nSHERY BULLETIN: VOL 76. NO. 2 



which the regions are described and the inherent 

 separation between them. 



These regions are described mathematically by 

 multivariate probability density functions. 

 Fisher's il936i linear discriminant function 

 defines a linear decision surface ih>-perplanei de- 

 rived by describing these regions as multivariate 

 normal density distributions with common 

 variance-covariance matrices i Welch 1939 ». 

 Quadratic discriminant functions have been de- 

 veloped ' Smith 1947 '. The resulting decision sur- 

 faces are nonlinear. The quadratic discriminant 

 function does not require common variance- 

 covariance matrices. Anas and Murai > 1969 > com- 

 pared the classificatory abilities of the linear and 

 quadratic discriminant functions. They found ' in 

 agreement with Isaacson 1954' that even if the 

 assumption that the distributions have common 

 variance-covariance matrices is violated, the 

 linear discriminamt function would still give good 

 results for large sample sizes. But the quadratic 

 function gave slightly bener results. 



All investigators utilizing discriminant 

 analyses to separate races of Pacific salmon have 

 assumed that the density distributions of mea- 

 surements from a particular class of salmon were 

 multivariate normal. The frequency distributions 

 of scale characters in Major et al. ' 1975 > show that 

 multimodal and skewed distributions occilt for 

 chinook salmon scale characters even in the uni- 

 variate case. In many other cases, the underh-ing 

 distribution functions may be non-Gaussian. Dis- 

 criminant functions based upon non-Gaussian dis- 

 tributions or obtained by distribution-free 

 methods are preferable to those based upon an 

 unrealized assumption of normality. 



Nearly all of the discriminant function analyses 

 used in the investigations of Pacific salmon have 

 been two-class analyses designed to determine the 

 continent of origin of salmon taken on the high 

 seas. For the two-class situation only one discrim- 

 inant function need be calculated. These two-class 

 problems are a special case of the many-class prob- 

 lems in which a separate discriminant function is 

 calculated for each class. Bilton and Messinger 

 <1975> calculated discriminant functions for each 

 of several runs in a classification study on sockeye 

 salmon. If several stocks of salmon intermingle 

 and are to be classified, analyses of this t>-pe are 

 needed. 



Spechts 1966' polynomial discriminant 

 method does not require that the underhing den- 

 sity distributions be multivariate normal nor that 



they have common variance-covariance matrices. 

 Since this method is nonparametric. various scale 

 characteristics may be used for discrimination 

 with no particular regard to the underlying dis- 

 tributions. Thus, the method is flexible and practi- 

 cal. 



Specht 1966' uses an estimated probability 

 density function of the form described by Parzen 

 1 1962 1 and extended by Murthy > 1966 • to the mul- 

 tivariate case. The underh"ing multivariate den- 

 sity for each class is modeled by a sum of functions 

 that are multivariate Gaussian in form, one such 

 function for each fish in the learning sample for 

 that class. This set of functions is complete. There- 

 fore, for each class the underhnng continuous 

 probability density. Gaussian or not, may be ap- 

 proximated arbitrarily closely by such a sum. A 

 power series expansion of this estimated denisty 

 then results in a pohTiomial term in the density 

 function, the coefficients of which are functions of 

 the obser\"ations i fish i in the learning sample. One 

 such set of coefficients is computed for each class to 

 be considered. These polynomials determine the 

 nonlinear decision surfaces and are the basis for 

 discrimination. 



The indi\'idual multivariate Gaussian functions 

 I which when summed model the underlying mid- 

 tivariate distribution for that class' contain a 

 "'smoothing parameter." cr. which appears in the 

 place of a standard error. This parameter is then 

 incorporated in the estimates of the pol\*nomial 

 coefficients. The reader is referred to Specht ' 1966 1 

 for a discussion of the effect of this smoothing 

 parameter and for the algorithm for the calcula- 

 tion of the sets of pohTiomial coefficients { -Dfe  

 k- ... fefj ( • The f>olynomial discriminant func- 



tion is: 



P(X) = Do ^ Z)iXi - D0X2 - 



DpX, 



'11^1 



D^^X■^~ ' . . . + Df^^f^^Xf^^Xf^^ 



D^pX\ ^ 1^11 A' 



+ . . . + Dfe^;;2fe3^fei^fe2'^fe3 



"^ DpppX p 



Dk. 



kh ^fei 



-^fei 



416 



