using these a; . . . a,. One—step—ahead predictions for rolling and swaying and a 
10— steps—ahead prediction for swaying are shown here. 
7.2 EXAMPLES OF MODEL FITTING TECHNIQUE APPLIED 
TO THE SEAKEEPING DATA 
Figure 7.2*° shows examples of spectra of model seakeeping data obtained in a 
model basin for a model of a ship. On the left-hand side (a), the spectra calculated by the 
nonparametric method, the so—called Blackman—Tukey method, are shown and on the 
right-hand side (b), the same spectra calculated by the parametric method, i.e., the AR(n) 
model fitting technique. The spectra for wave height, heaving, and relative wave height 
are shown on a logarithmic scale, and the Nyquist frequency of 2.50 shows 
At = (1/2.50 X 2) = 0.2 sec. The total number of observations is N = 225, which is 
rather small, and the maximum lag number for the B—T method is 50, which is large com- 
pared to N. We can find the spectrum from the AR model, the order of each process being 
shown by the AR number in each figure. The spectra are very smooth and the peaks of 
the spectra are sharply defined even from these short records (small number of observa- 
tions). 
Figure 7.3% shows the behavior of AIC values that were used to find the order n of 
the AR models fitted to each set of these seakeeping data. From this figure, we found n to 
be 11, 9, and 10 for wave height, heave, and relative wave height, respectively. These 
values give the minimum AIC. 
Figure 7.4 shows the time series of an AR(2) model simulated by the difference 
equation, 
X- 0.5X,-1 + 0.7X,_2 = Ey 
when a, =—0.5, @2 = 0.7 in Eq. 5.59 for a general AR(2) model. There, €, is white noise 
with Gaussian distribution N(0, 1), i.e., with 0 mean and a variance 1. The time series are 
also shown in the same figure. The number of observations is N=1,000 for both time 
series €, and X;,. The theoretical spectrum of this AR(2) model is shown at the top of Fig. 
7.5.52 An AR(n) model was fitted to the simulated process, and from AIC criteria, the 
optimum n was estimated as n=2. The coefficients a), a2, and the variance ad of €; were 
estimated by the method described in Section 5.2.3, and the spectrum s(q@) was calculated 
using these parameters, as shown at the bottom of Fig. 7.5. Independent of this order n=2 
by minimum AIC criteria, spectra were also estimated for higher n; A(10) and AR(20) 
models were fitted and their spectra were obtained as shown in the same Fig. 7.5. 
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