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 $\frac{1}{n-k}$.

raw

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 $R_k=\frac{C_k}{C_0}$ where $C_k=\sum _{t=1}^{n-k}\left(X_t-\mathrm{\mu }\right)\left(X_{t+k}-\mathrm{\mu }\right)$ for $k=0..n-1$ and  $\mathrm{\mu }$ 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] which works strictly in hardware precision, and thus the accuracy of the autocorrelation is not purely determined by the setting of Digits.

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

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

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

$\left[\begin{array}{c}1.\\ \mathrm{-0.875000000009056}\\ 0.750000000020185\\ \mathrm{-0.625000000014873}\\ 0.500000000015000\\ \mathrm{-0.375000000015127}\\ 0.250000000009815\\ \mathrm{-0.125000000020944}\end{array}\right]$

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

$\left[\begin{array}{c}1.\\ \mathrm{-0.875000000009056}\\ 0.750000000020185\end{array}\right]$

$\mathrm{AutoCorrelation}\left(⟨1\,2\,1\,2\,1\,2\,1\,2⟩\,0..2\right)$

$\left[\begin{array}{c}1.\\ \mathrm{-0.875000000009056}\\ 0.750000000020185\end{array}\right]$

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

$\left[\begin{array}{c}\mathrm{-0.875000000009056}\\ 0.750000000020185\end{array}\right]$

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

$\left[\begin{array}{c}1.\\ \mathrm{-1.00000000001035}\\ 1.00000000002691\end{array}\right]$

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

$\left[\begin{array}{c}0.0624999999981250\\ \mathrm{-0.0546874999989254}\\ 0.0468749999998553\end{array}\right]$

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

$\left[\begin{array}{c}0.499999999985000\\ \mathrm{-0.437499999991403}\\ 0.374999999998843\end{array}\right]$

$t≔\mathrm{TimeSeriesAnalysis}:-\mathrm{TimeSeries}\left(\left[\left[1\,2\,1\,2\,1\,2\,1\,2\right]\,\left[8\,7\,6\,5\,4\,3\,2\,1\right]\right]\,\mathrm{header}=\left[''Sales''\,''Profits''\right]\,\mathrm{enddate}=''2012-01-01''\,\mathrm{frequency}=''monthly''\right)$

$t≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{Sales,\; Profits}\\ \mathrm{8\; rows\; of\; data:}\\ \mathrm{2011-06-01\; -\; 2012-01-01}\end{array}\right]$

$\mathrm{AutoCorrelation}\left(t\left[..,''Sales''\right]\,2\right)$

$\left[\begin{array}{c}1.\\ \mathrm{-0.875000000009056}\\ 0.750000000020185\end{array}\right]$

$R≔⟨\mathrm{seq}\left(\frac{1}{3}\left(\mathrm{evalf}\left(\sin \left(17.2i\right)\cos \left(13.8i\right)+1.17\right)+\frac{\mathrm{rand}\left(0..1\right)\left(\right)\cdot 2}{3}\right)\,i=1..500\right)⟩\:$

$\mathrm{LineChart}\left(R\,\mathrm{size}=\left[0.5\,''golden''\right]\right)$

$\mathrm{AutoCorrelationPlot}\left(R\,\mathrm{lags}=100\right)$

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

$\mathrm{Data}≔\mathrm{Import}\left(''datasets/sunspots.csv''\,\mathrm{base}=\mathrm{datadir}\,\mathrm{output}=\mathrm{Matrix}\right)$

$\mathrm{Data}≔\begin{array}{c}\left[\begin{array}{cc}''Date''& ''Mean\; Sunspot\; Number''\\ 1700& 5.0\\ 1701& 11.0\\ 1702& 16.0\\ 1703& 23.0\\ 1704& 36.0\\ 1705& 58.0\\ 1706& 29.0\\ 1707& 20.0\\ 1708& 10.0\\ ⋮& ⋮\end{array}\right]\\ \hfill \text{315 × 2 Matrix}\end{array}$

$\mathrm{tsData}≔\mathrm{TimeSeries}\left(\mathrm{Data}\left[265..310,2\right]\right)$

$\mathrm{tsData}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{data\; set}\\ \mathrm{46\; rows\; of\; data:}\\ \mathrm{1979\; -\; 2024}\end{array}\right]$

$S≔\mathrm{AutoCorrelation}\left(\mathrm{tsData}\right)$

$S≔\begin{array}{c}\left[\begin{array}{c}1.\\ 0.783890092874826\\ 0.351423428724205\\ \mathrm{-0.150321630266540}\\ \mathrm{-0.543639946257113}\\ \mathrm{-0.719092977250328}\\ \mathrm{-0.648508489493646}\\ \mathrm{-0.354789317272298}\\ 0.0388542261132316\\ 0.444115234353966\\ ⋮\end{array}\right]\\ \hfill \text{46 element Vector[column]}\end{array}$

$\mathrm{AutoCorrelationPlot}\left(\mathrm{GetData}\left(\mathrm{tsData}\right)\right)$

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.