Estimating Fish Movement
(o )
1
(o )
2
5% 1%
Ph.D. thesis
Anders Nielsen Royal Veterinary and Agricultural University June 2004
ii
Estimating Fish Movement Ph.D. thesis
Anders Nielsen Department of Mathematics and Physics Royal Veterinary and Agricultural University
Thesis advisors:
Ib Michael Skovgaard, Royal Veterinary and Agricultural University Peter Lewy, Danish Institute for Fisheries Research Henrik Gislason, University of Copenhagen
Anders Nielsen Department of Mathematics and Physics Royal Veterinary and Agricultural University Thorvaldsensvej 40 DK–1871 Frederiksberg C Denmark Fax: (+45) 3528–2350 Email:
[email protected] URL: http://www.dina.kvl.dk/~anielsen
ISBN: 87–7611–065–6
Preface
This thesis has been submitted to the Royal Veterinary and Agricultural University (KVL) as part of the requirements for obtaining the Ph.D. degree. The work presented in this thesis has been carried out at KVL and at the Danish Institute for Fisheries Research (DIFRES) with Ib Michael Skovgaard, Peter Lewy and Henrik Gislason as supervisors. The research was supported by the SLIP research school under the Danish Network for Fisheries and Aquaculture Research (www.fishnet.dk) financed by the Danish Ministry for Food, Agriculture and Fisheries and the Danish Agricultural and Veterinary Research Council. I would like to thank my three supervisors for excellent supervision, for massive encouragement and for showing great interest in the project. I would also like to thank Peter Setoft (KVL) for dedicated planning and supervising of the computer science part of my Ph.D. education, Uffe Høgsbro Thygesen (DIFRES) for always being willing to share his profound insights in almost all imaginable areas of mathematical modeling in an intuitive and enthusiastic manner, Mats Rudemo for helpful discussions about the particle filter and Bjarke Feenstra for being the perfect office mate and for sharing my interest in advances in statistical computing. Furthermore, I would like to thank Claus Reedtz Sparrevohn (DIFRES) and Josianne G. Støttrup (DIFRES) for a very enjoyable collaboration in the turbot study. During the work on this thesis I visited manager John Sibert of the Pelagic Fisheries Research Program (PFRP), the Joint Institute for Marine and Atmospheric Research (JIMAR) at the University of Hawai’i (UH). I would like to thank all who made this visit such a wonderful experience, but that seems to include half the population of Oahu, so I have to restrict myself. John Sibert has been a true source of inspiration, both through his published work and through our many many hours of enjoyable discussions of various modeling and programming issues. I thank him for that and for hosting my stay. For making the visit absolutely free from practical hassles I thank Dodie Lau, and for being such a good friend and for answering tons of questions I thank my JIMAR office mate Shiham Adam. For making everyday life at the JIMAR office so pleasant I thank Johnoel Ancheta, Dave Itano and all above mentioned. During my stay I collaborated closely with a long list of outstanding researchers including Mike Musyl, Keith Bigelow, Molly Lutcavage, Yonat Swimmer and Lianne McNaughton I thank them all for sharing their data and ideas. Especially, I would like to thank Mike, Keith and John for
iv their efforts in the final preparation of our shared paper. Finally, I would like to thank Pierre Kleiber for showing me how the GMT stuff works, for our many lunch time meetings and for his great hospitality. To summarize: Mahalo! I am also grateful to Tina Dawson and Jesper B. Andersen for helpful comments on previous versions of this thesis, which improved the final result substantially, and to Helle Sørensen for helping with the layout of the thesis. Finally, I would like to thank my friends and family for supporting me and in particular Karina for her never failing love and support, and for never questioning my decisions to work late or travel far.
Copenhagen, June 2004
Anders Nielsen
Summary
Fish are mobile creatures, so it seems natural to include the spatial aspect when modeling fish populations, but most often heterogeneity and movement are completely ignored in fish stock assessment models. This is mainly because of the difficulties in estimating the movements. This thesis describes ways to estimate fish movements with the long term goal of improving stock assessment models. The thesis is divided into five introductory chapters and four papers. In the introductory chapters, the basics of fish stock assessment is explained and some selected models are described in detail to give an impression of how these fish stock assessment models work and how leaving out the spatial aspect can lead to huge errors (not just greater uncertainties) (chapter 1). The most important data source for estimating movement patterns are tagging studies and some classic models used for tagging data are presented (chapter 2). These models may seem too simple and they do rely on strict assumptions, but they are a valuable starting point for understanding the more modern approaches presented later in this thesis. Archival tags, also referred to as data storage tags, are a fairly new type of tags that are designed to be attached to, or surgically implanted into, an animal to record measurements that can (hopefully) later be used to reconstruct its track. A widely used and fairly successful approach to reconstruct these tracks is described and a few improvements are presented (chapter 3). There is a link between the individual based model presented for archival tags and modern population based models. This link is described along with two population based models (chapter 4). A simulation study is conducted in which fish are simulated and randomly assigned either a conventional or an archival tag. It is evaluated if more accurate parameter estimates can be expected from archival tags than from conventional tags. The individual model is extended to fit the same scenario as the population based model. Finally, the two separate models are combined into one coherent model using data from both tag types (chapter 5). The papers apply and extend the described models. The population based model is applied in a field study where 3529 turbots were released and followed for nine days to see how fast they colonized their new habitat. Furthermore, it was possible to estimate their survival rate (paper I). Premature detachment of the archival tags can be problematic, especially if the tag is unable to report when and where it came off. A model to estimate the most likely detachment point is introduced along with a method to quantify the significance of the proposed de-
vi tachment point (paper II). The model can also be useful for detecting changes in movement behavior. Archival tags use light measurements to estimate longitude (from local noon) and latitude (from local day length). This method gives very uncertain geolocations, especially around an equinox where day lengths are similar everywhere. Archival tags also store temperature measurements and these are used in combination with sea surface temperatures obtained from satellites to get highly improved (more accurate) reconstructed tracks (paper III). The reconstructed tracks from the basic individual based model from chapter 3 sometimes cross land areas, because such barriers cannot be built into that model type (a similar problem occurs with heterogeneous fishing pressure). A model is introduced where these extensions are possible (paper IV).
Dansk resumé (Danish Summary)
Fisk er mobile væsner, så det virker naturligt at inkludere det rumlige aspekt når fiskebestande modelleres, men oftest ignoreres heterogenitet og bevægelse helt i modeller til vurdering af fiskebestande. Dette er hovedsageligt på grund af problemer med at estimere bevægelser. Denne afhandling beskriver metoder til at estimere fisks bevægelser med det langsigtede mål at forbedre bestandsvurderingen. Afhandlingen er opdelt i fem introducerende kapitler og fire artikler. I de introducerende kapitler forklares grundlaget for at vurdere fiskebestande, og nogle udvalgte modeller beskrives i detaljer, for at give et indtryk af hvordan disse vurderingsmodeller virker, og hvordan udeladelse af det rumlige aspekt kan føre til store fejl (ikke bare store usikkerheder) (kapitel 1). Den vigtigste datakilde til estimation af bevægelsesmønstre er mærkningsforsøg. Nogle af de klassiske modeller der anvendes på mærknings data præsenteres (kapitel 2). Disse modeller kan virke for simple, og de bygger på strenge antagelser, men de udgør et værdifuldt udgangspunkt til at forstå de mere moderne tilgange, som præsenteres senere i denne afhandling. Arkivmærker er en ret ny type af mærker, som er designet til at blive monteret på, eller operereret ind i, et dyr for at optage målinger, som (forhåbentlig) senere kan bruges til at rekonstruere dets spor. En udbredt og ret succesfuld metode til at rekonstruere disse spor beskrives, og nogle få forbedringer præsenteres (kapitel 3). Der er en forbindelse mellem den præsenterede individ baserede model for arkivmærker og moderne populations baserede modeller. Denne forbindelse beskrives sammen med to populations baserede modeller (kapitel 4). Et simulationsstudie udføres, hvori simulerede fisk tilfældigt tildeles enten et konventionelt mærke eller et arkivmærke. Det evalueres om mere præcise parameterestimater kan forventes fra arkivmærker end fra konventionelle mærker. Individ modellen udvides, så den passer sammen med den populations baserede model. Til sidst kombineres de to separate modeller til en sammenhængende model, som gør brug af begge typer af mærker (kapitel 5). Artiklerne anvender og udvider de beskrevne modeller. Den populations baserede model anvendes i et feltstudie hvor 3529 pighvarre blev udsat og fulgt i ni dage, for at se hvor hurtigt de koloniserede deres nye omgivelser. Yderligere, var det muligt at estimere deres overlevelsesrate (artikel I). Arkivmærker, der afkobles for tidligt, er et problem, specielt hvis mærket ikke rapporterer hvor og hvornår det faldt af. En model til at estimere det mest sandsynlige afkoblings punkt introduceres sammen med en metode til at kvantificere signifikansen af det fores-
viii låede punkt (artikel II). Modellen kan også være nyttig til at påvise ændringer i bevægelsesmønsteret. Arkivmærker bruger lysmålinger til at estimere længdegraden (fra lokal middagstid) og breddegraden (fra lokal dagslængde). Denne metode giver meget usikre geolokaliseringer specielt omkring jævndøgn, hvor dagene har ens længde overalt. Arkiv mærker gemmer også temperaturmålinger, og disse bruges i kombination med overfladetemperaturer målt fra satellitter, til at få meget forbedrede (mere præcise) rekonstruerede spor (artikel III). De rekonstruerede spor fra den basale individ baserede model fra kapitel 3 krydser nogle gange landområder, fordi sådanne barrierer ikke kan indbygges i modeller af den type (et tilsvarende problem opstår ved heterogent fiskeritryk). En model introduceres hvor disse udvidelser er mulige (artikel IV).
Table of Contents
Table of Contents
ix
1 Fish Stock Assessment
1
1.1 Types of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 The Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3 Extended Survivors Analysis . . . . . . . . . . . . . . . . . . . . . .
3
1.4 Statistical Fish Stock Assessment Models . . . . . . . . . . . . . . .
5
1.5 Spawning Stock Biomass and Recruitment . . . . . . . . . . . . . .
7
1.6 Why Include Migration? . . . . . . . . . . . . . . . . . . . . . . . .
10
2 Classical Tagging Models
13
2.1 Estimating Population Size from Tagging Data . . . . . . . . . . . .
13
2.2 Geographically Stratified Populations . . . . . . . . . . . . . . . . .
16
2.3 Estimation of Movement Pattern Directly from Tagging Data: Simplest case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3 Individual Based Approach
19
3.1 Archival Tags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2 Kalman Filter Tracking
. . . . . . . . . . . . . . . . . . . . . . . .
20
3.3 Extended Kalman Filter Tracking . . . . . . . . . . . . . . . . . . .
23
3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4 Population Based Approach
29
4.1 Scaling from Individuals to a Population . . . . . . . . . . . . . . .
29
4.2 Advection–Diffusion–Reaction Model . . . . . . . . . . . . . . . . .
30
4.3 Markov Model
31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
Table of Contents 5 Combining Individual and Population Based Models
37
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5.2 Simulation and Estimation . . . . . . . . . . . . . . . . . . . . . . .
38
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
Bibliography
45
Appendices A Kalman Filter Tracking
51
B Simulation of Numerous Individuals
53
Papers I
Diffusion of Fish from a Single Release Point
55
II Locating Point of Premature Detachment of Archival Tags or a Sudden Change in Movement Pattern 75 III Improving Light-based Geolocation by Including Sea Surface Temperature 87 IV Tracking Archival-tagged Individuals via the Particle Filter: The Repulsion from Dry Land and Fishing 103
1 Fish Stock Assessment
The background for this Ph.D. study is fish stock assessment. Fish stock assessment models are primarily used to answer one important question: How many fish are left in the ocean? Reliable fish stock assessment is the key element in successful management of fisheries. This chapter is intended as an introduction to this important, interesting, but rather specialized field in applied statistics.
1.1
Types of Data
The optimal data for the task would naturally be direct observations of the actual number of fish in the ocean or in some sub-area of the ocean, but direct observations of the number of fish are never available. The only thing known is what has been removed from the fish stock. Thus fish stock assessment can be very difficult.
