AUTOCORRELATION ANALYSIS 



A visual observation of the data suggests statistically fitting some 

 theoretical function which oscillates with period 1 year. Further justification is 

 provided by the autocorrelation function: 



C^ = GOV {TiTi^ k) / [VAR (T,) VAR (T,+ fe)] '''^ for lags fe = 0, 1, 2,... 



The variable T, is the sea-surface temperature on day i, T,- _^ ^ is the tempera- 

 ture k days later, and GOV and VAR are the covariance and variance of the 

 variables as indicated. 



Figures 3, 4, and 5 present the results of an autocorrelation analysis for 

 the six time-series. The upper figure for each station is the autocorrelation 

 function of the daily temperatures. The peaks in the autocorrelation functions 

 have magnitudes and spacings indicating so strongly the existence of an annual 

 oscillation in the time-series that any statistical test of hypothesis is super- 

 fluous. 



The middle set of figures presents the autocorrelations of the residuals 

 (or anomalies) after removing the 12-month oscillatory terms from the original 

 data (discussed in the next section). An obvious feature of these figures is the 

 peaks and troughs in the autocorrelation function at intervals of 6 months for 

 PAPA, EGHO, Cape St. James, and Triple Island, and the lack of this oscilla- 

 tion for Langara Island and Scripps Pier. 



The lower set of figures presents the autocorrelation of the residuals 

 after the annual and semiannual oscillatory terms are removed. The autocorre- 

 lation function has a form typical for such residuals, decreasing as a negative 

 exponential function for small lags. 



The autocorrelation function for Scripps Pier was computed out to a lag 

 of 1800 days, an arbitrary figure slightly over 10 percent of the total sample 

 length. Since conclusions based only on sampling variability of the autocorrela- 

 tion function must be avoided, values of the function significantly different from 

 zero at some probability level are of primary interest. If the standard deviation 

 of the autocorrelation coefficients were known, and normality assumptions made, 

 then significance levels could be determined. The Scripps Pier autocorrelation 

 coefficients with lags from 400 to 1800 days provide an estimate of a^ , the 

 standard deviation of the nonsignificant correlations. The estimate is based on 

 a large sample of the autocorrelation coefficients, and the maximum lag involved 

 is still only a small fraction of the total time-series length. The PAPA and 

 ECHO series are much too short to supply such an estimate, and the records for 

 the other stations are of marginal length for this purpose. 



This standard deviation is ac = 0.0293. The 95 percent significance levels for 

 Scripps Pier are ± 1.96x0.0293 = ±0.0574, for a nominal series length of 13140. 



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