Returns the Box-Cox transformation of the input data point(s).
Syntax
BOXCOX(X, Alpha, Lambda, Return_type)
- X
- the real number for which we compute the transformation.
- Alpha
- is the input shift parameter for X. If omitted, the default value is 0.
- Lambda
- is the input power parameter of the transformation, on a scale from 1 to 0. If omitted, the default value of 0 is assumed.
- Return_type
- 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_TYPE NUMBER RETURNED 1 or omitted Box-Cox Transform 2 Inverse of Box-Cox transform 3 LLF of Box-Cox transform
Remarks
- Box-Cox transform is perceived as a useful data (pre)processing technique used to stabilize variance and make the data more normally distributed.
- 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 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.
- The BOXCOX function accepts a single value or an array of values for X.
- The shift parameter must be large enough to make all the values of X positive.
- To compute the log-likelihood function (LLF), the BoxCox function assumes a Gaussian distribution which parameters ($\mu,\sigma^2$) are calculated using the maximum-likelihood estimate (MLE) method.
$$LLF_{\textrm{BoxCox}} = \frac{-N}{2}\times\left( \ln\left( 2\pi\hat{\sigma}^2\right)+1\right) $$
$$\hat{\sigma}^2=\frac{\sum_{t=1}^N{\left(y_t-\mu\right )^2}}{N}$$
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:
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| Formula | Description (Result) |
|---|---|
| =BOXCOX($B$2:$B$30,3,0.5,3) | LLF Boxcoc (-33.35) |
Files Examples
References
- James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896
- Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740
- D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
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