To further condense these data, we averaged 

 for each month the deviations from the linear 

 regressions for the Marsden squares in the 

 albacore area. These deviations are plotted in 

 figure 6. We note from figure 6 that there are, 

 as one would expect, both positive and negative 

 deviations and that these deviations tend to be 

 inversely proportional to time (on a per-annum 

 scale). An index of the magnitude of each 

 year's deviations for the peak fishing period 

 was formed by "integrating" each year's De- 

 cember, January, February, and March devia- 

 tions. The integration was accomplished by 

 triangular approximation. The indices for each 

 year are given in table 2. We note from table 2 

 that the greatest negative deviation was obtained 

 in 1950 and the greatest positive deviation in 

 1952. Furthermore, positive deviations do not 

 appear to be distributed randomly among the 

 negative deviations: In this respect, there are 

 two runs of three negative deviations, one run 

 of three positive deviations, and one run of two 

 negative deviations. These groupings indicate 

 that above-average and below-average years do 

 not occur at random. This phenomenon is most 

 easily explained by strong year classes appear- 

 ing for more than 1 year in the fishery. It is 

 interesting to observe, however, that the initial 

 year of a run of positive deviations does not 

 correspond with decreased average size of the 

 fish in the longline fishery (fig. 5). 



SPATIAL STATISTICS 



The spatial distribution of fish in the ocean 

 is customarily studied with the aid of carto- 

 graphic plots of apparent abundance (see. for 

 example, Nankai Regional Fisheries Research 

 Laboratory, 1959). In some instances, patterns 

 of movement from cartographic plots may be 

 difficult to discern. We have tried to avoid this 

 difficulty in our study of the albacore by exam- 

 ining statistics related to the spatial moments 

 of effort, albacore catch, and albacore CPUE. 

 Thus, we have computed for effort, catch, and 

 CPUE for each month statistics derived from, 

 or equivalent to, the spatial moments--the lon- 

 gitudinal mean, the latitudinal mean, the longi- 

 tudinal standard deviation, latitudinal standard 

 deviation, and the correlation coefficient be- 

 tween latitudinal and longitudinal positions; 

 these five statistics are called, for convenience, 

 spatial statistics. 



The method of computing spatial statistics 

 (i.e., moments or statistics derived from mo- 



ments) is given in any text on mathematical 

 statistics (e.g.. Parzen, 1960: pp. 199-210) and 

 need not be considered in detail here. We 

 simply outline a typical calculation: the latitu- 

 dinal mean for catch was computed by 

 z 



I 



4 



J = 1 



where j = 1, 2, z degrees of north latitude (z 



is an arbitrary number greater than the maxi- 

 mum latitude at which albacore are caught); C; 

 is the total catch (in numbers) of albacore taken 

 in the j-th latitudinal strip; and T is the total 

 number of albacore taken in all of the latitudi- 

 nal strips. The latitudinal variance of catch is 

 computed simply by squaring j in the above 

 formula and subtracting from this result the 

 square of the first latitudinal moment of catch. 

 The standard deviation is. of course, the square 

 root of the variance , etc. 



We now consider several aspects of the North 

 Pacific longline fishery for albacore based on 

 the spatial statistics. These aspects include 

 interpretation of the spatial statistics, periodi- 

 cities and trends in the spatial statistics, the 

 average location of the spatial statistics, and 

 apparent movement. But first we note that al- 

 though the spatial statistics were computed for 

 effort, albacore catch, and albacore CPUE, it 

 became apparent that the spatial statistics for 

 catch and CPUE were quite similar and there- 

 fore did not, for the North Pacific longline fish- 

 ery for albacore, in most cases, warrant sep- 

 arate detailed study. It also became apparent 

 that some of the effort included in our data was 

 not, by nature of its spatial location, directly 

 intended to catch albacore. Therefore, because 

 of the similarity between catch and CPUE spa- 

 tial statistics and the nature of the reported 

 effort, emphasis--in our discussion--is placed 

 on the spatial statistics for CPUE while those 

 for effort are reported mostly to indicate the 

 spatial-temporal nature of the effort distribu- 

 tion included in our data. 



Interpretation 



A degree of caution must be maintained in 

 the interpretation of spatial statistics. For 

 example, a well-known difficulty in the inter- 

 pretation of moments is the possibility that 

 they may exist at a point that has no observa- 

 tions. Thus the first moment can, as in a nor- 

 mal distribution, be coincident with the modal 

 value of the distribution, or, on the other hand. 



