Application of the Moving Averaging Technique in Surplus Production Models

WANG Yu, and LIU Qun

College of Fisheries, Ocean University of China, Qingdao 266003, P. R. China

Application of the Moving Averaging Technique in Surplus Production Models

WANG Yu, and LIU Qun*

College of Fisheries, Ocean University of China, Qingdao 266003, P. R. China

Surplus production models are the simplest analytical methods effective for fish stock assessment and fisheries management. In this paper, eight surplus production estimators (three estimation procedures) were tested on Schaefer and Fox type simulated data in three simulated fisheries (declining, well-managed, and restoring fisheries) at two white noise levels. Monte Carlo simulation was conducted to verify the utility of moving averaging (MA), which was an important technique for reducing the effect of noise in data in these models. The relative estimation error (REE) of maximum sustainable yield (MSY) was used as an indicator for the analysis, and one-way ANOVA was applied to test the significance of the REE calculated at four levels of MA. Simulation results suggested that increasing the value of MA could significantly improve the performance of the surplus production model (low REE) in all cases when the white noise level was low (coefficient of variation (CV) = 0.02). However, when the white noise level increased (CV= 0.25), adding the value of MA could still significantly enhance the performance of most models. Our results indicated that the best model performance occurred frequently whenMAwas equal to 3; however, some exceptions were observed whenMAwas higher.

moving averaging; surplus production model; Monte Carlo simulation

1 Introduction

Surplus production models are flexible tools for fishery data analysis due to their low data requirement and easy interpretation of model output (Schaefer, 1954; Pella and Tomlinson, 1969; Walter and Hilborn, 1976; Fox, 1970; Prager, 1994). Since Schaefer (1954) put forward the production model, the surplus production model had undergone a long history of development. Prager (1994) reviewed a set of extensions of the simple surplus production model and detailed descriptions of several fundamental equations relating to the population dynamics. In recent years, computer packages such as Catch Effort Data Analysis (CEDA) and A Surplus-Production Model Incorporating Covariates (ASPIC) provided an efficient approach to fit nonequilibrium surplus production models with the ‘observation error’ method (Hoggarthet al., 2006; Prager, 2005; Panhwaret al., 2012; Mesnil, 2012; Punt, 2012). In this study, the relative estimation error (REE) of the maximum sustainable yield (MSY) was used as an evaluation indicator of the performance of the surplus production model. Moving averaging (MA) is a useful technique in reducing data noise, thus improving model performance. Although the application of MA has been reported (Gulland, 1983), the validity of MA in surplus production model estimations has not been detected. Meanwhile, a Monte Carlo simulation was conducted to verify the utility of MA in surplus production models by using the data of three typical fisheries. One-way ANOVA was used for the significance test for the REE calculated from different values of MA.

2 Data and Methods

2.1 Data

Three typical fishery exploitation histories were analyzed in this study according to Cuiet al. (2008). The first type of fishery,i.e., the ‘one-way trip’ fishery or ‘declining fishery’, which is the most common type wherein fishing effort, is continuously increasing while biomass is declining. The second type is the ‘well-managed fishery’, which has a strong contrast between biomass and effort. The third is ‘restoring fishery’, which has a low level of fishing effort and increasing biomass. Figs.1 to 3 show the biomass and effort for these three fisheries.

2.2 Models and Estimators

Two types of surplus production models, namely, the Schaefer and Fox surplus models, were used to simulate data for three fishery histories according to Cuiet al. (2008). These two models are expressed as Eqs. (1) and(2), whereB(t)andB(t+1)are the biomass in yeartand yeart+ 1, respectively;ris the intrinsic population growth rate;Kis the carrying capacity. The true parameters (r= 0.4,K= 1000,q= 0.01) and coefficient of variation (CV) of white noise (0.02 and 0.25) were set for three fishery histories in advance (Figs.1 to 3). We noted that the value ofB0was different among the three fisheries. For declining and well-managed fisheries,B0was equal to 800, whereas for restoring fishery,B0was equal to 200.

Fig.1 Biomass and effort data generated from the Schaefer (upper) and Fox (lower) model for the Declining Fishery.

Fig.2 Biomass and effort data generated from the Schaefer (upper) and Fox (lower) model for the Well-managed Fishery.

Fig.3 Biomass and effort data generated from the Schaefer (upper) and Fox (lower) model for the Restoring Fishery.

The parameters in the paper were estimated with the equilibrium, process error, and observation error methods. The equilibrium method relies on the assumption that each level of fishing effort corresponds to an equilibrium sustainable yield (Boerema and Gulland, 1973; Larkin, 1977). The forms of the Schaefer (Schaefer, 1954) and Fox (Fox, 1970) equilibrium surplus production models are expressed as Eqs. (5) and (6), whereYeis the catch at equilibrium condition;Bis the stock biomass;fis the fishing effort;a,b,c, anddare the regression parameters of the equilibrium estimators. The management parameters ofMSYandfMSYfor the two equilibrium estimators are calculated by using Eqs. (7) and (8), whereMSYis the maximum sustainable yield andfMSYis the optimum fishing effort.

