The Box-Cox transformation takes a time series and replaces each value $T_i$ with the value $\frac{T_i^{\mathrm{\lambda }}-1}{\mathrm{\lambda }}$. If the parameter $\mathrm{\lambda }$ is set to $0$, the resulting value is $\underset{\mathrm{\lambda }\to 0}{lim}\frac{T_i^{\mathrm{\lambda }}-1}{\mathrm{\lambda }}=\ln \left(T_i\right)$.

Apply this transformation to a time series using the Apply command. Translate transformed information, such as a forecast from transformed data, back to the original domain by using the Unapply command.

By default, the $\mathrm{\lambda }$ parameter has value $0$, and the transformation is the log transformation (the natural logarithm, with base $\ⅇ$). By supplying the $\mathrm{\lambda }$ option, you can set this parameter to a different value. The command LogTransform is the same as BoxCoxTransform, except you cannot supply the lambda parameter: it is fixed to always be $0$.

If $\mathrm{\lambda }=0$, then you can choose to use a different base for the logarithm by supplying the $\mathrm{base}$ option. For example, to obtain the common logarithm, supply the option $\mathrm{base}=10$. It is an error to supply the $\mathrm{base}$ option if $\mathrm{\lambda }$ is set to a value different from $0$. The default base is $\ⅇ$.

Negative data points result in undefined values. To prevent this from happening, you can supply the $\mathrm{shift}=s$ option. The value $s$ is added to each data point before the transformation is applied. By default, $s$ is $0$.

Applying a Box-Cox transformation may have the transformed time series being of a different order of magnitude than the original time series, and Box-Cox transformations with different values for the parameter $\mathrm{\lambda }$ may also have different orders of magnitude. If this is undesirable, for example when comparing an absolute measure of error involving the results of different Box-Cox transformations, you can supply the geometricmean = true option. This computes the geometric mean of each data set after the shift is applied but before the transformation is applied; then after applying the transformation, each data value is multiplied by this geometric mean, raised to the power $1-\mathrm{\lambda }$. This can be viewed as preserving the units of measurement that the time series data is expressed in.

$\mathrm{with}\left(\mathrm{TimeSeriesAnalysis}\right)\:$

$\mathrm{sales}≔\mathrm{TimeSeries}\left(\left[150\,147\,114\,113\,91\,164\,56\,39\,32\,86\right]\,\mathrm{startdate}=''2010-01-01''\,\mathrm{frequency}=''weekly''\,\mathrm{header}=''Weekly\; Sales''\right)$

$\mathrm{sales}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales}\\ \mathrm{10\; rows\; of\; data:}\\ \mathrm{2010-01-01\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{sales}\right)\left[..4\right]$

$\left[\begin{array}{c}150.\\ 147.\\ 114.\\ 113.\end{array}\right]$

$\mathrm{log\_sales}≔\mathrm{Apply}\left(\mathrm{LogTransform}\,\mathrm{sales}\right)$

$\mathrm{log\_sales}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Logarithm\; of\; Weekly\; Sales}\\ \mathrm{10\; rows\; of\; data:}\\ \mathrm{2010-01-01\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{log\_sales}\right)\left[..4\right]$

$\left[\begin{array}{c}5.01063529409626\\ 4.99043258677874\\ 4.73619844839450\\ 4.72738781871234\end{array}\right]$

$\mathrm{original}≔\mathrm{Unapply}\left(\mathrm{LogTransform}\,\mathrm{log\_sales}\right)$

$\mathrm{original}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales}\\ \mathrm{10\; rows\; of\; data:}\\ \mathrm{2010-01-01\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{original}\right)\left[..4\right]$

$\left[\begin{array}{c}150.000000000000\\ 147.000000000000\\ 114.000000000000\\ 113.000000000000\end{array}\right]$

$\mathrm{common\_log\_sales}≔\mathrm{Apply}\left(\mathrm{LogTransform}\left(\mathrm{base}=10\right)\,\mathrm{sales}\right)$

$\mathrm{common\_log\_sales}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Logarithm\; of\; Weekly\; Sales}\\ \mathrm{10\; rows\; of\; data:}\\ \mathrm{2010-01-01\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{common\_log\_sales}\right)\left[..4\right]$

$\left[\begin{array}{c}2.17609125905568\\ 2.16731733474818\\ 2.05690485133647\\ 2.05307844348342\end{array}\right]$

$\mathrm{original\_2}≔\mathrm{Unapply}\left(\mathrm{LogTransform}\left(\mathrm{base}=10\right)\,\mathrm{common\_log\_sales}\right)$

$\mathrm{original\_2}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales}\\ \mathrm{10\; rows\; of\; data:}\\ \mathrm{2010-01-01\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{original\_2}\right)\left[..4\right]$

$\left[\begin{array}{c}150.000000000000\\ 147.000000000000\\ 114.000000000000\\ 113.000000000000\end{array}\right]$

$\mathrm{boxcox}≔\mathrm{Apply}\left(\mathrm{BoxCoxTransform}\left(\mathrm{\lambda }=\frac{2}{3}\,\mathrm{geometricmean}=\mathrm{true}\right)\,\mathrm{sales}\right)$

