Partial autocorrelation

Examine the partial autocorrelation function. The high partial autocorrelations will indicate the order of the differencing needed. 

Figure 2 Partial Autocorrelation Function of Potential Data for Urethane Coating...

Lag I Autocorrelation Function IBox/PrcI Partial Autocorrelations X... 

Autocorrelation, Autoregression, Partial Autocorrelation, and Cross-correlation Function... 

Very useful tools for analyzing time series are the autocorrelation function, the crosscorrelation function, and the partial autocorrelation function [SCHLITTGEN and STREITBERG, 1989 DOERFFEL and WUNDRACK, 1986], The interpretation of the patterns of these functions provides the experienced user with substantially more information about the time series than plotting methods. 

First, the autocorrelation function must be computed. In the example plot a strong seasonal effect could be seen in the explanatory variable (nitrate concentration in the feeder stream) as well as in the dependent variable (nitrate concentration in the drinking water reservoir) (Fig. 6-2). The autocorrelation function (Fig. 6-16) has, therefore, the expected exponentially decreasing shape and, because of the seasonal fluctuations, increasing values at x = 12, 24,. .. A better tool for determining the order is the partial autocorrelation function. This function shows the partial correlation between x(t) and x(t - x) and ignores the influences of other variables, e.g. x(t - x + 1). It reveals the order one by the spike at x = 1 in Fig. 6-17. 

Fig. 6-17. Partial autocorrelation function of the nitrate time series at the drinking water reservoir...

The search for the right order of the AR and the MA elements requires the computation of the autocorrelation function and the partial autocorrelation function. Patterns of these functions give hints to the order of these processes. Here are some general rules (Figs. 6-21 to 6-23) ... 

Autoregressive processes have an exponentially decreasing autocorrelation function and one or more spikes in the partial autocorrelation function. The number of spikes in the partial autocorrelation function indicates the order of autoregression. 

Fig. 6-21. Autocorrelation and partial autocorrelation functions for ARIMA(1,0,0) and ARIMA(0,0,1)...

Moving average processes have one or more spikes in the autocorrelation function, the number indicates the order of the moving average process. The partial autocorrelation functions have an exponentially decreasing shape. 

In reality, these functions are more complex and the operator has to use the trial and error mode. Practical criteria which improve the likelihood of correct selection of the parameters of the ARIMA model are the autocorrelation and the partial autocorrelation function of the errors of the resulting ARIMA fit. If they do not have significant spikes the model is satisfactory. 

Next, one calculates the autocorrelation function, ACF, (Fig. 6-16) and the partial autocorrelation function, PACF, (Fig. 6-17) from the original nitrate time series. The autocorrelation function leads to the following conclusions ... 

Fig. 6-28. Partial autocorrelation function of errors from ARIMA(0,0,0)(1,0,0)...

Estimates of the autocorrelation function and the partial autocorrelations for the rotational growth curve in Figure 2 are presented in Figures 3 and 4. The autocorrelation function often depicts dependence of a measurement value on recent previous values whereas the partial autocorrelations show the significance... 

Figure 4. Partial autocorrelation of a rotational growth series.

In partial autocorrelation, the conditioning is performed on all values located between the two variables of interest. This implies that the effect of the intermediate variables is removed from the computed autocorrelation between the two variables. In order to compute this function, it is first necessary to develop the appropriate models for time series analysis. Further information about computing the values are provided in Sect. 5.4.1.2. 

The autocorrelation plot is shown in Fig. 5.2, the partial autocorrelation plot in Fig. 5.3, and the cross-correlation between the mean summer and spring temperatures in Edmonton in Fig. 5.4. For the autocorrelation plot shown in Fig. 5.2, there are some salient features that need to be considered. Firstly, it can be noted that at a lag of zero, the autocorrelation is, as expected, 1. Secondly, it can be seen that all of the autocorrelations are located above the 95% confidence interval for significance. Note that the confidence intervals are equal to 2/a/121 = 0.18. This suggests that all of the observed correlations are significant. Finally, there seems to be a weak, but noticeable, 8-lag oscillation. 

The partial autocorrelation plot for the mean summer temperature in Edmonton, shown in Eig. 5.3, has the same format as the autocorrelation plot. Unlike in the autocorrelation plot, here, there are values located both inside and outside of the confidence region. A similar pattern to that previously observed can be seen here, that is, the values are significant at multiples of some constant. In this case, the significant partial autocorrelation values are located at lags of 1, 2, 3, and 8. This suggests a potential 2-year seasonal component (with values at 2, 4, 6, and 8). 

Fig. 5.3 Partial autocorrelation plot for the mean summer temperature in Edmonton. The thick dashed lines show the 95% confidence intervals for the given data set...

Fig. 5.7 Partial autocorrelation plot for (left) AR(1) and (right) MA(2) processes...

Simulate an integrating process for 2,000 samples and compare it with an AR (1) process with i = —0.98. Compute the sample autocorrelation and partial autocorrelation functions. Compare and suggest ways to distinguish the two cases. The simulation results are shown in Fig. 5.6. The Gaussian noise for both processes is the same (Fig. 5.8). 

The autocorrelation function for an ARMA process can be computed exactly (for details, see Appendix A3 of Shardt 2012a). In general, the determination of the orders can be estimated by examining the autocorrelation and partial autocorrelation plots. In most cases, it is desired to obtain an approximate value for these parameters to be used as an initial estimate for the identification procedure. 

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