Partial autocorrelation function

Partial autocorrelation of the time series of the depths of the Huron Sea

The partial autocorrelation function (PAKF, English PACF) is like the autocovariance function and the autocorrelation function an instrument to identify dependencies between the values ​​of a time series at different times. The PAKF measures the linear relationship between and by eliminating the influence of the variables in between. ${\displaystyle Y_{t}}$ ${\displaystyle Y_{t+k}}$

In the case of autocorrelated stationary processes, the observations to contain information about the expected amount and sign of the size (with ). The partial autocorrelation then expresses the additional information about the expression of , which is obtained if one also knows the state of the process at the time . ${\displaystyle Y_{t}}$ ${\displaystyle Y_{T-1}}$ ${\displaystyle Y_{T}}$ ${\displaystyle t<T\in \mathbb {N} }$ ${\displaystyle Y_{T}}$ ${\displaystyle Y_{t-1}}$ ${\displaystyle t-1}$

The formal definition is for centered stationary time series

${\displaystyle \varphi _{kk}:=\operatorname {Corr} (Y_{t},Y_{t-k}|Y_{t-1},\cdots ,Y_{t-k+1})\quad k=0,\pm 1,\pm 2,\cdots }$

The operation describes the conditional correlation , formed with the conditional expectation values and conditional variances ${\displaystyle \operatorname {Corr} (\cdot ,\cdot |\cdot )}$

${\displaystyle \operatorname {Corr} (W,Z|{\mathcal {F}}_{t}):={\frac {E(WZ|{\mathcal {F}}_{t})-E(W|{\mathcal {F}}_{t})E(Z|{\mathcal {F}}_{t})}{\sqrt {\operatorname {Var} (W|{\mathcal {F}}_{t})\operatorname {Var} (Z|{\mathcal {F}}_{t})}}}.}$

The function is symmetric in and its values ​​lie in the interval . It applies . ${\displaystyle k}$ ${\displaystyle [-1,1]}$ ${\displaystyle \varphi _{00}=1}$

There are various methods of determining the PAHF:

• Yule-Walker equations (after George Udny Yule ),
• Durbin-Levinson algorithm .

The latter method works recursively . It can also be used to calculate an empirical PAKF (estimated PAKF). An approximation of the standard deviation of the empirical PAKF is possible with the Quenouille approximation :

${\displaystyle {\hat {\sigma }}({\hat {\varphi }}_{kk})\approx {\frac {1}{\sqrt {T}}}}$.

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• Box, GEP, Jenkins, GM, and Reinsel, GC (1994). Time Series Analysis, Forecasting and Control , 3rd ed. Prentice Hall, Englewood Clifs, NJ.
• Brockwell, Peter J. and Davis, Richard A. (1987). Time Series: Theory and Methods , Springer-Verlang.
• Rinne H. (2003). Pocket book of statistics , published by Harri Deutsch.