scale

scale parameter, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dgumbel allows for automatic differentiation with RTMB.

f(x; μ,σ)=1σexp⁡ ⁣(−(z+e−z)),z=x−μσ.f(x;\,\mu,\sigma) = \frac{1}{\sigma}\exp\!\bigl(-(z + e^{-z})\bigr), \quad z = \frac{x-\mu}{\sigma}.

Value

dgumbel gives the density, pgumbel gives the distribution function, qgumbel gives the quantile function, and rgumbel generates random deviates.

Examples

x <- rgumbel(1, 0.5, 2)
d <- dgumbel(x, 0.5, 2)
p <- pgumbel(x, 0.5, 2)
q <- qgumbel(p, 0.5, 2)

Inverse Chi-squared distribution

Description

Density, distribution function, quantile function, and random generation for the inverse Chi-squared distribution.

Usage

dinvchisq(x, df, scale = 1/df, log = FALSE)

pinvchisq(q, df, scale = 1/df, lower.tail = TRUE, log.p = FALSE)

qinvchisq(p, df, scale = 1/df, lower.tail = TRUE, log.p = FALSE)

rinvchisq(n, df, scale = 1/df)

Arguments

x, q

vector of quantiles, must be positive.
df

degrees of freedom (ν>0\nu > 0)
scale

optional positive scale parameter. Default value of 1/df corresponds to standard inverse Chi-squared
log, log.p

logical; if TRUE, probabilities/densities are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

If X∼Chisq(ν)X \sim \text{Chisq}(\nu), then 1/X∼invChisq(ν)1/X \sim \text{invChisq}(\nu).

The inverse Chi-squared distribution with ν\nu degrees of freedom has density

f(x)=(νs/2)ν/2Γ(ν/2)x−(ν/2+1)exp⁡ ⁣(−νs2x),x>0,f(x) = \frac{(\nu s/2)^{\nu/2}}{\Gamma(\nu/2)} x^{-(\nu/2+1)} \exp\!\left(-\frac{\nu s}{2x}\right), \quad x>0,

This implementation of dinvchisq, pinvchisq, and qinvchisq allows for automatic differentiation with RTMB.

Value

dinvchisq gives the density, pinvchisq gives the distribution function, qinvchisq gives the quantile function, and rinvchisq generates random deviates.

Examples

x <- rinvchisq(1, df = 5)
d <- dinvchisq(x, df = 5)
p <- pinvchisq(x, df = 5)
q <- qinvchisq(p, df = 5)

Inverse Gamma distribution

Description

Density, distribution function, and random generation for the inverse Gamma distribution.

Usage

dinvgamma(x, shape, rate, scale = 1/rate, log = FALSE)

pinvgamma(q, shape, rate, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)

qinvgamma(p, shape, rate, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)

rinvgamma(n, shape, rate, scale = 1/rate)

Arguments

x, q

vector of quantiles, must be positive.
shape, rate, scale

positive parameters of corresponding gamma distribution
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dinvgamma, pinvgamma, and qinvgamma allows for automatic differentiation with RTMB.

If X∼Γ(α,β)X \sim \Gamma(\alpha, \beta), then 1/X∼InvGamma(α,β)1/X \sim \text{InvGamma}(\alpha, \beta).

f(x; α,s)=sαΓ(α) x−(α+1)exp⁡ ⁣(−sx),x>0,f(x;\,\alpha,s) = \frac{s^\alpha}{\Gamma(\alpha)}\, x^{-(\alpha+1)}\exp\!\left(-\frac{s}{x}\right), \quad x > 0,

where s=scale=1/rates = \text{scale} = 1/\text{rate}.

Value

dinvgamma gives the density, pinvgamma gives the distribution function, qinvgamma gives the quantile function, and rinvgamma generates random deviates.

Examples

x <- rinvgamma(1, 1, 0.5)
d <- dinvgamma(x, 1, 0.5)
p <- pinvgamma(x, 1, 0.5)
q <- qinvgamma(p, 1, 0.5)

Inverse Gaussian distribution

Description

Density, distribution function, and random generation for the inverse Gaussian distribution.

Usage

dinvgauss(x, mean = 1, shape = 1, log = FALSE)

pinvgauss(q, mean = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)

qinvgauss(p, mean = 1, shape = 1, lower.tail = TRUE, log.p = FALSE, ...)

rinvgauss(n, mean = 1, shape = 1)

Arguments

x, q

vector of quantiles, must be positive.
mean

location parameter
shape

shape parameter, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
...

additional parameter passed to statmod::qinvgauss for numerical evaluation of the quantile function.
n

number of random values to return

Details

This implementation of dinvgauss allows for automatic differentiation with RTMB. qinvgauss and rinvgauss are imported from the statmod package.

f(x; μ,λ)=λ2πx3exp⁡ ⁣(−λ(x−μ)22μ2x),x>0.f(x;\,\mu,\lambda) = \sqrt{\frac{\lambda}{2\pi x^3}} \exp\!\left(-\frac{\lambda(x-\mu)^2}{2\mu^2 x}\right), \quad x > 0.

Value

dinvgauss gives the density, pinvgauss gives the distribution function, qinvgauss gives the quantile function, and rinvgauss generates random deviates.

Examples

x <- rinvgauss(1, 1, 0.5)
d <- dinvgauss(x, 1, 0.5)
p <- pinvgauss(x, 1, 0.5)
q <- qinvgauss(p, 1, 0.5)

Kumaraswamy distribution

Description

Density, distribution function, quantile function, and random generation for the Kumaraswamy distribution.

Usage

dkumar(x, a, b, log = FALSE)

pkumar(q, a, b, lower.tail = TRUE, log.p = FALSE)

qkumar(p, a, b, lower.tail = TRUE, log.p = FALSE)

rkumar(n, a, b)

Arguments

x, q

vector of quantiles in (0,1)(0,1)
a, b

positive shape parameters
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return.

Details

f(x; a,b)=ab xa−1(1−xa)b−1,x∈(0,1).f(x;\,a,b) = a b\, x^{a-1}(1-x^a)^{b-1}, \quad x \in (0,1).

Value

dkumar gives the density, pkumar gives the distribution function, qkumar gives the quantile function, and rkumar generates random deviates.

Examples

x <- rkumar(1, a = 1, b = 2)
d <- dkumar(x, a = 1, b = 2)
p <- pkumar(x, a = 1, b = 2)
q <- qkumar(p, a = 1, b = 2)

Laplace distribution

Description

Density, distribution function, quantile function, and random generation for the Laplace distribution.

Usage

dlaplace(x, mu = 0, b = 1, eps = NULL, log = FALSE)

plaplace(q, mu = 0, b = 1, lower.tail = TRUE, log.p = FALSE)

qlaplace(p, mu = 0, b = 1, lower.tail = TRUE, log.p = FALSE)

rlaplace(n, mu = 0, b = 1)

Arguments

x, q

vector of quantiles
mu

location parameter
b

scale parameter, must be positive.
eps

optional smoothing parameter for dlaplace to smooth the absolute value function. See abs_smooth for details. It is recommended to set this to a small constant like 1e-6 for numerical optimisation.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dlaplace allows for automatic differentiation with RTMB.

f(x; μ,b)=12bexp⁡ ⁣(−∣x−μ∣b).f(x;\,\mu,b) = \frac{1}{2b}\exp\!\left(-\frac{|x-\mu|}{b}\right).

Value

dlaplace gives the density, plaplace gives the distribution function, qlaplace gives the quantile function, and rlaplace generates random deviates.

Examples

x <- rlaplace(1, 1, 1)
d <- dlaplace(x, 1, 1)
p <- plaplace(x, 1, 1)
q <- qlaplace(p, 1, 1)

Log-logistic distribution

Description

Density, distribution function, quantile function, and random generation for the log-logistic distribution.

Usage

dllogis(x, alpha = 1, beta = 1, log = FALSE)

pllogis(q, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)

qllogis(p, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)

rllogis(n, alpha = 1, beta = 1)

Arguments

x, q

vector of quantiles (x>0x > 0).
alpha

scale parameter (α>0\alpha > 0); equal to the median.
beta

shape parameter (β>0\beta > 0); controls tail heaviness.
log

logical; if TRUE, densities are returned on the log scale.
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
log.p

logical; if TRUE, probabilities are returned on the log scale.
p

vector of probabilities.
n

number of random values to return.

Details

The log-logistic distribution has density

f(x; α,β)=(β/α) (x/α)β−1(1+(x/α)β)2,x>0,f(x;\,\alpha,\beta) = \frac{(\beta/\alpha)\,(x/\alpha)^{\beta-1}}{\bigl(1 + (x/\alpha)^\beta\bigr)^2}, \quad x > 0,

where α>0\alpha > 0 is the scale parameter and β>0\beta > 0 is the shape parameter. The scale parameter equals the median. Larger β\beta gives lighter tails; the distribution has a finite mean only when β>1\beta > 1 and a finite variance only when β>2\beta > 2.

The log-logistic arises naturally as the distribution of X=eYX = e^Y where Y∼Logistic(log⁡α, 1/β)Y \sim \mathrm{Logistic}(\log\alpha,\, 1/\beta), which yields the numerically convenient log-density

log⁡f(x)=log⁡β−log⁡x+log⁡flogistic ⁣(βlog⁡(x/α)).\log f(x) = \log\beta - \log x + \log f_{\mathrm{logistic}}\!\bigl(\beta\log(x/\alpha)\bigr).

