Propagation of measurement uncertainty in spatial characterisation of recreational fishing catch rates using logistic transform indicator kriging

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1 st International Congress on Modelling and Simlation, Gold Coast, Astralia, 9 Nov to 4 Dec 05 Propagation of measrement ncertainty in spatial characterisation of recreational fishing catch rates sing logistic transform indicator riging E.N. Aidoo a, U. Meller a, G.A. Hyndes b and K.L. Ryan c a School of Engineering, Edith Cowan University, Joondalp, Astralia b Centre for Marine Ecosystem Research, School of Natral Sciences, Edith Cowan University, Joondalp, Astralia c Western Astralian Fisheries and Marine Research Laboratories, Hillarys, Astralia en.aidoo@yahoo.com Abstract: Geostatistical estimation techniqes sch as riging have been widely accepted and applied to characterise the spatial distribtion of natral phenomena. In fisheries science, these techniqes have been applied for compting indices inclding catch per nit effort catch rate sed for stoc assessment. The nonparametric riging approach, nown as indicator riging, is particlarly helpfl for the estimation of catch rates sing data observed from recreational fishing srveys becase it can also handle other featres of the data distribtion sch as ero-inflation, high sewness and class-specific spatial patterns. The problem considered in this paper is the se of indicator riging for estimation of catch rates associated with an ncertainty de to mltiple measrements observed at some locations. This measrement ncertainty is often non-negligible and needs to be propagated to prodce accrate estimates. In addition, the ncertainty might be spatially atocorrelated and correlated with the data and so will restrict the se of a parametric approach of ncertainty propagation. Using catch rate data for Astralian herring Arripis georgians from a recreational fishing srvey, this stdy presents a soft indicator riging approach that ses a logistic fnction transformation to allow the propagation of measrement ncertainty in the estimation. The performance of the ncertainty propagated model was evalated based on the leave-one-ot cross-validation method. The accracy plot and goodness statistic indicate agreement between the expected and empirical proportions of the observed catch rates falling within probability intervals of increasing sie. These sggest a good estimation performance of the approach for Astralian herring catch rate data. The spatial distribtions of catch rate estimates with propagated ncertainty can be sed for qantifying location-specific patterns to assist stoc assessment or validation of policy spporting stoc assessment models. The measrement ncertainty is considered to be potentially valable for estimation of catch rate. Keywords: Logistic transform, soft indicator riging, catch rate, spatial characterisation 45

2 . INTRODUCTION Globally, poplations of many fish species are either flly or overexploited, creating a challenge for managers of fisheries resorces Cooe and Cowx, 004. Since it is becoming widely accepted that a significant proportion of the catch from some fisheries is taen by recreational sector Cooe and Cowx, 006, there is a need to establish appropriate techniqes to assess the recreational sector. The ability to tae a proactive approach towards fisheries management reqires data collection and stoc assessment inclding catch per nit effort catch rate, which is freqently sed as a relative measre of species abndance. Mapping the distribtion of catch rates can assist in delineating high priority areas and improve or nderstanding of the spatial patterns of recreational fishing at appropriate spatial scales for fisheries management Aidoo et al., 05. Geostatistical estimation techniqes sch as riging have been widely applied to characterise the spatial distribtion of natral resorces, inclding observed catch rates from recreational fishing Aidoo et al., 05. The nonparametric riging approach, nown as indicator riging, is particlarly helpfl for estimating catch rate becase it can also handle other featres of the data distribtion sch as ero-inflation, high sewness and class-specific spatial patterns. In the traditional indicator riging approach, measred vales are transformed into hard indicator variables with the assmption that the data are precise Goovaerts, 997. However, observed catch rates from recreational fishing are sally obtained throgh srveys and may be associated with high levels of ncertainty. For any particlar location, observed catch rates are highly variable becase sill levels of fishers and motivations for catching fish can differ greatly within any fishery National Research Concil, 998. For example, in any given trip, only a few fishers catch many fish, while most fishers catch fewer fish National Research Concil, 998; and the time of fishing within the year and the nmber of fishers at a particlar location may vary reslting in the existence of measrement ncertainty in the observed catch rates. In addition, the ncertainty might be spatially atocorrelated and correlated with the observed catch rates and so will restrict the se of a parametric approach of ncertainty propagation. There have been limited attempts to propagate measrement ncertainty in riging estimation sch as riging with measrement error Hamehpor et al., 03 and soft indicator riging Saito and Goovaerts, 00, bt only on physico-chemical properties of sediments. The latter approach is more sitable for catch rate data becase of the distribtional properties of those data. However, the choice of an indicator transformation based on a normal distribtion may not be sitable for catch rate data. This paper aims to present a soft indicator riging approach that ses a logistic fnction transformation to allow the propagation of the measrement ncertainty in the estimation. We illstrate this approach, with catch rates for Astralian herring, Arripis georgians, which is one of the poplar nearshore species commonly targeted by recreational fishers Smith et al., 03.. MATERIAL AND METHODS.. Data The data were collected throgh a phone-diary srvey of boat-based recreational fishers in Western Astralia Ryan et al., 03 spplied by the Department of Fisheries, Western Astralia. The stdy is limited to the West Coast bioregion of Astralia extending from latitde 6 30 S to 5 30 E sing observed catch rates for Astralian herring. Estimates of catch rate and associated ncertainty were calclated for individal 0 0 natical mile NM spatial blocs location sing the ratio of mean estimator Jones et al., 995. For each location bloc centroid, the catch rate was defined as: c cr c d r r= r= = = = d d r r r = r = and the standard error s of the estimator was approximated as: s d r= [ c r th where c r, v and d r denote respectively the nmber of fish caght by the r boat, the nmber of boats that visited the location and the effort in terms of the boat fishing trip dration hors mltiplied by d r ] 46

