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Statistics

  

AutoCorrelation

  

compute sample autocorrelations of a real Vector

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

AutoCorrelation(X)

AutoCorrelation(X, lags)

Parameters

X

-

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.

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 first entry is 1 when scaling=unbiased or scaling=none.

Description

• 

For a discrete time series X, the AutoCorrelation command computes the autocorrelations Rk=CkC0 where Ck=t=1nkXtμXt+kμ for k=0..n1 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.

Examples

with(Statistics):

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

1.−0.8750000000090560.750000000020185−0.6250000000148730.500000000015000−0.3750000000151270.250000000009815−0.125000000020944

(1)

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

1.−0.8750000000090560.750000000020185

(2)

AutoCorrelation(<1,2,1,2,1,2,1,2>, 0..2);

1.−0.8750000000090560.750000000020185

(3)

AutoCorrelation(<1,2,1,2,1,2,1,2>, 1..2);

−0.8750000000090560.750000000020185

(4)

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

1.−1.000000000010351.00000000002691

(5)

AutoCorrelation(<1,2,1,2,1,2,1,2>, 2, scaling=biased);

0.0624999999981250−0.05468749999892540.0468749999998553

(6)

AutoCorrelation(<1,2,1,2,1,2,1,2>, 2, raw);

0.499999999985000−0.4374999999914030.374999999998843

(7)

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");

tTime seriesSales, Profits8 rows of data:2011-06-01 - 2012-01-01

(8)

AutoCorrelation(t[.., "Sales"], 2);

1.−0.8750000000090560.750000000020185

(9)

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)>:

LineChart(R, size=[0.5,"golden"]);

AutoCorrelationPlot(R, lags=100);

Periodicity in a time series can be observed with Autocorrelation.

with(TimeSeriesAnalysis):

Data := Import( "datasets/sunspots.csv", base=datadir, output=Matrix );

(10)

tsData := TimeSeries( Data[265..310, 2] );

tsDataTime seriesdata set46 rows of data:1974 - 2019

(11)

S := AutoCorrelation(tsData);

(12)

AutoCorrelationPlot(GetData(tsData));

Compatibility

• 

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.

See Also

ColumnGraph

Statistics[Correlogram]

Statistics[CrossCorrelation]

TimeSeriesAnalysis