find_lambda_rcpp: Selecting the Box-Cox parameter for general d using Rcpp
In rust: Ratio-of-Uniforms Simulation with Transformation
View source: R/box_cox_functions_rcpp.R
find_lambda_rcpp R Documentation
Selecting the Box-Cox parameter for general d using Rcpp
Description
Finds a value of the Box-Cox transformation parameter lambda for which
the (positive) random variable with log-density \log f has a
density closer to that of a Gaussian random variable.
In the following we use theta (\theta) to denote the argument
of logf on the original scale and phi (\phi) on the
Box-Cox transformed scale.
Usage
find_lambda_rcpp(
logf,
...,
d = 1,
n_grid = NULL,
ep_bc = 1e-04,
min_phi = rep(ep_bc, d),
max_phi = rep(10, d),
which_lam = 1:d,
lambda_range = c(-3, 3),
init_lambda = NULL,
phi_to_theta = NULL,
log_j = NULL,
user_args = list()
)
Arguments
logf
A pointer to a compiled C++ function returning the log
of the target density f.
...
further arguments to be passed to logf and related
functions.
d
A numeric scalar. Dimension of f.
n_grid
A numeric scalar. Number of ordinates for each variable in
phi. If this is not supplied a default value of
ceiling(2501 ^ (1 / d)) is used.
ep_bc
A (positive) numeric scalar. Smallest possible value of
phi to consider. Used to avoid negative values of phi.
min_phi, max_phi
Numeric vectors. Smallest and largest values
of phi at which to evaluate logf, i.e., the range of values
of phi over which to evaluate logf. Any components in
min_phi that are not positive are set to ep_bc.
which_lam
A numeric vector. Contains the indices of the components
of phi that ARE to be Box-Cox transformed.
lambda_range
A numeric vector of length 2. Range of lambda over
which to optimise.
init_lambda
A numeric vector of length 1 or d. Initial value
of lambda used in the search for the best lambda. If init_lambda
is a scalar then rep(init_lambda, d) is used.
phi_to_theta
A pointer to a compiled C++ function returning
(the inverse) of the transformation from theta to phi 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.
log_j
A pointer to a compiled C++ function returning the log of
the Jacobian of the transformation from theta to phi, i.e.,
based on derivatives of phi with respect to theta. Takes
theta as its argument.
user_args
A list of numeric components providing arguments to
the user-supplied functions phi_to_theta and log_j.
Details
The general idea is to evaluate the density f on a
d-dimensional grid, with n_grid ordinates for each of the
d variables.
We treat each combination of the variables in the grid as a data point
and perform an estimation of the Box-Cox transformation parameter
lambda, in which each data point is weighted by the density
at that point. The vectors min_phi and max_phi define the
limits of the grid and which_lam can be used to specify that only
certain components of phi are to be transformed.
Value
A list containing the following components
lambda
A numeric vector. The value of lambda.
gm
A numeric vector. Box-cox scaling parameter, estimated by the
geometric mean of the values of phi used in the optimisation to find
the value of lambda, weighted by the values of f evaluated at phi.
init_psi
A numeric vector. An initial estimate of the mode of the
Box-Cox transformed density
sd_psi
A numeric vector. Estimates of the marginal standard
deviations of the Box-Cox transformed variables.
phi_to_theta
as detailed above (only if phi_to_theta is
supplied)
log_j
as detailed above (only if log_j is supplied)
user_args
as detailed above (only if user_args is supplied)
References
Box, G. and Cox, D. R. (1964) An Analysis of Transformations.
Journal of the Royal Statistical Society. Series B (Methodological), 26(2),
211-252.
Andrews, D. F. and Gnanadesikan, R. and Warner, J. L. (1971)
Transformations of Multivariate Data, Biometrics, 27(4).
Eddelbuettel, D. and Francois, R. (2011). Rcpp: Seamless
R and C++ Integration. Journal of Statistical Software,
40(8), 1-18. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v040.i08")}
Eddelbuettel, D. (2013). Seamless R and C++ Integration
with Rcpp, Springer, New York. ISBN 978-1-4614-6867-7.