The CDF is

F(x; α,β)=11+(x/α)−β,F(x;\,\alpha,\beta) = \frac{1}{1 + (x/\alpha)^{-\beta}},

and the quantile function is

Q(p; α,β)=α(p1−p)1/β.Q(p;\,\alpha,\beta) = \alpha \left(\frac{p}{1-p}\right)^{1/\beta}.

dllogis allows for automatic differentiation with RTMB.

Value

dllogis gives the density, pllogis gives the distribution function, qllogis gives the quantile function, and rllogis generates random deviates.

Examples

x <- rllogis(5, alpha = 1, beta = 2)
dllogis(x, alpha = 1, beta = 2)
pllogis(c(0.5, 1, 2), alpha = 1, beta = 2)
qllogis(c(0.25, 0.5, 0.75), alpha = 1, beta = 2)

Sample parameters from approximate Gaussian posterior distribution

Description

Efficient Monte Carlo sampling of parameters (and REPORTed quantities) from the approximate posterior of an (R)TMB model. See sdreport for details on posterior variance-covariance in random effects models.

Usage

mcreport(
  obj,
  nSamples = 1000,
  include_random_pars = TRUE,
  report = FALSE,
  Q = NULL,
  ...
)

Arguments

obj

Optimised RTMB object generated by MakeADFun
nSamples

Number of samples to draw
include_random_pars

Logical; Should random parameters be included in the output?
report

Logical; Should REPORTed quantities be sampled as well? Defaults to FALSE because this may be slow depending on your model.
Q

Optional precalculated sparse precision matrix returned by sdreport(..., getJointPrecision = TRUE)$jointPrecision. If not provided, computed internally using sdreport. Only used for models with random effects.
...

For internal use only

Details

Caution: This has nothing to do with Bayesian posterior sampling. It is simple Monte Carlo sampling from a Gaussian distribution with mean at the MLE and covariance given by the inverse of the Hessian (for fixed effects models) or the joint precision matrix (for random effects models). This is a common approach to get an approximate idea of parameter uncertainty around the MLE, but it relies on large-sample asymptotics.

Sampling random effects from their posterior as compared to calculating marginal standard devitions like sdreport is particularly useful for multidimensional random effects (e.g. for locations xx and yy) where pointwise confidence intervals (e.g. along a path) based on standard deviations are not possible.

Value

A list structured like the original parameter list used in the MakeADFun call (potentially including additional REPORTed quantities). Each entry is a list with nSamples entries.

Examples

set.seed(123)

### Example 1: Without random effects ###

# simulate data
x <- rgumbel(100, location = 5, scale = 2)

# negative log-likelihood function
nll <- function(par) {
  getAll(par)
  x <- OBS(x) # mark x as the response
  scale <- exp(log_scale)
  ADREPORT(scale); REPORT(scale)
  -sum(dgumbel(x, loc, scale, log = TRUE))
}

# RTMB AD object
par <- list(loc = 5, log_scale = log(2))
obj <- MakeADFun(nll, par, silent = TRUE)

# model fitting using AD gradient
opt <- nlminb(obj$par, obj$fn, obj$gr)

# sample from asymptotic distribution of the MLE
samples <- mcreport(obj, report = TRUE)
# parameters
mean(unlist(samples$loc)); sd(unlist(samples$loc))
# reported transformed parameters
mean(unlist(samples$scale)); sd(unlist(samples$scale))

# compare to delta method sd from sdreport
sdr <- sdreport(obj)
summary(sdr)



### Example 2: With random effects ###

# simulate random effects
id <- rep(1:10, each = 100) # 10 groups with 100 observations each
true_gamma <- rnorm(10, 0, 2) # random coefficients
true_probs <- plogis(1 + true_gamma) # linear predictor

# simulate Bernoulli data conditional on random coefficients
x <- rbinom(1000, 1, true_probs[id])

# negative log-likelihood function
nll <- function(par) {
  getAll(par)
  x <- OBS(x) # mark x as the response

  # random effect likelihood
  sd_gamma <- exp(log_sd_gamma)
  ADREPORT(sd_gamma); REPORT(sd_gamma)
  nll <- -sum(dnorm(gamma, 0, sd_gamma, log = TRUE))

  # data likelihood
  probs <- plogis(beta0 + gamma) # linear predictor
  ADREPORT(probs); REPORT(probs)
  nll <- nll - sum(dbinom(x, 1, probs[id], log = TRUE))
  return(nll)
}

# RTMB Laplace approximation
par <- list(beta0 = 1, log_sd_gamma = log(2), gamma = rep(0,10))
obj <- MakeADFun(nll, par, random = "gamma", silent = TRUE)

# model fitting using AD gradient
opt <- nlminb(obj$par, obj$fn, obj$gr)

# draw samples
samples <- mcreport(obj, report = TRUE)

# run sdreport
sdr <- sdreport(obj)

# standard deviation using delta method
as.list(sdr, "Std", report = TRUE)$probs

# standard deviation from samples
apply(simplify2array(samples$probs), 1, sd)

Multivariate t distribution

Description

Density and and random generation for the multivariate t distribution

Usage

dmvt(x, mu, Sigma, df, log = FALSE)

rmvt(n, mu, Sigma, df)

Arguments

x

vector or matrix of quantiles
mu

vector or matrix of location parameters (mean if df > 1)
Sigma

positive definite scale matrix (proportional to the covariance matrix if df > 2)
df

degrees of freedom; must be positive
log

logical; if TRUE, densities pp are returned as log⁡(p)\log(p).
n

number of random values to return.

Details

This implementation of dmvt allows for automatic differentiation with RTMB.

Note: for df ≤1\le 1 the mean is undefined, and for df ≤2\le 2 the covariance is infinite. For df > 2, the covariance is df/(df-2) * Sigma.

f(x; μ,Σ,ν)=Γ((ν+d)/2)Γ(ν/2) (νπ)d/2 ∣Σ∣1/2(1+(x−μ)⊤Σ−1(x−μ)ν)−(ν+d)/2,f(\mathbf{x};\,\boldsymbol{\mu},\Sigma,\nu) = \frac{\Gamma((\nu+d)/2)}{\Gamma(\nu/2)\,(\nu\pi)^{d/2}\,|\Sigma|^{1/2}} \left(1 + \frac{(\mathbf{x}-\boldsymbol{\mu})^\top \Sigma^{-1}(\mathbf{x}-\boldsymbol{\mu})}{\nu}\right)^{-(\nu+d)/2},

where dd is the dimension.

Value

dmvt gives the density, rmvt generates random deviates.

Examples

# single mu
mu <- c(1,2,3)
Sigma <- diag(c(1,1,1))
df <- 5
x <- rmvt(2, mu, Sigma, df)
d <- dmvt(x, mu, Sigma, df)
# vectorised over mu
mu <- rbind(c(1,2,3), c(0, 0.5, 1))
x <- rmvt(2, mu, Sigma, df)
d <- dmvt(x, mu, Sigma, df)

Reparameterised negative binomial distribution

Description

Probability mass function, distribution function, quantile function, and random generation for the negative binomial distribution reparameterised in terms of mean and size.

Usage

dnbinom2(x, mu, size, log = FALSE)

pnbinom2(q, mu, size, lower.tail = TRUE, log.p = FALSE)

qnbinom2(p, mu, size, lower.tail = TRUE, log.p = FALSE)

rnbinom2(n, mu, size)

Arguments

x, q

vector of quantiles
mu

mean parameter, must be positive.
size

size parameter, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

pnbinom is an AD-compatible implementation of the standard parameterisation of the CDF, missing from RTMB.

Reparameterises the negative binomial by mean μ\mu via p=r/(r+μ)p = r/(r+\mu):

P(X=k; μ,r)=(k+r−1k)(rr+μ)r(μr+μ)k,k=0,1,2,…P(X=k;\,\mu,r) = \binom{k+r-1}{k}\left(\frac{r}{r+\mu}\right)^r\left(\frac{\mu}{r+\mu}\right)^k, \quad k = 0,1,2,\ldots

Value

dnbinom2 gives the density, pnbinom2 gives the distribution function, qnbinom2 gives the quantile function, and rnbinom2 generates random deviates.

Examples

set.seed(123)
x <- rnbinom2(1, 1, 2)
d <- dnbinom2(x, 1, 2)
p <- pnbinom2(x, 1, 2)
q <- qnbinom2(p, 1, 2)

One-inflated beta distribution

Description

Density, distribution function, and random generation for the one-inflated beta distribution.

Usage

doibeta(x, shape1, shape2, oneprob = 0, log = FALSE)

poibeta(q, shape1, shape2, oneprob = 0, lower.tail = TRUE, log.p = FALSE)

roibeta(n, shape1, shape2, oneprob = 0)

Arguments

x, q

vector of quantiles
shape1, shape2

non-negative shape parameters of the beta distribution
oneprob

one-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

f(x; a,b,p1)=p1 1[x=1]+(1−p1) fBeta(x; a,b) 1[x∈(0,1)],f(x;\,a,b,p_1) = p_1\,\mathbf{1}[x=1] + (1-p_1)\,f_{\mathrm{Beta}}(x;\,a,b)\,\mathbf{1}[x\in(0,1)],

where fBetaf_{\mathrm{Beta}} is the beta density and p1p_1 is oneprob.

Value

doibeta gives the density, poibeta gives the distribution function, and roibeta generates random deviates.