For simplicity, the normally distributed random variables were generated based on the Box-Muller scheme (Eqs. (3) and (4)), which used two uniformly distributed random numbersU1andU2(between 0 and 1) (Hilborn and Mangel, 1997). Thus,Z1andZ2are the normally distributed random numbers with a mean of 0 and a variance of 1. Eqs. (9) to (12) are the transformations of the simple surplus production models (Eqs. (1) and (2)). Eqs. (9) and (10) are Schaefer-type models (Schnute, 1977; Walters and Hilborn, 1976), and Eqs. (11) and (12) are Fox-type models (Fox, 1970; Yoshimoto and Clarke, 1993). Here,Utis the catch per unit effort (CPUE) or abundance index in yeart, andqis the catchability coefficient.

The observation error method is a nonlinear technique and is currently the most popular method. This method is used to minimize the squared deviations between observed and predicted CPUE. The equations of the discrete form of the Schaefer and Fox models are expressed as Eqs. (13) to (16), whereis the estimated biomass in yeart,is the estimated catch in yeart, andis the estimated CPUE or abundance index in yeart. The corresponding management parameters ofMSYandfMSYfor the process error and observation error estimators are calculated by Eqs. (17) and (18). Eq. (17) is for the Schaefer nonequilibrium models (Eqs. (9), (10), and (13)). Eq. (18) is for the Fox nonequilibrium models (Eqs. (11), (12), and (14)).

2.3 Monte Carlo Simulation and Moving Averaging

The Monte Carlo simulation is a calculation-intensive method that can be used to test a specific hypothesis and to answer ‘what if’ type questions, including projections into the future (Polachecket al., 1993; Kinas, 1996). This method is one of the most common ways to assess thequality of an estimator and is currently extensively applied in risk assessment and management strategy evaluation in fisheries (Francis, 1992; Haddon, 2011). For the simulation study, we followed the procedure of Hilborn and Walters (1992). The initial step was generating numerous data sets by using the Schaefer and Fox production models (Eqs. (1) and (2)), representing simulated fished populations under two levels of CV. Uncertainty consideration is important for fishery stock assessments and management decisions (Zhu et al., 2012). Thus, white noise levels (normally distributed random numbers) are used in the Monte Carlo simulation. The values of CV were set at 0.02 and 0.25 following Prager (2002). A simple unweighed MA technique was used to smooth the original data to reduce the effect of noise. The MA levels were one, three, five, and seven years. The formulation of this technique is expressed as Eq. (19), where C1is the catch after MA, C0is the original data, N is the number of data points, and n is the number of MA. MA was applied to the data of the six simulated fisheries, and eight models were used to conduct the assessment. The biological reference points, such as MSY and its REE, were estimated (Eq. (20)). Finally, 1000 repetitions were generated by Monte Carlo simulation, and the average of the REE at four MAs was used to evaluate the performance of the eight models.

All works in this study were performed by using Visual Basic for Applications in Microsoft Office Excel 2007, and the significance test was conducted by using SPSS (V.17).

2.4 Functions

3 Results

3.1 Declining Fishery

Table 1 shows that for both Schaefer-type and Foxtype simulated data of declining fishery, increasing the value of MA improved the model performance for all the cases, and the significance levels of the average REE estimated under four levels of MA were all <0.01. Table 2 shows the average REE of 15 out of 16 cases, with the exception of the Schaefer model with Schaefer simulated data, decreased with increasing value of MA when CV increased. The significance level of three cases was >0.01. In general, for the three parameter estimation methods, the value of REE estimated from the observation error method was the smallest, except for the Obs.Schaefer model under the Schaefer simulated data. Fig.4 shows that the accuracy of REE was high when CV was equal to 0.02 for all the models, but began to decline when CV was equal to 0.25. With the increasing value of MA, REE decreased significantly.

Table 1 Average REE (Relative Estimation Error) computed from eight models by changing the value of MA from 1 to 7 using Schaefer and Fox type simulated data of the three fisheries when CV=0.02

Table 2 Average REE computed from eight models by changing the value of MA from 1 to 7 using Schaefer and Fox type simulated data of the three fisheries when CV=0.25

(continued)

Fig.4 REE of maximum sustainable yield (MSY) for the eight estimators under four moving averagings and two CV levels using the Schaefer and Fox simulated data of declining fishery. REE values higher than 100% were abandoned.

3.2 Well-Managed Fishery

With regard to the well-managed fishery (Fig.2), Table 1 shows that the value of REE decreased whenMAincreased, with only one exception (I-Fox model in Schaefer simulated data). This result indicated thatMAcould generally improve the model performance for this fishery. However, whenCVwas equal to 0.25 (Table 2), all the average REE computed from the eight models had the smallest value whenMAwas >1. The REE from the four levels ofMAwere significantly different, with the exceptions of Schaefer (1954) and Fox (1970). As shown in Fig.5, whenCVwas equal to 0.02, the observation error method showed the smallest REE at all four levels ofMA.

