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Fig. 5.1. Theoretical autocovariance of a pure random process. 
Fig. 5.2. Theoretical spectrum of a pure random process. 
52.1.1 Example of a Pure Random Process. Figure 5.3 shows an example of a 
pure random Gaussian process generated fort = 1 — 600, with average 0 and variance 
1.0. At the bottom, a part (t = 100 — 250) of the record is shown expanded on the time 
axis. Its readings are listed in Appendix Al as Table Al.1; pp. 251, 252, and 253, for 
reference. This process was generated by the method of “Random number generation and 
testing,” generally popular as “‘multiplication type residual method.” 
It was found that by this method it is hard to get a really white process with precise- 
ly designed variance of 02 = 1.0 from this short record (N = 600), and the sample 
variance appeared as G2 = 1.046490. Figure 5.4a shows the theoretical autocorrelation 
coefficient 9(0) = R(0)/R(0) = 1, o(r) = R(r)/R(O) = 0, r = 0 by Eq. 5.5. Figure 5.4b 
shows the estimated Q(r) = R(r) /R(0) from the generated process. From the model fitting 
technique and order determination that will be mentioned in Section 5.5, the order ap- 
peared to be 0, and AR(0) appeared to be the most appropriate model to fit this process. 
The estimated autocorrelation of this model fitted by Eq. 5.4 is just the same with the 
theoretical o(r) shown as Fig. 5.4a because the difference is only ing, values, that its 
drawing was omitted. Figure 5.5a shows the theoretical spectrum function of this process 
designed by Eq. 5.6, that is, the white noise. Figure 5.5b shows the estimated spectrum of 
the model fitted by Eq. 5.6. This is also the white noise and looks very similar to the 
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