Examples

set.seed(123)
x <- roibeta(1, 2, 2, 0.5)
d <- doibeta(x, 2, 2, 0.5)
p <- poibeta(x, 2, 2, 0.5)

Reparameterised one-inflated beta distribution

Description

Density, distribution function, and random generation for the one-inflated beta distribution reparameterised in terms of mean and concentration.

Usage

doibeta2(x, mu, phi, oneprob = 0, log = FALSE)

poibeta2(q, mu, phi, oneprob = 0, lower.tail = TRUE, log.p = FALSE)

roibeta2(n, mu, phi, oneprob = 0)

Arguments

x, q

vector of quantiles
mu

mean parameter, must be in the interval from 0 to 1.
phi

concentration parameter, must be positive.
oneprob

one-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Uses the same density as oibeta with a=μϕa = \mu\phi and b=(1−μ)ϕb = (1-\mu)\phi:

f(x; μ,ϕ,p1)=p1 1[x=1]+(1−p1) fBeta(x; μϕ, (1−μ)ϕ) 1[x∈(0,1)].f(x;\,\mu,\phi,p_1) = p_1\,\mathbf{1}[x=1] + (1-p_1)\,f_{\mathrm{Beta}}(x;\,\mu\phi,\,(1-\mu)\phi)\,\mathbf{1}[x\in(0,1)].

Value

doibeta2 gives the density, poibeta2 gives the distribution function, and roibeta2 generates random deviates.

Examples

set.seed(123)
x <- roibeta2(1, 0.6, 2, 0.5)
d <- doibeta2(x, 0.6, 2, 0.5)
p <- poibeta2(x, 0.6, 2, 0.5)

Pareto distribution

Description

Density, distribution function, quantile function, and random generation for the pareto distribution.

Usage

dpareto(x, mu = 1, log = FALSE)

ppareto(q, mu = 1, lower.tail = TRUE, log.p = FALSE)

qpareto(p, mu = 1, lower.tail = TRUE, log.p = FALSE)

rpareto(n, mu = 1)

Arguments

x, q

vector of quantiles
mu

location parameter, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dpareto and ppareto allows for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package. See gamlss.dist::PARETO for more details.

f(x; μ)=μxμ+1,x>1.f(x;\,\mu) = \frac{\mu}{x^{\mu+1}}, \quad x > 1.

Value

dpareto gives the density, ppareto gives the distribution function, qpareto gives the quantile function, and rpareto generates random deviates.

References

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

set.seed(123)
x <- rpareto(1, mu = 5)
d <- dpareto(x, mu = 5)
p <- ppareto(x, mu = 5)
q <- qpareto(p, mu = 5)

Power generalized Weibull distribution

Description

Survival, hazard, cumulative distribution, density, quantile and sampling function for the power generalized Weibull (PgW) distribution with parameters scale, shape and powershape.

Usage

spgweibull(q, scale = 1, shape = 1, powershape = 1, log = FALSE)

hpgweibull(x, scale = 1, shape = 1, powershape = 1, log = FALSE)

ppgweibull(q, scale = 1, shape = 1, powershape = 1,
           lower.tail = TRUE, log.p = FALSE)

dpgweibull(x, scale = 1, shape = 1, powershape = 1, log = FALSE)

qpgweibull(
  p,
  scale = 1,
  shape = 1,
  powershape = 1,
  lower.tail = TRUE,
  log.p = FALSE
)

rpgweibull(n, scale = 1, shape = 1, powershape = 1)

Arguments

scale

positive scale parameter
shape

positive shape parameter
powershape

positive power shape parameter
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
x, q

vector of quantiles
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of observations

Details

The survival function of the PgW distribution is:

S(x)=exp⁡{1−[1+(xθ)ν]1γ}.S(x) = \exp \left\{ 1 - \left[ 1 + \left(\frac{x}{\theta}\right)^{\nu}\right]^{\frac{1}{\gamma}} \right\}.

The hazard function is

νγθν⋅xν−1⋅[1+(xθ)ν]1γ−1\frac{\nu}{\gamma\theta^{\nu}}\cdot x^{\nu-1}\cdot \left[ 1 + \left(\frac{x}{\theta}\right)^{\nu}\right]^{\frac{1}{\gamma-1}}

The cumulative distribution function is then F(x)=1−S(x)F(x) = 1 - S(x) and the density function is S(x)⋅h(x)S(x)\cdot h(x).

If both shape parameters equal 1, the PgW distribution reduces to the exponential distribution (see dexp) with rate=1/scale\texttt{rate} = 1/\texttt{scale} If the power shape parameter equals 1, the PgW distribution simplifies to the Weibull distribution (see dweibull) with the same parametrization.

Value

dpgweibull gives the density, ppgweibull gives the distribution function, qpgweibull gives the quantile function, and rpgweibull generates random deviates. spgweibull gives the survival function and hpgweibull gives the hazard function.

Examples

x <- rpgweibull(1, 2, 2, 3)
d <- dpgweibull(x, 2, 2, 3)
p <- ppgweibull(x, 2, 2, 3)
q <- qpgweibull(p, 2, 2, 3)
s <- spgweibull(x, 2, 2, 3)
h <- hpgweibull(x, 2, 2, 3)

Power Exponential distribution (PE and PE2)

Description

Density, distribution function, quantile function, and random generation for the Power Exponential distribution (two versions).

Usage

dpowerexp(x, mu = 0, sigma = 1, nu = 2, log = FALSE)

ppowerexp(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

qpowerexp(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

rpowerexp(n, mu = 0, sigma = 1, nu = 2)

dpowerexp2(x, mu = 0, sigma = 1, nu = 2, log = FALSE)

ppowerexp2(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

qpowerexp2(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

rpowerexp2(n, mu = 0, sigma = 1, nu = 2)

Arguments

x, q

vector of quantiles
mu

location parameter
sigma

scale parameter, must be positive
nu

shape parameter (real)
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p)
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x]
p

vector of probabilities
n

number of random values to return

Details

This implementation of the densities and distribution functions allow for automatic differentiation with RTMB while the other functions are imported from gamlss.dist package.

For powerexp, mu is the mean and sigma is the standard deviation while this does not hold for powerexp2.

See gamlss.dist::PE for more details.

For powerexp (PE), σ\sigma is the standard deviation; the density is

f(x; μ,σ,ν)=ν2c σ Γ(1/ν)exp⁡ ⁣(−12∣x−μcσ∣ν),f(x;\,\mu,\sigma,\nu) = \frac{\nu}{2c\,\sigma\,\Gamma(1/\nu)} \exp\!\left(-\tfrac{1}{2}\left|\frac{x-\mu}{c\sigma}\right|^\nu\right),

where c=[2−2/νΓ(1/ν)/Γ(3/ν)]1/2c = [2^{-2/\nu}\Gamma(1/\nu)/\Gamma(3/\nu)]^{1/2}.

For powerexp2 (PE2), σ\sigma is a scale parameter; the density is

f(x; μ,σ,ν)=ν2σ Γ(1/ν)exp⁡ ⁣(−∣x−μσ∣ν).f(x;\,\mu,\sigma,\nu) = \frac{\nu}{2\sigma\,\Gamma(1/\nu)} \exp\!\left(-\left|\frac{x-\mu}{\sigma}\right|^\nu\right).

Value

dpowerexp gives the density, ppowerexp gives the distribution function, qpowerexp gives the quantile function, and rpowerexp generates random deviates.

References

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

# PE
x <- rpowerexp(1, mu = 0, sigma = 1, nu = 2)
d <- dpowerexp(x, mu = 0, sigma = 1, nu = 2)
p <- ppowerexp(x, mu = 0, sigma = 1, nu = 2)
q <- qpowerexp(p, mu = 0, sigma = 1, nu = 2)

# PE2
x <- rpowerexp2(1, mu = 0, sigma = 1, nu = 2)
d <- dpowerexp2(x, mu = 0, sigma = 1, nu = 2)
p <- ppowerexp2(x, mu = 0, sigma = 1, nu = 2)
q <- qpowerexp2(p, mu = 0, sigma = 1, nu = 2)

Sample from a multivariate Gaussian with a sparse precision matrix

Description

Draw samples from a multivariate Gaussian distribution specified by a sparse precision matrix. This is numerically efficient for high-dimensional but sparse systems.

Usage

rgmrf(n, mean = 0, Q)

Arguments

n

Number of samples to draw.
mean

Mean vector (or scalar, which will be recycled to match the dimension of Q).
Q

Sparse precision matrix (Σ−1\Sigma^{-1}).

Value

A matrix of samples with rows corresponding to samples and columns to dimensions.

Examples

rgmrf(3, mean = c(1, 2, 3), Q = Matrix::Diagonal(3))

Skellam distribution

Description

Probability mass function, distribution function, and random generation for the Skellam distribution.

Usage

dskellam(x, mu1, mu2, log = FALSE)

rskellam(n, mu1, mu2)

Arguments

x

integer vector of counts
mu1, mu2

Poisson means
log

logical; return log-density if TRUE
n

number of random values to return.

Details

The Skellam distribution is the distribution of the difference of two Poisson random variables. Specifically, if X1∼Pois(μ1)X_1 \sim \text{Pois}(\mu_1) and X2∼Pois(μ2)X_2 \sim \text{Pois}(\mu_2), then X1−X2∼Skellam(μ1,μ2)X_1 - X_2 \sim \text{Skellam}(\mu_1, \mu_2).

This implementation of dskellam allows for automatic differentiation with RTMB.

Value

dskellam gives the probability mass function and rskellam generates random deviates.