1.1.1
Commercial Catches and Effort Information
The fish stock under consideration may be exploited by one or more commercial fleets. Fleets are typically distinguished by their type of fishing gear and nationality. For each of these fleets the total catch is divided into length-classes. Samples are taken from each length-class and age-determined in the laboratory by studying the otoliths from each fish. The age-frequencies found in the sample are scaled to match the total number of caught fish, such that an estimate is obtained of the total number of fish from each age-class and by each fleet. This is done on an annual basis for most exploited stocks. These numbers are denoted C f ,a,y . Here f = 1, . . . , n f is the fleet index, a = 1, . . . , A is the age-class index, and y = 1, . . . ,Y is the year index. Only a small fraction of the total catch is sampled, so some sampling error must be expected. The commercial fleets also report the amount of “effort” they have used to obtain their catch. These numbers are denoted e f ,y .
2
Chapter 1. Fish Stock Assessment The reported effort can be “days at sea”, “hours fishing” or some other number which is proportional to the relevant time spent to obtain the catch. It is common that effort information is missing for some fraction of the total catches. These catches are known as “residual catches” and denoted Cres,a,y .
1.1.2
Scientific Surveys
A number of scientific surveys are often available to supplement the commercial catch data. Each year these surveys try to make an index of the fish stock by fishing the same time of year, in the same area and with the same type of gear. These indices are denoted Is,a,y . Here s = 1, . . . , ns is the survey fleet number. The number of fish caught by the survey fleets is often very small (negligible) compared to the numbers caught by the commercial fleets, but the data quality can be expected to be much higher. Additional data used to compute additional outputs (explained below) include the proportion of fish in a given age-class that has reached sexual maturity, which is determined by examining the gonads of a sample of the catch at a specific time of year. This “proportion mature” is denoted pa,y . Furthermore, the average weight in each age-class is determined and denoted wa,y .
1.2
The Basic Equations
As data only consists of information about fish that are removed from the population, some structural assumptions are needed in order to be able to estimate the remaining population’s size. The following equations are common ground in stock assessment models ranging from deterministic virtual population methods (e.g., Fry 1949; Murphy 1964), to real stochastic models based on the maximum likelihood principle (e.g., Doubleday 1976; Fournier and Archibald 1982), and recent approaches based on the Bayesian paradigm (e.g., Ianelli and Fournier 1998; Lewy and Nielsen 2003). The total number of fish in a population at time t is denoted Nt . Consider a closed area where no new individuals enter or leave the population, then the number of individuals in the population is expected to decrease according to the following differential equation: d Nt = − (Ft + Mt ) Nt | {z } dt Zt
(1.1)
3
1.3. Extended Survivors Analysis Here Ft is the part of the mortality rate that is due to fishing, Mt is the natural mortality rate and Zt is the total mortality rate. Let Na,y denote the number of fish in age-class a at the beginning of year y. Na,y is expected to follow an equation similar to (1.1). Assuming constant mortality rates within each year the expected number of survivors after exactly one year is according to (1.1) Na+1,y+1 = Na,y e−Za,y .
(1.2)
This equation is known as the stock equation. Now let Ca,y denote the fishermen’s catch from age-class a during year y and Da,y denote the number of fish dying from other causes in the same period. Consequently Na,y = Na+1,y+1 + Da,y +Ca,y . | {z } | {z } died survived
(1.3)
The number of fish caught by the fishermen can be expressed as a fraction of those that did not survive, namely Ca,y =
Fa,y (1 − e−Za,y )Na,y . Za,y
(1.4)
This equation is known as the catch equation. Notice that Ca,y is the number of fish caught during the year, whereas Na,y is the number of fish remaining in the population at the beginning of the year. Assessment models are built from these basic equations by assuming further structure on the model parameters and matching observed catch with the expected catch from the catch equation.
1.3
Extended Survivors Analysis
To give an impression of the methods currently used to estimate fish stock parameters, the extended survivors analysis (XSA) method, which is a standard method used by the International Council for the Exploration of the Sea (ICES), will briefly be described. For further details see Shepherd (1999). The total catch C·,a,y from all commercial fleets including the residual fleet is assumed known. Here y = 1, . . . ,Y denotes the year the fish are caught, and a = 1, . . . , A denotes the age of the fish. For each commercial fleet with effort information and for all survey fleets the catch per unit effort CPUE f 0 = C f 0 ,a,y /e f 0 ,a,y is calculated. Here the fleet index is f 0 = 1, . . . , n f , 1, . . . , ns , as this method does not distinguish between commercial fleets and scientific surveys. Survey fleets are simply considered as fleets with constant effort, which can conveniently be set to one. The natural mortality M is assumed to be a known constant.
4
Chapter 1. Fish Stock Assessment
1.3.1
The XSA-algorithm
0: Guess the number of survivors. For each cohort initial values must be supplied for the number of fish surviving in each cohort (Na,y )a=A+1;y=1,...,Y +1 and (Na,y )a=1,...,A;y=Y +1 . 1: Calculate N from the survivors. Assuming that the number of survivors Na,y in a cohort is known it is possible to calculate the size of the one year younger stock of the previous year Na−1,y−1 . Isolate Fa−1,y−1 in the stock equation (1.2) Na,y − M. Fa−1,y−1 = − log Na−1,y−1 Substitute this into the catch equation (1.4) M C·,a−1,y−1 = 1 − (Na−1,y−1 − Na,y ). log(Na−1,y−1 ) − log(Na,y ) This equation can be solved numerically for Na−1,y−1 , but often the following approximation is used instead for simplicity Na−1,y−1 ≈ Na,y eM +C·,a−1,y−1 eM/2 .
(1.5)
This is called Pope’s approximation (Pope 1972). It is derived by pretending that the catch is simply taken exactly in the middle of the year. The remaining time the stock develops according to the stock equation (1.2) with Z = M (figure 1.1). The entire N-matrix is calculated from the survivors by applying this equation repeatedly in all cohorts. This procedure (0–1) is known as the Virtual Population Analysis (VPA) and it has previously been used in stock assessment (e.g., Fry 1949; Murphy 1964). 2: Calculate q−1 from N and C. For each fleet it is assumed that CPUE is proportional to the stock size, calculate q−1 f 0 ,a,y =
Na,y . CPUE f 0 ,a,y
3: Smooth the q−1 estimates. Assume q−1 f 0 ,a,y is constant over years and find smoothed estimates simply by averaging qˆ−1 f 0 ,a,y =
1 Y −1 ∑ q f 0,a,y. Y y=1
(Other smoothing methods are available, but they will not be presented here.)
1.4. Statistical Fish Stock Assessment Models
−M/2 −C·,a−1,y−1 e−M/2 6Nay ≈ Na−1,y−1 e Na−1,y−1
( C·,a−1,y−1 Nay -
|
{z One year
}
Figure 1.1: Pope’s approximation: The entire annual catch is removed exactly in the middle of the year. 4: Calculate Nˆ from qˆ−1 . Find new estimates of the stock size by Nˆ f 0 ,a,y =
CPUE f 0 ,a,y . qˆ f 0 ,a,y
Notice how this gives different stock size estimates for each fleet. 5: Re-estimate survivors from Nˆ f 0 . Combine the fleet specific estimate of the stock size with qˆ and re-calculate the number of survivors via the stock equation (1.2). The (weighted) average of these survivor-estimates is the new survivor-estimate. 6: Repeat 1–6 until convergence. Convergence is diagnosed when the logarithm of the ratio between the old and the new estimated number of survivors is below some predefined threshold. This method is purely deterministic and provides no information about the precision of the estimates. Furthermore, the actual assumptions underlying this algorithm are not very transparent.
1.4
Statistical Fish Stock Assessment Models
The term statistical model, as opposed to deterministic model, is here used about models where observations are treated as stochastic variables subject to measurement noise and where this noise term is a part of the specified model. From the
5
6
Chapter 1. Fish Stock Assessment description of catch and survey data in section 1.1 it should be clear that many different sources can result in considerable measurement noise. Even besides the whole sampling issue, it is reasonable to assume that the actual catches are influenced by factors not included in the model, and hence subject to measurement noise. These random factors could for instance include weather, abundance of predators/prey and spatial distribution. This section gives an example of a statistical fish stock assessment model. The presented model (Nielsen and Lewy 2002) is among the simplest imaginable, but it is sufficient to illustrate the general principles and problems in statistical fish stock assessment and differences between the deterministic and statistical approach. The model includes catch-at-age and effort data from commercial fleets and scientific surveys. All log-catches are assumed to follow normal distributions with mean values originating from the equations in section 1.2. Log-catches are used instead of catches, as catches are positive and because it is not reasonable to assume that all catches, even within the same fleet, have the same variance. Assuming common variance of the log-catches corresponds to assuming a common coefficient of variation (CV), which seems more reasonable. For the commercial catches with corresponding effort information e f ,y Q f ,a −Za,y 2 1−e Na,y , σ f . logC f ,a,y ∼ N log Za,y
(1.6)
Here f = 1, . . . , n f is the fleet index, a = 1, . . . , A is the age-class index, and y = 1, . . . ,Y is the year index. Notice that the mean value is the logarithm of the catch equation (1.4), except that the fishing mortality rate is assumed proportional to the effort e f ,y within each age-class. These constants of proportionality are known as catchabilities and denoted Q f ,a . Catchabilities are fleet specific, because different fleets may use different fishing gear, and hence catch different age-classes with different efficiencies. For the residual catch, the model includes additional parameters Ey for the missing effort Ey Qres,a −Za,y 2 1−e Na,y , σres . (1.7) logCres,a,y ∼ N log Za,y Here for the effort parameter the first years “effort” Ey=1 is fixed to one to avoid over-parameterization. For the indices from scientific surveys, each conducted on the same day-of-year with same effort each year DOYs log Is,a,y ∼ N log Qs,a e−Za,y 365 Na,y , σs2 . (1.8) Here s = 1, . . . , ns is the survey fleet index, DOYs is the day-of-year where the survey s was conducted and exp(−Za,y DOY 365 )Na,y is the actual stock size the day the survey
7
1.5. Spawning Stock Biomass and Recruitment was conducted according to the solution of (1.1). The joint catches from these scientific surveys are assumed sufficiently small, such that their effect on the total mortality can be ignored. The total mortality can be expressed as nf
Za,y = Ma,y + Ey Qres,a +
∑ e f ,yQ f ,a.
(1.9)
f =1
The initial stock sizes (N1,y )y=1,...,Y and (Na,1 )a=2,...,A are selected as model parameters. All other stock sizes (Na,y )a,y≥2 are not considered as model parameters, but as functions of previously defined model parameters given recursively by the stock equation (1.2) Na,y = Na−1,y−1 e−Za−1,y−1 ,
a ≥ 2, y ≥ 2.
(1.10)
The complete set of parameters for this model is {(N1,y )y=1,...,Y , (Na,1 )a=2,...,A , (Q f ,a ) f =1,...,n f ;a=1,...,A , (Qres,a )a=1,...,A , 2 (Ey )y=2,...,A , (Qs,a )s=1,...,ns ;a=1,...,A , (σ 2f ) f =1,...,n f , σres , (σs2 )s=1,...,ns }.
(1.11)
For plaice in the North Sea (Lewy and Nielsen 2003) this amounts to around 70 parameters. Note that this is a very simple model. It is fairly common that fish stock assessment models have hundreds of model parameters. Models with hundreds of parameters often exceed the capabilities of most standard software for optimization in non-linear models. Standard software for optimization is frequently based on finite difference approximations of the gradient, which is not precise enough for these large models. High performance optimization tools using automatic computer generated derivatives (e.g. Otter Research Ltd 2003) are frequently used to estimate parameters in these high dimensional assessment models. Notice also that the parameters describing the natural mortality Ma,y are left out of the set of parameters to be estimated (1.11). The structure of these age-structured assessment models is such, that data contain very little or no information about the natural mortality, or put differently, it is hardly ever possible to estimate both catchabilities and natural mortalities (best illustrated in Gavaris and Ianelli 2002). The common “solution” to this problem is to fix the mortality parameters to some reasonable numbers. This approach should be supplemented with a sensitivity analysis to see how key parameters are influenced by different natural mortality assumptions. The model presented here also assumes a fixed constant natural mortality rate.
