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Correlation Analysis
Figure 10.6 Autocorrelation function of a basic TES process with (L, R) ¼ ( 0:05, 0:05).
or counterclockwise? In that case the circular distance remains small (not exceeding 0.01) but the linear distance can be arbitrarily close to 1. In other words, the sample path when plotted on the real line would have an abrupt “jump.” Furthermore, since both F and its inverse, F 1 , are monotone nondecreasing (this is a property inherent in distribution functions), these abrupt “jumps” are inherited by the foreground TES process. Consequently, although the marginal distribution and autocorrelation function of the resultant foreground process may well fit their empirical counterparts, the model's sample paths may be qualitatively different from the empirical data. The foregoing discussion motivates the need for a mathematical means of “smoothing” the sample paths of background TES processes while simultaneously retaining their uniform distribution. Fortunately there is a simple way to achieve both goals, using a so-called stitching transformation, which maps the interval [0, 1] onto itself. The family of stitching transformations is defined by 8u > if 0 u x < , x (10:9) Sx (u) ¼ > : 1 u , if x u 1 1 x