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Fishery Bulletin 93(2), 1995 



point and provide a probabilistic framework within 

 which management decisions may be based. 



Statistical methods 



Estimation of abundance indices 



Bottom trawl surveys have been conducted in the 

 autumn since 1963 and in the spring since 1968 by 

 the Northeast Fisheries Science Center. The surveys 

 are based on a stratified random sampling design 

 and are used to develop time series of abundance 

 indices that are not subject to biases inherent in fish- 

 ery-dependent data. A detailed review of the survey 

 sampling design, methodology, and application has 

 been provided by Grosslein (1969), Clark ( 1979), and 

 Almeida et al. (1986). Following Cochran (1977), the 

 stratified mean catch per tow is expressed as 



y t 



-s 



h=l 



N h y h 

 N 



(1) 



where N = total area of all strata; N h = area of stra- 

 tum h; y h = sample mean in stratum h. The true abun- 

 dance of a fish stock can be modeled as a population 

 process with a linear stochastic difference equation 

 of the form 



time series of survey abundance estimates, y t , are 

 assumed to be directly proportional to the true popu- 

 lation size, z t , and it is further assumed that the sur- 

 vey index is measured with error, that is y, = z t + e t 

 (where the e t 's are iid MO, o e 2 )), then the survey time 

 series may be represented by 



<p(B)y t = ri(B)c t , 



(4) 



where y t is the survey abundance estimates, B is the 

 backshift operator, and c ( 's are iid MO, e> 2 ). The 

 autoregressive parameter, <f>, remains unchanged in 

 Equation 4 while r\ is the new integrated moving 

 average parameter that reflects error in the survey 

 abundance estimates. Appropriate model specifica- 

 tion is determined by examining the autocorrelation 

 and partial autocorrelation functions, by estimating 

 appropriate parameters, and by checking for model 

 adequacy (Box and Jenkins, 1976). However, this 

 formal procedure of specifying and adequately esti- 

 mating parameters is typically hampered by short 

 time series of data such as those from fishery-inde- 

 pendent surveys. Pennington (1985, 1986) has pro- 

 posed an approach based on a priori specification of 

 the model which addresses: 1) the limitation in the 

 length of the survey time series; and 2) changes in 

 the population availability or catchability. Following 

 Pennington (1986), the true population can be rep- 

 resented as 



2* =*A-1 + 02^-2 •••QpZt-p 



-6 1 a t _ 1 -6 2 a t _ 2 .--9 q a t _ q , 



(2) 



where z t is the population abundance at equally 

 spaced points in time, <fx and 9i are autoregressive 

 and moving average parameters, respectively, and 

 a t 's are independent identically distributed (iid) nor- 

 mal random MO, a 2 ) errors. The autoregressive com- 

 ponent represents "memory," while the moving av- 

 erage component represents past "shocks" or pertur- 

 bations in the system. A principal objective of time- 

 series analysis is to filter the effects of measurement 

 error in the raw survey abundance indices from "true" 

 or process variability resulting from changing popu- 

 lation levels. Box and Jenkins (1976) described a 

 general class of models that estimate the parameters 

 in Equation 2 that represent the autoregressive in- 

 tegrated moving average process. The model can be 

 expressed in more compact form as 



(t>(B)z t =6(B)a t , 



(3) 



where B is the backward shift operator, and all other 

 parameters are defined as above in Equation 2. If a 



z t =z t _ l e a ' or(l-B)lne, 



(5) 



Here the a/s represent the process variability or those 

 factors which cause changes in the population from 

 year t—1 to year t (such as recruitment, fishing mor- 

 tality, migrations, etc.). Pennington (1985) demon- 

 strated that if the model (Eq. 5) and the ratio of vari- 

 ances, a e 2 /o c 2 , are known, then z. and the variance of 

 the estimator can be estimated. If we again assume 

 the survey index, y t , to be an estimate of the true 

 population abundance, z„ and that the measurement 

 errors of the index are multiplicative, then 



lny, =lnz, +e t . 



(6) 



Assuming the e ( 's are iid MO, a 2 ) and independent 

 of the a t 's, then y. can be represented by the inte- 

 grated-moving average model: 



(l-S)lny, =(l-6B)c t , 



(7) 



where the c,'s are iid MO, o c 2 ) and represent the re- 

 siduals generated by fitting the model to the observed 

 data. For the model (Eq. 7) 



