bcPower {car}R Documentation

Box-Cox, Box-Cox with Negatives Allowed, Yeo-Johnson and Basic Power Transformations

Description

Transform the elements of a vector or columns of a matrix using, the Box-Cox, Box-Cox with negatives allowed, Yeo-Johnson, or simple power transformations.

Usage

bcPower(U, lambda, jacobian.adjusted=FALSE, gamma=NULL)

bcnPower(U, lambda, jacobian.adjusted = FALSE, gamma)

bcnPowerInverse(z, lambda, gamma)

yjPower(U, lambda, jacobian.adjusted = FALSE)

basicPower(U,lambda, gamma=NULL)

Arguments

U

A vector, matrix or data.frame of values to be transformed

lambda

Power transformation parameter with one element for each column of U, usuallly in the range from 2-2 to 22.

jacobian.adjusted

If TRUE, the transformation is normalized to have Jacobian equal to one. The default FALSE is almost always appropriate.

gamma

For bcPower or basicPower, the transformation is of U + gamma, where gamma is a positive number called a start that must be large enough so that U + gamma is strictly positive. For the bcnPower, Box-cox power with negatives allowed, see the details below.

z

a numeric vector the result of a call to bcnPower with jacobian.adjusted=FALSE

.

Details

The Box-Cox family of scaled power transformations equals (xλ1)/λ(x^{\lambda}-1)/\lambda for λ0\lambda \neq 0, and log(x)\log(x) if λ=0\lambda =0. The bcPower function computes the scaled power transformation of x=U+γx = U + \gamma, where γ\gamma is set by the user so U+γU+\gamma is strictly positive for these transformations to make sense.

The Box-Cox family with negatives allowed was proposed by Hawkins and Weisberg (2017). It is the Box-Cox power transformation of

z=.5(U+U2+γ2))z = .5 (U + \sqrt{U^2 + \gamma^2)})

where for this family γ\gamma is either user selected or is estimated. gamma must be positive if UU includes negative values and non-negative otherwise, ensuring that zz is always positive. The bcnPower transformations behave similarly to the bcPower transformations, and introduce less bias than is introduced by setting the parameter γ\gamma to be non-zero in the Box-Cox family.

The function bcnPowerInverse computes the inverse of the bcnPower function, so U = bcnPowerInverse(bcnPower(U, lambda=lam, jacobian.adjusted=FALSE, gamma=gam), lambda=lam, gamma=gam) is true for any permitted value of gam and lam.

If family="yeo.johnson" then the Yeo-Johnson transformations are used. This is the Box-Cox transformation of U+1U+1 for nonnegative values, and of U+1|U|+1 with parameter 2λ2-\lambda for UU negative.

The basic power transformation returns UλU^{\lambda} if λ\lambda is not 0, and log(λ)\log(\lambda) otherwise for UU strictly positive.

If jacobian.adjusted is TRUE, then the scaled transformations are divided by the Jacobian, which is a function of the geometric mean of UU for skewPower and yjPower and of U+gammaU + gamma for bcPower. With this adjustment, the Jacobian of the transformation is always equal to 1. Jacobian adjustment facilitates computing the Box-Cox estimates of the transformation parameters.

Missing values are permitted, and return NA where ever U is equal to NA.

Value

Returns a vector or matrix of transformed values.

Author(s)

Sanford Weisberg, <sandy@umn.edu>

References

Fox, J. and Weisberg, S. (2019) An R Companion to Applied Regression, Third Edition, Sage.

Hawkins, D. and Weisberg, S. (2017) Combining the Box-Cox Power and Generalized Log Transformations to Accomodate Nonpositive Responses In Linear and Mixed-Effects Linear Models South African Statistics Journal, 51, 317-328.

Weisberg, S. (2014) Applied Linear Regression, Fourth Edition, Wiley Wiley, Chapter 7.

Yeo, In-Kwon and Johnson, Richard (2000) A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.

See Also

powerTransform, testTransform

Examples

U <- c(NA, (-3:3))
## Not run: bcPower(U, 0)  # produces an error as U has negative values
bcPower(U, 0, gamma=4)
bcPower(U, .5, jacobian.adjusted=TRUE, gamma=4)
bcnPower(U, 0, gamma=2)
basicPower(U, lambda = 0, gamma=4)
yjPower(U, 0)
V <- matrix(1:10, ncol=2)
bcPower(V, c(0, 2))
basicPower(V, c(0,1))

[Package car version 3.1-2 Index]