1.5
Spawning Stock Biomass and Recruitment
The annual spawning stock biomass SSBy is one of the most important figures used in fisheries management. It is an estimate of the mass of sexually mature fish in
8
Chapter 1. Fish Stock Assessment the stock. The spawning stock biomass is defined by: A
SSBy =
∑ Na,ywa,y pa ,
(1.12)
a=1
where wa is the mean weight of a fish in age group a, pa is the maturity proportion in age group a, and Na,y is the estimated number of fish in age group a in year y. Recruitment is simply defined as the number of fish in the first age group. One may assume that the number of recruits in a given year depends on the number of spawning mature fish the year before. Biologists have designed quite a few different models to explain this connection between spawning stock biomass SSBy−1 and recruitment Ry , but the two most commonly used are the BevertonHolt model (Beverton and Holt 1957) Ry =
αSSBy−1 1+
SSBy−1 K
(1.13)
,
and the Ricker model (Ricker 1954) Ry = αSSBy−1 exp −β SSBy−1 .
(1.14)
These are both based on biologically plausible arguments. The Beverton-Holt model assumes the mortality rate to be a linear function of population size, whereas the Ricker model assumes the mortality rate to be time-invariant (only a function of initial stock size). Ricker vs. Beverton−Holt 600 Ricker Beverton−Holt
Recruitment in millions
500
400
300
200
100
0 0
500
1000
1500
2000
SSB in thousand tonnes
Figure 1.2: Stock-recruitment for North Sea plaice (1989-1997)
9
1.5. Spawning Stock Biomass and Recruitment These models are often difficult to verify from data and even harder to distinguish between. A typical example of stock-recruitment data can be seen in Figure 1.2. Data is only available in the region where the models are very similar. Even though it is not possible to identify the connection between spawning stock biomass and recruitment, it is often included in the models anyway. This is done because this connection is needed to do predictions of future stock sizes and because the models should include the fact that if SSB is zero, then the future number of recruits will be zero too. The structure of the age-based stock development is illustrated in figure 1.3.
# # # U U U N11
N12
N13
p
p
p
N21
N22
N23
p
p
p
@ @ @ @ @ @ R @ R @ R @ @ @ @ @ @ @ R @ R @ R @ N31
N32
N33
@ @ @ @ @ @ R @ R @ R @ p p p p p p p p p
p
p
p p NA3 NA1 NA2 "! "! "!
p
p
# U N1Y
p
p
p
N2Y
p
p
p
@ @ R @ @ @ R @ N3Y
@ @ R @ p p p
U Stock-Recruitment
p
p
p p NAY "!
p
@ @ R @ Stock equation
p
Figure 1.3: Structure of the age-based stock development The population structure in the model presented here is very rigid. This is because only the initial stock sizes in each cohort are chosen as free parameters and the rest are determined by deterministic survival according to (1.10). A more natural and less rigid approach is to consider the entire N-matrix (figure 1.3) as an unobserved stochastic process (e.g., Gudmundsson 1994; Ianelli and Fournier 1998; Lewy and Nielsen 2003). The resulting models have fewer parameters to estimate, but more unobserved states to predict. Deterministic approaches are still used in fish stock assessment, but few seriously believe that stock sizes computed from the guessed number of survivors, arbitrary natural mortality and adjusted by data, have much practical value. Statistical fish stock assessment models provide a coherent way of dealing with measurement
10
Chapter 1. Fish Stock Assessment noise and provide estimates of uncertainties of parameters of interest. Model assumptions can be investigated via standard methods such as residual plots.
1.6
Why Include Migration?
Fish are mobile creatures, so it seems natural to include the spatial aspect when modeling fish populations, but all models are simplified versions of their corresponding real system, so maybe it is wise to leave out the added spatial complexity from these already fairly complex models. Each of the following four different assumptions could justify that: 1) The spatial distribution of the fish stock is uniform in the entire exploited area. 2) The spatial distribution of the fishing effort for each fleet is uniform over the entire exploided area. 3) The spatial distribution of the fish stock is the same every year and the spatial distribution of the fishing effort is the same every year. 4) The spatial distribution of fishing effort follows the spatial distribution of the fish stock. If any of these four assumptions (figure 1.4) are fulfilled, then it does not add bias to the stock assessment to leave out the spatial component. 2) Uniform effort density
3) Different densities, but 4) Effort density same as same every year stock density
Stock
Effort
1) Uniform stock density
Exploited area (1D)
Figure 1.4: Four cases (one in each column) where non-spatial fish stock assessment would not add bias to the stock estimates. In most monitored fisheries assumptions 1)–3) can easily be ruled out. Both from biological knowledge and from reported effort and catch per unit effort. Assumption 4) could be reasonable, as fishermen probably spend more time fishing where the catch (per unit effort) is high and would rather quickly move on to another area if the catch is low. This behavior though, does not guarantee that the distribution of fishing effort is proportional to the distribution of the fish stock. If the fishermen are truly able to follow the distribution of the fish stock, they would likely spend as much time as possible in the high density areas. A worst-case scenario for a fish stock assessment model without the spatial component could be the following (illustrated in figure 1.5). Assume that the stock under consideration have some preferred area, but this area supports only a fraction of the total stock. The rest of the stock is scattered in the remaining area. The fishing effort is concentrated in the area where the fish concentration is high
11
1.6. Why Include Migration? (the preferred area), and the fishing effort is about the same every year. As fishing removes fish from the preferred area, the fish from the surrounding areas migrate into the preferred area. If the fishing pressure is too high the surrounding areas will gradually contain fewer fish, but non-spatial fish stock assessment will not show this until the stock in the preferred area starts to decline (year 4 in figure 1.5). Even when the stock in the preferred area starts to decline, the non-spatial fish stock assessment will detect a much smaller decline than what has really happened, because it is unaware that the surrounding area was drained first. Year 2
Year 3
Year 4
Stock
Effort
Year 1
Exploited area (1D)
Figure 1.5: Four years (one in each column) illustrating a worst-case scenario for non-spatial fish stock assessment. The larger part of the area is drained, but the model is unaware because there is no effort in that area. Leaving out the spatial component in fish stock assessment can lead to serious mistakes, but the existence of several fleets and scientific survey fleets reduces the risk. In the best imaginable case leaving out the spatial component will merely introduce additional variability into the catches from surveys and fleets. Results from applied non-spatial stock assessment models suggest that a little of both could be happening. Attempts have been made to divide a single stock assessment into two areas (Lewy personal correspondence) and surprisingly different estimates were obtained. This suggests that some bias is introduced by leaving out the spatial component. Observed catches most often have huge uncertanties, even though the models have hundreds of parameters. This could suggest that not all is explained by the models yet. The idea that the spatial component is really an important missing piece from these fish stock assessment models is certainly consistent with observed model behavior. A final more practical reason to consider adding the spatial component is to be able to predict the effect of a certain type of management advice. The current non-spatial fish stock assessment models are unable to predict the effect of closing an area for fishing, or restricting the effort in a spatial dependent way. Notice that in order to do that, it is not even sufficient to know the population density. The traffic in and out of the closed area must also be estimated.
12
Chapter 1. Fish Stock Assessment
2 Classical Tagging Models
Modern analysis of tagging experiments often involve lengthy computations including Monte Carlo simulations, numerical solvers and/or optimizers, but the history of tagging experiments predates the computer revolution by far. This chapter shows a selection of what can be calculated directly from tagging observations. The assumptions behind these models may seem strict, but these models are often a valuable starting point for more advanced models. Furthermore, deeper understanding of complicated models stems from fully understanding simplified versions.
2.1 2.1.1
Estimating Population Size from Tagging Data The Petersen Population Estimate
Estimating population size is among the first and simplest uses of tagging data. Assume M individuals from a population of unknown size N0 are caught, marked and released back into the population. After a while a new catch of C individuals are taken from the population and from these individuals R have a mark already (re-captives). These types of experiments are known as mark–recapture studies. Under certain “mixing” assumptions, which are explained below, it is reasonable to assume that the fraction of marked individuals in the second catch is a fair estimate of the fraction of marked individuals in the population. This leads to the intuitive Petersen population estimate b0 = MC N R
(2.1)
(Petersen 1896). This estimate is very simple, because the assumptions on which it is based are fairly strict. First it is assumed that the population size is constant between mark and recapture. No new individuals can enter the population (birth or immigration), and no individuals can leave the population (death or emigration). This assumption is only valid for fish populations, if the time between mark and recapture is short. Second it is assumed that all individuals, marked as well as
14
Chapter 2. Classical Tagging Models unmarked, have equal probability of being caught. This assumption requires that the population had time to mix, which might be problematic according to the first assumption. Furthermore, this second assumption requires that the actual marking mechanism does not affect the probability of being caught. Anderson and Bagge (1963) reported that plaice marked with the Petersen discs were caught in meshes they would otherwise have escaped, which is a clear violation of this assumption. Finally, it is assumed that no individual looses its mark and that all marked animals are reported. The Petersen population estimate is problematic if the number of recaptures R is zero, then the Petersen estimate is infinity. This problem can be avoided by using the adjusted population estimate b0 = (M + 1)(C + 1) − 1 N (R + 1)
(2.2)
(Chapman 1951). This population estimate also has less bias for small sample sizes. The uncertainty associated with the adjusted Petersen population estimate can be quantified by the variance estimate b0 ) = c N var(
(M + 1)(C + 1)(M − R)(C − R) (R + 1)2 (R + 2)
(2.3)
(Seber 1982). This variance estimate can be used to construct a confidence interval based on the normal distribution approximation q b0 ± z1− α var( b0 ) c N N (2.4) 2
A confidence interval, which is more appropriate for small sample sizes, but requires an iterative solver, is described in Seber (1982).
2.1.2
The Schnabel Population Estimate
The Petersen method can be extended to handle multiple mark and recapture rounds. Consider the situation where first M0 individuals are caught, marked and released back into the population. After a while a new catch of C1 individuals are taken from the population and from these individuals R1 have a mark already. The caught individuals, that were not already marked, are now marked and all caught individuals are released back into the population. The total number of marked individuals in the population is now M1 = M0 +C1 − R1 . After a while a new catch C2 is taken, and this continues for n rounds. After n rounds the observations can be summarized by the numbers M0 , . . . , Mn−1 , C0 , . . . ,Cn , and R1 , . . . , Rn . Notice that Mi−1 is the number of marked individuals in the population just before the catch Ci is taken.
15
2.1. Estimating Population Size from Tagging Data Assuming that each catch Ci is sufficiently small compared to the population size N0 to ignore complications of sampling without replacement, it is possible to write down the maximum likelihood equation for the population estimate n (Ci − Ri )Mi−1 = ∑ Ri ∑ i=1 i=1 N0 − Mi−1 n
(2.5)
(Schnabel 1938). This maximum likelihood equation can be solved numerically, but with the additional assumption that the number of marked individuals Mn−1 is negligible compared to the population size the Schnabel population estimate is obtained n c0 = ∑i=1nCi Mi−1 N ∑i=1 Ri
(2.6)
(Schnabel 1938). The assumptions regarding closed population, complete mixing, and all marks reported are the same for this method as for the Petersen method. b0 than of N b0 It turns out to be simpler to obtain an estimate of the variance of 1/N directly. The variance estimate is 1 ∑n Ri c var = n i=1 (2.7) b0 N (∑i=1 Ci Mi−1 )2 This variance estimate can be used to construct an approximative confidence inb0 terval for the Schnabel population estimate N
1 1 b0 N
r + t1− α ;n−1 2
c var
1 b0 N
;
1 1 b0 N
r − t1− α ;n−1 2
c var
1 b0 N
(2.8)
(Schnabel 1938).
