Sensitivities and Variances of Virtual Population Analysis As Applied to the Mackerel, Scomber japonicus

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Abstract
Virtual population analysis (VPA) is widely used in fish stock assessment. However, VPA results are generally presented as point estimates, without error variance. Using numerical methods, we estimated the total variance of historical (1929–65) biomass estimates of mackerel, Scomber japonicus, off southern California. In the years before 1940, coefficients of variation (CV's) approached 100%; later, when weights at age and the age structure of the catch were better known, the CV's were about 25%. Most of the variability derives from uncertainties in estimates of natural mortality (M) and of weights at age. We also developed dimensionless coefficients (sensitivities) to examine the effects of errors in the inputs on the VPA biomass estimates. The largest sensitivities were to M and the total catch and varied substantially from year to year. As expected, sensitivity to M decreased with increasing exploitation, and sensitivity to catch increased with increasing exploitation. Using such sensitivities, one could estimate the error in a biomass estimate for a past year when M (or any other input) was thought to be unusually high or low. Thus, retrospective corrections can be made. Also, such sensitivities form an analytic tool for examining the properties of VPA, or any quantitative model.

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