Partial autocorrelation function: Difference between revisions

Revision as of 03:30, 12 July 2022

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==Definition==

Given a time series <math>z_t</math>, the partial autocorrelation of lag <math>k</math>, denoted <math>\phi_{k,k}</math>, is the [[autocorrelation]] between <math>z_t</math> and <math>z_{t+k}</math> with the linear dependence of <math>z_t</math> on <math>z_{t+1}</math> through <math>z_{t+k-1}</math> removed. Equivalently, it is the autocorrelation between <math>z_t</math> and <math>z_{t+k}</math> that is not accounted for by lags <math>1</math> through <math>k-1</math>, inclusive.<math display="block">\phi_{1,1} = \operatorname{corr}(z_{t+1}, z_{t}),\text{ for }k= 1,</math><math display="block">\phi_{k,k} = \operatorname{corr}(z_{t+k} - \hat{z}_{t+k},\, z_{t} - \hat{z}_{t}),\text{ for }k\geq 2,</math>where <math>\hat{z}_{t+k}</math> =and <math>\beta_1 z_hat{t+k-1} + \beta_2 z_{t+k-2} + ... + \beta_{k-1} z_{t+1z}_t</math> is theare [[linearLinear combination|linear combinations]] of <math>\{z_{t+k-1}, z_{t+k-2}, ..., z_{t+k-1}\}</math> that minimizesminimize the [[mean squared error]], <math>\Epsilon[z_{t+k}of - \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 minimizingand <math>\Epsilon[z_t - \hat{z}_t]^2</math> respectively. For [[Stationary process|stationary processes]], the coefficients <math>\beta_1,hat{z}_{t+k}</math> \beta_2, ...,and <math>\beta_hat{k-1z} _t</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>

== Calculation ==