geoR (version 1.6-21)

boxcox: The Box-Cox Transformed Normal Distribution

Description

Functions related with the Box-Cox family of transformations. Density and random generation for the Box-Cox transformed normal distribution with mean equal to mean and standard deviation equal to sd, in the normal scale.

Usage

rboxcox(n, lambda, lambda2 = NULL, mean = 0, sd = 1)
dboxcox(x, lambda, lambda2 = NULL, mean = 0, sd = 1)

Arguments

lambda

numerical value(s) for the transformation parameter $\lambda$.

lambda2

logical or numerical value(s) of the additional transformation (see DETAILS below). Defaults to NULL.

n

number of observations to be generated.

x

a vector of quantiles (dboxcox) or an output of boxcox.fit (print, plot, lines).

mean

a vector of mean values at the normal scale.

sd

a vector of standard deviations at the normal scale.

Value

• The functions returns the following results:
• rboxcox
  a vector of random deviates.
• dboxcox
  a vector of densities.

concept

Box-Cox trransformation

Details

Denote $Y$ the variable at the original scale and $Y'$ the transformed variable. The Box-Cox transformation is defined by: $$Y' = \left{ $$\begin{array}{l}log(Y)\text{ , if }\lambda =0\\ \frac{Y^{\lambda }-1}{\lambda }\text{ , otherwise}\end{array}$$ \right.$$.Anadditionalshiftingparameter\${\lambda }_2\$canbeincludedinwhichcasethetransformationisgivenby:$$Y' = \left{ $$\begin{array}{l}log(Y+{\lambda }_2)\text{ , }\lambda =0\\ \frac{(Y+{\lambda }_2{)}^{\lambda }-1}{\lambda }\text{ , otherwise}\end{array}$$ \right.$$.

The function rboxcox samples $Y'$ from the normal distribution using the function rnorm and backtransform the values according to the equations above to obtain values of $Y$. If necessary the back-transformation truncates the values such that $Y' \geq \frac{1}{\lambda}$ results in $Y = 0$ in the original scale. Increasing the value of the mean and/or reducing the variance might help to avoid truncation.

References

Box, G.E.P. and Cox, D.R.(1964) An analysis of transformations. JRSS B 26:211--246.

See Also

The parameter estimation function boxcox.fit, the function boxcox in the package MASS and the function boxcox in the package car.

Examples

## Simulating data
simul <- rboxcox(100, lambda=0.5, mean=10, sd=2)
##
## Comparing models with different lambdas,
## zero  means and unit variances
curve(dboxcox(x, lambda=-1), 0, 8)
for(lambda in seq(-.5, 1.5, by=0.5))
  curve(dboxcox(x, lambda), 0, 8, add = TRUE)