Generalized ratio-of-uniforms sampling

Uses the generalized ratio-of-uniforms method to simulate from a distribution with log-density $\log f$ (up to an additive constant). The density $f$ must be bounded, perhaps after a transformation of variable.

ru(
  logf,
  ...,
  n = 1,
  d = 1,
  init = NULL,
  mode = NULL,
  trans = c("none", "BC", "user"),
  phi_to_theta = NULL,
  log_j = NULL,
  user_args = list(),
  lambda = rep(1L, d),
  lambda_tol = 1e-06,
  gm = NULL,
  rotate = ifelse(d == 1, FALSE, TRUE),
  lower = rep(-Inf, d),
  upper = rep(Inf, d),
  r = 1/2,
  ep = 0L,
  a_algor = if (d == 1) "nlminb" else "optim",
  b_algor = c("nlminb", "optim"),
  a_method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  b_method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  a_control = list(),
  b_control = list(),
  var_names = NULL,
  shoof = 0.2
)

Arguments

logf

A function returning the log of the target density $f$ evaluated at its first argument. This function should return -Inf when the density is zero. It is better to use logf = explicitly, for example, ru(logf = dnorm, log = TRUE, init = 0.1), to avoid argument matching problems. In contrast, ru(dnorm, log = TRUE, init = 0.1) will throw an error because partial matching results in logf being matched to log = TRUE.

...

Further arguments to be passed to logf and related functions.

n

A non-negative integer scalar. The number of simulated values required. If n = 0 then no simulation is performed but the component box in the returned object gives the ratio-of-uniforms bounding box that would have been used.

d

A positive integer scalar. The dimension of $f$.

init

A numeric vector of length d. Initial estimate of the mode of logf. If trans = "BC" or trans = "user" this is after Box-Cox transformation or user-defined transformation, but before any rotation of axes. If init is not supplied then rep(1, d) is used. If length(init) = 1 and d > 1 then init <- rep(init, length.out = d) is used.

mode

A numeric vector of length d. The mode of logf. If trans = "BC" or trans = "user" this is after Box-Cox transformation or user-defined transformation, but before any rotation of axes. Only supply mode if the mode is known: it will not be checked. If mode is supplied then init is ignored.

trans

A character scalar. trans = "none" for no transformation, trans = "BC" for Box-Cox transformation, trans = "user" for a user-defined transformation. If trans = "user" then the transformation should be specified using phi_to_theta and log_j and user_args may be used to pass arguments to phi_to_theta and log_j. See Details and the Examples.

phi_to_theta

A function returning (the inverse) of the transformation from theta ($\theta$) to phi ($\phi$) that may be used to ensure positivity of $\phi$ prior to Box-Cox transformation. The argument is phi and the returned value is theta. If phi_to_theta is undefined at the input value then the function should return NA. See Details. If lambda$phi_to_theta (see argument lambda below) is supplied then this is used instead of any function supplied via phi_to_theta.

log_j

A function returning the log of the Jacobian of the transformation from theta ($\theta$) to phi ($\phi$), i.e., based on derivatives of $\phi$ with respect to $\theta$. Takes theta as its argument. If lambda$log_j (see argument lambda below) is supplied then this is used instead of any function supplied via log_j.

user_args

A list of numeric components. If trans = "user" then user_args is a list providing arguments to the user-supplied functions phi_to_theta and log_j.

lambda

Either

A numeric vector. Box-Cox transformation parameters, or
A list with components
lambda

A numeric vector. Box-Cox parameters (required).

gm

A numeric vector. Box-Cox scaling parameters (optional). If supplied this overrides any gm supplied by the individual gm argument described below.

init_psi

A numeric vector. Initial estimate of mode after Box-Cox transformation (optional).

sd_psi

A numeric vector. Estimates of the marginal standard deviations of the Box-Cox transformed variables (optional).

phi_to_theta

As above (optional).

log_j

As above (optional).

