Partial autocorrelation function: Difference between revisions

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: <math>\alpha(1) = \operatorname{corr}(z_{t+1}, z_{t}),\text{ for }k= 1,</math>

: <math>\alpha(k) = \operatorname{corr}(z_{t+k} - P_\hat{t,kz}(z__{t+k}),\, z_{t} - P_\hat{t,kz}(z__{t})),\text{ for }k\geq 2,.</math>

where <math>\hat{z}_{t+k} = \beta_1 z_{t+k-1} + \beta_2 z_{t+k-2} + ... + \beta_{k-1} z_{t+1}</math> is the [[linear combination]] of <math>\{z_{t+k-1}, z_{t+k-2}, ..., z_{t+1}\}</math> that minimizes the [[mean squared error]], <math>\Epsilon[z_{t+k} - \hat{z}_{t+k}]^2</math>. Similarly, <math>\hat{z}_t = \beta_1 z_{t+1} + \beta_2 z_{t+2} + ... + \beta_{k-1} z_{t+k-1} </math> is a linear combination minimizing <math>\Epsilon[z_t - \hat{z}_t]^2</math>. For [[Stationary process|stationary processes]], the coefficients <math>\beta_1, \beta_2, ..., \beta_{k-1} </math> are the same.<ref>{{Cite book |last=Shumway |first=Robert H. |url=http://link.springer.com/10.1007/978-3-319-52452-8 |title=Time Series Analysis and Its Applications: With R Examples |last2=Stoffer |first2=David S. |date=2017 |publisher=Springer International Publishing |isbn=978-3-319-52451-1 |series=Springer Texts in Statistics |___location=Cham |pages=97-98 |language=en |doi=10.1007/978-3-319-52452-8}}</ref>

where <math>P_{t,k}(x)</math> is the surjective operator of orthogonal projection of <math>x</math> onto the linear subspace of Hilbert space spanned by <math> z_{t+1}, \dots, z_{t+k-1}</math>.

There are algorithms for estimating the partial autocorrelation based on the sample autocorrelations.<ref name=":0">{{Cite book |last=Box |first=George E. P. |title=Time Series Analysis: Forecasting and Control |last2=Reinsel |first2=Gregory C. |last3=Jenkins |first3=Gwilym M. |publisher=John Wiley |year=2008 |isbn=9780470272848 |edition=4th |___location=Hoboken, New Jersey |language=en}}</ref><ref>{{Cite book |last=Brockwell |first=Peter J. |title=Time Series: Theory and Methods |last2=Davis |first2=Richard A. |publisher=Springer |year=1991 |isbn=9781441903198 |edition=2nd |___location=New York, NY |language=en}}</ref> These algorithms derive from the exact theoretical relation between the partial autocorrelation function and the autocorrelation function.