COOK and LORD: IDENTIFICATION OF STOCKS OF SOCKE'i'E SALMON 



where p = dimension of the vector X (set of scale 

 characters ) 



1 <kj<p 



j = 1,2, ..., /2 



h = the degree of the variable portion of the term. 



The decision on an unknown X iset of scale mea- 

 surements from a salmon of unknown origin i is 

 thus: 



Choose d{X) = 9^ so that h^P^X) > h,P' (X) 

 for all s =f r 



where d(X) 

 p'iX) 



/2; = 



the decision on an unknown X 

 the classes (origins) 



the polynomial value for X 

 calculated using the discrimi- 

 nant function for class B; 

 the a priori probability, the 

 uses of which will be de- 

 scribed later. 



APPLICATION OF THE METHOD 



Three scale sample sets are required to imple- 

 ment the polynomial discriminant method: learn- 

 ing samples, test samples, and onknowTi samples. 

 The learning and testing samples are collected 

 from each subpopulation when they are segre- 

 gated (i.e.. in the rivers of origin i. Scale characters 

 to be measured in the unknown sample for the 

 required discrimination are determined by 

 evaluating characters measured in the learning 

 samples. The learning samples and the characters 

 selected are used to calculate the coefficients in the 

 pohTiomial discriminant functions. To calculate 

 these coefficients, the value for the smoothing 

 parameter and the point at which the discrimin- 

 ant function should be truncated must be deter- 

 mined. Various circumstances will dictate differ- 

 ent choices. When a smoothing parameter of 1.5 

 was chosen, all terms in the discriminant function 

 greater than the fourth order contributed negligi- 

 bly to polynomial values and so were truncated in 

 our applications. Often, polynomial discriminant 

 functions of lower order yield adequate results.^ 



*A poKiiomial discriminant function with six variables and of 

 the fourth order will contain 210 terms. Since our calculations 

 were performed by computer, we chose not to delete the third or 

 fourth degree terms. However, if more than six vso-iables are 

 used, it would be wise to truncate further in order to keep the 

 number of terms down. 



The fish comprising the test samples are classified 

 to test the effectiveness of the polynomial discrim- 

 inant method and to determine the a priori prob- 

 abilities. 'Each test sample consists offish from 

 one class. I Finally, fish collected from the zone of 

 intermingling are classified to determine the de- 

 gree of intermingling in the area of interest. 



Appraisal of the method using scale samples of 

 sockeye salmon collected from the 1967 escape- 

 ment in five Bristol Bay rivers showed large per- 

 centages of fish comprising the test samples were 

 correcth- classified. However, misclassified fish in 

 the test group • set of test samples from all rivers 

 being considered i were not assigned to the rivers 

 in proportion to the known relative test sample 

 sizes. To balance these misclassifications. wher- 

 ever a greater number of fish comprising the test 

 group was assigned to a particular river than 

 should have been i according to the relative test 

 sample sizes ). the a priori probability for that river 

 w-as lowered. Corresponding increases were made 

 for those classes with insufficient assignment. By 

 alternatively using the decision procedure of the 

 polynomial discriminant method and adjusting 

 the a priori probabilities, we obtained solutions so 

 that the number offish belonging to a certain river 

 that were misassigned to all other rivers approxi- 

 mately equaled the number of fish misassigned to 

 that certain river from all other rivers. Thus the a 

 priori probabilities were not used in the manner 

 their name suggests, but a priori knowledge may 

 dictate test sample sizes. The relative test sample 

 sizes in the test group may be in the relative pro- 

 portions to be expected in the unknown sample 

 ( i.e.. historical relative run sizes >. The adjustment 

 procedure, then, shifts the nonlinear decision sur- 

 faces between the probability- densities so that the 

 incorrectly identified samples are assigned to the 

 various rivers in the proportions dictated by the 

 test sample sizes in the test group. However, the 

 primary purpose of the adjustment procedtire is 

 not to balance the misclassifications but to 

 maximize the number of correct classifications. As 

 the misclassifications are balanced, the number of 

 correct classifications generally increases. At this 

 point the result is a classification method that 

 maximizes the total number of correct classifica- 

 tions and balances misclassification rates for a test 

 group in which the test sample sizes are in particu- 

 lar proportions. 



However, it is obvious that the proportions of 

 fish from the various classes in the test group 

 would rareh- be identical to those proportions in 



417 