See Also
ru_rcpp to perform ratio-of-uniforms sampling.
find_lambda_one_d_rcpp to produce (somewhat)
automatically a list for the argument lambda of ru for the
d = 1 case.
Examples
# Log-normal density ===================
# Note: the default value max_phi = 10 is OK here but this will not always
# be the case
ptr_lnorm <- create_xptr("logdlnorm")
mu <- 0
sigma <- 1
lambda <- find_lambda_rcpp(logf = ptr_lnorm, mu = mu, sigma = sigma)
lambda
x <- ru_rcpp(logf = ptr_lnorm, mu = mu, sigma = sigma, d = 1, n = 1000,
trans = "BC", lambda = lambda)
# Gamma density ===================
alpha <- 1
# Choose a sensible value of max_phi
max_phi <- qgamma(0.999, shape = alpha)
# [Of course, typically the quantile function won't be available. However,
# In practice the value of lambda chosen is quite insensitive to the choice
# of max_phi, provided that max_phi is not far too large or far too small.]
ptr_gam <- create_xptr("logdgamma")
lambda <- find_lambda_rcpp(logf = ptr_gam, alpha = alpha, max_phi = max_phi)
lambda
x <- ru_rcpp(logf = ptr_gam, alpha = alpha, d = 1, n = 1000, trans = "BC",
lambda = lambda)
# Generalized Pareto posterior distribution ===================
n <- 1000
# 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)
n <- 1000
# Sample on original scale, with no rotation ----------------
ptr_gp <- create_xptr("loggp")
for_ru_rcpp <- c(list(logf = ptr_gp, init = init, d = 2, n = n,
lower = c(0, -Inf)), ss, rotate = FALSE)
x1 <- do.call(ru_rcpp, for_ru_rcpp)
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$xm)
summary(x1)
# Sample on original scale, with rotation ----------------
for_ru_rcpp <- c(list(logf = ptr_gp, init = init, d = 2, n = n,
lower = c(0, -Inf)), ss)
x2 <- do.call(ru_rcpp, for_ru_rcpp)
plot(x2, xlab = "sigma", ylab = "xi")
abline(a = 0, b = -1 / ss$xm)
summary(x2)
# Sample on Box-Cox transformed scale ----------------
# Find initial estimates for phi = (phi1, phi2),
# where phi1 = sigma
# and phi2 = xi + sigma / max(x),
# and ranges of phi1 and phi2 over over which to evaluate
# the posterior to find a suitable value of lambda.
temp <- do.call(gpd_init, ss)
min_phi <- pmax(0, temp$init_phi - 2 * temp$se_phi)
max_phi <- pmax(0, temp$init_phi + 2 * temp$se_phi)
# Set phi_to_theta() that ensures positivity of phi
# We use phi1 = sigma and phi2 = xi + sigma / max(data)
# Create an external pointer to this C++ function
ptr_phi_to_theta_gp <- create_phi_to_theta_xptr("gp")
# Note: log_j is set to zero by default inside find_lambda_rcpp()
lambda <- find_lambda_rcpp(logf = ptr_gp, ss = ss, d = 2, min_phi = min_phi,
max_phi = max_phi, user_args = list(xm = ss$xm),
phi_to_theta = ptr_phi_to_theta_gp)
lambda
# Sample on Box-Cox transformed, without rotation
x3 <- ru_rcpp(logf = ptr_gp, ss = ss, d = 2, n = n, trans = "BC",
lambda = lambda, rotate = FALSE)
plot(x3, xlab = "sigma", ylab = "xi")
abline(a = 0, b = -1 / ss$xm)
summary(x3)
# Sample on Box-Cox transformed, with rotation
x4 <- ru_rcpp(logf = ptr_gp, ss = ss, d = 2, n = n, trans = "BC",
lambda = lambda)
plot(x4, xlab = "sigma", ylab = "xi")
abline(a = 0, b = -1 / ss$xm)
summary(x4)
def_par <- graphics::par(no.readonly = TRUE)
par(mfrow = c(2,2), mar = c(4, 4, 1.5, 1))
plot(x1, xlab = "sigma", ylab = "xi", ru_scale = TRUE,
main = "mode relocation")
plot(x2, xlab = "sigma", ylab = "xi", ru_scale = TRUE,
main = "mode relocation and rotation")
plot(x3, xlab = "sigma", ylab = "xi", ru_scale = TRUE,
main = "Box-Cox and mode relocation")
plot(x4, xlab = "sigma", ylab = "xi", ru_scale = TRUE,
main = "Box-Cox, mode relocation and rotation")
graphics::par(def_par)
rust documentation built on Sept. 2, 2023, 5:06 p.m.
