compute sample autocorrelations of a real Vector
discrete univariate real time series given as a Vector, list, DataSeries object, Matrix with one column, DataFrame with one column, or TimeSeries object with one dataset.
(optional) maximal lag to return, or a range of lags to return. By default all possible lags are returned.
One of biased, unbiased, or none. Default is none. scaling=biased computes Rk=Ckn. scaling=unbiased scales each Ck by 1n−k.
If this option is given, the output is not normalized so that the first entry is 1 when scaling=unbiased or scaling=none.
For a discrete time series X, the AutoCorrelation command computes the autocorrelations Rk=CkC0 where Ck=∑t=1n−k⁡Xt−μ⁢Xt+k−μ for k=0..n−1 and μ is the mean of X.
For efficiency, all of the lags are computed at once using a numerical discrete Fourier transform. Therefore all data provided must have type realcons and all returned solutions are floating-point, even if the problem is specified with exact values.
Note: AutoCorrelation 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.
AutoCorrelation(<1,2,1,2,1,2,1,2>, 2, scaling=unbiased);
AutoCorrelation(<1,2,1,2,1,2,1,2>, 2, scaling=biased);
AutoCorrelation(<1,2,1,2,1,2,1,2>, 2, raw);
t := TimeSeriesAnalysis:-TimeSeries([[1,2,1,2,1,2,1,2],[8,7,6,5,4,3,2,1]], header=["Sales", "Profits"], enddate="2012-01-01", frequency="monthly");
t≔Time seriesSales, Profits8 rows of data:2011-06-01 - 2012-01-01
AutoCorrelation(t[.., "Sales"], 2);
Autocorrelation can be used to create correlograms which are useful for detecting periodicity in signals.
R := <seq((1/3*(evalf(sin(17.2*i)*cos(13.8*i)+1.17)+rand(0..1)()*2/3)), i=1..500)>:
Periodicity in a time series can be observed with Autocorrelation.
Data := Import( "datasets/sunspots.csv", base=datadir, output=Matrix );
tsData := TimeSeries( Data[265..310, 2] );
tsData≔Time seriesdata set46 rows of data:1974 - 2019
S := AutoCorrelation(tsData);
The Statistics[AutoCorrelation] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
The Statistics[AutoCorrelation] command was updated in Maple 2015.
The X parameter was updated in Maple 2015.
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