Examples

x <- rskellam(1, 2, 3)
d <- dskellam(x, 2, 3)

Skew normal distribution

Description

Density, distribution function, quantile function, and random generation for the skew normal distribution.

Usage

dskewnorm(x, xi = 0, omega = 1, alpha = 0, log = FALSE)

pskewnorm(q, xi = 0, omega = 1, alpha = 0, lower.tail = TRUE, log.p = FALSE)

qskewnorm(p, xi = 0, omega = 1, alpha = 0, lower.tail = TRUE, log.p = FALSE)

rskewnorm(n, xi = 0, omega = 1, alpha = 0)

Arguments

x, q

vector of quantiles
xi

location parameter
omega

scale parameter, must be positive.
alpha

skewness parameter, +/- Inf is allowed.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dskewnorm allows for automatic differentiation with RTMB while the other functions are imported from the sn package. See sn::dsn for more details.

f(x; ξ,ω,α)=2ω ϕ ⁣(x−ξω)Φ ⁣(αx−ξω),f(x;\,\xi,\omega,\alpha) = \frac{2}{\omega}\,\phi\!\left(\frac{x-\xi}{\omega}\right)\Phi\!\left(\alpha\frac{x-\xi}{\omega}\right),

where ϕ\phi and Φ\Phi are the standard normal PDF and CDF.

Value

dskewnorm gives the density, pskewnorm gives the distribution function, qskewnorm gives the quantile function, and rskewnorm generates random deviates.

See Also

skewnorm2, skewt, skewt2

Examples

# alpha is skew parameter
x <- rskewnorm(1, alpha = 1)
d <- dskewnorm(x, alpha = 1)
p <- pskewnorm(x, alpha = 1)
q <- qskewnorm(p, alpha = 1)

Reparameterised skew normal distribution

Description

Density, distribution function, quantile function and random generation for the skew normal distribution reparameterised in terms of mean, standard deviation and skew magnitude

Usage

dskewnorm2(x, mean = 0, sd = 1, alpha = 0, log = FALSE)

pskewnorm2(q, mean = 0, sd = 1, alpha = 0, lower.tail = TRUE, log.p = FALSE)

qskewnorm2(p, mean = 0, sd = 1, alpha = 0, lower.tail = TRUE, log.p = FALSE)

rskewnorm2(n, mean = 0, sd = 1, alpha = 0)

Arguments

x, q

vector of quantiles
mean

mean parameter
sd

standard deviation, must be positive.
alpha

skewness parameter, +/- Inf is allowed.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return

Details

This implementation of dskewnorm2 allows for automatic differentiation with RTMB while the other functions are imported from the sn package.

Uses the same density as skewnorm with location ξ\xi, scale ω\omega, and shape α\alpha reparameterised from the mean mm, standard deviation ss, and skewness parameter α\alpha:

δ=α1+α2,ω=s1−2δ2/π,ξ=m−ωδ2/π.\delta = \frac{\alpha}{\sqrt{1+\alpha^2}}, \quad \omega = \frac{s}{\sqrt{1 - 2\delta^2/\pi}}, \quad \xi = m - \omega\delta\sqrt{2/\pi}.

Value

dskewnorm2 gives the density, pskewnorm2 gives the distribution function, qskewnorm2 gives the quantile function, and rskewnorm2 generates random deviates.

See Also

skewnorm, skewt, skewt2

Examples

# alpha is skew parameter
x <- rskewnorm2(1, alpha = 1)
d <- dskewnorm2(x, alpha = 1)
p <- pskewnorm2(x, alpha = 1)
q <- qskewnorm2(p, alpha = 1)

Skewed students t distribution

Description

Density, distribution function, quantile function, and random generation for the skew t distribution (type 2).

Usage

dskewt(x, mu = 0, sigma = 1, skew = 0, df = 100, log = FALSE)

pskewt(q, mu = 0, sigma = 1, skew = 0, df = 100,
       method = 0, lower.tail = TRUE, log.p = FALSE)

qskewt(p, mu = 0, sigma = 1, skew = 0, df = 100,
       tol = 1e-8, method = 0, lower.tail = TRUE, log.p = FALSE)

rskewt(n, mu = 0, sigma = 1, skew = 0, df = 100)

Arguments

x, q

vector of quantiles
mu

location parameter
sigma

scale parameter, must be positive.
skew

skewness parameter, can be positive or negative.
df

degrees of freedom, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
method

an integer value between 0 and 5 which selects the computing method; see ‘Details’ in the pst documentation below for the meaning of these values. If method=0 (default value), an automatic choice is made among the four actual computing methods, depending on the other arguments.
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
tol

a scalar value which regulates the accuracy of the result of qsn, measured on the probability scale.
n

number of random values to return.

Details

This corresponds to the skew t type 2 distribution in GAMLSS (ST2), see pp. 411-412 of Rigby et al. (2019) and the version implemented in the sn package. This implementation of dskewt allows for automatic differentiation with RTMB while the other functions are imported from the sn package. See sn::dst for more details.

Caution: In a numerial optimisation, the skew parameter should NEVER be initialised with exactly zero. This will cause the initial and all subsequent derivatives to be exactly zero and hence the parameter will remain at its initial value.

f(x; μ,σ,λ,ν)=2σ ft ⁣(x−μσ; ν)Ft ⁣(λν+1ν+z2⋅z; ν+1),f(x;\,\mu,\sigma,\lambda,\nu) = \frac{2}{\sigma}\, f_t\!\left(\frac{x-\mu}{\sigma};\,\nu\right) F_t\!\left(\lambda\sqrt{\frac{\nu+1}{\nu + z^2}}\cdot z;\,\nu+1\right),

where z=(x−μ)/σz=(x-\mu)/\sigma, ft(⋅;ν)f_t(\cdot;\nu) is the Student-tt PDF, and Ft(⋅;ν+1)F_t(\cdot;\nu+1) is its CDF.

Value

dskewt gives the density, pskewt gives the distribution function, qskewt gives the quantile function, and rskewt generates random deviates.

See Also

skewt2, skewnorm, skewnorm2

Examples

x <- rskewt(1, 1, 2, 5, 2)
d <- dskewt(x, 1, 2, 5, 2)
p <- pskewt(x, 1, 2, 5, 2)
q <- qskewt(p, 1, 2, 5, 2)

Moment-parameterised skew t distribution

Description

Density, distribution function, quantile function, and random generation for the skew t distribution reparameterised so that mean and sd correspond to the true mean and standard deviation.

Usage

dskewt2(x, mean = 0, sd = 1, skew = 0, df = 100, log = FALSE)

pskewt2(q, mean = 0, sd = 1, skew = 0, df = 100,
        method = 0, lower.tail = TRUE, log.p = FALSE)

qskewt2(
  p,
  mean = 0,
  sd = 1,
  skew = 0,
  df = 100,
  tol = 1e-08,
  method = 0,
  lower.tail = TRUE,
  log.p = FALSE
)

rskewt2(n, mean = 0, sd = 1, skew = 0, df = 100)

Arguments

x, q

vector of quantiles
mean

mean parameter
sd

standard deviation parameter, must be positive.
skew

skewness parameter, can be positive or negative.
df

degrees of freedom, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
method

an integer value between 0 and 5 which selects the computing method; see ‘Details’ in the pst documentation below for the meaning of these values. If method=0 (default value), an automatic choice is made among the four actual computing methods, depending on the other arguments.
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
tol

a scalar value which regulates the accuracy of the result of qsn, measured on the probability scale.
n

number of random values to return.

Details

This corresponds to the skew t type 2 distribution in GAMLSS (ST2), see pp. 411-412 of Rigby et al. (2019) and the version implemented in the sn package. However, it is reparameterised in terms of a standard deviation parameter sd rather than just a scale parameter sigma. Details of this reparameterisation are given below. This implementation of dskewt allows for automatic differentiation with RTMB while the other functions are imported from the sn package. See sn::dst for more details.

Caution: In a numerial optimisation, the skew parameter should NEVER be initialised with exactly zero. This will cause the initial and all subsequent derivatives to be exactly zero and hence the parameter will remain at its initial value.

For given skew =α= \alpha and df = ν\nu, define

δ=α/1+α2,bν=ν/π Γ((ν−1)/2)/Γ(ν/2),\delta = \alpha / \sqrt{1 + \alpha^2}, \qquad b_\nu = \sqrt{\nu / \pi}\, \Gamma((\nu-1)/2)/\Gamma(\nu/2),

then

E(X)=μ+σδbν,Var(X)=σ2(νν−2−δ2bν2).E(X) = \mu + \sigma \delta b_\nu,\quad Var(X) = \sigma^2 \left( \frac{\nu}{\nu-2} - \delta^2 b_\nu^2 \right).

Value

dskewt2 gives the density, pskewt2 gives the distribution function, qskewt2 gives the quantile function, and rskewt2 generates random deviates.

See Also

skewt, skewnorm, skewnorm2

Examples

x <- rskewt2(1, 1, 2, 5, 5)
d <- dskewt2(x, 1, 2, 5, 5)
p <- pskewt2(x, 1, 2, 5, 5)
q <- qskewt2(p, 1, 2, 5, 5)

Student t distribution with location and scale

Description

Density, distribution function, quantile function, and random generation for the t distribution with location and scale parameters.

