TM No. 377 



deviation also equal to unity); the sum of these Gaussian distributed 

 variables can be represented as; 



This sum (always a positive quantity) is defined as a chi~square distri» 

 but ion with F degrees of freedom, and is designated "oy DF. Defining the 

 variability of ")(£ by the ratio of the standard deviation to ( the mean value, 

 (both as function of k) this parameter is equivalent to (_ 2 /k)' /i . Thus, as k 

 increases, the variability of "^p 1 becomes less. This applies also to any 

 multiples of ~%p , The variable DF can therefore be used to describe the 

 stability of a given statistical estimate; assuming, of course, that the 

 distribution is approximately Gaussian,, The stability or reliability of 

 the statistical estimate may then be obtained as a function of the number 

 of points in the particular sample „ 



Tukey (19^9) has shown that with certain assumptions, the samples 

 provide values of the spectra function (designated as <j>*(f)) which can 

 be represented as a chi-square distribution for each value of DF. The 

 desirable length of the record may be determined by calculating the DF 

 for each spectral estimate (for DF = 1,2,3, ••• M=l) from the following 

 relations : 



DF = 2 § - r for n = 1,2,3 5 ' » M-l. 



DF = £ - * 

 M k 



(111-10) 



for m ~ o, M. 



n is the number of data points in the record, and M is as previously defined. 

 The larger the value of n, the larger is the value of DFj hence, the precision 

 of the spectral estimate. With a chi-square distribution the gain in the pre- 

 cision of the estimate is great up to values of DF around k-0 to 50. Beyond 50 

 the precision gains become rapidly less. 



An unequivical interpretation of the results requires that the process 

 measured by quasi-stationary, as previously discussed. However, in measuring 

 ocean waves , which are an uncontrolled phenomenon in the field, the apparent 

 gain in precision (resulting from making very long records or, equivalently, 

 from obtaining a very large value lor N in equation (III-10)) is often offset 

 in a complex manner by the changes in the nature of the process being measured. 

 In other words, the longer the sampling continues, the smaller the probability 

 that the record will be stationary. 



Figure III-3 shows the behavior of the chi-square distribution as a func- 

 tion of DF* Within the upper and lower boundary curves, the probability is 

 80$ that the ratio of an obtained estimate to its mean value will fall within 

 the limits defined by the intersection of the line of constant value of DF 

 with the upper and lower limit curves. Likewise, there is a 40$ probability 

 for a given DF that the ratios will fall between' the middle and upper, or 



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