3 the nmber of persons fishing on each boat. The mean catch and mean effort are represented by c and d respectively Cochran, 977; Smner and Williamson, Soft Indicator Kriging The indicator riging approach is a non-parametric modelling of the conditional cmlative distribtion fnction ccdf of a random fnction i.e. catch rate sing a vector of K indicator variables. At each location the indicator variables i ; are obtained by defining a set of K threshold vales,,, K and determining whether the threshold vale is exceeded or not Goovaerts, 997: if i ; = =,, K 3 0 otherwise In eqation 3, the observed catch rate is assmed to be precise withot any measrement ncertainty leading to a binary hard indicator variable i ; for each threshold. To accont for measrement ncertainty, a logistic fnction was sed to transform the observed catch rate and associated ncertainty s into soft indicators defined as: j ; =, s > 0 4 [ ] + exp s At locations with ero measrement ncertainty hard indicators were sed. The spatial variability of each of the soft indicator variables is described by the indicator semivariogram γ h; describe as: γ h; = N h N h = [ j ; j + h; ] 5 where j ; and j + h; represent the soft indicators at locations, and + h and N h is the nmber of pairs of samples separated by the distance vector h Goovaerts, 997. The soft indicator semivariograms were modelled as linear combinations of a ngget and a spherical model Goovaerts, 997 to describe the spatial continity. The models were fitted sing an ordinary least sqares procedre. The estimates of the K soft indicators at each nsampled location were obtained via ordinary riging of the neighboring data: * j ; = n λ ; = j ;, n = λ ; = 6 The optimal riging weights λ ; were obtained as the soltion of a system of linear eqations with [ n + ] nnowns Goovaerts, 997. The estimates allow the constrction of a conditional cmlative distribtion fnction ccdf F ; n at each grid node with expected vale of the ccdf E-type estimate *, an estimator for catch rate, was approximated by as Goovaerts, 997: K + * = [ F ; n F ; n] 7 and the spread of the distribtion was measred by the inter-qartile range: q R = F ;0.75 n F ;0.5 n 8 where is the mean of the class, ], which depends on the within-class interpolation model e.g. linear. Linear extrapolation was sed to obtain ccdf vales beyond the largest threshold. 47