2.1.3
The Jolly–Seber Population Estimate
Assuming a closed population is seldom realistic for more than very short time intervals. The following method is closely related to the Schnabel method, but is designed for an open population, where new individuals can enter the population (birth or immigration) and individuals can leave the population (death or emigration). The setup is a multiple mark and recapture experiment, as for the Schnabel method. The individuals are caught, tagged and released in a number of rounds, but contrary to the previous setup it is necessary here to keep track of when a marked individual was last caught. The method involves a bit more bookkeeping. Let Ct be the total catch in round t from which Rt individuals have a mark. St denotes the number of individuals that are released with marks from round t
16
Chapter 2. Classical Tagging Models (should be the same as Ct , but some could have died during the catch–release process). Furthermore, let Rr,t denote the number of individuals with a mark caught in round t, which were last caught in round r, and Wt denote the number of the St individuals released in round t and caught again in a later round. Finally, let Zt denote the number of individuals marked before round t, but not caught before after round t. The population is considered open, so marked animals can leave the the population. Seber (1982) showed that the number of marked animals in the population at time t can be estimated by bt = St Zt + Rt M Wt
(2.9)
This estimate is inserted into the Petersen estimate to obtain an estimate of the population at time t b bt = Mt Ct N Rt
(2.10)
(Seber 1982). This is known as the Jolly–Seber population estimate. Besides the population estimate Seber (1982) also provides formulas for estimating the survival rate (“deaths” include tag-losses and emigration), the dilution rate and associated variance estimates. Fixed population size is not assumed here, but assumptions regarding complete mixing and all marks reported, are the same for this method as for the Petersen method. Furthermore, it is assumed that all individuals marked as well as unmarked have the same probability of surviving.
2.2
Geographically Stratified Populations
The assumption that all individuals “blend” between marking rounds is often not plausible. If the population is geographically stratified this assumption may be justified within each stratum, but not for the entire population. If individuals also move between strata, but not enough to blend completely, the probability of these moves should also be part of the model. Consider again the simple mark and recapture experiment with only two rounds, but now with several geographical strata s = 1, . . . , S. In the first round a number of individuals within each stratum are marked M = (M1 , . . . , MS )0 . In the second round a catch is taken from each stratum C = (C1 , . . . ,CS )0 . Assuming that the fish are individually marked (tagged), or at least marked differently (for instance a different color) in each stratum, the recapture matrix R can be constructed. The (i, j)’th element of R is the number of individuals marked in stratum i and recaptured in stratum j.
2.3. Estimation of Movement Pattern Directly from Tagging Data: Simplest case It is necessary to assume that the matrix R is non–singular, which has the implication that it is not possible to include strata where no fish are marked, or strata where no marked fish are recaptured. If it happens that no marked fish are recaptured in a stratum, a practical solution is often to join a few strata. The maximum likelihood estimate for the total population size is b0 = C0 R−1 M N | {z }
(2.11)
Pb
(Darroch 1961). This estimator is very similar to the Petersen estimator (2.1), and it is known as the stratified Petersen estimator. The assumptions of this model are fairly similar to the assumptions of the Petersen method. The population is closed. All individuals within the same stratum, marked as well as unmarked, have equal probability of being caught. Individuals move and are caught independently of one another. The probability that an individual within the j’th stratum is caught is denoted p j , and the maximum likelihood estimates of these probabilities are given in (2.11). With these estimates in place it is also possible to estimate the probabilities of moving from one stratum i to another j. This matrix of movement probabilities is denoted Θ. The estimate is b = (diag(M))−1 R diag(P) Θ
(2.12)
(Darroch 1961). This matrix Θ is the first example of the kind of estimates needed to include movement information in the management of fish stocks. If for instance stratum j was declared a marine protected area (no fishing allowed), it would be possible to estimate how many individuals would stay, enter, or leave the protected area and hence evaluate the effect of closing the area. A benefit of the stratified Petersen estimator is that it can be calculated directly from data, no numerical computations are needed, but the data requirements are really high. In marine systems the tag retention rate is often very low and to mark enough individuals to be sure to get recaptures in more than a few strata would be an enormous effort.
2.3
Estimation of Movement Pattern Directly from Tagging Data: Simplest case
Consider the situation where M individuals are caught, marked, and released from a specific point (x0 , y0 ) at time t0 = 0. At different later times t1 , . . . ,tn some of these individuals are caught at positions (x1 , y1 ),. . .,(xn , yn ). The rest of the individuals are never seen again. Assuming individuals move independent of each other and
17
18
Chapter 2. Classical Tagging Models that natural mortality and the probability of getting caught are uniform in both time and space, it is possible to estimate the mean movement vector α = (u, v)0 and the isotropic diffusion coefficient D. These formulas are usually presented (e.g., Rijnsdorp 1995; Jones 1959) in polar coordinates to get slightly simpler formulas, but this author prefers to avoid the conversion and continue in Cartesian coordinates. Define ∆xi = xi − x0 and ∆yi = yi − y0 . The mean movement vector is estimated by 0 ∑ ∆x ∑ ∆y b= α , (2.13) ∑t ∑t If the time unit is days, this vector indicates the average daily displacement of the population. The isotropic diffusion coefficient is estimated by ! 2 2 2 2 ∆x ( ∆x) ∆y ( ∆y) 1 ∑ ∑ b= − − +∑ (2.14) D 4(n − 1) ∑ t t ∑t ∑t This is the unbiased estimate. The maximum likelihood estimate has 4n in the denominator instead. The diffusion coefficient will reappear later in this thesis both in individual and population based models. In this example it can best be described in relation to the variance of the population. The population variance after t days will be 2DtI2×2 .
3 Individual Based Approach
The focus in individual based models is primarily on reconstructing the movements of a single individual. Using only a single individual it may seem that there is a long way to describing the movements of the entire population, and there is. However, studying the detailed movements of a few individuals can give some idea about the large scale population movements. After all, the population consists of individuals, and far more important, studying few individuals in detail can potentially reveal features of the movement pattern, for instance delimit spawning locations or migration corridors, that would require an enormous amount of mark and recapture studies to detect.
3.1
Archival Tags
Archival tags, also referred to as data storage tags (DSTs) or pop-up satellite archival tags (PSATs), are electronic devices designed to be mounted on, or surgically implanted into, an individual to record measurements about the environment as the individual travels through it. These measurements typically consist of lightlevel, temperature, and pressure (depth). A few different archival tags can be seen in figure 3.1. Data from these archival tags are retrieved when the individual is caught or in the case of PSATs. at a pre-programmed release time. Most PSATs also have a release mechanism that is triggered by pressure in case the individual dies and drops below a certain depth (1200m) or remains at the same depth for a long time (four days). After the tag has been released it starts transmitting data to the tag’s manufacturer via satellite. The measurements are used to obtain raw geo-location estimates either by the tag’s manufacturer or via proprietary software supplied by the tag’s manufacturer. Details of these algorithms are unfortunately unknown, but basically estimates of dawn and dusk from the measured light intensities are used to calculate longitude (from local noon) and latitude (from local day length) (Musyl et al. 2001). Naturally, these raw light-based geo-locations are not very accurate and raw geolocations are often mistakenly placed hundreds of kilometers from their actual location (Musyl et al. 2001; Sibert et al. 2003), which can easily be on dry land.
20
Chapter 3. Individual Based Approach
Figure 3.1: A pop-up satellite archival tag on tagging pole, 2 small archival tags beneath it and a pop-up satellite tag rigged for a shark harness. Also shown are 2 small plastic spaghetti tags. These small plastic tags were once the mainstay of fisheries tagging programs and are still useful depending on the research application. Photo by M. K. Musyl, University of Hawaii/Joint Institute for Marine and Atmospheric Research. For some demersal species it is possible to obtain more accurate geo-location estimates by matching the pressure measurements while the fish is resting at the sea-bed to tidal information (Hunter et al. 2003).
3.2
Kalman Filter Tracking
To make sense out of these light based geo-locations Sibert et al. (2003) suggested a state-space model and estimation via the Kalman filter. This method has become widely used among biologists because of two advantages: 1) It provides a parametric representation of the entire track of the individual and these parameters can be given a large scale interpretation in advection diffusion population models. 2) It provides a more resonable and less variable track than produced by the raw geo-locations from the tag. Much of the work in this Ph.D. study has been devoted to study, use, improve and further develop this approach.
3.2.1
The State-space Model
The basic model for an observed track of an individual is a state-space model, where the transition equation describes the movements of the fish in nautical miles
21
3.2. Kalman Filter Tracking from longitude zero and latitude zero. A biased random walk model is assumed αi = αi−1 + ci + ηi ,
i = 1, . . . , T
(3.1)
Here αi is a two dimensional vector containing the coordinates at time ti . ci is the drift (or bias) vector describing the deterministic part of the movement, and finally ηi is the noise vector describing the random part of the movement. The deterministic part of the movement is assumed to be proportional to time ci = (u∆ti , v∆ti )0 , and the random part ηi is assumed to be serially uncorrelated and follow a two dimensional Gaussian distribution with mean vector 0 and covariance matrix Qi = 2D∆ti I2×2 . The measurement equation of the state-space model is a non-linear mapping of the nautical miles coordinates to the usual sphere coordinates in degrees of longitude and latitude. The measurement equation describing the actual raw geo-locations is given as yi = z(αi ) + di + εi ,
i = 1, . . . , T
Here z is the coordinate change function given by z(αi ) =
αi,1 60 cos(αi,2 π/(180·60)) αi,2 60
(3.2)
(3.3)
The idea behind this transformation between nautical miles and the usual sphere coordinates in degrees is that a random walk would not make much sense in longitude/latitude coordinates, because the distance between the same two degrees of longitude is much greater near equator than near the poles. The observation bias di = (blon , blat )0 describes systematic geo-location error (for instance if the internal clock in the archival tag is slightly off). The measurement error εi is assumed to follow a Gaussian distribution with mean vector 0 and covariance matrix 2 0 σ (3.4) Hi = lon 2 0 σ lati The variance of light based latitude measurements is closely related to the equinox, because measurements close to the equinox have large latitude errors, as days have almost the same length everywhere. Sibert et al. (2003) suggests three different variance structures for the latitude geo-location error. 1) Constant, which is mainly usable for fairly short tracks. 2) A parametric structure depending on the time since last solstice, which is the general recommendation (described below). 3) Daily derivations, but limited by a fixed
22
Chapter 3. Individual Based Approach prior, which is mainly used to investigate the variance structure by plotting the estimates. The parametric latitude variance structure is constructed such that observations near an equinox are considered highly uncertain and observations near a solstice are considered most accurate. The structure is given by . 2 2 (3.5) cos2 2π(Ji + b0 )/365.25 = σlat σlat i
0
Here Ji is the number of days since last solstice prior to all observations, b0 is a parameter to be estimated expressing the number of days prior to the equinox, where the latitude uncertainty is maximal. σ 2 is the baseline latitude variance lat0 parameter.
3.2.2
The Kalman Filter
The Kalman filter (Harvey 1990) is used to predict the states and calculate the likelihood of a linear state-space model, but the state-space model described here is not linear due to the function z described in (3.3). Sibert et al. (2003) suggest the following ad-hoc approximation. Writing the function z as 1 60 cos(αi,2 ·π/(180·60)) 0 z(αi ) = (3.6) αi 1 0 60 | {z } Zei
and then inserting the optimal estimator of αi into Zei gives a linear approximation. The Kalman filter equations are written as bi|i−1 = α bi−1 + ci α
(3.7)
Pi|i−1 = Pi−1 + Qi Fi = Zei Pi|i−1 Zei0 + Hi
(3.8)
bi|i−1 − di wi = yi − Zei α bi = α bi|i−1 + Pi|i−1 Zei0 F −1 wi α
i 0 −1 e e Pi = Pi|i−1 − Pi|i−1 Zi Fi Zi Pi|i−1
(3.9) (3.10) (3.11) (3.12)
(Harvey 1990). The filter is started by calculating α0 = Ze0−1 y0 and assuming this position to be known without error (P0 = 02×2 ).
3.2.3
The Likelihood Function
The model parameters of this model with the parametric equinox dependent latitude variance structure (3.5) are θ = (u, v, D, blon , blat , σ 2 , σ 2 , b0 )0 . The negalon lat0
23
3.3. Extended Kalman Filter Tracking tive log-likelihood function based on the Kalman filter is given by T
T
i=1
i=1
`(θ ) = T log(2π) + 0.5 ∑ log(|Fi |) + 0.5 ∑ w0i Fi−1 wi .
(3.13)
(Harvey 1990). Estimated values of model parameters are found by minimizing ` with respect to θ θb = arg min `(θ )
(3.14)
θ ∈Θ
Here Θ is the domain of the model parameters expressing lower and upper bounds.
