The differencing transformation takes a time series $T_1,T_2,T_3,\mathrm{...}$ and returns the time series $T_2-T_1,T_3-T_2,\mathrm{...}$.

Apply this transformation to a time series using the Apply command. Translate differenced information such as forecasts back to the original domain by using the Unapply command.

By using the $\mathrm{lag}$ option, one can difference a time series with a given lag. In particular, the transformation $\mathrm{Difference}\left(\mathrm{lag}=2\right)$ transforms the time series $T_1,T_2,T_3,\mathrm{...}$ into $T_3-T_1,T_4-T_2,T_5-T_2,\mathrm{...}$. The default lag is 1.

If there are missing data points (represented in Maple by the value $\mathrm{undefined}$ in the time series), the differencing transformation skips over them while leaving the undefined value in place. For example, if $T_4$ is undefined, then the result of differencing with $\mathrm{lag}=2$ is

By using the $\mathrm{times}$ option, one can specify the transformation that consists of differencing multiple times. The default is $\mathrm{times}=1$. Specifying $\mathrm{times}=2$, for example, describes the transformation that takes the time series $T_1,T_2,T_3,\mathrm{...}$ to $T_1+T_3-2T_2,T_2+T_4-2T_3,\mathrm{...}$.

Every time differencing is applied, the resulting time series has $k$ data points fewer than the input. This is reflected in the timestamps; the first timestamp is omitted from the resulting time series. In other words, the timestamp for the difference $T_2-T_1$ is taken to be the time at which $T_2$ occurred, because it is only at this time that both $T_1$ and $T_2$ are available and the first difference can be computed.

For the Unapply call, to transform $T_2-T_1,T_3-T_2,\mathrm{...}$ back into the original $T_1,T_2,T_3,\mathrm{...}$, Maple needs to know the value $T_1$, which will be referred to as the origin. If the forecast is created using TimeSeriesAnalysis commands, then Unapply can generally find the time series object from which the original differences were taken, and use this object to infer the origin. Otherwise, or to override this choice, the origin can be specified explicitly using the $\mathrm{origin}=o$ option.

A Matrix can be specified as a list or Vector of the elements needed.

A Matrix can be specified as the list or Vector of the elements of its first row; this row will be repeated $k$ times.

A Matrix can be specified as a single number, which the $kxm$ Matrix will be filled with.

Instead of a list of Matrices, you can specify a single $nxm$ Matrix, each row of which will be expanded into a $kxm$ by repeating it $k$ times.

Finally, if you specify a single number, it will be used to fill all $n$ of the $kxm$ 

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

$\mathrm{sales\_numbers}≔⟨150\,147\,114\,113\,91\,164\,56\,39\,32\,86⟩$

$\mathrm{sales\_numbers}≔\left[\begin{array}{c}150\\ 147\\ 114\\ 113\\ 91\\ 164\\ 56\\ 39\\ 32\\ 86\end{array}\right]$

$\mathrm{sales}≔\mathrm{TimeSeries}\left(\mathrm{sales\_numbers}\,\mathrm{startdate}=''2010-01-01''\,\mathrm{frequency}=''weekly''\,\mathrm{header}=''Weekly\; Sales''\right)$

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

$\mathrm{GetData}\left(\mathrm{sales}\right)\left[..5\right]$

$\left[\begin{array}{c}150.\\ 147.\\ 114.\\ 113.\\ 91.\end{array}\right]$

$\mathrm{differenced}≔\mathrm{Apply}\left(\mathrm{Difference}\,\mathrm{sales}\right)$

$\mathrm{differenced}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales\; (differenced)}\\ \mathrm{9\; rows\; of\; data:}\\ \mathrm{2010-01-08\; -\; 2010-03-05}\end{array}\right]$

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

$\left[\begin{array}{c}\mathrm{-3.}\\ \mathrm{-33.}\\ \mathrm{-1.}\\ \mathrm{-22.}\end{array}\right]$

$\mathrm{original}≔\mathrm{Unapply}\left(\mathrm{Difference}\,\mathrm{differenced}\right)$

$\mathrm{original}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales}\\ \mathrm{9\; rows\; of\; data:}\\ \mathrm{2010-01-08\; -\; 2010-03-05}\end{array}\right]$

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

$\left[\begin{array}{c}147.\\ 114.\\ 113.\\ 91.\end{array}\right]$

$\mathrm{differenced\_twice}≔\mathrm{Apply}\left(\mathrm{Difference}\,\mathrm{differenced}\right)$

$\mathrm{differenced\_twice}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales\; (differenced)\; (differenced)}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{differenced\_twice}\right)\left[..3\right]$