4 .3. Model Evalation The performance of the spatial model was evalated via the goodness statistic compted from leave-one-ot cross-validation Detsch, 996. At any location, a series of symmetric p-probability intervals PI bonded by the p/ and + p/ qantiles was calclated from the nown ccdf F ; n for all p [0,]. The proportion of the tre vales falling into the symmetric p-pi was defined as Detsch, 996: with N ξ p = ξ ; p for all p [0,] 9 N = ξ if F ; p / < F ; + p / ; p = 0 otherwise The accracy of the model was assessed visally sing accracy plot, a scattergram of the estimated verss theoretical fractions. The closeness of the estimated and theoretical fractions was qantified by the goodness statistics G defined as Detsch, 996: G = [3a p ][ ξ p p] dp 0 0 where a p is eqal to if ξ p p and to 0 otherwise. For maximm goodness, G = i.e. ξ p = p for all p [0,] and G = 0 when ξ p = 0 for all p [0, ]. 3. RESULTS AND DISCUSSION The distribtion of observed herring catch rates was positively sewed and ranged from 0 to 0.63 fish per angler hor, with a mean of 0.4 Figre A. The data showed spatial variation with a greater prevalence of higher catch rate estimates in the soth and soth-central areas and lower estimates in the north of the stdy area Figre B. The measrement ncertainty associated with catch rate varied across the stdy area Figre C and showed a distribtion similar to that of the observed catch rates. The observed catch rate and associated ncertainty were transformed into soft indicator variables sing eqation 4 with 5 threshold vales. Fewer threshold vales were sed de to small sample sie and to ensre that the occrrence of order relation deviations was minimised see Goovaerts, 997 for discssion. The first threshold was set to ero to accont for observed catch rates eqal to ero i.e. nil catch and the remaining threshold vales correspond to 0 th, 40 th, 60 th and 80 th percentiles of the non-ero observed catch rates. These threshold vales are catch rates of 0, 0.05, 0.08, 0.66 and Figre. Histogram of observed catch rates A, and spatial distribtion of observed catch rate B and measrement ncertainty C across 0 0 NM locations. 48

5 Figre. Indicator semivariogram dots and the fitted model solid line. The semivariogram models Figre of the soft indicator variables were fitted sing an ordinary least sqares procedre. De to the elongated shape of the stdy area, an isotropic process was assmed. The range parameter representing the distance of spatial inflence varied between 6 and 00 NM Table. The ngget and sill parameters represent the semivariance vales at a short distance and very large distance, respectively. The degree of spatial correlation, as qantified by the ratio NSR of the ngget to the total sill i.e. inclding the ngget Cambardella et al., 994, varied between 0.67 and 0.69 Table. The spatial atocorrelation for thresholds 3 to 5 is strong relative to thresholds and which show moderate spatial atocorrelation. The differences in the semivariograms models and the NSR indicate the presence of classspecific spatial patterns in the observed catch rates. Hence, sing an indicator approach and different models for distinct thresholds is appropriate for catch rate data. The accracy plot sggests that the model is satisfactory and so sitable for prediction Figre 3. The plot indicates that the model is accrate for probability intervals above 0., bt less accrate below 0.. In comparison to hard indicator riging, the deviation from the 45 bisector line was higher for soft IK with smaller goodness statistic. Table. Threshold vales of observed catch rate and corresponding cmlative percentages with the parameters of the fitted spherical models and corresponding ngget to sill ratio NSR. Spherical model Percentage Threshold ngget range sill NSR Figre 3. Accracy plot obtained from cross-validation based on the soft and the traditional hard indicator riging approaches. 3.. Spatial Distribtion of Estimated Catch Rates Based on the fitted semivariogram models, ordinary riging estimates of the indicators were obtained at nodes of sie 5 5 NM. Possible order relation deviations were corrected sing an pward/downward averaging method Goovaerts, 997. For soft and hard indicators ccdfs the expected vale was sed as the estimator of catch rate Figre 4. The estimates from both approaches show globally similar spatial distribtion patterns bt differ locally, especially in areas where measrement ncertainty is high. These differences may be attribted to the ncertainty propagated in the model. High catch rate estimates are concentrated in the soth and there is some patchiness in the central of the stdy region. The predicted vales 49

6 at the nsampled locations have lower mean and variance than the observed samples Table, with hard indicator estimates having a slightly higher mean. Part of these differences may be attribted to the ncertainty propagation in the model. The ncertainty associated with the predicted catch rates was indicated by high inter-qartile range IQR in areas where high and low estimates of catch rates were intermingled, and the occrrence was more prononced in the soth of the stdy region Figre 4B. The spatial patterns of the IQR are similar for both approaches bt differ in areas where measrement ncertainty is high. Figre 4. Maps of estimated catch rate A and inter-qartile range B from soft and hard indicator riging and measrement ncertainty C. Table. Smmary statistics of the observed catch rates and the E-type estimates from the soft indicator riging. Estimator Cont Min Max Mean Variance Sewness Observed Estimate from soft IK Estimate from hard IK CONCLUSIONS Mapping the distribtion of catch rates from recreational fishing is essential to fisheries managers for delineating high priority areas Aidoo et al., 05. Incorporation of measrement ncertainty is particlarly important to prodce accrate and reliable reslts Woille et al., 009. In this paper, soft indicator riging was shown to be appropriate for incorporating measrement ncertainty into maps of estimated catch rates. This method does not reqire any assmptions regarding the measred ncertainty in contrast to other approaches Hamehpor et al., 03; Saito and Goovaerts, 00. It is a promising for informing stoc assessment and management, taing into accont the class-specific spatial patterns and highly sewed distribtions common to the catch rates from recreational fishing. The reslts show that inclsion of measrement error provides better discrimination in areas where ncertainty is high. For Astralian herring highest catch rates occr near the metropolitan area, where the majority of the Western Astralian poplation resides. Understanding the spatial patterns in catch rates of this species is important, as there are specific sstainability concerns related to resorce sharing between commercial and recreational fishers, potential shift of fishing activity towards targeting near shore species and increasing hman poplation growth Smith et al., 03, and the predominance of immatre fish in the catches Smith and Brown, 04. This stdy only considers catch rates for boat-based fishers, as shore-based fishers who also target the species Smith and Brown, 04; Smith et al., 03 were ot of scope of the phone-diary srvey Ryan et al., 03, nor are seasonal differences in fishing effort acconted for Ryan et al., 03; Smith et al., 03. The measrement ncertainty is considered to be potentially valable inpt for estimating catch rates since it can inflence the estimation when ignored. In addition to measrement ncertainty, it mst be noted that variogram and spatial ncertainty may be potentially valable. Ftre research will be focsed on the extension of the approach to other fish species in a simlation stdy and to assess the effect of measrement ncertainty on catch rate mapping and area delineation. 50

