BoxCox - Box-Cox Transform

Returns the Box-Cox transformation of the input data point(s).

Syntax

BOXCOX(X, Lo, Hi, $\lambda$, Return)

X

is the real value(s) for which we compute the transformation of a single value or a one-dimensional array of cells (e.g., rows or columns)).

Lo

is the x-domain lower limit. If missing, Lo is assumed to be 0.

Hi

is the x-domain upper limit. If missing, Hi is assumed to be infinity.

$\lambda$

is the input power parameter of the transformation ($\lambda\in[0,1)$). If omitted, the default value of 0 is assumed.

Return

is a number that determines the type of return value: 1 (or missing) = Box-Cox, 2 = Inverse Box-Cox, 3 = LLF of Box-Cox.

Return	Description
0 or omitted	Box-Cox transform
1	The inverse of Box-Cox transform
2	Log-likelihood function of the transform

Remarks

1. BOXCOX() transform function converts a one-bound domain (e.g., $x\in(a,\infty)$, $x\in(-\infty, b)$) into an unbounded $(-\infty,\infty)$ domain.
2. If both values of Lo and Hi arguments are given, the BOXCOX() returns #VALUE!.
3. Box-Cox transform is perceived as a useful data (pre)processing technique used to stabilize variance and make the data more normally distributed.
4. The Box-Cox transformation is defined as follows:
   $$ T\left ( x_{t}; \lambda, \alpha \right ) = \begin{cases} \dfrac{\left ( x_{t} + \alpha \right )^{\lambda}-1}{\lambda} & \text{ if } \lambda \neq 0 \\ \log \left ( x_t + \alpha \right ) & \text{ if } \lambda= 0 \end{cases} $$ Where:
   
   • $x_{t}$ is the input value of the input time series at time $t$.
   • $\lambda$ is the input scalar value of the Box-Cox transformation.
   • $\alpha$ is the shift parameter.
   • $\left(x_t +\alpha \right) \gt 0$ for all $t$ values.
5. Using the negative values of \{x_t\}, we can use the Box-Cox transform for a domain with an upper bound.
   $$F(x_t;\lambda,b)=\begin{matrix}\frac{(b-x)^\lambda-1}{\lambda}&\lambda\neq0\\\ln{(b-x_t)}&\lambda=0\\\end{matrix}$$
6. To calculate the inverse of the BoxCox transform:
   • Domain with lower bound (a):
     $$x=a+e^\frac{\ln{(\lambda y+1)}}{\lambda}$$
   • Domain with upper bound (b):
     $$x=b-e^\frac{\ln{(\lambda y+1)}}{\lambda}$$
7. To compute the log-likelihood function (LLF), the Box-Cox function assumes a Gaussian distribution in which parameters ($\mu,\sigma^2$) are calculated using the maximum-likelihood estimate (MLE) method.
   $$LLF_{\textit{BoxCox}} = \frac{-N}{2}\times( \ln( 2\pi\hat{\sigma}^2)+1)$$ $$\hat{\sigma}^2=\frac{\sum_{t=1}^N{(y_t-\mu)^2}}{N}$$ Where:
   • $\hat{\sigma}^2$ is the biased estimate of the variance.
   • $N$ is the number of non-missing values in the sample data.
   • $y_t$ is the t-th transformed observation.

Examples

Example 1:

Related Links

• Wikipedia - Power transform
• Wikipedia - Maximum Likelihood (MLE)

References

• Box, George E. P.; Cox, D. R. (1964). "An analysis of transformations". Journal of the Royal Statistical Society, Series B. 26 (2): 211–252. JSTOR 2984418.
• Sakia, R. M. (1992), "The Box–Cox transformation technique: a review", The Statistician, 41 (2): 169–178.