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View source: R/box_cox_functions_rcpp.R
| find_lambda_rcpp | R Documentation |
Selecting the Box-Cox parameter for general d using Rcpp
Description
Finds a value of the Box-Cox transformation parameter lambda for which
the (positive) random variable with log-density \log f has a
density closer to that of a Gaussian random variable.
In the following we use theta (\theta) to denote the argument
of logf on the original scale and phi (\phi) on the
Box-Cox transformed scale.
Usage
find_lambda_rcpp(
logf,
...,
d = 1,
n_grid = NULL,
ep_bc = 1e-04,
min_phi = rep(ep_bc, d),
max_phi = rep(10, d),
which_lam = 1:d,
lambda_range = c(-3, 3),
init_lambda = NULL,
phi_to_theta = NULL,
log_j = NULL,
user_args = list()
)
Arguments
logf |
A pointer to a compiled C++ function returning the log
of the target density |
... |
further arguments to be passed to |
d |
A numeric scalar. Dimension of |
n_grid |
A numeric scalar. Number of ordinates for each variable in
|
ep_bc |
A (positive) numeric scalar. Smallest possible value of
|
min_phi, max_phi |
Numeric vectors. Smallest and largest values
of |
which_lam |
A numeric vector. Contains the indices of the components
of |
lambda_range |
A numeric vector of length 2. Range of lambda over which to optimise. |
init_lambda |
A numeric vector of length 1 or |
phi_to_theta |
A pointer to a compiled C++ function returning
(the inverse) of the transformation from |
log_j |
A pointer to a compiled C++ function returning the log of
the Jacobian of the transformation from |
user_args |
A list of numeric components providing arguments to
the user-supplied functions |
Details
The general idea is to evaluate the density f on a
d-dimensional grid, with n_grid ordinates for each of the
d variables.
We treat each combination of the variables in the grid as a data point
and perform an estimation of the Box-Cox transformation parameter
lambda, in which each data point is weighted by the density
at that point. The vectors min_phi and max_phi define the
limits of the grid and which_lam can be used to specify that only
certain components of phi are to be transformed.
Value
A list containing the following components
lambda |
A numeric vector. The value of |
gm |
A numeric vector. Box-cox scaling parameter, estimated by the geometric mean of the values of phi used in the optimisation to find the value of lambda, weighted by the values of f evaluated at phi. |
init_psi |
A numeric vector. An initial estimate of the mode of the Box-Cox transformed density |
sd_psi |
A numeric vector. Estimates of the marginal standard deviations of the Box-Cox transformed variables. |
phi_to_theta |
as detailed above (only if |
log_j |
as detailed above (only if |
user_args |
as detailed above (only if |
References
Box, G. and Cox, D. R. (1964) An Analysis of Transformations. Journal of the Royal Statistical Society. Series B (Methodological), 26(2), 211-252.
Andrews, D. F. and Gnanadesikan, R. and Warner, J. L. (1971) Transformations of Multivariate Data, Biometrics, 27(4).
Eddelbuettel, D. and Francois, R. (2011). Rcpp: Seamless R and C++ Integration. Journal of Statistical Software, 40(8), 1-18. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v040.i08")}
Eddelbuettel, D. (2013). Seamless R and C++ Integration with Rcpp, Springer, New York. ISBN 978-1-4614-6867-7.
See Also
ru_rcpp to perform ratio-of-uniforms sampling.
find_lambda_one_d_rcpp to produce (somewhat)
automatically a list for the argument lambda of ru for the
d = 1 case.
