PART I 
MODEL FITTING TECHNIQUES 
(PARAMETRIC SPECTRAL ANALYSIS) 
CHAPTER 5 
DISCRETE MODEL, MODEL FITTING, 
CORRELATION AND SPECTRUM FUNCTIONS 
5.1. INTRODUCTION 
In Part I, the procedures for estimating the spectrum nonparametnically through 
autovariance from a sample observation were summarized and discussed. We noted that 
the statistical consideration of each step of the computations are important in getting 
reliable results, and the author suggested several ways to improve reliability. In Part I, 
another approach to getting reliable results from a single or a short observation of a 
process will be discussed. The method introduced here is called a parametric approach, 
because some type of model is fitted to the sample observation, and then the parameters 
of the model are estimated statistically. In looking for the models, we can use the knowl- 
edge that we already have about the process, such as the degrees of freedom, and the 
physical characteristics of the equations of motion that govern the behavior of the system 
or about the inputs. A few methods which use the parametric approach, such as the 
maximum likelihood method (MLM) and the maximum entropy method (MEM) are 
essentially the same with the one mentioned here, and will be explained later. 
In this analysis, the criteria for deciding the fitness of the model are very important. 
The method introduced here, called the “‘“MAIC Method,” was introduced by Dr. H. 
Akaike of Japan and provides a powerful guide in finding the properly fitted model, prop- 
er from a statistical point of view. In this analysis, the time domain expression of this 
process, the time histories themselves, and the correlation of the processes play a big role 
as was pointed out in Sections 1.3 and 2.5.8 and in the conclusion to Part I. 
This parametric approach is not yet well known in the field of naval architecture, 
although a few books by Priestley,2> Pandit and Wu,*° Box and Jenkins,?’ and others*® 
deal with it in part or in depth. For this reason Sections 5.2.1 to 5.2.5 give several statisti- 
cal models of stationary time series in some detail with simulated examples by this 
author. Then criteria for choosing the model, estimating the parameters, and deriving 
the spectrum will be given at the end of Chapter 5. 
Chapter 6 discusses the application of this method to a two—variate process, an 
input/output system, and the usefulness of this method for the analysis of a response 
system with feedback is shown. 
In Chapter 7, examples of the application of this method to the analysis of seakeep- 
ing data and a comparison with analysis by the nonparametric method are shown. 
Finally, Chapter 8 provides conclusions and summaries for Part II. 
OF, 
