find_lambda_one_d: Selecting the Box-Cox parameter in the 1D case
In rust: Ratio-of-Uniforms Simulation with Transformation

find_lambda_one_d

R Documentation

Selecting the Box-Cox parameter in the 1D case

Description

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.

Usage

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.

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.

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)

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).

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
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
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
x <- ru(logf = dgamma, shape = alpha, log = TRUE, d = 1, n = 1000,
        trans = "BC", lambda = lambda)


plot(x)
plot(x, ru_scale = TRUE)

rust documentation built on Sept. 11, 2024, 6:43 p.m.