between Models I and II. The value of r^ using Model I (observed data only) 

 is often referred to as the reduction in variance of the estimator made possible 

 by using the apparent association between X and Y. A value of r^ = 

 indicates that knowledge of the X-values makes no improvement in the prediction 

 of Y and using the mean value of the y's as the estimator would not increase 

 the sum of the squares of the deviations. At the other extreme if r^ = 1, all 

 sample points lie on a sloping straight line implying a strong predictive value. 

 Similarly with Model II, higher r^ values indicate improved fit of the data; 

 but comparing r^ values between Models I and II does not reveal which is 

 correct or even preferable. There is a slight conceptual and a substantial 

 computational difference between the r^ values for the two models. The two 

 values should not be compared; both indicate the relative fit of various data 

 to their own particular model. Either value can be used to measure "goodness 

 of fit" in particular applications; or even to indicate the usefulness of several 

 versions of the particular model chosen. For example comparison of r-values 

 would indicate whether taking logs of the measurements, or raising them to a 

 given power prior to regression, improved the fit. But comparison of the r- 

 value would not be a valid basis for choosing between Models I and II. 



V. EXAMPLES 



The following problems illustrate a frequent need to constrain the regres- 

 sion line in coastal engineering applications. The problems also illustrate 

 the usefulness of r^ to rank different predictors in terms of how well they 

 fit data. Before initially applying the described method to an actual problem, 

 it may be helpful to reanalyze one of the small data sets used in these examples 

 and compare the results with those published in this report. 



**************** EXAMPLE PROBLEM 1*************** 



Consider the requirement to simulate a long-term history of wave-induced 

 longshore currents for a particular coastal site. Assume hindcasted wave data 

 are available, but that current measurements were not made over the period of 

 interest. According to the Shore Protection Manual (U.S. Army, Corps of 

 Engineers, Coastal Engineering Research Center, 1977), the longshore current 

 (v) can be calculated as a function of the beach slope (m) , the gravitational 

 acceleration (g) , and the angle and height of breaking waves (a^^, H^,, 

 respectively) . 



V = 20.7 m (gHt,)^/^ sin 2c(b (1) 



The coefficient of proportionality (20.7) is based on typical mixing and fric- 

 tional factors for the surf zone. Empirical formulas, like equation (1) can be 

 adjusted by regression analysis of test data from the specific site of intended 

 application. This will customize the formula to fit site-sensitive conditions. 

 The longshore velocity also varies laterally within the surf zone. The problem 

 of estimating the spatial structure of flow across the surf zone may be avoided 

 by obtaining current measurements at the exact point where the long-term flow 

 must be reconstructed, then regressing the test measurements against simul- 

 taneously determined breaker conditions. Steps in such an analysis are given 

 below. Only a few data points are used in the example to encourage the reader 

 to go through the computations and check the results. The data are taken from 

 a frequently referenced field study done at Nags Head, North Carolina (Galvin 

 and Savage, 1966) . 



15 