3.3
Extended Kalman Filter Tracking
During a visit to John Sibert’s department at University of Hawai’i/Joint Institute for Marine and Atmospheric Research the Kalman filter approach was further developed, as part of this Ph.D. study. These contributions are parts of a manuscript in preparation (Sibert et al. in prep.). This manuscript did not meet the deadline for this thesis, but for the purpose of this thesis the contributions made by this Ph.D. study have been extracted. The minor adjustments of the method are described in this section, and a method to locate the most likely position of a premature tag detachment is presented in a separate manuscript (paper II).
3.3.1
Latitude Variance Structure
The latitude variance structure (3.5) is inspired by findings from the mooring experiment (Musyl et al. 2001). This experiment showed that the latitude error is highest near an equinox, but not exactly at the equinox. The highest latitude error usually occurs a few days (often ten) before or after the equinox. This was built into (3.5) by including a model parameter b0 that would estimate a shift of the variance curve. This was the best solution for individuals that were only at liberty during one equinox. However, closer investigation of the mooring experiment indicated that this shift was alternating, such that the latitude variance was highest a few days towards the winter side of the equinox. Possible reasons for this is given in Hill and Braun (2001). The latitude variance structure was updated to . 2 2 2 si σlat = σlat cos 2π(Ji + (−1) b0 )/365.25 + a0 (3.15) i 0 where Ji is the number of days since last solstice prior to all observations, si is the season number since the beginning of the track (one for the first 182.625 days, then two for the next 182.625, then three and so on). a0 , b0 and σ 2 are model lat0 parameters. The additional parameter a0 builds in an upper limit to the latitude variance, after all, it is pretty safe to assume that the tagged individual is still on this planet. The two latitude variance structures are illustrated in figure 3.2.
24
Latitude variance
Chapter 3. Individual Based Approach
0
200
400
600
0
200
400
600
Days
Figure 3.2: The vertical fat lines indicate times of equinox. The old latitude variance structure (left) is shifted slightly to the same side every time, and the maximal variance size is unlimited. The new latitude variance structure (right) is shifted towards the winter side, and a maximum variance size build in.
3.3.2
Extended Kalman filter
As the model described in section 3.2 is non-linear, because of the coordinate change function z (3.3), an approximation is needed to apply the Kalman filter. The extended Kalman filter (Harvey 1990) uses a first order Taylor approximation bi|i−1 around the optimal estimator α bi|i−1 ) + Zbi (αi − α bi|i−1 ) z(αi ) ≈ z(α
(3.16)
Here Zbi is the first derivative (or the Jacobi matrix) of the function z, which is calculated as z0 (αi ) =
∂ z(αi ) = ∂ αi
1 60 cos(αi,2 ·π/(180·60))
αi,1 π sin(αi,2 ·π/(180·60)) 180(60 cos(αi,2 ·π/(180·60)))2
0
1 60
(3.17)
bi|i−1 is inserted into the matrix in (3.17) to obtain Zbi . The optimal estimator α Notice how this approximation only differs from the ad-hoc approximation Ze in (3.6) in the last element of the first row. This element ∂ zi,1 /∂ αi,2 is left out by using the plug-in approximation Zei . The updated extended Kalman filter equations
25
3.3. Extended Kalman Filter Tracking are written as bi|i−1 = α bi−1 + ci α
(3.18)
Pi|i−1 = Pi−1 + Qi Fi = Zbi Pi|i−1 Zbi0 + Hi
(3.19)
bi|i−1 ) − di wi = yi − z(α bi = α bi|i−1 + Pi|i−1 Zbi0 F −1 wi α
i 0 Pi = Pi|i−1 − Pi|i−1 Zbi Fi−1 Zbi Pi|i−1
(3.20) (3.21) (3.22) (3.23)
b0 = z−1 (y0 ) and assuming this position to be The filter is started by calculating α known without error (P0 = 02×2 ). As seen above the linear approximation is used over and over again in the Kalman equations, which probably makes it fairly important to use the best possible approximation.
3.3.3
Fixed Detachment/Recapture Point
The last position observed in the track is either the recapture position, or the point where the tag detached and started to transmit to the satellite. In both cases this position is known with great accuracy either from the fisherman’s notes, or from the communication with the satellite. If the last position is known without error, or with an error term which is negligible compared to the other observations, it can be used as a fixed point. This corresponds to setting the measurement error and the measurement bias to zero for the last observation (HT = 02×2 and dT = 02×1 ). This results in a predicted (and most probable) track, ending in the fixed pop-off location.
3.3.4
Most Probable Track
The Kalman filter and the maximum likelihood principle supply estimates of the model parameters and the predicted track of the individual. A point on the predicted track at any given time point is calculated using all observations available at bi = E(αi |y1 , . . . , yi ). Once the entire track is known it is possible that time, that is α to obtain better estimates with smaller variance. What is here denoted “the most probable track” is calculated using all observations, after the parameters have been bi|T = E(αi |y1 , . . . , yT ). estimated. A point on the most probable track is α The actual computation of the most probable track is done in a single backwards updating sweep of the predicted track. The last point of the most probable track is identical to the last point of the predicted track, as all observations were available to the predicted track at the final point. The last point of the most probable track bT |T = α bT and the prediction error of the last point is PT |T = PT . The is known α
26
Chapter 3. Individual Based Approach following equations are used to recursively compute the previous points of the most probable track. ai|T = ai + Pi? (ai+1|T − ai − ci−1 )
(3.24)
Pi|T = Pi + Pi? (Pi+1|T − Pi+1|i )Pi? 0 ,
where
−1 Pi? = Pi Pi+1|i
(3.25)
In textbooks on the Kalman filter (Harvey 1990) this technique is known as smoothing, as the resulting track most often is smoother than the predicted track.
3.4
Examples
The two examples in this section are taken from two large studies. This Ph.D study has contributed to these studies by analyzing numerous tracks via the extended Kalman filter method. The first is a study of the horizontal movements of Olive Ridley sea turtles in the eastern tropical Pacific (Swimmer et al. in prep.). The second is a study of temporal and spatial movement patterns in relation to oceanographic conditions for epipelagic sharks in the central Pacific ocean (Laurs et al. in prep.). Both of these studies are ongoing.
3.4.1
Olive Ridley Sea Turtle Raw geo−locations
Prediction from previous
Prediction from all
18
Latitude
16 14 12 10 8 266
268
270
272
274
266
268
270
272
274
266
268
270
272
274
Longitude
Figure 3.3: The raw light based geo-locations connected with lines (left), the prediction track produced by the extended Kalman filter (middle), and the most probable track (right) of an Olive Ridley sea turtle off the coast of Costa Rica. Deployment point (∇) and detachment point (∆) are known positions. The Olive Ridley sea turtle (figure 3.3) was accidently caught in longline fishing gear off the coast of Costa Rica on 29 November 2001. An archival tag was
27
3.4. Examples attached and the tag recorded for almost two months before it was detached and transmitted on 27 January 2002. The raw light based geo-locations (figure 3.3 left frame) are often on land and the apparent movement pattern is highly unlikely. The predicted (figure 3.3 middle frame) and most probable track (figure 3.3 right frame) from the extended Kalman filter is a remarkable improvement. The daily movements now have a realistic magnitude and the track now remains in the water. Notice that the model is unaware of any land areas and only uses the raw light based geo-locations to estimate the track. In all honesty, estimated tracks from the extended Kalman filter do sometimes intersect with land areas, but typically the estimated track is far more plausible than the raw geo-locations.
3.4.2
Oceanic White-tip Shark
30
202
25
198 194
50
100
150
200 Latitude
0
20
10
−20
Date
22Jun2003
3May2003
14Mar2003
23Jan2003
4Dec2002
−60
Latitude
15
60
Days at liberty
20
Longitude
206
Estimated track of Ocean White Tip Shark
194
196
198
200
202
204
Longitude
Figure 3.4: The raw light based geo-locations (×) are included on the longitude figure (upper left) and the latitude figure (lower left), but left out of the geographic plot of the track (right) for better scaling. The prediction track (light line) and the most probable track (dark line) are in all frames. Deployment point (∇) and detachment point (∆) are known positions. The Oceanic White-tip shark (figure 3.4) was caught and tagged north-east of the island Oahu (Hawai’i) on 4 December 2002. The tag recorded for more than seven months before it was detached and transmitted on 11 July 2003. This track is an excellent example of two issues when tracking archival tagged individuals. First,
28
Chapter 3. Individual Based Approach it illustrates the importance of the equinox adjusted latitude variance. Considering the latitude coordinate plot of figure 3.4 it is evident that the measurement uncertainty increases dramatically near the equinox (20 March). The estimated track ignores these wild north-south excursions, because this is accounted for in the variance structure. Second, it illustrates that an archival tag provides a lot more insight than a conventional tag. With a conventional tag on this shark only the deployment point (∇) and the detachment point (∆) would be observed (provided it was recaptured). These two points alone only suggest a very short route, but the archival tagged track is much more interesting.
4 Population Based Approach
Population based models are really what is needed in order to use estimated movement patterns in fish stock assessment models. This chapter describes two modern approaches, but first it describes the link between the individual and the population based models.
4.1
Scaling from Individuals to a Population
Backtracking to the individual based approach the basic movement model (3.1) for an individual is u αi = αi−1 + ∆ti + ηi , i = 1, . . . , T (4.1) v the random part ηi is assumed to be serially uncorrelated and follow a two dimensional Gaussian distribution with mean vector 0 and covariance matrix 2D∆ti I2×2 . This model is a discrete time version of the Itô-sense continuous time stochastic differential equation (SDE) dαt =
u v |{z} f
√ dt + |
2D √0 dBt 0 2D {z }
(4.2)
g
Here Bt is a standard 2-dimensional Brownian motion, implying that (Bt+∆t − Bt ) ∼ N (0, ∆tI2×2 ). More generally the individual based movement model can be expressed as a SDE of the form dαt = f (αt ,t, θ )dt + g(αt ,t, θ )dBt
(4.3)
Here αt is the n-dimensional state of the individual at time t. The state is typically a position, but could also include velocity or other relevant quantities. f is a vector function of dimension n, g is a n × k-valued matrix function, θ is a vector of model parameters and Bt is a standard k-dimensional Brownian motion.
30
Chapter 4. Population Based Approach The Fokker–Planck equation (Gardiner 1985), also known as the forward Kolmogorov equation, describes the time evolution of the density N(αt ,t, θ ) of the individual(s) via the advection–diffusion equation ∂ N = −∇ · ( f − ∇ 21 gg0 )N − 12 gg0 ∇N ∂t
(4.4)
(arguments of N, f and g are left out for compactness). If the “population” consists of one individual, the Fokker–Planck equation (4.4) describes the probability density of the state of that individual. If the population consists of several independent individuals, each following the SDE (4.3), the integral of N over some state-volume expresses the expected number of individuals in that volume. Returning to the simple individual based 2-dimensional model (4.2) with constant drift parameters and constant isotropic diffusion, the Fokker–Planck equation simplifies to ∂N ∂N ∂ 2N ∂ 2N ∂ N = −u −v +D 2 +D 2 ∂t ∂ αt,1 ∂ αt,2 ∂ αt,1 ∂ αt,2
(4.5)
If the population to be described is a number of individuals released at a single point in an unbounded area the solution to this partial differential equation (PDE) is the density of a Gaussian distribution, scaled to the number of individuals released. In general this equation must be solved numerically though.