$\mathrm{boxcox}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Box-Cox\; transform\; of\; Weekly\; Sales}\\ \mathrm{10\; rows\; of\; data:}\\ \mathrm{2010-01-01\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{boxcox}\right)\left[..4\right]$

$\left[\begin{array}{c}181.145998663384\\ 178.633601318965\\ 149.746747925208\\ 148.830791928663\end{array}\right]$

$\mathrm{original\_3}≔\mathrm{Unapply}\left(\mathrm{BoxCoxTransform}\left(\mathrm{\lambda }=\frac{2}{3}\,\mathrm{geometricmean}=\mathrm{true}\right)\,\mathrm{boxcox}\right)$

$\mathrm{original\_3}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales}\\ \mathrm{10\; rows\; of\; data:}\\ \mathrm{2010-01-01\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{original\_3}\right)\left[..4\right]$

$\left[\begin{array}{c}150.000000000000\\ 147.000000000000\\ 114.000000000000\\ 113.000000000000\end{array}\right]$

$\mathrm{average\_temperature\_matrix}≔⟨⟨-12|-5⟩\,⟨-11|-2⟩\,⟨-6|2⟩\,⟨1|11⟩\,⟨9|18⟩\,⟨12|23⟩\,⟨16|29⟩\,⟨14|26⟩\,⟨10|21⟩\,⟨4|14⟩\,⟨1|10⟩\,⟨-4|3⟩\,⟨-7|1⟩\,⟨-5|2⟩\,⟨0|12⟩\,⟨0|12⟩\,⟨9|22⟩\,⟨13|25⟩\,⟨15|29⟩\,⟨13|26⟩\,⟨9|21⟩\,⟨5|13⟩\,⟨-2|6⟩\,⟨-3|3⟩⟩$

$\mathrm{average\_temperature\_matrix}≔\begin{array}{c}\left[\begin{array}{cc}\mathrm{-12}& \mathrm{-5}\\ \mathrm{-11}& \mathrm{-2}\\ \mathrm{-6}& 2\\ 1& 11\\ 9& 18\\ 12& 23\\ 16& 29\\ 14& 26\\ 10& 21\\ 4& 14\\ ⋮& ⋮\end{array}\right]\\ \hfill \text{24 × 2 Matrix}\end{array}$

$\mathrm{average\_temperatures}≔\mathrm{TimeSeries}\left(\mathrm{average\_temperature\_matrix}\,\mathrm{startdate}=''2011-01-01''\,\mathrm{frequency}=''monthly''\,\mathrm{headers}=\left[''average\; minimum\; temperature''\,''average\; maximum\; temperature''\right]\right)$

$\mathrm{average\_temperatures}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{average\; minimum\; temperature,\; average\; maximum\; temperature}\\ \mathrm{24\; rows\; of\; data:}\\ \mathrm{2011-01-01\; -\; 2012-12-01}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{average\_temperatures}\right)\left[..4\right]$

$\left[\begin{array}{cc}\mathrm{-12.}& \mathrm{-5.}\\ \mathrm{-11.}& \mathrm{-2.}\\ \mathrm{-6.}& 2.\\ 1.& 11.\end{array}\right]$

$\mathrm{transformation}≔\mathrm{BoxCoxTransform}\left(\mathrm{shift}=20\,\mathrm{\lambda }=\frac{1}{2}\,'\mathrm{geometricmean}'\right)$

$\mathrm{transformation}≔\mathrm{<<\; Box-Cox\; transformation\; with\; parameter\; 1/2\; >>}$

$\mathrm{transformed\_temperatures}≔\mathrm{Apply}\left(\mathrm{transformation}\&comma;\mathrm{average\_temperatures}\right)$

$\mathrm{transformed\_temperatures}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Box-Cox\; transform\; of\; average\; minimum\; temperature,\; Box-Cox\; transform\; of\; average\; maximum\; temperature}\\ \mathrm{24\; rows\; of\; data:}\\ \mathrm{2011-01-01\; -\; 2012-12-01}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{transformed\_temperatures}\right)\left[..4\right]$

$\left[\begin{array}{cc}17.0113783673777& 32.3535475864755\\ 18.6076635345684& 36.5163724025921\\ 25.5079190900768& 41.5589049817995\\ 33.3316815594305& 51.4389969826016\end{array}\right]$

$\mathrm{original\_temperatures}≔\mathrm{Unapply}\left(\mathrm{transformation}\&comma;\mathrm{transformed\_temperatures}\right)$

$\mathrm{original\_temperatures}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{average\; minimum\; temperature,\; average\; maximum\; temperature}\\ \mathrm{24\; rows\; of\; data:}\\ \mathrm{2011-01-01\; -\; 2012-12-01}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{original\_temperatures}\right)\left[..4\right]$

$\left[\begin{array}{cc}\mathrm{-12.0000000000000}& \mathrm{-5.00000000000000}\\ \mathrm{-11.}& \mathrm{-2.00000000000001}\\ \mathrm{-6.}& 1.99999999999999\\ 0.999999999999989& 11.0000000000000\end{array}\right]$

The TimeSeriesAnalysis[BoxCoxTransform] and TimeSeriesAnalysis[LogTransform] commands were introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.