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)
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 |
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 |
.
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 + √[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 a vector or matrix of transformed values.
Sanford Weisberg, <sandy@umn.edu>
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.
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))
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