with the noise content of the data, it becomes more understandable that 

 the total io SS reduction by all 13 variables simultaneously was only 

 69.97'^. In actual fact it would be slightly lower than the 69.97 figure 

 if variables X3 and X5, which exhibit considerable "data reduxidancy" 

 with variables X2 and Hh, had been left out of the analysis. Data re- 

 dundancy is brought up again later. 



Discussion 



Table 2 shows that the variable X5, wave steepness, has the greatest 

 influence upon longshore- current velocity, followed by the wave factors of 

 Lo^ T, and Hq. As noted earlier, the fundamental variables of T (X2) and 

 H(x4) could be expected from theoretical (Putman, Munk, and Traylor, 19^9) 

 and laboratory (Brehner and Kamphuis, I963) studies to be among the most- 

 influential in determining longshore- current velocity. Also of expected 

 importance would be beach slope and angle of wave approach, but both of 

 these variables are outranked by wind velocity onshore and offshore when 

 the variables are taken one at a time. 



As stated, when the independent variables are taken one at a time, 

 their relative rank cannot, in general, be determined directly from the 

 degree to which they reduce the sum of squares of the dependent variable^ 

 Some of the variables may be redimdant. That such redundancies are pre- 

 sent in the longshore- current example can be seen by adding the individual 

 reductions in the sum of squares of Xq in table 2. This sum is 135- ^1*^, 

 indicating that some variables, when taken one at a time, show stronger 

 relations than they show when taken in combination with other independent 

 variables. Thus, one may say that there must be, as a minimum, at least 

 35"^ of data redundancy and (or) noise in these observations. 



Before going further it is well to note that variables X3 and X5 

 (table 1) will exhibit a certain amount of artifically introduced data 

 redundancy because they repeat information found in variables X2 and ih. 

 Thus, wave length L, being directly proportional to the square of the 

 wave period T, might be expected to exhibit a closely similar % SS re- 

 duction to T. Because X3 gives a k.hQio SS reduction (table 2) over that 

 for X2, however, the relation between V and T may be non-linear. More 

 will be said of non-linearity later. Variable X5, then, repeats informa- 

 tion found in the "fundamental" variables X2 and ih. It is stronger 

 than either, however, and by combining these two attributes wave steep- 

 ness be comes a useful variable to measure in studies involving prediction 

 of V. Its non-dimensionality may also enter, inasmuch as Hq/Lq is free 

 of any scale factor. 



The next step is to examine the computer output for the strongest 

 combinations of variables when they are taken in combinations of two, 

 three, four, and so on. Table 3 summarizes certain of the strongest 

 combinations for Xs two through nine at a time; it reveals that wave 

 steepness retains its dominant role in all of the strongest combinations. 

 The strongest doublet is not X5 and X3 (Hq/Lq and Lq), as might be ex- 



20 



