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scaling

One of biased, unbiased, or none. Default is none. scaling=biased computes $R_k=\frac{C_k}{n}$. scaling=unbiased scales each $C_k$ by $\frac{1}{n-\left|k\right|}$.

raw

If this option is given, the output is not normalized so that the middle entry (corresponding to $R_0$) is 1 when scaling=unbiased or scaling=none.

For a discrete time series X1 and X2, the CrossCorrelation command computes the cross-correlations $R_k=\frac{C_k}{C_0}$ where $C_k=\sum _{t=1}^{n-k}{\mathrm{X1}}_t\stackrel{\&conjugate0;}{{\mathrm{X2}}_{t+k}}$ for $k=-n+1..n-1$.

For efficiency, all of the lags are computed at once using a numerical discrete Fourier transform. Therefore all data provided must have type complexcons and all returned solutions are floating-point, even if the problem is specified with exact values.

If the inputs are not the same length, the shorter is padded with zeros at the end.

Note: CrossCorrelation makes use of DiscreteTransforms[FourierTransform] which works strictly in hardware precision, and thus the accuracy of the cross-correlation is not purely determined by the setting of Digits.

For more time series related commands, see the TimeSeriesAnalysis package.

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

$\mathrm{CrossCorrelation}\left(⟨1\,2\,1\,2⟩\,⟨1\,2\,1\,2⟩\right)$

$\left[\begin{array}{c}0.200000000062240\\ 0.500000000030599\\ 0.599999999986719\\ 1.\\ 0.599999999986719\\ 0.500000000030599\\ 0.200000000062240\end{array}\right]$

$\mathrm{CrossCorrelation}\left(⟨1\,2\,1\,2⟩\,⟨1\,2\,1\,2⟩\,2\right)$

$\left[\begin{array}{c}0.500000000030599\\ 0.599999999986719\\ 1.\\ 0.599999999986719\\ 0.500000000030599\end{array}\right]$

$\mathrm{CrossCorrelation}\left(⟨1\,2\,1\,2⟩\,⟨1\,2\,1\,2⟩\,-2..2\right)$

$\left[\begin{array}{c}0.500000000030599\\ 0.599999999986719\\ 1.\\ 0.599999999986719\\ 0.500000000030599\end{array}\right]$

$\mathrm{CrossCorrelation}\left(⟨1\,2\,1\,2⟩\,⟨1\,2\,1\,2⟩\,2\,\mathrm{scaling}=\mathrm{unbiased}\right)$

$\left[\begin{array}{c}1.00000000006120\\ 0.799999999982292\\ 1.\\ 0.799999999982292\\ 1.00000000006120\end{array}\right]$

$\mathrm{CrossCorrelation}\left(⟨1\,2\,3⟩\,⟨4\,5⟩\right)$

$\left[\begin{array}{c}0.857142857196174\\ 1.64285714296768\\ 1.\\ 0.357142857174354\\ 2.32142559164592\times {10}^{\mathrm{-11}}\end{array}\right]$

$\mathrm{CrossCorrelation}\left(⟨1\,2\,3⟩\,⟨4\,5\,0⟩\right)$

$\left[\begin{array}{c}0.857142857196174\\ 1.64285714296768\\ 1.\\ 0.357142857174354\\ 2.32142559164592\times {10}^{\mathrm{-11}}\end{array}\right]$

$\mathrm{CrossCorrelation}\left(⟨4\,5⟩\,⟨1\,2\,3⟩\right)$

$\left[\begin{array}{c}2.32142559164592\times {10}^{\mathrm{-11}}\\ 0.357142857174354\\ 1.\\ 1.64285714296768\\ 0.857142857196174\end{array}\right]$

$\mathrm{CrossCorrelation}\left(⟨4\,5\,0⟩\,⟨1\,2\,3⟩\right)$

$\left[\begin{array}{c}2.32142559164592\times {10}^{\mathrm{-11}}\\ 0.357142857174354\\ 1.\\ 1.64285714296768\\ 0.857142857196174\end{array}\right]$

$\mathrm{t1}≔\mathrm{TimeSeriesAnalysis}:-\mathrm{TimeSeries}\left(\left[4\,5\,0\right]\,\mathrm{enddate}=''2012年01月01日''\,\mathrm{frequency}=''monthly''\right)$

$\mathrm{t1}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{data\; set}\\ \mathrm{3\; rows\; of\; data:}\\ \mathrm{2011年11月01日\; -\; 2012年01月01日}\end{array}\right]$

$\mathrm{t2}≔\mathrm{TimeSeriesAnalysis}:-\mathrm{TimeSeries}\left(\left[1\,2\,3\right]\,\mathrm{enddate}=''2015年09月30日''\,\mathrm{frequency}=''daily''\right)$

$\mathrm{t2}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{data\; set}\\ \mathrm{3\; rows\; of\; data:}\\ \mathrm{2015年09月28日\; -\; 2015年09月30日}\end{array}\right]$

$\mathrm{CrossCorrelation}\left(\mathrm{t1}\,\mathrm{t2}\right)$

$\left[\begin{array}{c}2.32142559164592\times {10}^{\mathrm{-11}}\\ 0.357142857174354\\ 1.\\ 1.64285714296768\\ 0.857142857196174\end{array}\right]$

$\mathrm{t3}≔\mathrm{TimeSeriesAnalysis}:-\mathrm{TimeSeries}\left(\left[\left[4\,5\,0\right]\,\left[1\,2\,3\right]\right]\,\mathrm{headers}=\left[''Sales''\,''Profits''\right]\,\mathrm{enddate}=''2013年05月01日''\,\mathrm{frequency}=''weekly''\right)$

$\mathrm{t3}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Sales,\; Profits}\\ \mathrm{3\; rows\; of\; data:}\\ \mathrm{2013年04月17日\; -\; 2013年05月01日}\end{array}\right]$

$\mathrm{CrossCorrelation}\left(\mathrm{t3}\left[..,''Sales''\right]\,\mathrm{t3}\left[..,''Profits''\right]\right)$

$\left[\begin{array}{c}2.32142559164592\times {10}^{\mathrm{-11}}\\ 0.357142857174354\\ 1.\\ 1.64285714296768\\ 0.857142857196174\end{array}\right]$

$L≔\mathrm{LinearAlgebra}:-\mathrm{RandomVector}\left(1000\,\mathrm{datatype}=\mathrm{float}\right)\:$

$S≔\mathrm{CrossCorrelation}\left(\frac{1}{3}\left(2L\left[101..1000\right]+L\left[51..950\right]\right)\,L\left[1..900\right]\,150\,\mathrm{scaling}=\mathrm{unbiased}\,\mathrm{raw}\right)\:$

$\mathrm{ColumnGraph}\left(S\,\mathrm{offset}=-151\,\mathrm{color}=''Gray''\,\mathrm{style}=\mathrm{polygon}\right)$

The Statistics[CrossCorrelation] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15 .

The Statistics[CrossCorrelation] command was updated in Maple 2015.

The X1 parameter was updated in Maple 2015.