This list may be created using find_lambda_one_d (for d = 1) or find_lambda (for any d).

lambda_tol

A numeric scalar. Any values in lambda that are less than lambda_tol in magnitude are set to zero.

gm

A numeric vector. Box-Cox scaling parameters (optional). If lambda$gm is supplied in input list lambda then lambda$gm is used, not gm.

rotate

A logical scalar. If TRUE (d > 1 only) use Choleski rotation. If d = 1 and rotate = TRUE then rotate will be set to FALSE with a warning. See Details.

lower, upper

Numeric vectors. Lower/upper bounds on the arguments of the function after any transformation from theta to phi implied by the inverse of phi_to_theta. If rotate = FALSE these are used in all of the optimisations used to construct the bounding box. If rotate = TRUE then they are use only in the first optimisation to maximise the target density.` If trans = "BC" components of lower that are negative are set to zero without warning and the bounds implied after the Box-Cox transformation are calculated inside ru.

r

A numeric scalar. Parameter of generalized ratio-of-uniforms.

ep

A numeric scalar. Controls initial estimates for optimisations to find the $b$-bounding box parameters. The default (ep = 0) corresponds to starting at the mode of logf small positive values of ep move the constrained variable slightly away from the mode in the correct direction. If ep is negative its absolute value is used, with no warning given.

a_algor, b_algor

Character scalars. Either "nlminb" or "optim". Respective optimisation algorithms used to find $a(r)$ and ($b$_i^-(r), $b$_i⁺(r)).

a_method, b_method

Character scalars. Respective methods used by optim to find $a(r)$ and ($b$_i^-(r), $b$_i⁺(r)). Only used if optim is the chosen algorithm. If d = 1 then a_method and b_method are set to "Brent" without warning.

a_control, b_control

Lists of control arguments to optim or nlminb to find $a(r)$ and ($b$_i^-(r), $b$_i⁺(r)) respectively.

var_names

A character (or numeric) vector of length d. Names to give to the column(s) of the simulated values.

shoof

A numeric scalar in [0, 1]. Sometimes a spurious non-zero convergence indicator is returned from optim or nlminb). In this event we try to check that a minimum has indeed been found using different algorithm. shoof controls the starting value provided to this algorithm. If shoof = 0 then we start from the current solution. If shoof = 1 then we start from the initial estimate provided to the previous minimisation. Otherwise, shoof interpolates between these two extremes, with a value close to zero giving a starting value that is close to the current solution. The exception to this is when the initial and current solutions are equal. Then we start from the current solution multiplied by 1 - shoof.

Value

An object of class "ru" is a list containing the following components:

sim_vals

An n by d matrix of simulated values.

box

A (2 * d + 1) by d + 2 matrix of ratio-of-uniforms bounding box information, with row names indicating the box parameter. The columns contain

column 1: values of box parameters.
columns 2 to (2+d-1): values of variables at which these box parameters are obtained.
column 2+d: convergence indicators.

Scaling of f within ru and relocation of the mode to the origin means that the first row of box will always be c(1, rep(0, d)).

pa

A numeric scalar. An estimate of the probability of acceptance.

r

The value of r.

d

The value of d.

logf

A function. logf supplied by the user, but with f scaled by the maximum of the target density used in the ratio-of-uniforms method (i.e. logf_rho), to avoid numerical problems in contouring f in plot.ru when d = 2.

logf_rho

A function. The target function actually used in the ratio-of-uniforms algorithm.

sim_vals_rho

An n by d matrix of values simulated from the function used in the ratio-of-uniforms algorithm.

logf_args

A list of further arguments to logf.

f_mode

The estimated mode of the target density f, after any Box-Cox transformation and/or user supplied transformation, but before mode relocation.

trans_fn

An R function that performs the inverse transformation from the transformed variable $\rho$, on which the generalised ratio-of-uniforms method is performed, back to the original variable $\theta$. Note: trans_fn is not vectorised with respect to $\rho$.

Details

For information about the generalised ratio-of-uniforms method and transformations see the Introducing rust vignette. This can also be accessed using vignette("rust-a-vignette", package = "rust").