Usage

dt2(x, mu, sigma, df, log = FALSE)

pt2(q, mu, sigma, df, lower.tail = TRUE, log.p = FALSE)

rt2(n, mu, sigma, df)

qt2(p, mu, sigma, df, lower.tail = TRUE, log.p = FALSE)

pt(q, df)

Arguments

x, q

vector of quantiles
mu

location parameter
sigma

scale parameter, must be positive.
df

degrees of freedom, must be positive.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities

Details

This implementation of dt2 allows for automatic differentiation with RTMB.

f(x; μ,σ,ν)=1σ ft ⁣(x−μσ; ν),f(x;\,\mu,\sigma,\nu) = \frac{1}{\sigma}\,f_t\!\left(\frac{x-\mu}{\sigma};\,\nu\right),

where ft(⋅;ν)f_t(\cdot;\nu) is the Student-tt PDF with ν\nu degrees of freedom.

Value

dt2 gives the density, pt2 gives the distribution function, qt2 gives the quantile function, and rt2 generates random deviates.

Examples

x <- rt2(1, 1, 2, 5)
d <- dt2(x, 1, 2, 5)
p <- pt2(x, 1, 2, 5)
q <- qt2(p, 1, 2, 5)

Truncated normal distribution

Description

Density, distribution function, quantile function, and random generation for the truncated normal distribution.

Usage

dtruncnorm(x, mean = 0, sd = 1, min = -Inf, max = Inf, log = FALSE)

ptruncnorm(q, mean = 0, sd = 1, min = -Inf, max = Inf,
           lower.tail = TRUE, log.p = FALSE)

qtruncnorm(p, mean = 0, sd = 1, min = -Inf, max = Inf,
           lower.tail = TRUE, log.p = FALSE)

rtruncnorm(n, mean = 0, sd = 1, min = -Inf, max = Inf)

Arguments

x, q

vector of quantiles
mean

mean parameter, must be positive.
sd

standard deviation parameter, must be positive.
min, max

truncation bounds.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return.

Details

This implementation of dtruncnorm allows for automatic differentiation with RTMB.

f(x; μ,σ,a,b)=ϕ((x−μ)/σ)/σΦ((b−μ)/σ)−Φ((a−μ)/σ),x∈[a,b],f(x;\,\mu,\sigma,a,b) = \frac{\phi((x-\mu)/\sigma)/\sigma}{\Phi((b-\mu)/\sigma) - \Phi((a-\mu)/\sigma)}, \quad x \in [a, b],

where ϕ\phi and Φ\Phi are the standard normal PDF and CDF.

Value

dtruncnorm gives the density, ptruncnorm gives the distribution function, qtruncnorm gives the quantile function, and rtruncnorm generates random deviates.

Examples

x <- rtruncnorm(1, mean = 2, sd = 2, min = -1, max = 5)
d <- dtruncnorm(x, mean = 2, sd = 2, min = -1, max = 5)
p <- ptruncnorm(x, mean = 2, sd = 2, min = -1, max = 5)
q <- qtruncnorm(p, mean = 2, sd = 2, min = -1, max = 5)

Truncated t distribution

Description

Density, distribution function, quantile function, and random generation for the truncated t distribution.

Usage

dtrunct(x, df, min = -Inf, max = Inf, log = FALSE)

ptrunct(q, df, min = -Inf, max = Inf, lower.tail = TRUE, log.p = FALSE)

qtrunct(p, df, min = -Inf, max = Inf, lower.tail = TRUE, log.p = FALSE)

rtrunct(n, df, min = -Inf, max = Inf)

Arguments

x, q

vector of quantiles
df

degrees of freedom parameter, must be positive.
min, max

truncation bounds.
log, log.p

logical; if TRUE, probabilities/densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return.

Details

This implementation of dtrunct allows for automatic differentiation with RTMB.

f(x; ν,a,b)=ft(x; ν)Ft(b; ν)−Ft(a; ν),x∈[a,b],f(x;\,\nu,a,b) = \frac{f_t(x;\,\nu)}{F_t(b;\,\nu) - F_t(a;\,\nu)}, \quad x \in [a, b],

where ft(⋅;ν)f_t(\cdot;\nu) and Ft(⋅;ν)F_t(\cdot;\nu) are the Student-tt PDF and CDF.

Value

dtrunct gives the density, ptrunct gives the distribution function, qtrunct gives the quantile function, and rtrunct generates random deviates.

Examples

x <- rtrunct(1, df = 5, min = -1, max = 5)
d <- dtrunct(x, df = 5, min = -1, max = 5)
p <- ptrunct(x, df = 5, min = -1, max = 5)
q <- qtrunct(p, df = 5, min = -1, max = 5)

Truncated t distribution with location and scale

Description

Density, distribution function, quantile function, and random generation for the truncated t distribution with location mu and scale sigma.

Usage

dtrunct2(x, df, mu = 0, sigma = 1, min = -Inf, max = Inf, log = FALSE)

ptrunct2(q, df, mu = 0, sigma = 1, min = -Inf, max = Inf,
         lower.tail = TRUE, log.p = FALSE)

qtrunct2(p, df, mu = 0, sigma = 1, min = -Inf, max = Inf,
         lower.tail = TRUE, log.p = FALSE)

rtrunct2(n, df, mu = 0, sigma = 1, min = -Inf, max = Inf)

Arguments

x, q

vector of quantiles
df

degrees of freedom parameter, must be positive.
mu

location parameter.
sigma

scale parameter, must be positive.
min, max

truncation bounds.
log, log.p

logical; if TRUE, probabilities/densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
p

vector of probabilities
n

number of random values to return.

Details

This implementation of dtrunct2 allows for automatic differentiation with RTMB.

f(x; ν,μ,σ,a,b)=ft((x−μ)/σ; ν)/σFt((b−μ)/σ; ν)−Ft((a−μ)/σ; ν),x∈[a,b].f(x;\,\nu,\mu,\sigma,a,b) = \frac{f_t((x-\mu)/\sigma;\,\nu)/\sigma}{F_t((b-\mu)/\sigma;\,\nu) - F_t((a-\mu)/\sigma;\,\nu)}, \quad x \in [a, b].

Value

dtrunct2 gives the density, ptrunct2 gives the distribution function, qtrunct2 gives the quantile function, and rtrunct2 generates random deviates.

Examples

x <- rtrunct2(1, df = 5, mu = 2, sigma = 3, min = -1, max = 5)
d <- dtrunct2(x, df = 5, mu = 2, sigma = 3, min = -1, max = 5)
p <- ptrunct2(x, df = 5, mu = 2, sigma = 3, min = -1, max = 5)
q <- qtrunct2(p, df = 5, mu = 2, sigma = 3, min = -1, max = 5)

von Mises distribution

Description

Density, distribution function, and random generation for the von Mises distribution.

Usage

dvm(x, mu = 0, kappa = 1, log = FALSE)

pvm(
  q,
  mu = 0,
  kappa = 1,
  from = NULL,
  tol = 1e-20,
  lower.tail = TRUE,
  log.p = FALSE
)

rvm(n, mu = 0, kappa = 1, wrap = TRUE)

Arguments

x, q

vector of angles measured in radians at which to evaluate the density function.
mu

mean direction of the distribution measured in radians.
kappa

non-negative numeric value for the concentration parameter of the distribution.
log

logical; if TRUE, densities are returned on the log scale.
from

value from which the integration for CDF starts. If NULL, is set to mu - pi.
tol

the precision in evaluating the distribution function
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
log.p

logical; if TRUE, probabilities are returned as log⁡(p)\log(p).
n

number of random values to return.
wrap

logical; if TRUE, generated angles are wrapped to the interval from -pi to pi.

Details

This implementation of dvm allows for automatic differentiation with RTMB. rvm and pvm are simply wrappers of the corresponding functions from circular.

f(x; μ,κ)=exp⁡(κcos⁡(x−μ))2π I0(κ),f(x;\,\mu,\kappa) = \frac{\exp(\kappa\cos(x-\mu))}{2\pi\, I_0(\kappa)},

where I0I_0 is the modified Bessel function of the first kind of order 0.

Value

dvm gives the density, pvm gives the distribution function, and rvm generates random deviates.

Examples

set.seed(1)
x <- rvm(10, 0, 1)
d <- dvm(x, 0, 1)
p <- pvm(x, 0, 1)

von Mises-Fisher distribution

Description

Density, distribution function, and random generation for the von Mises-Fisher distribution.

Usage

dvmf(x, mu, kappa, log = FALSE)

rvmf(n, mu, kappa)

Arguments

x

unit vector or matrix (with each row being a unit vector) of evaluation points
mu

unit mean vector
kappa

non-negative numeric value for the concentration parameter of the distribution.
log

logical; if TRUE, densities are returned on the log scale.
n

number of random values to return.

Details

This implementation of dvmf allows for automatic differentiation with RTMB. rvmf is a reparameterised import from movMF::rmovMF.

f(x; μ,κ)=Cp(κ) exp⁡(κ μ⊤x),f(\mathbf{x};\,\boldsymbol{\mu},\kappa) = C_p(\kappa)\,\exp(\kappa\,\boldsymbol{\mu}^\top\mathbf{x}),

where Cp(κ)=κp/2−1/((2π)p/2Ip/2−1(κ))C_p(\kappa) = \kappa^{p/2-1} / ((2\pi)^{p/2} I_{p/2-1}(\kappa)) is the normalising constant and IνI_\nu is the modified Bessel function of the first kind of order ν\nu.

Value

dvmf gives the density and rvm generates random deviates.

