Partial Autocorrelation Function (PACF)

When calculating the autocorrelation function for a given signal and delay, this calculation does not take into account the indirect effects on the correlation caused by other lags.

PACF allows us to only calculate the direct effects of a given lag by taking the partial correlation.

We do this by isolating the coefficients for each lag. This is done by writing a regression function along the lines of the following, using an example of \(k=2\), where \(k\) is the lag extent:

\[ x_t = \phi_{21} x_{t-1} + \phi_{22} x_{t-2} + \epsilon_t \]

The coefficient \(\phi_{22}\) quantifies the direct effects of the lag of 2 on the time series. In fact, it is \(\phi_{22}\) is the PACF for this particular case.

Each time we are interested in a new PACF, i.e. for a different lag, we need to build a new regression model including more or less terms, to mirror what is done above.

Eventually, we may collect the PACF for a series of lags, and produce a PACF plot

This plot allows us to observe at a glance which lags have a significant correlation with our time series to warrant an autoregressive model.

Useful youtube video

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