FISHERY BULLETIN: VOL. 69, NO. 1 



QUALITY OF FIT 



A superficial inspection of the figures in ap- 

 pendixes B and D shows that the harmonic 

 functions follow the trend of the observed val- 

 ues very well. Closer inspection, however, re- 

 veals that there are cases where the fitted curves 

 depart from the observed trend. An example 

 occun-ed when the Koko Head salinity function 

 (appendix B) for 19.56 fluctuated about the ob- 

 served values from day 145 to day 180. The 

 fluctuations were caused by a data gap between 

 these days. A 15-day data gap is too large when 

 harmonic analysis resolves a period of 1 month. 

 Another example of deviations occurred in the 

 Christmas Island temperature function (appen- 

 dix D) for 1968 between day 240 and day 275. 

 Again, a 30-day data gap is too large when har- 

 monic analysis resolves a period of 19 days. 



These examples illustrate that the sampling 

 interval in harmonic analysis may vary only 

 within limits and that the interval of permissible 

 sampling gaps depends upon the period resolved 

 by the analysis. In cases such as were cited, 

 where the fundamental period of analysis is 

 much longer than the sampling gap, it is pos- 

 sible to constrain the harmonic function by in- 

 serting "dummy" values based on linear inter- 

 polation of the last sample before, and the first 

 sample after, the data gap. 



There are cases where the fitted curve fails 

 to follow the observed trend. When the devi- 

 ations from the fitted curves are relatively large, 

 there is a tendency to reject the observed values 

 during quality control procedures, blaming the 

 deviations on erroneous sampling. Temperature 

 deviations of this type occurred at Koko Head 

 during days 65 to 90 of 1967. First the ob- 

 served temperatures fell to 0.6° C belov/ the 

 fitted curve and then rose abruptly 1.3° to 0.6° C 

 above the fitted curve. Erroneous sampling is 

 ruled out since more than one sample was in- 

 volved in establishing the trend that was abrupt- 

 ly broken and, in addition, the salinity showed 

 similar variability during the same time interval. 

 First the observed salinity rose to 0.15',,, above 

 the expected value and then dropped abruptly 

 0.37';, to 0.16;,, below the expected value. 

 In the Hawaiian region the temperature in- 



creases and the salinity decreases southward. 

 Thus, northward-southward displacements of 

 the water that would result in the observed 

 temperature and salinity changes were the pro- 

 bable cause for the large deviations rather than 

 sampling error. 



In order to assess the quality of fit quantita- 

 tively, we will consider several aspects of the 

 standard error of estimate (root mean square 

 deviations of the observed from the expected 

 values). This statistical parameter is listed in 

 three tables for each function, with harmonic 

 analysis carried out for the fundamental peri- 

 od, the first harmonic, the second harmonic, etc. 

 in = 1,2,3, . . .). Table 1 applies to Koko Head 

 temperatures. Table 2 to Koko Head salinities, 

 and Table 3 to Christmas Island temperatures. 



In each case the listed standard error of esti- 

 mate decreases or reaches a constant value with 

 increasing n. The fit of the function therefore 

 improves or levels off as the analysis is carried 

 out to higher harmonics. Exceptions to this 

 trend occurred in 1956, 1959, and 1961 when 

 the standard errors of estimate for the Koko 

 Head salinity functions (Table 2) increase as 

 the highest n values are reached. Prior to May 

 1961, only four or five samples per month were 

 obtained at Koko Head and therefore the high- 

 est n value permitted by the sampling frequency 

 had been reached. In addition, sampling gaps 

 occurred in 1956, as mentioned before, and in 

 1961 between days 220 and 241. 



The fit of the Koko Head temperature func- 

 tions (Table 1) improves most rapidly during 

 the first few harmonics and with analysis car- 

 ried out to M = 6, the standard error of estimate 

 is near or below 0.3° C. With analysis carried 

 out to 7? = 13, the standard error of estimate 

 is below 0.2° C for all years excepting 1963 and 

 1965-68. 



Greatest improvement of fit for the Koko 

 Head salinity functions (Table 2) does not al- 

 ways occur during the first few harmonics but 

 continues as analysis is carried beyond n = 6. 

 In 1960, for example, the standard error of esti- 

 mate with analysis to n = 1, w = 6, and n = 

 13, is 0.090',,, 0.075'^, and 0.038'rV, respectively. 

 The standard error of estimate at the « value 

 of best fit in Table 2 is below 0.04^. except 



186 



