# bcPower: Box-Cox, Box-Cox with Negatives Allowed, Yeo-Johnson and... In car: Companion to Applied Regression

### 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

 ```1 2 3 4 5 6 7``` ```bcPower(U, lambda, jacobian.adjusted=FALSE, gamma=NULL) bcnPower(U, lambda, jacobian.adjusted = FALSE, 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 eacul column of U, usuallly in the range from -2 to 2, or if `U` `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.

### Details

The Box-Cox family of scaled power transformations equals (U^(lambda)-1)/lambda for lambda not equal to zero, and log(U) if lambda = 0. If gamma is not specified, it is set equal to zero. `U + gamma` must be strictly positive to use this family.

The Box-Cox family with negatives allowed was proposed by Hawkins and Weisberg (2017). It is the Box-Cox power transformation of z = .5 * (y + (y^2 + γ^2)^{1/2}), where gamma is strictly positive if y includes negative values and non-negative otherwise. The value of z is always positive. The bcnPower transformations behave very similarly to the bcPower transformations, including much less bias than is introduced by setting the parameter gamma to be non-zero in the Box-Cox family.

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^{λ} if λ is not zero, and \log(λ) 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.

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. (2011) An R Companion to Applied Regression, Second Edition, Sage.

Hawkins, D. and Weisberg, S. (2017) Combining the Box-Cox Power and Generalized Log Transformations to Accomodate Negative Responses In Linear and Mixed-Effects Linear Models, submitted for publication.

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.

`powerTransform`, `testTransform`
 ```1 2 3 4 5 6 7 8 9``` ```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) 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)) ```