boxCoxTransform

Performs a forward or an inverse Box-Cox (power) transformation.

Synopsis

boxCoxTransform (z, power)

Required Arguments

float z[] (Input)
Array of length nObservations containing the observations.
float power (Input)
Exponent parameter in the Box-Cox (power) transformation.

Return Value

An array of length nObservations containing the transformed data. If no value can be computed, then None is returned.

Optional Arguments

shift, float (Input)

Shift parameter in the Box-Cox (power) transformation. Parameter shift must satisfy the relation min (z(i)) + shift > 0.

Default: shift = 0.0.

inverseTransform (Input)
If inverseTransform is specified, the inverse transform is performed.

Description

Function boxCoxTransform performs a forward or an inverse Box-Cox (power) transformation of n = nObservations observations \(\{Z_t\}\) for \(t=1,2,\ldots,n\).

The forward transformation is useful in the analysis of linear models or models with nonnormal errors or nonconstant variance (Draper and Smith 1981, p. 222). In the time series setting, application of the appropriate transformation and subsequent differencing of a series can enable model identification and parameter estimation in the class of homogeneous stationary autoregressive-moving average models. The inverse transformation can later be applied to certain results of the analysis, such as forecasts and prediction limits of forecasts, in order to express the results in the scale of the original data. A brief note concerning the choice of transformations in the time series models is given in Box and Jenkins (1976, p. 328).

The class of power transformations discussed by Box and Cox (1964) is defined by

\[\begin{split}X_t = \begin{cases} \frac{\left(Z_t + \xi\right)^{\lambda} - 1}{\lambda} & \lambda \neq 0 \\ \ln \left(Z_t + \xi\right) & \lambda = 0 \\ \end{cases}\end{split}\]

where \(Z_t+\xi>0\) for all t. Since

\[\lim_{\lambda \to 0} \frac{\left(Z_t + \xi\right)^{\lambda} - 1}{\lambda} = \ln \left(Z_t - \xi\right)\]

the family of power transformations is continuous.

Let λ = power and ξ = shift; then, the computational formula used by boxCoxTransform is given by

\[\begin{split}X_t = \begin{cases} \left(Z_t + \xi\right)^{\lambda} & \lambda \neq 0 \\ \ln \left(Z_t + \xi\right) & \lambda = 0 \\ \end{cases}\end{split}\]

where \(Z_t+\xi>0\) for all t. The computational and Box-Cox formulas differ only in the scale and origin of the transformed data. Consequently, the general analysis of the data is unaffected (Draper and Smith 1981, p. 225).

The inverse transformation is computed by

\[\begin{split}X_t = \begin{cases} Z_t^{1/\lambda} + \xi & \lambda \neq 0 \\ \exp \left(Z_t\right) - \xi & \lambda = 0 \\ \end{cases}\end{split}\]

where \(\{Z_t\}\) now represents the result computed by boxCoxTransform for a forward transformation of the original data using parameters λ and ξ.

Examples

Example 1

The following example performs a Box-Cox transformation with power = 2.0 on 10 data points.

from numpy import *
from pyimsl.stat.boxCoxTransform import boxCoxTransform
from pyimsl.stat.writeMatrix import writeMatrix

power = 2.0
z = (1.0, 2.0, 3.0, 4.0, 5.0, 5.5, 6.5, 7.5, 8.0, 10.0)

# Transform Data using Box Cox Transform
y = boxCoxTransform(z, power)
writeMatrix("Transformed Data", y, writeFormat="%5.1f")

Output

 
                          Transformed Data
    1      2      3      4      5      6      7      8      9     10
  1.0    4.0    9.0   16.0   25.0   30.2   42.2   56.2   64.0  100.0

Example 2

This example extends the first example—an inverse transformation is applied to the transformed data to return to the original data values.

from numpy import *
from pyimsl.stat.boxCoxTransform import boxCoxTransform
from pyimsl.stat.writeMatrix import writeMatrix

power = 2.0
z = (1.0, 2.0, 3.0, 4.0, 5.0, 5.5, 6.5, 7.5, 8.0, 10.0)

# Transform Data using Box Cox Transform
x = boxCoxTransform(z, power)
writeMatrix("Transformed Data", x, writeFormat="%5.1f")

# Perform an Inverse Transform on the Transformed Data
y = boxCoxTransform(x, power, inverseTransform=True)
writeMatrix("Inverse Transformed Data", y, writeFormat="%5.1f")

Output

 
                          Transformed Data
    1      2      3      4      5      6      7      8      9     10
  1.0    4.0    9.0   16.0   25.0   30.2   42.2   56.2   64.0  100.0
 
                      Inverse Transformed Data
    1      2      3      4      5      6      7      8      9     10
  1.0    2.0    3.0    4.0    5.0    5.5    6.5    7.5    8.0   10.0

Fatal Errors

IMSLS_ILLEGAL_SHIFT shift” = # and the smallest element of “z” is “z[#]” = #. “shift” plus “z[#]” = #. “shift” + “z[i]” must be greater than 0 for i = 1,…, “nObservations”. “nObservations” = #.
IMSLS_BCTR_CONTAINS_NAN One or more elements of “z” is equal to NaN (Not a number). No missing values are allowed. The smallest index of an element of “z” that is equal to NaN is #.
IMSLS_BCTR_F_UNDERFLOW Forward transform. “power” = #. “shift” = #. The minimum element of “z” is “z[#]” = #. (“z[#]”+ “shift”) ^ “power” will underflow.
IMSLS_BCTR_F_OVERFLOW Forward transformation. “power” = #. “shift” = #. The maximum element of “z” is “z[#]” = #. (“z[#]” + “shift”) ^ “power” will overflow.
IMSLS_BCTR_I_UNDERFLOW Inverse transformation. “power” = #. The minimum element of “z” is “z[#]” = #. exp(“z[#]”) will underflow.
IMSLS_BCTR_I_OVERFLOW Inverse transformation. “power” = #. The maximum element of “z[#]” = #. exp(“z[#]”) will overflow.
IMSLS_BCTR_I_ABS_UNDERFLOW Inverse transformation. “power” = #. The element of “z” with the smallest absolute value is “z[#]” = #. “z[#]” ^ (1/ “power”) will underflow.
IMSLS_BCTR_I_ABS_OVERFLOW Inverse transformation. “power” = #. The element of “z” with the largest absolute value is “z[#]” = #. “z[#]” ^ (1/ “power”) will overflow.