$\left[\begin{array}{c}\mathrm{-30.}\\ 32.\\ \mathrm{-21.}\end{array}\right]$

$\mathrm{differenced\_twice\_2}≔\mathrm{Apply}\left(\mathrm{Difference}\left('\mathrm{times}'=2\right)\,\mathrm{sales}\right)$

$\mathrm{differenced\_twice\_2}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales\; (differenced\; twice)}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{differenced\_twice\_2}\right)\left[..3\right]$

$\left[\begin{array}{c}\mathrm{-30.}\\ 32.\\ \mathrm{-21.}\end{array}\right]$

$\mathrm{original\_2}≔\mathrm{Unapply}\left(\mathrm{Difference}\left('\mathrm{times}'=2\right)\,\mathrm{differenced\_twice}\right)$

$\mathrm{original\_2}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{original\_2}\right)\left[..3\right]$

$\left[\begin{array}{c}114.\\ 113.\\ 91.\end{array}\right]$

$\mathrm{original\_3}≔\mathrm{Unapply}\left(\mathrm{Difference}\left('\mathrm{times}'=2\right)\,\mathrm{differenced\_twice\_2}\right)$

$\mathrm{original\_3}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{original\_3}\right)\left[..3\right]$

$\left[\begin{array}{c}114.\\ 113.\\ 91.\end{array}\right]$

$\mathrm{intermediate}≔\mathrm{Unapply}\left(\mathrm{Difference}\,\mathrm{differenced\_twice}\right)$

$\mathrm{intermediate}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales\; (differenced)}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{intermediate}\right)\left[..3\right]$

$\left[\begin{array}{c}\mathrm{-33.}\\ \mathrm{-1.}\\ \mathrm{-22.}\end{array}\right]$

$\mathrm{original\_4}≔\mathrm{Unapply}\left(\mathrm{Difference}\,\mathrm{intermediate}\right)$

$\mathrm{original\_4}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{original\_4}\right)\left[..3\right]$

$\left[\begin{array}{c}114.\\ 113.\\ 91.\end{array}\right]$

$\mathrm{intermediate\_2}≔\mathrm{Unapply}\left(\mathrm{Difference}\,\mathrm{differenced\_twice\_2}\right)$

$\mathrm{intermediate\_2}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales\; (differenced)}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{intermediate\_2}\right)\left[..3\right]$

$\left[\begin{array}{c}\mathrm{-33.}\\ \mathrm{-1.}\\ \mathrm{-22.}\end{array}\right]$

$\mathrm{original\_5}≔\mathrm{Unapply}\left(\mathrm{Difference}\,\mathrm{intermediate\_2}\right)$

$\mathrm{original\_5}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{original\_5}\right)\left[..3\right]$

$\left[\begin{array}{c}114.\\ 113.\\ 91.\end{array}\right]$

$\mathrm{differenced\_lag2}≔\mathrm{Apply}\left(\mathrm{Difference}\left(\mathrm{lag}=2\right)\,\mathrm{sales}\right)$

$\mathrm{differenced\_lag2}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales\; (differenced)}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{differenced\_lag2}\right)\left[..3\right]$

$\left[\begin{array}{c}\mathrm{-36.}\\ \mathrm{-34.}\\ \mathrm{-23.}\end{array}\right]$

$\mathrm{sales\_numbers}\left[5\right]≔\mathrm{undefined}\:$$\mathrm{sales\_numbers}$

$\left[\begin{array}{c}150\\ 147\\ 114\\ 113\\ \mathrm{undefined}\\ 164\\ 56\\ 39\\ 32\\ 86\end{array}\right]$

$\mathrm{sales\_missing}≔\mathrm{TimeSeries}\left(\mathrm{sales\_numbers}\,\mathrm{startdate}=''2010-01-01''\,\mathrm{frequency}=''weekly''\,\mathrm{header}=''Weekly\; Sales''\right)$

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

$\mathrm{differenced\_lag2\_missing}≔\mathrm{Apply}\left(\mathrm{Difference}\left(\mathrm{lag}=2\right)\,\mathrm{sales\_missing}\right)$

$\mathrm{differenced\_lag2\_missing}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Weekly\; Sales\; (differenced)}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2010-01-15\; -\; 2010-03-05}\end{array}\right]$

$\mathrm{GetData}\left(\mathrm{differenced\_lag2\_missing}\right)$

$\left[\begin{array}{c}\mathrm{-36.}\\ \mathrm{-34.}\\ Float\left(\mathrm{undefined}\right)\\ 51.\\ \mathrm{-58.}\\ \mathrm{-125.}\\ \mathrm{-24.}\\ 47.\end{array}\right]$

$\mathrm{GetHeaders}\left(\mathrm{differenced\_lag2\_missing}\right)$

$\left[''Weekly\; Sales\; (differenced)''\right]$

The TimeSeriesAnalysis[Difference] command was introduced in Maple 18.

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