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Figure 12.6 Correlogram (autocovariance and autocorrelation; values from Table 12.5) for the artificial series of Fig. 12.2b, after detrending (i.e. periodic signal + noise components only).

A practical problem occurs when there are several periodic signals in a series; this may increase the complexity of the correlogram. Nevertheless, high positive values in a correlogram may generally be interpreted as indicative of the presence of periodic variability in the series. For the numerical example, Fig. 12.6 indicates that there is a major periodicity at k = 20, corresponding to period T = 20; this interpretation is supported by the low value of ryy(10). Period T = 20 is indeed the distance between corresponding peaks or troughs in the series of Fig. 12.2b. Other features of the correlogram would be indicative of additional periods (which is indeed the case here; see Fig. 12.13) or may simply be the result of random noise.

Under the hypothesis of normality, a confidence interval can be computed and drawn on a correlogram in order to identify the values that are significantly different from zero. The confidence interval is usually represented on the correlogram as a two-standard-error band. The test formula for the Pearson correlation coefficient (eq. 4.13) cannot be used here because the data are not independent of one another, being autocorrelated (Section 1.1). According to Bartlett (1946), the variance of each term in the correlogram is a function of all the autocorrelation values in the series:

[ (k)] « - y [p2 (h) + p2 (h - k)p2 (h + k) +2p2 (h)p2 (k) - 4 p2 (h)p2 (k)p2 (h - k)] yy n ¿—t h =

In practice, (a) the series has a finite length, so that the maximum lag is kmax < n/4; (b) what is known are not the parameters p but their estimates r; and (c) the last three terms in the above equation are small. For these reasons, the variance of the correlogram is generally estimated using the following simplified formula, where i (ryy) is the standard error of each autocorrelation coefficient ryy: