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Statistics

  

CrossCorrelation

  

compute sample cross-correlations of two time series

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

CrossCorrelation(X1, X2)

CrossCorrelation(X1, X2, lags)

Parameters

X1, X2

-

discrete univariate time series given as Vectors, lists, Matrices with one column, or TimeSeries objects with one dataset.

lags

-

(optional) maximal lag to return, or a range of lags to return. By default all possible lags are returned.

Options

• 

scaling

  

One of biased, unbiased, or none.  Default is none. scaling=biased computes Rk=Ckn. scaling=unbiased scales each Ck by 1nk.

• 

raw

  

If this option is given, the output is not normalized so that the middle entry (corresponding to R0) is 1 when scaling=unbiased or scaling=none.

Description

• 

For a discrete time series X1 and X2, the CrossCorrelation command computes the cross-correlations Rk=CkC0 where Ck=t=1nkX1tX2t+k&conjugate0; for k=n+1..n1.

• 

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.

Examples

with(Statistics):

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

0.2000000000622400.5000000000305990.5999999999867191.0.5999999999867190.5000000000305990.200000000062240

(1)

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

0.5000000000305990.5999999999867191.0.5999999999867190.500000000030599

(2)

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

0.5000000000305990.5999999999867191.0.5999999999867190.500000000030599

(3)

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

1.000000000061200.7999999999822921.0.7999999999822921.00000000006120

(4)

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

0.8571428571961741.642857142967681.0.3571428571743542.3214255916459210−11

(5)

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

0.8571428571961741.642857142967681.0.3571428571743542.3214255916459210−11

(6)

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

2.3214255916459210−110.3571428571743541.1.642857142967680.857142857196174

(7)

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

2.3214255916459210−110.3571428571743541.1.642857142967680.857142857196174

(8)

The CrossCorrelation command also accepts TimeSeries objects, but the date information is ignored.

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

t1Time seriesdata set3 rows of data:2011-11-01 - 2012-01-01

(9)

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

t2Time seriesdata set3 rows of data:2015-09-28 - 2015-09-30

(10)

CrossCorrelation(t1, t2);

2.3214255916459210−110.3571428571743541.1.642857142967680.857142857196174

(11)

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

t3Time seriesSales, Profits3 rows of data:2013-04-17 - 2013-05-01

(12)

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

2.3214255916459210−110.3571428571743541.1.642857142967680.857142857196174

(13)

CrossCorrelation can be used to create cross-correlograms

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);

Compatibility

• 

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.

See Also

ColumnGraph

Statistics[AutoCorrelation]

TimeSeriesAnalysis