Autocorrelation Plot

It is assumed that the residuals are independent, normally distributed, with mean zero and constant variance. These are standard assumptions for maximum likelihood estimation and can be tested using standard methods examination of histograms, autocorrelation plots (ith residual versus lag-1 residual), univariate analysis with a test for normality, etc. 

Figure 3, Typical Dynamic Light Scattering Autocorrelation Plot.

Appropriate statistical tests such as lag plot or autocorrelation plot should be used, which effectively show the randoirmess of the data residuals [11, 12], When the number of lifetimes is beheved to be known, the mechanism and rate matrix of the process can be estimated and the latter model can be applied. 

Fig. 7 Autocorrelation plot of the two-parameter (dotted line) and three-parameter (solid line) fits of the donor signal of the donor-acceptor pair...

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). 

The cross-correlation plot shown in Eig. 5.4 between the mean summer and spring temperatures has a similar format to the previously considered autocorrelation plot. The confidence interval is, as was previously noted, the same as for the autocorrelation plot. The salient feature is the 4 largest lags at —20, —16, 3, and 4. At this point, it would be useful to comment briefly about the meaning of these values. Since the formula for computing the cross-covariance can be written as or equivalently yt = Xf we can see that positive values correspond to a relationship between past values of x (or in our case, the mean spring temperature)... 

Fig. 5.2 Autocorrelation plot for the mean summer temperature in Edmonton. The thick dashed lines show the 95% confidence intervals for the given data set...

The result follows from the fact that the signal values are independent of each other. This implies that for two values e, and e, t 7 0, the expected value will be zero. This is a very useful property of a white noise signal. The autocorrelation will then be 0 for all ITI > 0 and 1 for t = 0. This implies that a white noise signal will have a single peak on a autocorrelation plot at t = 0. All other values will be zero. A pure, white noise signal is always invertible and causal. 

Consider the same moving-average process as in Example 5.1, which has been simulated for 2,000 samples. Examine the provided autocorrelation plot and compare it with the values obtained previously. The results are shown in Fig. 5.5. 

As expected, the autocorrelation plot has only three significant peaks (at T = 0, 1, and 2). The estimated values are close to the theoretical values. Notice that even though the values for t > 3 should be zero, we see some of them being on the boundary or slightly over. This will always be the case with estimated values, which makes the exact determination of the order slightly more complicated. Nevertheless, for a moving-average process, the autocorrelation plot does allow for the process order to be estimated. 

Fig. 5.5 (Left) Time series plot of the given moving-avtaage process and (right) autocorrelation plot for the same process...

As expected, the autocorrelation plot decays slowly to zero. The first three values are close to the theoretical values of 0.5,0.25, and 0.125. As well, note that the estimated autocorrelation values for large lags are relatively imprecise, since in reality the value could easily be close to zero. For comparison purposes, the partial correlation plot is shown in Fig. 5.7 for both the AR(1) process considered in this example and the MA(1) process considered in Example 5.1. Here it is quite obvious that the autoregressive process has a single spike at a lag of 1, while the moving-average process has at least... 

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

First, it can be noted that the integrating process has a much larger deviation from the mean value than the causal autoregressive process with uneven distribution about the mean. The autocorrelation plot shows that the value for both decays slowly. However, for the autoregressive process, it is much... 

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. 

The simulation results are shown in Figs. 5.9 and 5.10. First, it can be noted that both the autocorrelation and partial autocorrelation plots do not show any clear behaviour or cutoff points. The autocorrelation plot does not decay exponentially to zero rather around a lag of 4, there is some unexpected behaviour. Furthermore, both plots show values that alternate in sign. 

Fig. 5.10 Left) Autocorrelation plot and right) partial autocorrelation plot for the ARMA process...

Simulate the following seasonal processes for 2,000 samples and comment on their autocorrelation and partial autocorrelation plots ... 

Stationarity Testing determine if the data set is stationary, by examining the data set itself, its autocorrelation, and its partial autocorrelation plots. If there is evidence of an integrator, difference the data, and repeat the procedure with the differenced data until the data are stationary. This will give the value of d and D. Note that it may sometimes be necessary to transform the data by applying a nonlinear transformation, for example, y, = log(y,). 

Model Order Determination using the (differenced) data, determine the model orders for the process. Model orders are determined by examining the autocorrelation and partial autocorrelation plots for the data set combined with the information presented in Table 5.1. This will give the values of p, P, q, and Q. 

Consider the previously examined Edmonton temperature data series detailed in Sect. A5.1. For the purposes of this example, consider the problem of estimating a model for the mean summer temperature. The autocorrelation and the partial autocorrelation plots have already been shown and analysed previously (see Figs. 5.2 and 5.3). Using the results from there, obtain an initial model for the data. 

Since it has been shown that the Fourier transform contains the same information as the autocorrelation function, one may wonder what is the advantage of using it. Basically, the Fourier transform provides a different perspective on the same information allowing for different features to be more prominent. In the case of the Fourier transform, the periodicities are made clear, while in the (partial) autocorrelation plots, the different orders are emphasised. 

Fig. 5.21 (Left) Residual analysis for the final temperature model autocorrelation plot of the residuals and (right) normal probability plot of the residuals...

The autocorrelation plot can be used to determine the orders of the autoregressive component. 

The presence of an integrator can be detected by a slowly decaying term in the partial autocorrelation plot. 

If, when examining the autocorrelation plot of the residuals, out of 25 autocorrelations, 2 (including the zero-lag contribution) are located outside the 95% confidence bands, then it can be concluded that the residuals are Gaussian. 

Fig. 6.12 (Top) Autocorrelation plot for the residuals and (bottom) cross-correlation plots between the inputs (left) Ui and (right) U2 and the residuals for the initial linear model...

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