Examples

set.seed(123)
# single parameter set
mu <- rep(1, 3) / sqrt(3)
kappa <- 4
x <- rvmf(1, mu, kappa)
d <- dvmf(x, mu, kappa)

# vectorised over parameters
mu <- matrix(mu, nrow = 1)
mu <- mu[rep(1,10), ]
kappa <- rep(kappa, 10)
x <- rvmf(10, mu, kappa)
d <- dvmf(x, mu, kappa)

Reparameterised von Mises-Fisher distribution

Description

Density, distribution function, and random generation for the von Mises-Fisher distribution.

Usage

dvmf2(x, theta, log = FALSE)

rvmf2(n, theta)

Arguments

x

unit vector or matrix (with each row being a unit vector) of evaluation points
theta

direction and concentration vector. The direction of theta determines the mean direction on the sphere. The norm of theta is the concentration parameter of the distribution.
log

logical; if TRUE, densities are returned on the log scale.
n

number of random values to return.

Details

In this parameterisation, θ=κμ\theta = \kappa \mu, where μ\mu is a unit vector and κ\kappa is the concentration parameter.

dvmf2 allows for automatic differentiation with RTMB. rvmf2 is imported from movMF::rmovMF.

Value

dvmf gives the density and rvm generates random deviates.

Examples

set.seed(123)
# single parameter set
theta <- c(1,2,3)
x <- rvmf2(1, theta)
d <- dvmf2(x, theta)

# vectorised over parameters
theta <- matrix(theta, nrow = 1)
theta <- theta[rep(1,10), ]
x <- rvmf2(10, theta)
d <- dvmf2(x, theta)

Wishart distribution

Description

Density and random generation for the wishart distribution

Usage

dwishart(x, nu, Sigma, log = FALSE)

rwishart(n, nu, Sigma)

Arguments

x

positive definite p×pp \times p matrix or array of such matrices of dimension p×p×np \times p \times n (for nn density evaluations)
nu

degrees of freedom, needs to be greater than p - 1
Sigma

scale matrix, needs to be positive definite and match the dimension of x.
log

logical; if TRUE, densities pp are returned as log⁡(p)\log(p).
n

number of random deviates to return

Details

f(X; ν,Σ)=∣X∣(ν−p−1)/2exp⁡ ⁣(−12tr⁡(Σ−1X))2νp/2 ∣Σ∣ν/2 Γp(ν/2),f(X;\,\nu,\Sigma) = \frac{|X|^{(\nu-p-1)/2}\exp\!\left(-\tfrac{1}{2}\operatorname{tr}(\Sigma^{-1}X)\right)}{2^{\nu p/2}\,|\Sigma|^{\nu/2}\,\Gamma_p(\nu/2)},

where pp is the dimension and Γp\Gamma_p is the multivariate gamma function.

Value

dwishart gives the density, rwishart generates random deviates (matrix for n = 1, array with n slices for n > 1)

Examples

# single input: matrix-valued
x <- rwishart(1, nu = 5, Sigma = diag(3))
d <- dwishart(x, nu = 5, Sigma = diag(3))

# multiple inputs: array of matrices
x <- rwishart(4, nu = 5, Sigma = diag(3))
d <- dwishart(x, nu = 5, Sigma = diag(3))

# multiple inputs for x, nu and Sigma
nu <- c(7,5,8,9)
Sigma <- array(dim = c(3,3,4))
for(i in 1:4) Sigma[,,i] <- (i + 3) * diag(3)
x <- rwishart(4, nu, Sigma)
d <- dwishart(x, nu, Sigma)

wrapped Cauchy distribution

Description

Density and random generation for the wrapped Cauchy distribution.

Usage

dwrpcauchy(x, mu = 0, rho, log = FALSE)

rwrpcauchy(n, mu = 0, rho, wrap = TRUE)

Arguments

x

vector of angles measured in radians at which to evaluate the density function.
mu

mean direction of the distribution measured in radians.
rho

concentration parameter of the distribution, must be in the interval from 0 to 1.
log

logical; if TRUE, densities are returned on the log scale.
n

number of random values to return.
wrap

logical; if TRUE, generated angles are wrapped to the interval from -pi to pi.

Details

This implementation of dwrpcauchy allows for automatic differentiation with RTMB. rwrpcauchy is simply a wrapper for rwrappedcauchyimported from circular.

f(x; μ,ρ)=12π⋅1−ρ21+ρ2−2ρcos⁡(x−μ).f(x;\,\mu,\rho) = \frac{1}{2\pi}\cdot\frac{1-\rho^2}{1 + \rho^2 - 2\rho\cos(x-\mu)}.

Value

dwrpcauchy gives the density and rwrpcauchy generates random deviates.

Examples

set.seed(1)
x <- rwrpcauchy(10, 0, 0.5)
d <- dwrpcauchy(x, 0, 0.5)

Zero-inflated density constructer

Description

Constructs a zero-inflated density function from a given probability density function

Usage

zero_inflate(dist, discrete = NULL)

Arguments

dist

either a probability density function or a probability mass function
discrete

logical; if TRUE, the density for x = 0 will be zeroprob + (1-zeroprob) * dist(0, ...). Otherwise it will just be zeroprob. In standard cases, this will be determined automatically. For non-standard cases, set this to TRUE or FALSE depending on the type of dist. See details.

Details

The definition of zero-inflation is different for discrete and continuous distributions. For discrete distributions with p.m.f. ff and zero-inflation probability pp, we have

Pr⁡(X=0)=p+(1−p)⋅f(0),\Pr(X = 0) = p + (1 - p) \cdot f(0),

and

Pr⁡(X=x)=(1−p)⋅f(x),x>0.\Pr(X = x) = (1 - p) \cdot f(x), \quad x > 0.

For continuous distributions with p.d.f. ff, we have

fzinfl(x)=p⋅δ0(x)+(1−p)⋅f(x),f_{\text{zinfl}}(x) = p \cdot \delta_0(x) + (1 - p) \cdot f(x),

where δ0\delta_0 is the Dirac delta function at zero.

Value

zero-inflated density function with first argument x, second argument zeroprob, and additional arguments ... that will be passed to dist.

Examples

# Zero-inflated normal distribution
dzinorm <- zero_inflate(dnorm)
dzinorm(c(NA, 0, 2), 0.5, mean = 1, sd = 1)

# Zero-inflated Poisson distribution
zipois <- zero_inflate(dpois)
zipois(c(NA, 0, 1), 0.5, 1)

# Non-standard case: Zero-inflated reparametrised beta distribution
dzibeta2 <- zero_inflate(dbeta2, discrete = FALSE)

Zero-inflated beta distribution

Description

Density, distribution function, and random generation for the zero-inflated beta distribution.

Usage

dzibeta(x, shape1, shape2, zeroprob = 0, log = FALSE)

pzibeta(q, shape1, shape2, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzibeta(n, shape1, shape2, zeroprob = 0)

Arguments

x, q

vector of quantiles
shape1, shape2

non-negative shape parameters of the beta distribution
zeroprob

zero-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

f(x; a,b,p0)=p0 1[x=0]+(1−p0) fBeta(x; a,b) 1[x∈(0,1)],f(x;\,a,b,p_0) = p_0\,\mathbf{1}[x=0] + (1-p_0)\,f_{\mathrm{Beta}}(x;\,a,b)\,\mathbf{1}[x\in(0,1)],

where p0p_0 is zeroprob.

Value

dzibeta gives the density, pzibeta gives the distribution function, and rzibeta generates random deviates.

Examples

set.seed(123)
x <- rzibeta(1, 2, 2, 0.5)
d <- dzibeta(x, 2, 2, 0.5)
p <- pzibeta(x, 2, 2, 0.5)

Reparameterised zero-inflated beta distribution

Description

Density, distribution function, and random generation for the zero-inflated beta distribution reparameterised in terms of mean and concentration.

Usage

dzibeta2(x, mu, phi, zeroprob = 0, log = FALSE)

pzibeta2(q, mu, phi, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzibeta2(n, mu, phi, zeroprob = 0)

Arguments

x, q

vector of quantiles
mu

mean parameter, must be in the interval from 0 to 1.
phi

concentration parameter, must be positive.
zeroprob

zero-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities

Details

This implementation allows for automatic differentiation with RTMB.

Uses the same density as zibeta with a=μϕa = \mu\phi and b=(1−μ)ϕb = (1-\mu)\phi:

f(x; μ,ϕ,p0)=p0 1[x=0]+(1−p0) fBeta(x; μϕ, (1−μ)ϕ) 1[x∈(0,1)].f(x;\,\mu,\phi,p_0) = p_0\,\mathbf{1}[x=0] + (1-p_0)\,f_{\mathrm{Beta}}(x;\,\mu\phi,\,(1-\mu)\phi)\,\mathbf{1}[x\in(0,1)].

Value

dzibeta2 gives the density, pzibeta2 gives the distribution function, and rzibeta2 generates random deviates.

Examples

set.seed(123)
x <- rzibeta2(1, 0.5, 1, 0.5)
d <- dzibeta2(x, 0.5, 1, 0.5)
p <- pzibeta2(x, 0.5, 1, 0.5)

Zero-inflated binomial distribution

Description

Probability mass function, distribution function, and random generation for the zero-inflated binomial distribution.

Usage

dzibinom(x, size, prob, zeroprob = 0, log = FALSE)

pzibinom(q, size, prob, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzibinom(n, size, prob, zeroprob = 0)

Arguments

x, q

vector of quantiles
size

number of trials (zero or more).
prob

probability of success on each trial.
zeroprob

zero-inflation probability between 0 and 1
log, log.p

logical; return log-density if TRUE
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

P(X=k; n,p,p0)=p0 1[k=0]+(1−p0) PBin(k; n,p),P(X=k;\,n,p,p_0) = p_0\,\mathbf{1}[k=0] + (1-p_0)\,P_{\mathrm{Bin}}(k;\,n,p),

where p0p_0 is zeroprob.