7 ACKNOWLEDGEMENTS This stdy wold not be possible withot contribtions from the recreational fishers who participated in the recreational fishing srveys and the Srvey Research Centre Edith Cowan University for facilitating data collection and entry of the phone-srvey data. REFERENCES Aidoo, E. N., Meller, U., Goovaerts, P., and Hyndes, G. A. 05. Evalation of geostatistical estimators and their applicability to characterise the spatial patterns of recreational fishing catch rates. Fisheries Research, 68, 0-3. Cambardella, C. A., Moorman, T. B., Nova, J. M., Parin, T. B., Karlen, D. L., Trco, R. F., et al Field-scale variability of soil properties in central Iowa soils. Soil Science Society of America Jornal, 58, Cochran, W. G Sampling Techniqes 3 ed.. New Yor: Wiley. Cooe, S. J., and Cowx, I. G The role of recreational fishing in global fishing crises. Bioscience, 549, Cooe, S. J., and Cowx, I. G Contrasting recreational and commercial fishing: searching for common isses to promote nified conservation of fisheries resorces and aqatic environments. Biological Conservation, 8, Detsch, C. V Direct assessment of local accracy and precision. Paper presented at the 5th International Geostatistics Congress, Wollongong' 96. Goovaerts, P Geostatistics for natral resorces evalation. New Yor: Oxford University Press. Hamehpor, N., Eghbal, M. K., Bogaert, P., Toomanian, N., and Sooti, R. S. 03. Spatal prediction of soil salinity sing riging with measrement errors and probabilistic soft data. Arid Land Research and Management, 7, Jones, C. M., Robson, D. S., Lais, H. D., and Kressel, J Properties of catch rate sed in analysis of anglar srvey. Transactions of the American Fisheries Society, 4, National Research Concil Review of Northeast fishery stoc assessments. Washinton, DC: National Academy Press. Ryan, K. L., Wise, B. S., Hall, N. G., Polloc, K. H., Slin, E. H., and Gaghan, D. J. 03. An integrated system to srvey boat-based recreational fishing in Western Astralia 0/0. Fisheries Research Report No. 49, Department of Fisheries, Western Astralia. Saito, H., and Goovaerts, P. 00. Acconting for measrement error in ncertainty modeling and decisionmaing sing indicator riging and p-field simlation: application to a dioxin contaminated site. Environmetrics, 3, Smith, K., and Brown, J. 04. Biological synopsis of Astralian herring Arripis georgians. Fisheries Research Report No. 5. Department of Fisheries, Western Astralia. Smith, K., Brown, J., Lewis, P., Dowling, C., Howard, A., Lenanton, R., et al. 03. Stats of nearshore finfish stocs in soth-western Western Astralia Part : Astralian herring. Fisheries Research Report No. 46. Department of Fisheries, Western Astralia. Smner, N. R., and Williamson, P. C A -month srvey of coastal recreational boat fishing between Agsta and Kalbarri on the west coast of Western Astralia dring Fisheries Research Report No. 7, Department of Fisheries, Western Astralia, 5 p. Woille, M., Rivoirard, J., and Fernandes, P. G Evalating the ncertainty of abndance estimates from acostic srveys sing geostatistical simlations. ICES Jornal of Marine Science, 666,