Cross-correlation plot

Goal Given 2 data series yt and zt return the corresponding cross-correlation plot for 20 lags. 

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Figure 3. Cross correlation plots between PM2.5 levels at three stations.

Fig. 1.12 Complex data visualisation example a cross-correlation plot...

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)... 

For both the prediction error and the Unear, least-squares methods, the value of the time delay must be known beforehand, that is, neither of the methods can estimate the time delay as part of the regression problem. Estimating the time delay can be performed using various different methods. The most common include using the values obtained from the step tests and the cross-correlation plot between the inputs and outputs. 

Another approach to estimating the time delay is to use the cross-correlation plot between the input and outputs. The delay will then appear as a series of zero values between a lag of 0 and the time delay, %. A typical cross-correlation plot is shown in Fig. 6.5 (left). In this plot, the time delay would be estimated as being 4, since that is the last nonzero value before the significant peak. Note that using this approach requires that the input be a white noise signal. 

Fig. 6.5 Estimating the time delay using (left) the cross-correlation plot and (right) the impulse response method...

The auto- and cross-correlation plots are shown in Fig. 6.12. A comparison between the predicted and actual levels is shown in Fig. 6.13. Both figures use the validation data set for testing the model. From Fig. 6.12, it is clear that the residuals are not uncorrelated with each other or the inputs. Therefore, the initial model needs to be improved. Since there is a suggestion that the process model is incorrectly specified, it will first be changed. The best approach is to increase the order of the numerator and denominator (of the B- and F-polynomials) until either the cross-correlation plot shows the desired behaviour or the confidence intervals for the parameters cover zero. If the second case is reached, then this could be a suggestion that a linear model is insufficient/inappropriate for the given data set. Furthermore, the fit between the predicted and measured levels is not great (55.4%). 

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...

If the cross-correlation plot shows many nonzero correlations, then the likely problem is a misspecified process model. 

Fig. 6.18 Estimating time delay (left) cross-correlation plot and (right) impulse response coefficients...

Top) Autocorrelation plot for the residuals and bottom) cross-correlation plots between the inputs left) Ui... 

Suppose, a liquid flows through a pipeline and a valve position is changed at the beginning of the pipeline. If the temperature is measured at the end of the pipeline, the time lag, which is equal to the transportation time of the liquid from valve to measurement location, can easily be calculated. Figure 21.4 shows a typical cross correlation plot between a process input and a process output. 

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