Value

dzibinom gives the probability mass function, pzibinom gives the distribution function, and rzibinom generates random deviates.

Examples

set.seed(123)
x <- rzibinom(1, size = 10, prob = 0.5, zeroprob = 0.5)
d <- dzibinom(x, size = 10, prob = 0.5, zeroprob = 0.5)
p <- pzibinom(x, size = 10, prob = 0.5, zeroprob = 0.5)

Zero-inflated gamma distribution

Description

Density, distribution function, and random generation for the zero-inflated gamma distribution.

Usage

dzigamma(x, shape, scale, zeroprob = 0, log = FALSE)

pzigamma(q, shape, scale, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzigamma(n, shape, scale, zeroprob = 0)

Arguments

x, q

vector of quantiles
shape

positive shape parameter
scale

positive scale parameter
zeroprob

zero-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return

Details

This implementation allows for automatic differentiation with RTMB.

f(x; α,s,p0)=p0 1[x=0]+(1−p0) fGamma(x; α,s) 1[x>0],f(x;\,\alpha,s,p_0) = p_0\,\mathbf{1}[x=0] + (1-p_0)\,f_{\mathrm{Gamma}}(x;\,\alpha,s)\,\mathbf{1}[x>0],

where p0p_0 is zeroprob.

Value

dzigamma gives the density, pzigamma gives the distribution function, and rzigamma generates random deviates.

Examples

x <- rzigamma(1, 1, 1, 0.5)
d <- dzigamma(x, 1, 1, 0.5)
p <- pzigamma(x, 1, 1, 0.5)

Zero-inflated and reparameterised gamma distribution

Description

Density, distribution function, and random generation for the zero-inflated gamma distribution reparameterised in terms of mean and standard deviation.

Usage

dzigamma2(x, mean = 1, sd = 1, zeroprob = 0, log = FALSE)

pzigamma2(q, mean = 1, sd = 1, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzigamma2(n, mean = 1, sd = 1, zeroprob = 0)

Arguments

x, q

vector of quantiles
mean

mean parameter, must be positive.
sd

standard deviation parameter, must be positive.
zeroprob

zero-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x], otherwise P[X>x]P[X > x].
n

number of random values to return

Details

This implementation allows for automatic differentiation with RTMB.

Uses the same density as zigamma with shape=μ2/s2\text{shape} = \mu^2/s^2 and scale=s2/μ\text{scale} = s^2/\mu:

f(x; μ,s,p0)=p0 1[x=0]+(1−p0) fGamma(x; μ2/s2, s2/μ) 1[x>0].f(x;\,\mu,s,p_0) = p_0\,\mathbf{1}[x=0] + (1-p_0)\,f_{\mathrm{Gamma}}(x;\,\mu^2/s^2,\,s^2/\mu)\,\mathbf{1}[x>0].

Value

dzigamma2 gives the density, pzigamma2 gives the distribution function, and rzigamma2 generates random deviates.

Examples

x <- rzigamma2(1, 2, 1, 0.5)
d <- dzigamma2(x, 2, 1, 0.5)
p <- pzigamma2(x, 2, 1, 0.5)

Zero-inflated inverse Gaussian distribution

Description

Density, distribution function, and random generation for the zero-inflated inverse Gaussian distribution.

Usage

dziinvgauss(x, mean = 1, shape = 1, zeroprob = 0, log = FALSE)

pziinvgauss(q, mean = 1, shape = 1, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rziinvgauss(n, mean = 1, shape = 1, zeroprob = 0)

Arguments

x, q

vector of quantiles
mean

location parameter
shape

shape parameter, must be positive.
zeroprob

zero-probability, must be in [0,1][0, 1].
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return

Details

This implementation of zidinvgauss allows for automatic differentiation with RTMB.

f(x; μ,λ,p0)=p0 1[x=0]+(1−p0) fIG(x; μ,λ) 1[x>0],f(x;\,\mu,\lambda,p_0) = p_0\,\mathbf{1}[x=0] + (1-p_0)\,f_{\mathrm{IG}}(x;\,\mu,\lambda)\,\mathbf{1}[x>0],

where p0p_0 is zeroprob and fIGf_{\mathrm{IG}} is the inverse Gaussian density.

Value

dziinvgauss gives the density, pziinvgauss gives the distribution function, and rziinvgauss generates random deviates.

Examples

x <- rziinvgauss(1, 1, 2, 0.5)
d <- dziinvgauss(x, 1, 2, 0.5)
p <- pziinvgauss(x, 1, 2, 0.5)

Zero-inflated log normal distribution

Description

Density, distribution function, and random generation for the zero-inflated log normal distribution.

Usage

dzilnorm(x, meanlog = 0, sdlog = 1, zeroprob = 0, log = FALSE)

pzilnorm(q, meanlog = 0, sdlog = 1, zeroprob = 0,
         lower.tail = TRUE, log.p = FALSE)

rzilnorm(n, meanlog = 0, sdlog = 1, zeroprob = 0)

plnorm(q, meanlog = 0, sdlog = 1, lower.tail = TRUE, log.p = FALSE)

Arguments

x, q

vector of quantiles
meanlog, sdlog

mean and standard deviation of the distribution on the log scale with default values of 0 and 1 respectively.
zeroprob

zero-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return

Details

This implementation allows for automatic differentiation with RTMB.

f(x; μℓ,σℓ,p0)=p0 1[x=0]+(1−p0) fLN(x; μℓ,σℓ) 1[x>0],f(x;\,\mu_{\ell},\sigma_\ell,p_0) = p_0\,\mathbf{1}[x=0] + (1-p_0)\,f_{\mathrm{LN}}(x;\,\mu_\ell,\sigma_\ell)\,\mathbf{1}[x>0],

where p0p_0 is zeroprob, μℓ\mu_\ell = meanlog, σℓ\sigma_\ell = sdlog, and fLNf_{\mathrm{LN}} is the log-normal density.

Value

dzilnorm gives the density, pzilnorm gives the distribution function, and rzilnorm generates random deviates.

Examples

x <- rzilnorm(1, 1, 1, 0.5)
d <- dzilnorm(x, 1, 1, 0.5)
p <- pzilnorm(x, 1, 1, 0.5)

Zero-inflated negative binomial distribution

Description

Probability mass function, distribution function, quantile function, and random generation for the zero-inflated negative binomial distribution.

Usage

dzinbinom(x, size, prob, zeroprob = 0, log = FALSE)

pzinbinom(q, size, prob, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzinbinom(n, size, prob, zeroprob = 0)

Arguments

x, q

vector of (non-negative integer) quantiles
size

size parameter, must be positive.
prob

mean parameter, must be positive.
zeroprob

zero-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities

Details

This implementation allows for automatic differentiation with RTMB.

P(X=k; r,p,p0)=p0 1[k=0]+(1−p0) PNB(k; r,p),P(X=k;\,r,p,p_0) = p_0\,\mathbf{1}[k=0] + (1-p_0)\,P_{\mathrm{NB}}(k;\,r,p),

where p0p_0 is zeroprob.

Value

dzinbinom gives the density, pzinbinom gives the distribution function, and rzinbinom generates random deviates.

Examples

set.seed(123)
x <- rzinbinom(1, size = 2, prob = 0.5, zeroprob = 0.5)
d <- dzinbinom(x, size = 2, prob = 0.5, zeroprob = 0.5)
p <- pzinbinom(x, size = 2, prob = 0.5, zeroprob = 0.5)

Zero-inflated and reparameterised negative binomial distribution

Description

Probability mass function, distribution function, quantile function and random generation for the zero-inflated negative binomial distribution reparameterised in terms of mean and size.

Usage

dzinbinom2(x, mu, size, zeroprob = 0, log = FALSE)

pzinbinom2(q, mu, size, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzinbinom2(n, mu, size, zeroprob = 0)

Arguments

x, q

vector of (non-negative integer) quantiles
mu

mean parameter, must be positive.
size

size parameter, must be positive.
zeroprob

zero-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities

Details

This implementation allows for automatic differentiation with RTMB.

Uses the same density as zinbinom with p=r/(r+μ)p = r/(r+\mu):

P(X=k; μ,r,p0)=p0 1[k=0]+(1−p0) PNB ⁣(k; r,rr+μ).P(X=k;\,\mu,r,p_0) = p_0\,\mathbf{1}[k=0] + (1-p_0)\,P_{\mathrm{NB}}\!\left(k;\,r,\tfrac{r}{r+\mu}\right).

Value

dzinbinom2 gives the density, pzinbinom2 gives the distribution function, and rzinbinom2 generates random deviates.

Examples

set.seed(123)
x <- rzinbinom2(1, 2, 1, zeroprob = 0.5)
d <- dzinbinom2(x, 2, 1, zeroprob = 0.5)
p <- pzinbinom2(x, 2, 1, zeroprob = 0.5)

Zero-inflated Poisson distribution

Description

Probability mass function, distribution function, and random generation for the zero-inflated Poisson distribution.

