with the study of longshore-current velocity. Non-linearity can be detected 

 in several ways^ the simplest perhaps being the examination of scatter dia- 

 grams of each pair of variables in the entire set. Another search method 

 is to run the regression analysis first with the "raw" data and then with 

 all or some Xs transformed to logarithms. In this example we shall use an 

 extension of the linear model to seek for quadratic effects among the 

 variables. 



When the linear regression of Y on some single X is studied; the 

 computational model reduces to the following form: 



Y = Bo + P^X 



In t^rms of regression analysis, this involves the 2x2 matrix, vector-Y, 

 and P-vector discussed in connection with equation (2), and yields the "sum 

 of squares reductions" associated with one X at a time. However, this 

 model can be extended as follows to include higher powers of X: 



Y = Sq + PiX + %^ (3) 



where the coefficient Pp is now associated with the quadratic form of X. 

 The coefficients axe still linear, and hence the same general technique 

 may be used, even, if desirable, to include such powers as x3, X^, etc. 



The procedure used here is first to take each X by itself in terms 

 of Y, to obtain the values in table 2. Then, for each X, its square is 

 also included as in equation (3), to obtain a corresponding sum of squares 

 of Y attributable to the combined linear and quadratic effects of the X. 

 The difference between these two "sum of squares reductions" gives an 

 estimate of the second degree non-linearity associated with each X. This 

 was done with a computer program that computed the linear and quadratic 

 relations between Y and each of the 13 Xs, as well as all interlocks between 

 the Xs themselves. In this latter analysis the order of entry of the Xs 

 is involved, in that the expression: 



XI = Pq + p-]_X2 + ^5X2^ 



is not the same as the expression: 



X2 = Pq + p-]_Xl + PgXl^^ 



even though, as was mentioned earlier, i'xiX2 -^^ ^■'^^ same as ^x2Xl* ^''^^^ 



the complete quadratic output for a problem involving 12 Xs is volimiinous, 



and we shall here emphasize mainly the analysis in equation (3) that contains 

 Y directly. 



Table 2A shows the linear and quadratic counterpart of table 2. It 

 is apparent that in absolute terms, variables X5^ X6, and Xll, representing 

 wave steepness, wind velocity onshore, and water density, have the largest 

 quadratic components. In relative terms, the lower foreshore slope angle. 



28 