If trans = "none" and rotate = FALSE then ru implements the (multivariate) generalized ratio of uniforms method described in Wakefield, Gelfand and Smith (1991) using a target density whose mode is relocated to the origin (`mode relocation') in the hope of increasing efficiency.

If trans = "BC" then marginal Box-Cox transformations of each of the d variables is performed, with parameters supplied in lambda. The function phi_to_theta may be used, if necessary, to ensure positivity of the variables prior to Box-Cox transformation.

If trans = "user" then the function phi_to_theta enables the user to specify their own transformation.

In all cases the mode of the target function is relocated to the origin after any user-supplied transformation and/or Box-Cox transformation.

If d is greater than one and rotate = TRUE then a rotation of the variable axes is performed after mode relocation. The rotation is based on the Choleski decomposition (see chol) of the estimated Hessian (computed using optimHess of the negated log-density after any user-supplied transformation or Box-Cox transformation. If any of the eigenvalues of the estimated Hessian are non-positive (which may indicate that the estimated mode of logf is close to a variable boundary) then rotate is set to FALSE with a warning. A warning is also given if this happens when d = 1.

The default value of the tuning parameter r is 1/2, which is likely to be close to optimal in many cases, particularly if trans = "BC".

References

Wakefield, J. C., Gelfand, A. E. and Smith, A. F. M. (1991) Efficient generation of random variates via the ratio-of-uniforms method. Statistics and Computing (1991), 1, 129-133. doi:10.1007/BF01889987 .

Examples

# Normal density ===================

# One-dimensional standard normal ----------------
x <- ru(logf = function(x) -x ^ 2 / 2, d = 1, n = 1000, init = 0.1)

# Two-dimensional standard normal ----------------
x <- ru(logf = function(x) -(x[1]^2 + x[2]^2) / 2, d = 2, n = 1000,
        init = c(0, 0))

# Two-dimensional normal with positive association ----------------
rho <- 0.9
covmat <- matrix(c(1, rho, rho, 1), 2, 2)
log_dmvnorm <- function(x, mean = rep(0, d), sigma = diag(d)) {
  x <- matrix(x, ncol = length(x))
  d <- ncol(x)
  - 0.5 * (x - mean) %*% solve(sigma) %*% t(x - mean)
}

# No rotation.
x <- ru(logf = log_dmvnorm, sigma = covmat, d = 2, n = 1000, init = c(0, 0),
        rotate = FALSE)

# With rotation.
x <- ru(logf = log_dmvnorm, sigma = covmat, d = 2, n = 1000, init = c(0, 0))

# three-dimensional normal with positive association ----------------
covmat <- matrix(rho, 3, 3) + diag(1 - rho, 3)

# No rotation.  Slow !
x <- ru(logf = log_dmvnorm, sigma = covmat, d = 3, n = 1000,
        init = c(0, 0, 0), rotate = FALSE)

# With rotation.
x <- ru(logf = log_dmvnorm, sigma = covmat, d = 3, n = 1000,
        init = c(0, 0, 0))

# Log-normal density ===================

# Sampling on original scale ----------------
x <- ru(logf = dlnorm, log = TRUE, d = 1, n = 1000, lower = 0, init = 1)

# Box-Cox transform with lambda = 0 ----------------
lambda <- 0
x <- ru(logf = dlnorm, log = TRUE, d = 1, n = 1000, lower = 0, init = 0.1,
        trans = "BC", lambda = lambda)

# Equivalently, we could use trans = "user" and supply the (inverse) Box-Cox
# transformation and the log-Jacobian by hand
x <- ru(logf = dlnorm, log = TRUE, d = 1, n = 1000, init = 0.1,
        trans = "user", phi_to_theta = function(x) exp(x),
        log_j = function(x) -log(x))

# Gamma(alpha, 1) density ===================

# Note: the gamma density in unbounded when its shape parameter is < 1.
# Therefore, we can only use trans="none" if the shape parameter is >= 1.