4.2
Advection–Diffusion–Reaction Model
Advection–diffusion models are widely used in animal ecology (e.g. Skellam 1951 and Okubo 1980). The following advection–diffusion–reaction (ADR) model has been applied in population models of fish movements (e.g. Sibert and Fournier 1994; Sibert et al. 1999; Adam and Sibert 2001). The time development of the density N of the tagged population is assumed to follow the ADR equation ∂ uN ∂ vN ∂ ∂N ∂ ∂N ∂N =− − + D + D − ZN (4.6) ∂t ∂x ∂y ∂x ∂x ∂y ∂y | {z }| {z } | {z } advection reaction diffusion Here N is a function of the longitude x and latitude y coordinates, the time t, and finally of model parameters θ = (u, v, D, Q, M) and the fishing effort e. The additional model parameters Q and M are parts in the total mortality rate Z. Q is the catchability and M is the natural mortality rate. The ADR equation is solved numerically by a finite difference partial differential equation solver on a mesh consisting of a number of cells nc . Let j = 1, . . . , nc be
31
4.3. Markov Model the index of these cells. The total mortality rate in the j’th cell with center (x j , y j ) at time t is defined as Z j,t = M + ∑ Fj,t, f ,
where Fj,t, f = Q f e j,t, f
(4.7)
f
where f = 1, . . . , n f is the index of the different fleets targeting these fish. The expected number of tagged individuals caught in cell j during the time interval from t − 1 to t by fleet f is calculated from the catch equation (1.4) as Fj,t, f C j,t, f = 1 − exp(−Z j,t ) N j,t (4.8) Z j,t The actual observed catch C j,t, f is assumed to follow a Poisson distribution with mean equal to the expected catch. The negative log-likelihood function becomes `(θ |C, e) ∝ ∑ C j,t, f −C j,t, f log(C j,t, f ) (4.9) j,t, f
Alternatively the negative binomial distribution can be used to allow the variance of the observed catch to be greater than its mean. The population density of the tagged individuals is initialized, at the time of the first release t = 0, by setting the density of all cells to zero, except in the cell where the release took place. Here the density is set to the number of released individuals. All subsequent releases are added to the density simply by adding them to N at the right cell at the right time. Notice, that this model uses only one common population density for all tagged individuals. All information about when and where the individuals were released is lost as soon as they have been added to the density. The likelihood only depends on the number caught nothing else. Later applications of this model (Adam and Sibert 2001; Sibert, personal communication) have improved this by dividing the tagged population into release cohorts and modeling a density via the ADR model for each cohort. In Sibert et al. (1999) the movement parameters u, v and D are assumed constant within each combination of predefined seasons and regions. In a model with two seasons and ten regions this amounts to a total of 60 movement parameters. The fishing effort is separated into five fleets giving five catchability parameters and the natural mortality is assumed constant. Hence, the total number of parameters to be estimated is 66. Adam and Sibert (in press) extend this by using a multi– layer feed forward neural network to allow the movement parameters to depend on additional variables like for instance sea temperature and ocean depth in a very flexible way.
4.3
Markov Model
Solving the ADR equation numerically is a very general approach and the discrete approximation in time and space can be as fine as desired, provided that enough
Chapter 4. Population Based Approach
58
computer power is available. A different approach to estimate these movement parameters is offered by Deriso et al. (1991). The approach presented here differs in some ways from the approach in Deriso et al. (1991), but the concept is the same.
1
3
2
5
4
6
7
8
9 10
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 56
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82
54
Latitude
53 54 55 56 57 58 59 60 61 62 63
83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100101 102103104105
52
32
106107108 109110
−4
−2
0
2
4
6
8
Longitude
Figure 4.1: Numbered states for a Markov model for North Sea plaice. From each state it is possible in one step to stay put, move to one of the surrounding (max eight) states, or to move to one of the two states not on the map (dead D or caught C ). Consider again the situation where tagged individuals are released in a grid consisting of a number of grid cells nc (see figure 4.1). Let i = 1, . . . , nc be the index of (k) these cells, and let αt be the state of the k’th tagged individual at time t. At time t the individual can either be in one of the cells, it can be dead D from natural causes or it can be caught C. Hence the state space is (k)
αt
∈ {1, . . . , nc } ∪ {D} ∪ {C}
(4.10)
A Markov model is assumed, which implies that the future depends only on the present, not on the past. The transition matrix P describes the probabilities of moving from state i to another state j in one time step (k) (k) Pi, j = P αt+1 = j αt = i (4.11) and because a Markov model is assumed the corresponding n-step probabilities can be calculated as (Pn )i, j .
33
4.3. Markov Model If the time steps are chosen sufficiently small, it is reasonable to assume that in one step it is only possible to move to one of the (max eight) surrounding states. However, there is a balance to be aware of here. If the time steps are chosen too small compared to the size of the cells and to the velocity of the individuals it becomes impossible to move across a cell in two steps, which could contradict the Markov property that the probability of leaving a cell in any direction is independent of how long the individual stayed there. The probability of getting caught or dying from natural causes in each step can be calculated from the catch equation (1.4). The probability of staying in the same state can be calculated as the probability of surviving minus the probability of moving to a different cell. The probability of staying dead, or staying caught is one. The complete transition matrix becomes
Pi, j =
εE
, if j east of i ∈ {1, . . . , nc }
εNE .. .
, if j north-east of i ∈ {1, . . . , nc }
εSE
, if j south-east of i ∈ {1, . . . , nc }
Qei,t Qei,t +M (1 − exp(−Qei,t − M)) M Qei,t +M (1 − exp(−Qei,t − M))
, if i ∈ {1, . . . , nc } and j = {C}
exp(−Qei,t − M) − ∑ ε 1 0
(4.12)
, if i ∈ {1, . . . , nc } and j = {D} , if i = j ∈ {1, . . . , nc } , if i = j = {C} or i = j = {D} , otherwise
This transition matrix is presented in the case where only one fleet is fishing on the stock, but it is straightforward to extend this to include more fleets. The sum of the probabilities of moving to a surrounding cell ∑ ε only includes the directions where it is actually possible to move, and not the directions where the modeling area is delimited by choice or by land areas. Most elements in P are zero. (k)
The k’th individual is released at time t0 in cell i(k) . If the k’th individual is caught (k)
before time T where the study is ended, its recapture time tC and recapture cell j(k) are recorded. If the k’th individual is not caught before time T only its release information is known. The k’th individuals contribution to the likelihood is calculated by (k) (k) PtC −t0 −1 P (k) j ,{C} , if recaptured before T i(k) , j(k) (k) L (θ |X) = (4.13) (k) 1 − PT(k)−t0 , if not recaptured before T i ,{C} The probability of a recaptured individual is the probability of its movement times the probability of it getting caught. Notice that the probability of its movement include the probability of survival in each step.
34
Chapter 4. Population Based Approach All individuals are assumed independent and the total negative log-likelihood is calculated as the sum of all the individual contributions `(θ |X) = − ∑ log L(k) (θ |X)
(4.14)
k
An important practical note: The way the model is presented here is not the way it should be implemented (at least not for larger problems). An efficient implementation requires that the sparse nature of P is taken into account, and that many simultaneous releases are considered as one in order not to repeat the same calculations over and over. The model described here allows movement patterns not covered by the Advection– Diffusion (AD) equation, as the movement pattern is described by eight movement parameters, whereas the AD movement pattern is described by only three (u, v, D). To approximate the AD model by the Markov model, the probabilities εE , εNE , . . . , εSE must be selected to correspond with the AD movement pattern given by (u, v, D). The goal is to select the ε-probabilities such that the position of an individual moving in the grid cells according to these probabilities will follow a distribution similar to that of an individual following the SDE equation (4.2). This implies that the mean position after n steps should be α0 + n(u, v)0 and the covariance matrix should be n2DI2×2 .1 The following choice of ε-probabilities match both mean and 2 ), E(α 2 α ) covariance and furthermore some of the higher moments (E(αt,1 αt,2 t,1 t,2 2 2 and E(αt,1 αt,2 )). 2 0 2 εE 0 εNE 0 0 0 εN 2 0 εNW 1 0 0 0 εW = 4 −2 0 2 0 εSW 0 0 0 −2 0 εS 0 0 0 εSE
0 0 −2 0 0 1 1 1 2 0 0 −2 0 −1 −1 1 0 0 2 0 0 1 −1 −1 2 0 0 2 0 −1 1 −1
u −2 v 1 2 2D + u −2 2 2D + v 1 (4.15) uv −2 2 1 u(2D + u ) −2 v(2D + v2 ) 1 (2D + u2 )(2D + v2 )
This re-parameterization follows from a bit of tedious algebra including calculation of the moments of the SDE (4.2) and the Markov model after N steps, and solving for ε. Not all choices of u, v and D will result in sensible ε-probabilities, so care must be taken to ensure that each ε parameter is positive and that ∑ ε < 1. If this is not the case, it is most likely because the corresponding ADR cannot be described by a Markov model that only allows moves to the neighboring cells in each step, which implies an upper limit to the drift or diffusion of the population. 1A
Markov model where only transitions east, north, west and south are possible is not capable of matching both mean and covariance, which is why the more elaborate eight direction model is chosen.
4.3. Markov Model With this extension the Markov model now has the same model parameters as the ADR model θ = (u, v, D, Q, M), and similar to the ADR model they can be extended to depend on region and season. Notice that it is possible within the Markov model to test the AD movement pattern against the larger model with eight different movement vectors.
35
36
Chapter 4. Population Based Approach
5 Combining Individual and Population Based Models
The goal of the population based approach is to get the best possible information about the movement pattern of the entire population, which would be valuable in fish stock assessment models. The goal of the individual based approach is to reconstruct the best possible track for each individual, but this reconstruction includes estimation of the parameters describing the (population) movement pattern. In this chapter the possibility of combining the two approaches is investigated. After all, all tagged fish from the same population should be equal representatives of its movement pattern, no matter what type of tag they have been equipped with. However, it is expected, that the amount of information per tag is higher for archival tags than for conventional tags, but what and how much more is really learned from an archival tag, and why not use both?
5.1
Introduction
Combining the individual and population based approaches is an appealing idea, because the parameters of interest u, v and D are common in the two approaches, so for the sake of using all information available to estimate these parameters it would be worthwhile, which is also suggested by Sibert and Fournier (2001). Bolle et al. (2001) compare the migration pattern of North Sea plaice obtained from archival tags to those obtained from conventional tags and find consistence. In the following, the individual based approach is adjusted and extended to fit the same scenario as the population based approach and then combined to a coherent model. The approach used can be extended to include all extensions commonly used in the two separate approaches including different movement patterns in different areas and seasons, light-based variance corrections and adjustment for the curvature of the globe, but for the purpose of presenting the combined model the two separate approaches have been reduced to their simplest forms.
38
Chapter 5. Combining Individual and Population Based Models
5.2
Simulation and Estimation
58
Large parts of the modeling and estimation techniques used in this simulation study have been described in the two previous chapters and will not be repeated here. However, there are a few non-trivial extensions, especially for applying the individual based approach to multiple tracks and in an area with uneven fishing effort. The simulation approach will also be described.
57 58 59 60 61 62 63 64 41 42 43 44 45 46 47 48
56
33 34 35 36 37 38 39 40 25 26 27 28 29 30 31 32 17 18 19 20 21 22 23 24
55
Latitude
57
49 50 51 52 53 54 55 56
9 10 11 12 13 14 15 16
54
1
−4
−2
0
3
2
5
4
2
4
6
7
6
8
8
Longitude
Figure 5.1: The numbered area is the simulation and modeling area. The area is devided into 64 so called ICES squares. All releases are done at time t = 0 at the center of cell 28. Movement parameters are chosen such that all simulated individuals stay in the numbered area during the simulation time of one year. The effort in each cell is proportional to the radius of the white circles.
