Partial autocorrelation function

's depth with confidence interval (in blue, plotted around 0) In time series analysis, the partial autocorrelation function (PACF) gives the partial correlation of a stationary time series with its own lagged values, regressed the values of the time series at all shorter lags. It contrasts with the autocorrelation function, which does not control for other lags. This function plays an important role in data analysis aimed at identifying the extent of the lag in an autoregressive (AR) model. The use of this function was introduced as part of the Box–Jenkins approach to time series modelling, whereby plotting the partial autocorrelative functions one could determine the appropriate lags p in an AR (p) model or in an extended ARIMA (p,d,q) model.

's depth with confidence interval (in blue, plotted around 0) In time series analysis, the partial autocorrelation function (PACF) gives the partial correlation of a stationary time series with its own lagged values, regressed the values of the time series at all shorter lags. It contrasts with the autocorrelation function, which does not control for other lags. This function plays an important role in data analysis aimed at identifying the extent of the lag in an autoregressive (AR) model. The use of this function was introduced as part of the Box–Jenkins approach to time series modelling, whereby plotting the partial autocorrelative functions one could determine the appropriate lags p in an AR (p) model or in an extended ARIMA (p,d,q) model. ==Definition== Given a time series z_t , the partial autocorrelation of lag k , denoted \phi_{k,k} , is the autocorrelation between z_t and z_{t+k} with the linear dependence of z_t on z_{t+1} through z_{t+k-1} removed. Equivalently, it is the autocorrelation between z_t and z_{t+k} that is not accounted for by lags 1 through k-1 , inclusive. \phi_{1,1} = \operatorname{corr}(z_{t+1}, z_{t}),\text{ for }k= 1, \phi_{k,k} = \operatorname{corr}(z_{t+k} - \hat{z}_{t+k},\, z_{t} - \hat{z}_{t}),\text{ for }k\geq 2, where \hat{z}_{t+k} and \hat{z}_t are linear combinations of \{z_{t+1}, z_{t+2}, ..., z_{t+k-1}\} that minimize the mean squared error of z_{t+k} and z_t respectively. For stationary processes, the coefficients in \hat{z}_{t+k} and \hat{z}_t are the same, but reversed: \hat{z}_{t+k}=\beta_1z_{t+k-1}+\cdots+\beta_{k-1}z_{t+1}\qquad\text{and}\qquad\hat{z}_t=\beta_1z_{t+1}+\cdots+\beta_{k-1}z_{t+k-1}. The theoretical partial autocorrelation function of a stationary time series can be calculated by using the Durbin–Levinson Algorithm: \phi_{n,n} = \frac{\rho(n) - \sum_{k=1}^{n-1} \phi_{n-1, k} \rho(n - k)}{1 - \sum_{k=1}^{n-1} \phi_{n-1, k} \rho(k) } where \phi_{n,k} = \phi_{n-1, k} - \phi_{n,n} \phi_{n-1,n-k} for 1 \leq k \leq n - 1 and \rho(n) is the autocorrelation function. The formula above can be used with sample autocorrelations to find the sample partial autocorrelation function of any given time series. ==References==

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