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] and thus will work strictly in hardware precision, that is, its accuracy is independent of the setting of Digits.

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

with(Statistics):

CrossCorrelation(<1,2,1,2>,<1,2,1,2>);

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

CrossCorrelation(<1,2,1,2>,<1,2,1,2>, 2);

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

CrossCorrelation(<1,2,1,2>,<1,2,1,2>, -2..2 );

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

CrossCorrelation(<1,2,1,2>,<1,2,1,2>, 2, scaling=unbiased );

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

CrossCorrelation(<1, 2, 3>,<4, 5>);

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

CrossCorrelation(<1, 2, 3>,<4, 5, 0>);

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

CrossCorrelation(<4, 5>,<1,2,3>);

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

CrossCorrelation(<4, 5, 0>,<1,2,3>);

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

t1 := TimeSeriesAnalysis:-TimeSeries([4, 5, 0], enddate="2012-01-01", frequency="monthly");

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

t2 := TimeSeriesAnalysis:-TimeSeries([1, 2, 3], enddate="2015-09-30", frequency="daily");

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

CrossCorrelation(t1, t2);

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

t3 := TimeSeriesAnalysis:-TimeSeries([[4,5,0], [1,2,3]], headers=["Sales", "Profits"], enddate="2013-05-01", frequency="weekly");

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

CrossCorrelation(t3[.., "Sales"], t3[.., "Profits"]);

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

L := LinearAlgebra:-RandomVector(1000, datatype=float):

S := CrossCorrelation(1/3*(2*L[101..1000]+L[51..950]),L[1..900], 150, scaling=unbiased, raw):

ColumnGraph(S, offset=-151, color="Gray", style=polygon);

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.