# Sampling on original scale ----------------

alpha <- 10
x <- ru(logf = dgamma, shape = alpha, log = TRUE, d = 1, n = 1000,
        lower = 0, init = alpha)

alpha <- 1
x <- ru(logf = dgamma, shape = alpha, log = TRUE, d = 1, n = 1000,
        lower = 0, init = alpha)
#> Warning: The Hessian of the target log-density at its mode is not positive
#>             definite. This may not be a problem, but it may be that a mode
#>             at/near a parameter boundary has been found and/or that the target
#>             function is unbounded.
#>   It might be worth using the option trans = ``BC''. 

# Box-Cox transform with lambda = 1/3 works well for shape >= 1. -----------

alpha <- 1
x <- ru(logf = dgamma, shape = alpha, log = TRUE, d = 1, n = 1000,
        trans = "BC", lambda = 1/3, init = alpha)
summary(x)
#> ru bounding box:  
#>               box     vals1 conv
#> a        1.000000  0.000000    0
#> b1minus -1.051825 -1.609437    0
#> b1plus   1.096590  1.774103    0
#> 
#> estimated probability of acceptance:  
#> [1] 0.785546
#> 
#> sample summary 
#>        V1          
#>  Min.   :0.000494  
#>  1st Qu.:0.289279  
#>  Median :0.695389  
#>  Mean   :1.000469  
#>  3rd Qu.:1.387574  
#>  Max.   :6.480577  

# Equivalently, we could use trans = "user" and supply the (inverse) Box-Cox
# transformation and the log-Jacobian by hand

# Note: when phi_to_theta is undefined at x this function returns NA
phi_to_theta  <- function(x, lambda) {
  ifelse(x * lambda + 1 > 0, (x * lambda + 1) ^ (1 / lambda), NA)
}
log_j <- function(x, lambda) (lambda - 1) * log(x)
lambda <- 1/3
x <- ru(logf = dgamma, shape = alpha, log = TRUE, d = 1, n = 1000,
        trans = "user", phi_to_theta = phi_to_theta, log_j = log_j,
        user_args = list(lambda = lambda), init = alpha)
summary(x)
#> ru bounding box:  
#>               box     vals1 conv
#> a        1.000000  0.000000    0
#> b1minus -1.051825 -1.609437    0
#> b1plus   1.096590  1.774103    0
#> 
#> estimated probability of acceptance:  
#> [1] 0.7980846
#> 
#> sample summary 
#>        V1          
#>  Min.   :0.001001  
#>  1st Qu.:0.302703  
#>  Median :0.739617  
#>  Mean   :1.049657  
#>  3rd Qu.:1.425455  
#>  Max.   :9.801844  

# \donttest{
# Generalized Pareto posterior distribution ===================

# Sample data from a GP(sigma, xi) distribution
gpd_data <- rgpd(m = 100, xi = -0.5, sigma = 1)
# Calculate summary statistics for use in the log-likelihood
ss <- gpd_sum_stats(gpd_data)
# Calculate an initial estimate
init <- c(mean(gpd_data), 0)

# Mode relocation only ----------------
n <- 1000
x1 <- ru(logf = gpd_logpost, ss = ss, d = 2, n = n, init = init,
         lower = c(0, -Inf), rotate = FALSE)
plot(x1, xlab = "sigma", ylab = "xi")
# Parameter constraint line xi > -sigma/max(data)
# [This may not appear if the sample is far from the constraint.]
abline(a = 0, b = -1 / ss$xm)

summary(x1)
#> ru bounding box:  
#>                box      vals1      vals2 conv
#> a        1.0000000  0.0000000  0.0000000    0
#> b1minus -0.1331405 -0.2116871  0.1524397    0
#> b2minus -0.1010805  0.2842293 -0.1749433    0
#> b1plus   0.1650133  0.2927237 -0.1783390    0
#> b2plus   0.1090747 -0.2109005  0.1991069    0
#> 
#> estimated probability of acceptance:  
#> [1] 0.1316656
#> 
#> sample summary 
#>        V1               V2         
#>  Min.   :0.6692   Min.   :-0.7909  
#>  1st Qu.:0.9521   1st Qu.:-0.6055  
#>  Median :1.0293   Median :-0.5478  
#>  Mean   :1.0378   Mean   :-0.5484  
#>  3rd Qu.:1.1122   3rd Qu.:-0.4920  
#>  Max.   :1.4167   Max.   :-0.2628  

