Partial autocorrelation function

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...

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)...

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). 

Determine the causality and invertibility of this process. Simulate it for 2,000 samples and obtain estimates for the autocorrelation and partial autocorrelation functions. Can the true orders be determined ... 

Estimating the model parameter values is in general performed using one of two methods the method of moments leading to the Yule-Walker equations or the maximum-likelihood method. Although the Yule-Walker equations are simpler, they only provide an efficient estimator for autoregressive models. Also, the Yule-Walker equations are useful for estimating the partial autocorrelation function. Least-squares estimates are also possible, but they are difficult to solve analytically due to the complex nature of the models. 

In order to compute the partial autocorrelation function, assume that the process of interest can be modelled as a r-order autoregressive process. Note that it does not matter what model the true process has. Setup the r-order Yule-WaUcer equation in the form given by Eq. (5.76). It can be shown that the partial autocorrelation of lag t is equal to a, that is, the final parameter that is estimated (Franke et al. 2011)." Practically, rather than computing all the parameters, it is easier to simply compute the final desired value using Cramer s rule, that is,... 

This shows that, as previously mentioned, the partial autocorrelation function can be useful in identifying the order of the autoregressive function. 

The partial autocorrelation function can be estimated by replacing the true autocorrelations with the estimated ones. The statistical properties of the estimated partial autocorrelation function are identical to those of the estimated autocorrelation function, that is. 

The Durbin-Levinson Algorithm can be used to also compute the Yule-Walker parameters and the partial autocorrelation function, since both have a similar matrix form. 

For a causal AR(2) process, derive the autocorrelation and partial autocorrelation function. 

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