Fig.5 REE of maximum sustainable yield (MSY) for the eight estimators under four moving averaging and two CV levels using the Schaefer and Fox simulated data of well-managed fishery. REE values higher than 100% were abandoned.

3.3 Restoring Fishery

For the restoring fishery (Fig.3), Table 1 and 2 show that the increased value of MA could improve the model performance for all the cases. However, the REE values at four MAs for the Schnute (1977) model were not significantly different when CV was equal to 0.25. Table 1 shows that for the Schaefer simulated data, the W-H (1976) model with MA equal to 3 operated best, and the Obs. Fox model with MA = 5 gained the smallest REE for the Fox simulated data. The results shown in Table 2 are similar to those in Table 1. The D-Fox (1970) model with MA = 3 behaved excellently for the Fox simulated data. Fig.6 shows that when CV was equal to 0.02, the increasing value of MA could significantly decrease the value of REE.

Fig.6 Relative Estimation Error (REE) of maximum sustainable yield (MSY) for the eight estimators under four moving averaging and two CV levels using the Schaefer and Fox simulated data of restoring fishery. REE values higher than 100% were abandoned.

3.4 Confidence Interval

Table 3 shows the average MSY and their 95% confidence intervals (in brackets) for the global minimum of REE values (double underlined values in Tables 1 and 2). We observed that the confidence intervals were small for all the cases which offered a reliable result. All the values were close to the true value of MSY.

Table 3 Average MSY and their 95% confidence intervals (in brackets) for the global minimum of REE values, the double under-lined in Tables 1–2

4 Discussion

MA has been applied in science, with a long history of development. For example, Holt (2004) forecasted seasonal trends by using exponentially weighted MAs. In fishery science, Gulland (1983) and Haddon (2011) reported the effect of MA in smoothing noise in data. With 12000 generated artificial data sets, this study demonstrated that MA can reduce the effect of noise in the data effectively, thereby improving the accuracy of surplus production models significantly. Although the use of MA was not workable in all of the situations considered, results proved that MA was feasible for most of the cases. We noted that the effect of MA was sensitive to white noise. In some cases, when the value ofCVincreased, the values of REE at the four MA levels were not significantly different.

Tables 1 and 2 show that in Schaefer-type simulated fishery, Schaefer-type estimators, such as W-H (1976) and Obs.Schaefer generally behave better compared with other estimators. Similarly, Fox-type estimators such as I-Fox, D-Fox, and Obs. Fox, had better estimation accuracy compared with other estimators in Fox-type simulated fishery. This finding agrees well with practical experiences and proves the validity of using REE as an indicator. Therefore, choosing the right production model,i.e., Schaefer or Fox is important when analyzing fishery catch and effort data.

The result from the six simulated fisheries also indicated that the REE estimated from the equilibrium models was always larger than the values from nonequilibrium models. The two equilibrium estimators were not sensitive to white noise but sensitive to the value ofMA, whereas the six nonequilibrium estimators were sensitive to white noise andMA, especially for the observation error estimators. Therefore, we can conclude that equilibrium estimators are more stable than nonequilibrium estimators. However, this conclusion does not mean that equilibrium estimators are always perfect. Figs.4 to 6 show that observation error methods always possess significantly accurate estimates than the equilibrium and process error methods.

Polachecket al.(1993) stated that process error estimators should be applied only if simulation studies and practical experience suggest that they would be superior to observation error estimators. Our research proved that none of the estimators outperformed all of the other estimators for all the cases considered. However, the observation error estimator is considered the best choice because we should make a tradeoff between accuracy and precision. However, factors other than performance under simulated conditions should be considered when selecting a better estimator for assessment.

Computer packages programmed with nonlinear production models such as ASPIC and CEDA are extensively used today as parameter estimation tools (Prager, 2005; Hoggarthet al., 2006; Panhwaret al., 2012). However, these computer packages cannot be included in this Monte Carlo simulation study because repeating the computer packages 1000 times is difficult. The application of these two packages in this work will be interesting.

Given their simple concepts and assumptions, surplus production models generally cannot capture certain agedependent characteristics of a fish population. Therefore, if the data allow, age-structured production models that relate the present total number or total biomass of a fish population to its previous numbers through age structure should be used in future studies. The MA technique is simple and useful. However, research regarding the application of MA in the surplus production model in this paper was based only on simulated data. Therefore, additional tests on real fisheries data will yield optimal results. This research nonetheless provides a useful start in this area.

Acknowledgement

This work is supported by the special research fund of Ocean University of China (201022001).

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(Edited by Qiu Yantao)

(Received October 19, 2012; revised December 24, 2012; accepted December 2, 2013)

© Ocean University of China, Science Press and Spring-Verlag Berlin Heidelberg 2014

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E-mail: qunliu@ouc.edu.cn