# Rotation of axes plus mode relocation ----------------
x2 <- ru(logf = gpd_logpost, ss = ss, d = 2, n = n, init = init,
         lower = c(0, -Inf))
plot(x2, xlab = "sigma", ylab = "xi")
abline(a = 0, b = -1 / ss$xm)

summary(x2)
#> ru bounding box:  
#>                 box       vals1      vals2 conv
#> a        1.00000000  0.00000000  0.0000000    0
#> b1minus -0.04203948 -0.06214065  0.0359629    0
#> b2minus -0.06194203  0.04729381 -0.1072050    0
#> b1plus   0.11573546  0.26655126  0.1065524    0
#> b2plus   0.06684085  0.12990030  0.1220125    0
#> 
#> estimated probability of acceptance:  
#> [1] 0.3913894
#> 
#> sample summary 
#>        V1               V2         
#>  Min.   :0.6840   Min.   :-0.7967  
#>  1st Qu.:0.9523   1st Qu.:-0.6040  
#>  Median :1.0285   Median :-0.5497  
#>  Mean   :1.0346   Mean   :-0.5470  
#>  3rd Qu.:1.1144   3rd Qu.:-0.4928  
#>  Max.   :1.4894   Max.   :-0.2119  

# Cauchy ========================

# The bounding box cannot be constructed if r < 1.  For r = 1 the
# bounding box parameters b1-(r) and b1+(r) are attained in the limits
# as x decreases/increases to infinity respectively.  This is fine in
# theory but using r > 1 avoids this problem and the largest probability
# of acceptance is obtained for r approximately equal to 1.26.

res <- ru(logf = dcauchy, log = TRUE, init = 0, r = 1.26, n = 1000)

# Half-Cauchy ===================

log_halfcauchy <- function(x) {
  return(ifelse(x < 0, -Inf, dcauchy(x, log = TRUE)))
}

# Like the Cauchy case the bounding box cannot be constructed if r < 1.
# We could use r > 1 but the mode is on the edge of the support of the
# density so as an alternative we use a log transformation.

x <- ru(logf = log_halfcauchy, init = 0, trans = "BC", lambda = 0, n = 1000)
x$pa
#> [1] 0.7462687
plot(x, ru_scale = TRUE)


# Example 4 from Wakefield et al. (1991) ===================

# Bivariate normal x bivariate student-t
log_norm_t <- function(x, mean = rep(0, d), sigma1 = diag(d), sigma2 = diag(d)) {
  x <- matrix(x, ncol = length(x))
  log_h1 <- -0.5 * (x - mean) %*% solve(sigma1) %*% t(x - mean)
  log_h2 <- -2 * log(1 + 0.5 * x %*% solve(sigma2) %*% t(x))
  return(log_h1 + log_h2)
}

rho <- 0.9
covmat <- matrix(c(1, rho, rho, 1), 2, 2)
y <- c(0, 0)

# Case in the top right corner of Table 3
x <- ru(logf = log_norm_t, mean = y, sigma1 = covmat, sigma2 = covmat,
  d = 2, n = 10000, init = y, rotate = FALSE)
x$pa
#> [1] 0.2281282

# Rotation increases the probability of acceptance
x <- ru(logf = log_norm_t, mean = y, sigma1 = covmat, sigma2 = covmat,
  d = 2, n = 10000, init = y, rotate = TRUE)
x$pa
#> [1] 0.5216212

# Normal x log-normal: different Box-Cox parameters ==================
norm_lognorm <- function(x, ...) {
  dnorm(x[1], ...) + dlnorm(x[2], ...)
}
x <- ru(logf = norm_lognorm, log = TRUE, n = 1000, d = 2, init = c(-1, 0),
        trans = "BC", lambda = c(1, 0))
plot(x)

plot(x, ru_scale = TRUE)

# }