Extended survivors analysis

7.6.3 Extended survivors analysis

Since the terms P vpa (y,a)/u(y,a), may be regarded It is quite straightforward to extend the Survivors as individual estimates of reciprocal catchability, method to allow for multiple CPUE data sets, and this simply states that the best estimate of r(a) is a to derive a non-negative estimator for the sur- weighted geometric mean of the available esti- vivors themselves using least squares. We refer to mates of it. Doubleday then used equation (7.27) to the method so obtained (Shepherd 1999) as the estimate the population P(y,a) at all ages, and a sub- Extended Survivors Analysis (XSA for short). The tractive algorithm corresponding to VPA done basic procedure, involving iterative use of VPA, forwards in time (rather than retrospectively) to calibration of reciprocal catchability, calculation estimate the survivors corresponding to each of of estimated populations, and computation of these estimates, and combined these by a weighted weighted mean estimates of survivors, is the same, arithmetic average procedure. There are a number but there are important differences of detail. of difficulties with this procedure in practice. First,

In particular, a different algorithm is used for as is well known, the subtractive forward VPA the estimation of survivors, while the VPA phase algorithm can, and often does, lead to negative of the procedure (equations 7.14 through to 7.24) estimates of survivors. These infeasible estimates remains the same. Equation (7.25) is, however, could be included in the weighted mean, since this generalized to allow for several ‘fleets’ (or indices is an arithmetic mean, but they were in practice re- of abundance): placed by zeros, and it is not clear whether these zeros were, or should be, included in the mean, or uyaf ( ,, )=( qafPya , )( ,, )

(7.32) not. Tests with a Fortran re-implementation of the

procedure showed that this was a severe problem and similarly r(a) becomes r(a,f). The least squares on a number of data sets which had been analysed procedure for determining r generalizes without

without difficulty by ad hoc tuning procedures. difficulty, and the final estimate or r, correspond- Generation of infeasible negative estimates of ing to equation (7.31), is given by

survivors is clearly an undesirable feature, which undermines confidence in the results produced. It

can in fact be overcome without difficulty by using 2 Â ln [ P vpa ( yauyaf , )( ,, ) ] s ( af , ) the logarithm of survivors as the estimation vari- ln [ raf ( , ) ]= a 1 . s 2 ( , )

 af a

able, and determining it by the same least squares

procedure adopted for r(a), as described below. The (7.33) use of the subtractive algorithm is in fact incon-

sistent with the least squares approach, and this is the source of the problem.

The interpretation of this as a weighted geometric The performance of the original Survivors mean of all available estimates is unchanged. The

method has indeed been found to be rather variable analogue of equation (7.27) is in practice. It gives plausible results on some data sets, and thoroughly implausible ones on others. It Pyaf ( ,, )=( rafuyaf , )( ,,. )

(7.34) performed very poorly in the simulation tests

conducted by the ICES ‘Methods’ Working Group Each survey/CPUE datum now generates an esti- in 1988 (Anon. 1988), failing to give usable results mate of the true population, and thus an estimate on several of the more demanding data sets. The of the survivors of the appropriate cohort. If now essential process is well conceived, but some one differentiates S (as generalized for multiple

fleets) with respect to the logarithm of survivors, in order to obtain a non-negative estimate of that parameter, one can show (leaving out the indices for clarity) that

(7.35)

and thus

(7.36)

where w¢ = w/ECF. The interpretation of equation (7.36) is again quite straightforward. It simply asserts that the best estimate of survivors is a weighted geometric mean over all available data of the populations estimated from the CPUE/survey data, reduced by the estimated cumulative total mortality to the end of the final year. This is a com- monsense result, except that the weights are modi- fied by division by ECF. This term progressively reduces the weight attached to estimates based on older, earlier data, in addition to any explicit down- weighting of old data, and reflects the reduced utility of older data for determining the terminal population. This is not surprising, as the forward projection of the population is closely related to, but much more robust than, forward VPA, which is of course very sensitive to observation errors in old data.