10. 
11. 
12. 
13. 
14. 
15! 
16. 
After an appropriate AR, MA, or ARMA model has been fitted, it is easy to cal- 
culate its spectrum. The spectrum function can be estimated analytically from 
only a few parameters that characterize the fitted model. The spectrum thus ob- 
tained is smooth compared to the spectrum computed by the nonparametric 
method, or by the Fourier transform of the estimated autocorrelation. A sharp— 
peaked spectrum can be obtained by this method, free from the smoothing or 
blurring effects of the spectrum window that are inevitably applied in the 
nonparametric method. 
From the characteristics of the model fitting technique, we can get a fairly reli- 
able and sharp spectrum even from rather short records of observation, when 
reliable autocorrelation functions at large lag numbers are difficult to esumate. 
The AR model is effective in finding the peak frequency and in estimating the 
peak value of the spectrum. Usually the second order of an autoregressive model 
is necessary to identify one peak of the spectrum. 
When the purpose of the analysis is to estimate the characteristics of responses, 
the AR model that estimates the peak effectively is an appropriate approximation 
of the ARMA model. The ARMA model, however, is more reasonable for ex- 
pressing the response of a linear dynamic process to random excitations, as will 
be mentioned later. 
To express the flat valley of a spectrum, however, a large order becomes neces- 
sary if an AR model is to be fitted. A flat valley is more easily expressed by an 
MA model of low order. This tendency comes from the character of the trans- 
form function of the spectrum. 
The ARMA model that expresses peaks by an AR model and flat valleys by an 
MA will be the most appropriate in general cases. 
The model fitting technique relates all linear stationary processes to appropriate 
AR, MA, or ARMA models. All of these models assume that the process is the 
output of a pure random process or of white noise. The linear relations of this 
process to white noise can be derived as the Green’s function of these models. 
When a process is the linear output of some input that is not necessarily white 
but colored, we can get the response of this process to the real white noise if we 
assume that the colored input is the output of the real white noise. In this way we 
can relate all the colored input to a random process. 
Even when the linear response system has some feedback effects, we can fit a 
vector autoregressive process, inverting the input into a pure random process by 
applying an autoregressive process technique to the input, as mentioned in item 
(14). We can get the linear response characteristics of the system from the 
elements of the spectrum matrix of the vector process. In observations of sea- 
keeping data, sometimes the feedback effect is concealed, so this method can be 
applied effectively in the analysis of such data. With the conventional nonpara- 
metric method, the kind of system that has feedback effects is hard to handle. 
A second order autoregressive continuous process A(2), formulated by a second 
order linear differential equation of the damped mass spring vibration type, can 
be transformed into a discrete ARMA(2.1) process formulated by a difference 
equation and by the equivalent correlation theory, the coefficient of the 
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