TM No* 377 



range of frequency). Spectral analysis is associated directly with the 

 auto-covarianee and the covariance functions, which themselves provide much 

 information about the original time series (see Lee^ 19o0) Another impor- 

 tant factor is that confidence measure estimates or error can he determined 

 for the computed spectra by use of standard statistical parameters c 



It may be helpful to review briefly the basic assumptions and definitions 

 needed to understand the application of the pertinent statistical parameters 

 to time series data There are several extensive reviews of the methods used 

 in the so-called Tukey spectral estimates, including Blackman and Tukey (1958)^ 

 and Ellis and Collins (1964;. 



Basic Assumptions for Time Series Analysis — The time series data under 

 analysis were assumed to be representative of a random, or mixed process A 

 random (or stochastic) process is an ensemble of time functions given by 

 X^t)* where n = l s 2s,3s°<>° a -^ -&>* t <oo • so that the ensemble can be char- 

 acterized through its statistical properties „ A mixed process is a random 

 process containing time=periodic constituents e The mixed process is probably 

 quite well suited to 'describing the motions of wind waves and, in fact,, many 

 scales of oceanic flow / word of explanation may be required here regarding 

 the use of the term, ''ensemble :i o According to Kinsman (1965); this term can 

 denote an infinite or a finite collection of records of variables (associated. 

 with the same property) that are governed by identical phenomena and evaluated 

 for all timec In this sense o for simplicity ^ the ensemble is the collection 

 of time series measurements oi waves rie.de at different spatial positions in 

 the ocean t 



The classical theory for the analysis of time series data is based upon 

 three assumptions 3 that the process is stationary* ergodie s and of infinite 

 extento This latter requisite is, of coarse., experimentally inconceivable ; 

 however s it is important from a heuristic point of view to approximate this 

 as sumption o 



"Stationary" may be defined as the property of a time series in which 

 the probability of any particular event occurring during the series record 

 is constant,, To illustrate s let-Xp'be a sample at a particular time inter- 

 val P e If the statistical properties of ><p are the same as those of X^+a. , 

 where <a is any other sampled interval., then the record is stationary.. Ideally ^ 

 a system should be stationary daring the time interval of the samplings The 

 only factual justification for assuming stafeionarity, other than by an in- 

 tuitive inference from visual observations, is by an assessment of the vari- 

 ation of statistical properties at discreet intervals in the records 6 For 

 instance 5 the first and last 10 percent of a. particular wave record could 

 be analyzed for its auto«spectra Similarities in the two spectra would 

 thus be indicative of the degree of stationarity. 



The ergodic hypothesis is also a very important assumption related to the 

 study of time series* This theorem states that the ensemble average (i.=e»j 

 the average of the process over both a time and space domain from - 00 to -f-°° ) 

 may be replaced with unit probability by the time average of a single series 



49 



