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] 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{tsData}≔\left[\begin{array}{c}\mathrm{Time\; series}\\ \mathrm{data\; set}\\ \mathrm{46\; rows\; of\; data:}\\ \mathrm{1975\; -\; 2020}\end{array}\right]$

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

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

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

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.