bcPower function

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

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.

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 $-2$ to $2$.
• 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 $(x^(lambda)-1)/lambda$

for $lambda not equal to 0$, and $log(x)$ if $lambda = 0$. The bcPower

function computes the scaled power transformation of $x = U + gamma$, where $gamma$

is set by the user so $U + gamma$ 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

$$z = .5 (U + \sqrt{U^2 + \gamma^2)})z = .5 (U + \sqrt[U^2 + gamma^2])$$

where for this family $gamma$ is either user selected or is estimated. gamma must be positive if $U$ includes negative values and non-negative otherwise, ensuring that $z$ is always positive. The bcnPower transformations behave similarly to the bcPower transformations, and introduce less bias than is introduced by setting the parameter $gamma$ 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 $U+1$ for nonnegative values, and of $|U|+1$ with parameter $2-lambda$ for $U$ negative.

The basic power transformation returns $U^{lambda}$ if $lambda$ is not 0, and $log(lambda)$

otherwise for $U$ 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 $U$ for skewPower and yjPower and of $U + gamma$ 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.

Returns

Returns a vector or matrix of transformed values.

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.

Author(s)

Sanford Weisberg, sandy@umn.edu

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))