CHAPTER 15 
CONCLUSIONS FOR PART I THROUGH PART Ii 
Part I of this lecture note summarized the conventional procedures for analyzing the 
irregular time histories of observed data, like irregular ocean waves, and the responses of 
marine vehicles and structures on the ocean, that can be treated as weakly stationary er- 
godic stochastic processes. In these analyses, the so-called Wiener’s general harmonic 
analysis technique plays a large role, and the correlation and spectrum functions of the 
processes were important and included much information on the characteristics of the 
processes. When the responses of some dynamic systems to random excitation are to be 
treated, the cross relations of the input (excitation) and the output (response), i.e., the 
cross correlations and cross spectra, are very important. 
The theories of the analysis are complete and rather beautiful. However, in sample 
computations from practical data, that is, from discrete data sampled at some time inter- 
val of finite length, many statistical considerations are necessary in estimating the 
correlation and spectrum functions to get statistically reliable results. After the general 
procedures were summarized, the discussion concentrated on that point. In these proce- 
dures, this author made several suggestions for improving the reliability (actually the 
coherencies) and these results were summarized. This author stressed that we should pay 
more attention to the time—domain characteristics of the functions. For example, the cor- 
relogram, that is the diagram of the correlation function and is a function in the time 
domain, deserves as much attention as is now paid to the spectrum. 
Part II is concerned with the parametric analysis of a stationary process, which is 
really an alternative or more modern method for analyzing the sample random process. 
This author believes that the characteristics of the function in the time domain play bigger 
roles in this method than in the nonparametric conventional method, discussed in Part I. 
The parametric method fits a statistical model to an observed process and estimates 
the parameters from the observed data. In Part II, the autoregressive (AR) models, mov- 
ing average (MA) models, and mixed autoregressive moving average (ARMA) models 
were introduced. Since these models and the parametric approach are not familiar to most 
engineers, especially in the field of naval architecture, the author explained them in con- 
siderable detail. (He is afraid it was in too much detail.) 
Usually a moderate or rather low order finite AR(n), MA(m), or ARMA(n,m) model 
can be fitted to represent adequately most of the observed processes. The optimum order 
of these fitted models, n or m, can be estimated by a method called Akaike’s information 
criteria (AIC). This method is based on information theory and enables one to choose the 
order n or m that maximizes the statistical entropy of the estimate. It is therefore called 
the ‘Maximum Entropy Method,” although it is different from a similar method already 
published under the same name. The AIC method can give the optimum order to be 
adopted. In Part II, in explaining the characteristics of these models, this author generated 
several fundamental models, the AR(0) (pure random), AR(1), AR(2), ARMA(2.1), 
MA(2), MA(1), and ARMA(2.2) models, by simulation and analyzed them by the para- 
metric method. The orders were estimated by AIC and, for almost all processes, we 
succeeded to fit models that coincided with the models used to generate the simulated 
process. In these demonstrations, the simulated processes were also analyzed by the 
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