5.2.1
Simulation
The simulation area is chosen to be a 240 × 240 Nautical miles square in the North Sea (see figure 5.1). The North Sea is chosen because it is an area where both
39
5.2. Simulation and Estimation population based studies with conventional tags (e.g. De Veen 1961; 1962; 1965; 1970; 1978; Harden Jones et al. 1979) and individual based studies with archival tags (e.g. Arnold et al. 1997; Metcalfe and Arnold 1997; Hunter et al. 2003) have been carried out. A rectangular area is chosen to make the computations slightly simpler. The release position of all simulated individuals is at the center of cell 28 (figure 5.1) at longitude 3.5 and latitude 55.75. This point is chosen simply because it is near the center of the study area. The fishing effort (figure 5.1) is chosen constant in the entire simulation period and similar to the eleven year average (1990–2000) for North Sea plaice. Individual movements are simulated forward from the release point in daily steps following the model αi = αi−1 +
u v
∆ti + ηi ,
i = 1, . . . , T
(5.1)
The random part ηi is assumed to be serially uncorrelated and follow a two dimensional Gaussian distribution with mean vector 0 and covariance matrix 2D∆ti I2×2 . The movement parameters are chosen extremely moderate (u = 0.02Nm/day, v = 0Nm/day and D = 0.5Nm2 /day) for two reasons. Firstly, to ensure that all individuals stay in the simulation area during the entire simulation period of one year. Secondly, the very simple numerical partial differential equation solver used to estimate the parameters is more stable for small values of u and v. Constant movement pattern is chosen to simplify computations. An individual has in each step a probability of getting caught corresponding to an overall annual fishing mortality of 0.5, and a probability of dying from other (natural) causes corresponding to an annual mortality rate of 0.1. Before the simulation is started each individual is assigned either a conventional tag or an archival tag. If an individual is caught within the study period of one year the data from its tag is collected, otherwise it is lost. Conventional tags only “record” first and last time and position, so all data collected from the conventional tags can be represented by the number released nCT , the number recaptured rCT , the recapture times t1 , . . . ,trCT and the recapture grid-cells G1 , . . . , GrCT . Archival tags record daily geolocations, so all data from the archival tags can be represented by the number released nAT , the number recaptured rAT and a track from each recaptured tag (1) (1) (r ) (r ) (ti , αi )i=0,...,n(1) , . . ., (ti AT , αi AT )i=0,...,n(rAT ) . A Gaussian daily geolocation error with a standard deviation of σ = 10Nm/day is added to each coordinate of the observed track. This is a very small geolocation error, but for this area geolocation errors of this magnitude have been reported (Hunter et al. 2003). From these simulated data sets the movement parameters (and the few additional parameters) can be estimated from conventional tags, archival tags, or even from both.
40
Chapter 5. Combining Individual and Population Based Models
5.2.2
Estimation via Conventional Tags
Estimation via conventional tags is done by the advection–diffusion–reaction (ADR) model described in section 4.2. For the solution of the ADR equation a very simple (upwind) finite difference solver (Press et al. 1992) with daily time steps is used. The spatial grid used internally by the solver is nine times finer than the observational grid, or in other words each observational ICES square is divided into nine (10Nm × 10Nm) squares within the numerical solver. The negative log-likelihood is identical to (4.9), except that only one fleet is fishing on this simulated stock (5.2) `CT (θ |e) ∝ ∑ C j,t −C j,t log(C j,t ) j,t
C j,t is the number of caught conventional tags in ICES square number j at time t, C j,t is the corresponding expected catch calculated from the catch equation (1.4) and e is the fishing effort. The model parameters for this model are θ = (u, v, D, Q)0 , where Q is the catchability constant from the catch equation. The annual natural mortality rate M = 0.1 is assumed known from other sources. The effort numbers are normalized to have mean one, so estimates of Q should be around 0.5.
5.2.3
Estimation via Archival Tags
Estimation via archival tags is done almost as described in section 3.2, but some modifications are necessary to include the uneven effort pattern and the information from the individuals not caught by the end of the study. Furthermore, the details accounting for the curvature of the globe are omitted, as they are unimportant in this relatively small study area. The likelihood value should be based on all information observed. In each observed time step from an archival tag two things are actually observed: the position estimate from the tag and the fact the individual survived the step (except the last one). The probability of surviving each step can be ignored if fishing effort and natural mortality is uniform in the study area, because then it reduces to a constant in the likelihood function, but with a heterogeneous effort pattern it should be included. The problem is that the Kalman filter is based on assumptions of linearity and Gaussian distributions which are not fulfilled with a heterogeneous effort pattern. A solution to this, using the particle filter (Gordon et al. 1993), is described in PAPER IV, which also indicates that the reconstructed track was almost completely unaffected by moderately varying effort. This inspired the approximation used in this simulation study. For a set of model parameters θ = (u, v, D, Q, σ 2 ) the pure Kalman filter is used to estimate the track and to calculate its likelihood contribution. Stepping along this estimated track all the survival probabilities are multiplied and finally the catch
41
5.3. Results
probability in the last point. The negative log-likelihood contribution from the k’th observed archival tag becomes (k) (k) b (k) n −1 F α (k) (k) b (k) bi(k) /365 − log n 1 − exp −Z α /365 `1 (θ |e) ∝ `KF (θ ) + ∑ Z α n(k) (k) b (k) Z α |i=1 {z } n | {z } Survival Recapture (5.3) The Kalman filter likelihood contribution and the estimation of the track are described in section 3.2. The archival tagged individuals that were not recaptured before the study period ended (after T = 365 days) were either dead from natural causes, or still alive at time T . The probabilities of these two events are calculated via the numerical solution to the ADR model. The probability of dying from natural causes pM (θ ) is calculated by accumulating the fractions dying in each step of the numerical solver, and the probability of surviving longer than the study period pT (θ ) is calculated by accumulating the fraction in each grid cell at time T . The negative log-likelihood contribution becomes (k)
`2 (θ |e) = − log(pM (θ ) + pT (θ )) The total negative log-likelihood for all released archival tags becomes ( (k) nAT `1 (θ |e), if recaptured `AT (θ |e) ∝ ∑ (k) `2 (θ |e), if not recaptured. k=1
5.2.4
(5.4)
(5.5)
Estimation via both Archival and Conventional Tags
At this point it is simple to construct the negative log-likelihood for the combined data set of both archival and conventional tags. As all tagged individuals are assumed independent it is the sum of the separate `ACT (θ |e) ∝ `AT (θ |e) + `CT (θ |e).
5.3
(5.6)
Results
The two separate models are tested before combining them into a single coherent model (figure 5.2). The model for conventional tags is applied to 100 simulated sets of data with 5000 tagged releases in each. The estimates are distributed around the true parameter value in all cases, but with a slight indication of bias towards smaller values for the diffusion parameter D. The model for archival tags
Chapter 5. Combining Individual and Population Based Models
−0.02
0.02
0.06
25
12
20 15
8
10
6
5
4 −0.02
0.01
−0.02
0.01
0 0.40
0.50
0.60
0.40
0.50
0.60
0.4
0.6
0.4
0.6
0.00 0.05 0.10 0.15 0.20 0.25 0.30
10
5
35
3
8
2 1 0
0
0
0
5
2
5
10
4
10
15
6
20
15
25
4
20
30
25
12
0.06
0
2
20 0 0.02
σ2
Q
10
100 120 80 60 40
60 40 20 0 −0.02
D 30
v
80 100 120 140
u
Histogram of 100 estimations from 100 archival tags 5000 conventional tags
42
97
100
103
Figure 5.2: Histograms of the estimates from 100 simulated data sets each consisting of 5000 conventional tags and 100 archival tags. The top row is estimates based only on the conventional tags and the bottom row is estimates based only on the archival tags. The true parameter values are indicated by a thick vertical line. is applied to 100 simulated sets of data with 100 tagged releases in each. The estimates are distributed around the true parameter value in all cases and there is no indication of bias. As all individuals are assumed to be independent it is expected that the variance of each parameter estimate is inversely proportional to the number of tagged individuals. This is verified for both of the two separate models by running 100 simulations of each in four different scenarios. For the conventional tags with nCT =5000, 10000, 20000 and 30000 and for the archival tags with NAT =20, 50, 75 and 100. The estimates of D can be seen in figure 5.3. Since the precision (reciprocal variance) is a linear function of the number of tagged individuals it is possible to compare the amount of information per archival tag to the amount of information per conventional tag, or in other words, how many conventional tags should be used to get the same precision as one archival tag provides. Based on the simulations in figure 5.3 this ratio for the diffusion parameter is RC/A (D) = 20.6, which is calculated as the ratio between the two slopes in figure 5.3. Similarly for the remaining parameters which are present in both models RC/A (u) = 1.7, RC/A (v) = 2.0 and RC/A (Q) = 0.7.
43
5.3. Results
Archival tags
0.2
0.4
D 0.42
0.46
D
0.50
0.6
0.54
Conventional tags
5000
10000
15000
20000
25000
30000
20
40
80
100
80
100
600 200
400
1/V(D)
5000 2000
1/V(D)
60 Number of tags
8000
Number of tags
5000
10000
15000
20000
25000
30000
Number of tags
20
40
60 Number of tags
Figure 5.3: Estimates of the diffusion parameter D for a different number of conventional tags (first column) and archival tags (second column). For each different number of tags estimates are calculated from 100 simulated data sets. The last row shows the empirical precision (reciprocal variance) of the estimates. Parameter True name value u 0.02 v 0.00 D 0.50 Q 0.50 2 σ 100.00
Conventional tags bias std. dev. -0.00143 0.00259 -0.00232 0.00302 -0.02078 0.02403 -0.00228 0.01221
Archival tags bias std. dev. -0.00364 0.01467 0.00059 0.01150 -0.00244 0.03733 0.02835 0.08274 -0.13608 1.52063
All tags bias std. dev. -0.00188 0.00182 -0.00134 0.00299 -0.01695 0.01640 -0.00011 0.01164 0.09419 1.34204
Table 5.1: Empirical estimates of bias and standard deviations of the parameter estimates from the two separate and the combined approaches. This table is based on the simulations from figure 5.2 with 5000 conventional tags and 100 archival tags.
Combining the two separate models into one coherent model using all tags gives estimates with smaller standard deviations than any of the separate models for all model parameters (table 5.1). Even the model parameter σ 2 , which is only present in the model for archival tags is more precisely estimated by the combined model. All bias terms are small, but the slight bias on the diffusion parameter in the conventional tags model is reduced in the combined model. The estimates from the combined model are dominated by the estimates from conventional tags, as they are more abundant in this simulation.
44
Chapter 5. Combining Individual and Population Based Models
5.4
Discussion
This chapter demonstrates that it is indeed possible to combine the individual tracking model based on the Kalman filter with the population model based on the ADR equation. The result is better inference about all model parameters. It may be surprising that the drift parameters u and v are not estimated much more accurately by archival tags than by conventional tags, but even if every step of the track was observed without noise the first and last position would still be a sufficient statistic to estimate the drift parameters, which implies that no further knowledge about the track could improve the inference about the drift. The only reason that the estimates from the archival tags are slightly more accurate is most likely that the conventional tags model operates on a discrete grid. Archival tags provide a more accurate estimate of the diffusion parameter D, which comes as no surprise. The simulations also indicate that the catchability Q is estimated slightly more accurately from the conventional tags, but this is not plausible since an archival tag contains all data from a conventional tag (and lots more). However, the archival tag model does estimate one additional parameter σ 2 and this could explain less accuracy on Q, but most likely this small difference is a result of simulation uncertainty. The bottom line is, that with respect to the model parameters archival tags only really improve the estimation of D. The simulation study presented here is really only a prototype to illustrate that it is possible to combine the two sources of information. It would be interesting to investigate how these conclusions would change if the movement parameters were allowed to differ in for instance different regions and seasons. It is expected that archival tags would provide more accurate estimates of the drift than conventional tags in the case of different regions, as archival tags provide information about how long each tag spent in each region, but for different seasons the tags should still perform equally well. Also, it would be interesting to see how different values of D and σ 2 affect the ratio between the two tags. It is expected that larger values of σ 2 will reduce the difference between archival and conventional tags. Aside from generalizations, a few things could improve the model. A better numerical solver for the ADR model would be desirable. The finite difference solver applied works in this example, but it is suspected that the slight bias observed on the diffusion estimates could be a result of so called “numerical diffusion” in the solver. A finite volume solver on a finer triangulated mesh would probably do better and be better suited to fit irregular areas. To further improve the likelihood it could be considered to replace the approximation in the individual based part of the model with the particle filter model illustrated in paper IV, but this extension is expected to be of minor importance. Naturally, archival tags provide something much more important than just parameter estimates. The track itself can potentially reveal features of the movement pattern, for instance delimit spawning locations or migration corridors, that would require an enormous amount of conventional tags to detect. However, combining
5.4. Discussion the two models should also improve the reconstructed tracks from the individual based model, as the tracks rely on the estimated parameters. In a realistic setup the data from (cheap) conventional tags are much more abundant than data from archival tags, and hence the conventional tags could substantially improve the tracks from the archival tags, if the models are combined.
