REGRESSION AND AUTOCORRELATION ANALYSES 



A visual observation of the data suggests statistically 

 fitting some theoretical function which oscillates with a 

 period of one year. Further justification is provided by the 

 autocorrelation function 



a =COV(T.T. . )/VAR(T.), for lags h = 0, 1, 2, (1) 



In this application the variable T. is the sea-surface tem- 

 perature on day i, I^^is the temperature h days later, 

 and COV and VAR are the covariance and variance of the 

 variables as indicated. Computation of the autocorrelation 

 function yields peaks whose magnitudes and spacings 

 strongly indicate the existence of an annual oscillation in 

 the time series. 



The simplest model consisting of an oscillatory 

 function with period one year is 



T' = |S +asin [2ir {D - 8)/365 ] + e (2A) 



o 



= O + jS^in (277^/365) + 2 cos (27^/365) + € (2B) 



where D is time measured in days from some arbitrary 

 origin and T' is the fitted value of the surface temperature. 

 Fitting the function of equation (2B) to the observed sur- 

 face temperatures T using the method of least squares 

 yields estimates of the regression coefficients j3 , fi x and 

 j8 and an estimate of the variance of e. The amplitude a 

 and phase can be obtained from S and /3 3 . The quantity 

 € is the random, or error, or residual term. 



If the residuals T- T' are examined visually or by 

 computation of the autocorrelation function of the residual 

 time series, a fairly strong semiannual oscillation is dis- 

 covered for some of the stations. This suggests the model 



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