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)

Returns a vector or matrix of transformed values.

- 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`

.

Sanford Weisberg, <sandy@umn.edu>

The Box-Cox
family of *scaled power transformations*
equals \((x^{\lambda}-1)/\lambda\)
for \(\lambda \neq 0\), 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 $$z = .5 (U + \sqrt{U^2 + \gamma^2)})$$ 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`

.

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.

`powerTransform`

, `testTransform`

```
U <- c(NA, (-3:3))
if (FALSE) 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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