Selecting the Box-Cox parameter in the 1D case

Finds a value of the Box-Cox transformation parameter lambda (\(\lambda\)) for which the (positive univariate) random variable with log-density \(\log f\) has a density closer to that of a Gaussian random variable. Works by estimating a set of quantiles of the distribution implied by \(\log f\) and treating those quantiles as data in a standard Box-Cox analysis. In the following we use theta (\(\theta\)) to denote the argument of \(\log f\) on the original scale and phi (\(\phi\)) on the Box-Cox transformed scale.

find_lambda_one_d(
  logf,
  ...,
  ep_bc = 1e-04,
  min_phi = ep_bc,
  max_phi = 10,
  num = 1001,
  xdiv = 100,
  probs = seq(0.01, 0.99, by = 0.01),
  lambda_range = c(-3, 3),
  phi_to_theta = NULL,
  log_j = NULL
)

Arguments

logf

A function returning the log of the target density \(f\).

...

further arguments to be passed to logf and related functions.

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 scalars. 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.

num

A numeric scalar. Number of values at which to evaluate logf.

xdiv

A numeric scalar. Only values of phi at which the density \(f\) is greater than the (maximum of \(f\)) / xdiv are used.

probs

A numeric scalar. Probabilities at which to estimate the quantiles of that will be used as data to find lambda.

lambda_range

A numeric vector of length 2. Range of lambda over which to optimise.

phi_to_theta

A function returning (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.

log_j

A 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. If this is not supplied then a constant Jacobian is used.

Value

A list containing the following components

lambda

A numeric scalar. The value of lambda.

gm

A numeric scalar. Box-Cox scaling parameter, estimated by the geometric mean of the quantiles used in the optimisation to find the value of lambda.

init_psi

A numeric scalar. An initial estimate of the mode of the Box-Cox transformed density

sd_psi

A numeric scalar. 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)

Details

The general idea is to estimate quantiles of \(f\) corresponding to a set of equally-spaced probabilities in probs and to use these estimated quantiles as data in a standard estimation of the Box-Cox transformation parameter lambda.

The density f is first evaluated at num points equally spaced over the interval (min_phi, max_phi). The continuous density \(f\) is approximated by attaching trapezium-rule estimates of probabilities to the midpoints of the intervals between the points. After standardizing to account for the fact that \(f\) may not be normalized, (min_phi, max_phi) is reset so that values with small estimated probability (determined by xdiv) are excluded and the procedure is repeated on this new range. Then the required quantiles are estimated by inferring them from a weighted empirical distribution function based on treating the midpoints as data and the estimated probabilities at the midpoints as weights.

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).

See also

ru and ru_rcpp to perform ratio-of-uniforms sampling.

find_lambda and find_lambda_rcpp to produce (somewhat) automatically a list for the argument lambda of ru/ru_rcpp for any value of d.

find_lambda_one_d_rcpp for a version of find_lambda_one_d that uses the Rcpp package to improve efficiency.

Examples

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

# Note: the default value of max_phi = 10 is OK here but this will not
# always be the case.

lambda <- find_lambda_one_d(logf = dlnorm, log = TRUE)
lambda
#> $lambda
#> [1] 0.06564725
#> 
#> $gm
#> [1] 0.9535484
#> 
#> $init_psi
#> [1] -0.06345259
#> 
#> $sd_psi
#> [1] 0.9753502
#> 
x <- ru(logf = dlnorm, log = TRUE, 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)
# [I appreciate that typically the quantile function won't be available.
# 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.]

lambda <- find_lambda_one_d(logf = dgamma, shape = alpha, log = TRUE,
                            max_phi = max_phi)
lambda
#> $lambda
#> [1] 0.2727968
#> 
#> $gm
#> [1] 0.5689906
#> 
#> $init_psi
#> [1] -0.2016904
#> 
#> $sd_psi
#> [1] 0.7835109
#> 
x <- ru(logf = dgamma, shape = alpha, log = TRUE, d = 1, n = 1000,
        trans = "BC", lambda = lambda)

alpha <- 0.1
# NB. for alpha < 1 the gamma(alpha, beta) density is not bounded
# So the ratio-of-uniforms emthod can't be used but it may work after a
# Box-Cox transformation.
# find_lambda_one_d() works much better than find_lambda() here.

max_phi <- qgamma(0.999, shape = alpha)
lambda <- find_lambda_one_d(logf = dgamma, shape = alpha, log = TRUE,
                            max_phi = max_phi)
lambda
#> $lambda
#> [1] 0.06758891
#> 
#> $gm
#> [1] 0.008056577
#> 
#> $init_psi
#> [1] -0.0342618
#> 
#> $sd_psi
#> [1] 0.009372876
#> 
x <- ru(logf = dgamma, shape = alpha, log = TRUE, d = 1, n = 1000,
        trans = "BC", lambda = lambda)

# \donttest{
plot(x)

plot(x, ru_scale = TRUE)

# }