Examples
# Log-normal density ===================
# Note: the default value max_phi = 10 is OK here but this will not always
# be the case
ptr_lnorm <- create_xptr("logdlnorm")
mu <- 0
sigma <- 1
lambda <- find_lambda_rcpp(logf = ptr_lnorm, mu = mu, sigma = sigma)
lambda
x <- ru_rcpp(logf = ptr_lnorm, mu = mu, sigma = sigma, d = 1, n = 1000,
trans = "BC", lambda = lambda)
# Gamma density ===================
alpha <- 1
# Choose a sensible value of max_phi
max_phi <- qgamma(0.999, shape = alpha)
# [Of course, typically the quantile function won't be available. However,
# In practice the value of lambda chosen is quite insensitive to the choice
# of max_phi, provided that max_phi is not far too large or far too small.]
ptr_gam <- create_xptr("logdgamma")
lambda <- find_lambda_rcpp(logf = ptr_gam, alpha = alpha, max_phi = max_phi)
lambda
x <- ru_rcpp(logf = ptr_gam, alpha = alpha, d = 1, n = 1000, trans = "BC",
lambda = lambda)
# Generalized Pareto posterior distribution ===================
n <- 1000
# 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)
n <- 1000
# Sample on original scale, with no rotation ----------------
ptr_gp <- create_xptr("loggp")
for_ru_rcpp <- c(list(logf = ptr_gp, init = init, d = 2, n = n,
lower = c(0, -Inf)), ss, rotate = FALSE)
x1 <- do.call(ru_rcpp, for_ru_rcpp)
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)
# Sample on original scale, with rotation ----------------
for_ru_rcpp <- c(list(logf = ptr_gp, init = init, d = 2, n = n,
lower = c(0, -Inf)), ss)
x2 <- do.call(ru_rcpp, for_ru_rcpp)
plot(x2, xlab = "sigma", ylab = "xi")
abline(a = 0, b = -1 / ss$xm)
summary(x2)
# Sample on Box-Cox transformed scale ----------------
# Find initial estimates for phi = (phi1, phi2),
# where phi1 = sigma
# and phi2 = xi + sigma / max(x),
# and ranges of phi1 and phi2 over over which to evaluate
# the posterior to find a suitable value of lambda.
temp <- do.call(gpd_init, ss)
min_phi <- pmax(0, temp$init_phi - 2 * temp$se_phi)
max_phi <- pmax(0, temp$init_phi + 2 * temp$se_phi)
# Set phi_to_theta() that ensures positivity of phi
# We use phi1 = sigma and phi2 = xi + sigma / max(data)
# Create an external pointer to this C++ function
ptr_phi_to_theta_gp <- create_phi_to_theta_xptr("gp")
# Note: log_j is set to zero by default inside find_lambda_rcpp()
lambda <- find_lambda_rcpp(logf = ptr_gp, ss = ss, d = 2, min_phi = min_phi,
max_phi = max_phi, user_args = list(xm = ss$xm),
phi_to_theta = ptr_phi_to_theta_gp)
lambda
# Sample on Box-Cox transformed, without rotation
x3 <- ru_rcpp(logf = ptr_gp, ss = ss, d = 2, n = n, trans = "BC",
lambda = lambda, rotate = FALSE)
plot(x3, xlab = "sigma", ylab = "xi")
abline(a = 0, b = -1 / ss$xm)
summary(x3)
# Sample on Box-Cox transformed, with rotation
x4 <- ru_rcpp(logf = ptr_gp, ss = ss, d = 2, n = n, trans = "BC",
lambda = lambda)
plot(x4, xlab = "sigma", ylab = "xi")
abline(a = 0, b = -1 / ss$xm)
summary(x4)
def_par <- graphics::par(no.readonly = TRUE)
par(mfrow = c(2,2), mar = c(4, 4, 1.5, 1))
plot(x1, xlab = "sigma", ylab = "xi", ru_scale = TRUE,
main = "mode relocation")
plot(x2, xlab = "sigma", ylab = "xi", ru_scale = TRUE,
main = "mode relocation and rotation")
plot(x3, xlab = "sigma", ylab = "xi", ru_scale = TRUE,
main = "Box-Cox and mode relocation")
plot(x4, xlab = "sigma", ylab = "xi", ru_scale = TRUE,
main = "Box-Cox, mode relocation and rotation")
graphics::par(def_par)
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