Usage

dzipois(x, lambda, zeroprob = 0, log = FALSE)

pzipois(q, lambda, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rzipois(n, lambda, zeroprob = 0)

Arguments

x, q

integer vector of counts
lambda

vector of (non-negative) means
zeroprob

zero-inflation probability between 0 and 1
log, log.p

logical; return log-density if TRUE
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

P(X=k; λ,p0)=p0 1[k=0]+(1−p0) PPois(k; λ),P(X=k;\,\lambda,p_0) = p_0\,\mathbf{1}[k=0] + (1-p_0)\,P_{\mathrm{Pois}}(k;\,\lambda),

where p0p_0 is zeroprob.

Value

dzipois gives the probability mass function, pzipois gives the distribution function, and rzipois generates random deviates.

Examples

set.seed(123)
x <- rzipois(1, 0.5, 1)
d <- dzipois(x, 0.5, 1)
p <- pzipois(x, 0.5, 1)

Zero-inflated Weibull distribution

Description

Density, distribution function, and random generation for the zero-inflated Weibull distribution.

Usage

dziweibull(x, shape, scale, zeroprob = 0, log = FALSE)

pziweibull(q, shape, scale, zeroprob = 0, lower.tail = TRUE, log.p = FALSE)

rziweibull(n, shape, scale, zeroprob = 0)

Arguments

x, q

vector of quantiles
shape

positive shape parameter
scale

positive scale parameter
zeroprob

zero-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return

Details

This implementation allows for automatic differentiation with RTMB.

f(x; k,λ,p0)=p0 1[x=0]+(1−p0) fWeibull(x; k,λ) 1[x>0],f(x;\,k,\lambda,p_0) = p_0\,\mathbf{1}[x=0] + (1-p_0)\,f_{\mathrm{Weibull}}(x;\,k,\lambda)\,\mathbf{1}[x>0],

where p0p_0 is zeroprob, kk = shape, λ\lambda = scale.

Value

dziweibull gives the density, pziweibull gives the distribution function, and rziweibull generates random deviates.

Examples

x <- rziweibull(1, 1, 1, 0.5)
d <- dziweibull(x, 1, 1, 0.5)
p <- pziweibull(x, 1, 1, 0.5)

Zero- and one-inflated beta distribution

Description

Density, distribution function, and random generation for the zero-one-inflated beta distribution.

Usage

dzoibeta(x, shape1, shape2, zeroprob = 0, oneprob = 0, log = FALSE)

pzoibeta(q, shape1, shape2, zeroprob = 0, oneprob = 0,
         lower.tail = TRUE, log.p = FALSE)

rzoibeta(n, shape1, shape2, zeroprob = 0, oneprob = 0)

Arguments

x, q

vector of quantiles
shape1, shape2

non-negative shape parameters of the beta distribution
zeroprob

zero-inflation probability between 0 and 1.
oneprob

one-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

f(x; a,b,p0,p1)=p0 1[x=0]+(1−p0−p1) fBeta(x; a,b) 1[x∈(0,1)]+p1 1[x=1],f(x;\,a,b,p_0,p_1) = p_0\,\mathbf{1}[x=0] + (1-p_0-p_1)\,f_{\mathrm{Beta}}(x;\,a,b)\,\mathbf{1}[x\in(0,1)] + p_1\,\mathbf{1}[x=1],

where p0p_0 = zeroprob and p1p_1 = oneprob.

Value

dzoibeta gives the density, pzoibeta gives the distribution function, and rzoibeta generates random deviates.

Examples

set.seed(123)
x <- rzoibeta(1, 2, 2, 0.2, 0.3)
d <- dzoibeta(x, 2, 2, 0.2, 0.3)
p <- pzoibeta(x, 2, 2, 0.2, 0.3)

Reparameterised zero- and one-inflated beta distribution

Description

Density, distribution function, and random generation for the zero-one-inflated beta distribution reparameterised in terms of mean and concentration.

Usage

dzoibeta2(x, mu, phi, zeroprob = 0, oneprob = 0, log = FALSE)

pzoibeta2(q, mu, phi, zeroprob = 0, oneprob = 0,
         lower.tail = TRUE, log.p = FALSE)

rzoibeta2(n, mu, phi, zeroprob = 0, oneprob = 0)

Arguments

x, q

vector of quantiles
mu

mean parameter, must be in the interval from 0 to 1.
phi

concentration parameter, must be positive.
zeroprob

zero-inflation probability between 0 and 1.
oneprob

one-inflation probability between 0 and 1.
log, log.p

logical; if TRUE, probabilities/ densities pp are returned as log⁡(p)\log(p).
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

Uses the same density as zoibeta with a=μϕa = \mu\phi and b=(1−μ)ϕb = (1-\mu)\phi.

Value

dzoibeta2 gives the density, pzoibeta2 gives the distribution function, and rzoibeta2 generates random deviates.

Examples

set.seed(123)
x <- rzoibeta2(1, 0.6, 2, 0.2, 0.3)
d <- dzoibeta2(x, 0.6, 2, 0.2, 0.3)
p <- pzoibeta2(x, 0.6, 2, 0.2, 0.3)

Zero-truncated Binomial distribution

Description

Probability mass function, distribution function, and random generation for the zero-truncated Binomial distribution.

Usage

dztbinom(x, size, prob, log = FALSE)

pztbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)

rztbinom(n, size, prob)

Arguments

x, q

integer vector of counts
size

number of trials
prob

success probability in each trial
log, log.p

logical; return log-density if TRUE
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

By definition, this distribution only has support on the positive integers (1, ..., size). Any zero-truncated distribution is defined as

P(X=x∣X>0)=P(X=x)/(1−P(X=0)),P(X=x | X>0) = P(X=x) / (1 - P(X=0)),

where P(X=x)P(X=x) is the probability mass function of the corresponding untruncated distribution.

Value

dztbinom gives the probability mass function, pztbinom gives the distribution function, and rztbinom generates random deviates.

Examples

set.seed(123)
x <- rztbinom(1, size = 10, prob = 0.3)
d <- dztbinom(x, size = 10, prob = 0.3)
p <- pztbinom(x, size = 10, prob = 0.3)

Zero-truncated Negative Binomial distribution

Description

Probability mass function, distribution function, and random generation for the zero-truncated Negative Binomial distribution.

Usage

dztnbinom(x, size, prob, log = FALSE)

pztnbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)

rztnbinom(n, size, prob)

Arguments

x, q

integer vector of counts
size

target for number of successful trials, or dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer.
prob

probability of success in each trial. 0 < prob <= 1.
log, log.p

logical; return log-density if TRUE
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

By definition, this distribution only has support on the positive integers (1, 2, ...). Any zero-truncated distribution is defined as

P(X=x∣X>0)=P(X=x)/(1−P(X=0)),P(X=x | X>0) = P(X=x) / (1 - P(X=0)),

where P(X=x)P(X=x) is the probability mass function of the corresponding untruncated distribution.

Value

dztnbinom gives the probability mass function, pztnbinom gives the distribution function, and rztnbinom generates random deviates.

Examples

set.seed(123)
x <- rztnbinom(1, size = 2, prob = 0.5)
d <- dztnbinom(x, size = 2, prob = 0.5)
p <- pztnbinom(x, size = 2, prob = 0.5)

Reparameterised zero-truncated negative binomial distribution

Description

Probability mass function, distribution function, quantile function, and random generation for the zero-truncated negative binomial distribution reparameterised in terms of mean and size.

Usage

dztnbinom2(x, mu, size, log = FALSE)

pztnbinom2(q, mu, size, lower.tail = TRUE, log.p = FALSE)

rztnbinom2(n, mu, size)

Arguments

x, q

integer vector of counts
mu

mean parameter, must be positive
size

size/dispersion parameter, must be positive
log, log.p

logical; return log-density if TRUE
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

By definition, this distribution only has support on the positive integers (1, 2, ...). Any zero-truncated distribution is defined as

P(X=x∣X>0)=P(X=x)/(1−P(X=0)),P(X=x | X>0) = P(X=x) / (1 - P(X=0)),

where P(X=x)P(X=x) is the probability mass function of the corresponding untruncated distribution.

Value

dztnbinom2 gives the probability mass function, pztnbinom2 gives the distribution function, and rztnbinom2 generates random deviates.

Examples

set.seed(123)
x <- rztnbinom2(1, mu = 2, size = 1)
d <- dztnbinom2(x, mu = 2, size = 1)
p <- pztnbinom2(x, mu = 2, size = 1)

Zero-truncated Poisson distribution

Description

Probability mass function, distribution function, and random generation for the zero-truncated Poisson distribution.

Usage

dztpois(x, lambda, log = FALSE)

pztpois(q, lambda, lower.tail = TRUE, log.p = FALSE)

rztpois(n, lambda)

Arguments

x, q

integer vector of counts
lambda

vector of (non-negative) means
log, log.p

logical; return log-density if TRUE
lower.tail

logical; if TRUE, probabilities are P[X≤x]P[X \le x], otherwise, P[X>x]P[X > x].
n

number of random values to return.

Details

This implementation allows for automatic differentiation with RTMB.

By definition, this distribution only has support on the positive integers (1, 2, ...). Any zero-truncated distribution is defined as

P(X=x∣X>0)=P(X=x)/(1−P(X=0)),P(X=x | X>0) = P(X=x) / (1 - P(X=0)),

where P(X=x)P(X=x) is the probability mass function of the corresponding untruncated distribution.

Value

dztpois gives the probability mass function, pztpois gives the distribution function, and rztpois generates random deviates.

Examples

set.seed(123)
x <- rztpois(1, 0.5)
d <- dztpois(x, 0.5)
p <- pztpois(x, 0.5)