45
46
Chapter 5. Combining Individual and Population Based Models
Bibliography
Adam, M. S., and Sibert, J. R. 2002. Population dynamics and movements of skipjack tuns(Katsuwonus pelamis) in the Maldivian fishery: analysis of tagging data from an advection–diffusion–reaction model. Aquat. Living Resour. 15:13– 23. Adam, M. S., and Sibert, J. R. (in press). Uses of neural networks with advection– diffusion–reaction models to estimate large scale movements of skipjack tuna from tagging data. Pelagic Fisheries Research Program, Honolulu, Hawaii. Anderson, K. P., and Bagge, O. 1963. The bennefit of plaice transportation as estimated by tagging experiments. Spec. Publ. ICNAF 4:164–171. Arnold, G. P., Metcalfe, J.D., Holford, B.H., and Buckley, A.A. 1997. Availability and accessibility of demersal fish to survey gears: new observations of ’natural’ behaviour with electronic data storage tags. ICES CM W:11. Beverton, R. J. H., and Holt, S. J. 1957. On the dynamics of exploited fish populations. Fish. Invest. Ser. II. Mar. Fish. G. B. Minst. Agric. Fish. Food 19. Bolle, L. J., Hunter, E., Rijnsdorp, A.D., Pastoors, M.A., Metcalfe, J.D., and Reynolds, J.D. 2001. Do tagging experiments tell the truth? Using electronic tags to evaluate conventional tagging data. ICES CM O:02. Darroch, J. N. 1961. The two–sample capture–recapture census when tagging and sampling are stratified. Biometrika 48:241–260. De Veen, J. F. 1961. The 1960 tagging experiments on mature plaice in different spawning areas in the southern North Sea. ICES CM Near North. Seas Comm. 44. De Veen, J. F. 1962. On the subpopulation of plaice in the southern North Sea. ICES CM Near North. Seas Comm. 94. De Veen, J. F. 1965. On the homing ability of plaice. ICES CM Near North. Seas Comm. 61. De Veen, J. F. 1970. On the orientation of the plaice (Pleuronectes platessa L.). I. Evidence for orientating factors derived from ICES transplantation experiments in the years 1904-1909. J. Cons. int. Explor. Mer 33: 192-227.
48
BIBLIOGRAPHY De Veen, J. F. 1978. On selective tidal transport in the migration of North Sea plaice (Pleuronectes platessa) and other flatfish species. Neth. J. Sea Res. 12: 115-147. Deriso, R. B., Punsly, R. G., and Bayliff, W. H. 1991. A Markov movement model of yellowfin tuna in the Eastern Pacific Ocean and some analysis for international management. Fisheries Research 11:375–395. Chapman, D. H. 1951. Some properties of the hypergeometric distribution with applications to zoological censuses. Univ. Calif. Public. Stat. 2:131–160. Doubleday, W. G. 1976. A least squares approach to analyzing catch at age data. Int. Comm. Northwest Atl. Fish. Res. Bull. 12:69–81. Fournier, D., and Archibald, C. P. 1982 A general theory for analyzing catch at age data. Can. J. Fish. Aquat. Sci. 39:1195–1207. Fry, E. E. J. 1949. Statistics of a lake trout fishery. Biometrics 5:27–67. Gardiner, C. W. 1985. Handbook of Stochastic Models. Springer. Gavaris, S., and Ianelli, J. N. 2002. Statistical issues in fisheries’ stock assessments. Scand. J. Stat. 29(2):245–267. Gordon, N. J., Salmond, D. J., and Smith, A. F. M. 1993. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE-Proceedings-F 140:107–113. Gudmundsson, G. 1994. Time series analysis of catch-at-age observations. Applied Statistics 43:1117–1126. Harden Jones, F. R., Arnold, G.P., Greer Walker, M., and Scholes, P. 1979. Selective tidal stream transport and the migration of plaice (Pleuronectes platessa L.) in the southern North Sea. J. Cons. int. Explor. Mer 38(3): 331-337. Harvey, A. C. 1990. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press, Cambridge. Hill, R. D., and Braun, M. J. 2001. Geolocation by Light Level - The Next Step: Latitude. In J.R. Sibert and J. Nielsen (Eds.), Electronic Tagging and Tracking in Marine Fisheries (pp. 315-330). Kluwer Academic Publishers, The Netherlands. Hunter, E., Aldridge, J. N., Metcalfe, J. D. and Arnold, G. P. 2003. Geolocation of free-ranging fish on the European continental shelf as determined from environmental variables. Marine Biology 142:601–609. Ianelli, J. N., and Fournier, D. A. 1998. Alternative analysis of NRC simulated stock assessment data. In Analysis of simulated data sets in support of NRC study on stock assessment methods. Edited by V. R. Restrepo. NOAA Technical Memorandum NMFS-F/SPO-30. U. S. Department of Commerce, Washington, D. C.:81– 96.
49
BIBLIOGRAPHY Jones, R. 1959. A method J. Cons. CIEM 25(1):59–72.
of
analysis
of
some
haddock
returns.
Kloeden, P. E., and Platen, E. 1992. Numerical solutions of stochastic differential equations. Springer–Verlag, Berlin. Laurs, M., Foley, D., Nielsen, A., Bigelow, K. A., Musyl, M. K., Brill, R., and McNaughton, L. in prep. Temporal and Spatial Movement Patterns in Relation to Oceanographic Conditions for Epipelagic Sharks As revealed by Pop-up Satellite Archival Tags (PSATs) in the Central Pacific Ocean: I. Oceanic White-tip shark (Carcharhinus falciformes). Lewy, P., and Nielsen, A. 2003. Modeling stochastic fish stock dynamics using Markov chain Monte Carlo. ICES J. Mar. Sci. 60(4):743–752. Metcalfe, J. D., Arnold, G.P. 1997. Tracking fish with electronic tags. Nature 387: 665-666. Murphy, G. I. 1964. A solution J. Fish. Res. Bd. Canada 22(1):191–201.
to
the
catch
equation.
Musyl, M. K., Brill, R. W., Curran, D. S., Gunn, J. S., Hartog, J. R., Hill, R. D., Welch, D. W., Eveson, J. P., Boggs, C. H., and Brainard, R.E. 2001. Ability of archival tags to provide estimates of geographical position based on light intensity. In Electronic Tagging and Tracking in Marine Fisheries Reviews: Methods and Technologies in Fish Biology and Fisheries. J.R. Sibert & J.L. Nielsen (eds) Dordrecht: Kluwer Academic Press, pp. 343–368. Nielsen, A., and Lewy, P. 2002. Comparison of the frequentist properties of Bayes and the maximum likelihood estimators in an age-structured fish stock assessment model. Can. J. Fish. Aquat. Sci. 59:136–143. Okubo, A. 1980. Diffusion and ecological problems: Mathematical models. Springer–Verlag. New York. Otter Research Ltd 2003. AD Model Builder Documentation. Otter Research Ltd, Sidney, British Columbia. http://otter-rsch.com. Petersen, C. G. J. 1896. The yearly immigration of young plaice into the Limfjiord from the German Sea. Report of Danish Biological Station 6:1–48. Pope, J. G. 1972. An investigation of the accuracy og virtual population analysis using cohort analysis. ICNAF Research Bulletin 9:65–74. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. 1992. Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. Cambridge Univ. Press. Ricker, W. E. 1954. Stock and recruitment. J. Fish. Res. Board Can. 11:559–623.
50
BIBLIOGRAPHY Rijnsdorp, A. D., and Pastoors, M. A. 1995. Modelling the spatial dynamics and fisheries of North Sea plaice (Pleuronectes platessa L.) based on tagging data. ICES J. Mar. Sci. 52:963–980. Schnabel, Z. E. 1938. The estimation of the total fish population of a lake. Fish. Bull. USFWS 52(69):191–203. Seber. G. A. F. 1982. The estimation of animal abundance, and related parameters. Charles Griffin and Company Limited. London. Shepherd, J. G. 1999. Extended sorvivors analysis: An improved method for the analysis of catch–at–age data and aboundance indices. ICES J. Mar. Sci. 56:584– 591. Sibert, J., and Fournier, D. A. 1994. Evaluation of advection-diffusion equations for estimation of movement parameters from tag recapture data. In Interactions of Pacific tuna fisheries, Vol. 1 — Summary report and papers on interaction, FAO Fisheries Technical Paper 336/1, pp. 108–121. Sibert, J., and Fournier D. A. 2001. Possible models for combining tracking data with conventional tagging data. In J.R. Sibert and J. Nielsen (Eds.), Electronic Tagging and Tracking in Marine Fisheries (pp. 443-456). Kluwer Academic Publishers, The Netherlands. Sibert, J., Hampton, J., Fournier, D. A. and Bills, P. J. 1999. An advection– diffusion–reaction model for estimation of fish movement patterns from tagging data, with application to skipjack tuna (Katsuwonus pelamis). Can. J. Fish. Aquat. Sci. 56:925–938. Sibert, J., Musyl, M. K. and Brill, R. W. 2003. Horizontal movements of bigeye tuna (Thunnus obesus) near Hawaii determined by Kalman filter analysis of archival tagging data. Fish. Oceanogr. 12(3):141–151. Sibert, J., Nielsen, A., Lutcavage, M. and Bril, R. W. in prep. Large-scale movement, distribution and changes in behavior of Atlantic bluefin tuna determined from analysis of pop-up archival tag data. Skellam, J. G. 1951. Biometrika 38:196–218.
Random
dispersal
in
theoretical
populations.
Swimmer, Y., Musyl, M., Arauz, R., Nielsen, A., McNaughton, L., and Brill., R. In prep. Horizontal Movements of Olive ridley turtles (Lepidochelys olivacea) in the Eastern Tropical Pacific.
A Kalman Filter Tracking
KFTrack (Kalman Filter Tracking) is an add-on package for the statistical environment R to efficiently estimate movement parameters and predict “the most probable track” from raw geolocations of archival tagged individuals. Its basic use is to fit and plot the models described in chapter 3. For instance, to get a plot similar to figure 3.4 all commands necessary are:
library("kftrack") WT38575 0 can be described as the density of a normal distribution ϕαt,σ 2t with mean αt and variance σ 2t. The expected number of tagged fish in a trawl-width w around a position p at time t > 0 is denoted N w,p,t N w,p,t = E(Nw,p,t ) Z p+w/2
= N0,0
ϕαt,σ 2t (u)du p + w/2 − αt p − w/2 − αt √ √ = N0,0 Φ −Φ , σ 2t σ 2t p−w/2
(I.2)
where Φ is the distribution function of a standard normal distribution and N0,0 is the number of fish initially released. The expected catch is the fraction q ∈ (0, 1) of the expected number of fish that has survived to time = t which can be caught Cw,p,t = qN w,p,t (I.5)
(I.3)
60
Paper I. Diffusion of Fish from a Single Release Point √ If the standard deviation of the distribution of the tagged fish σ 2t is large compared to the trawl-width we can assume the number of caught fish to follow a poisson distribution (I.4) C p,t ∼ pois qN w,p,t (α, σ ) . Another approximation which is also valid if the standard deviation is large compared to the trawl-width is: N w,p,t ≈ N0,0 wϕαt,σ 2t (p).
(I.5)
This can be useful if it is too time consuming to evaluate Φ. The negative log likelihood function of the observed catches given the parameter values is given by l(c; q, α, σ ) =
∑
log(c p,t !) + qN w,p,t (α, σ ) − c p,t log(qN w,p,t (α, σ ))
∑
qN w,p,t (α, σ ) − c p,t log(qN w,p,t (α, σ )) .
(p,t)
∝
(I.6)
(p,t)
Partial Differential Equation Let us again assume the x-position of each fish to follow (I.1) and to be independent. The development of the stock density can be described by a partial differential equation (Okubo 1980) ∂ σ2 ∂2 ∂ Nx,t = −α Nx,t + Nx,t ∂t ∂x 2 ∂ x2
&
N·,0 = 1{0} N0,0 .
(I.7)
This model is exactly the same as the NDA if we simply solve the equation. Besides solving the PDE analytically where the solution is the NDA, an equation like (I.7) can be solved numerically by iterating forward in time from the initial condition N·,0 = 1{0} N0,0 . This way of solving a PDE can be utilized to incorporate the effect of fishing into the solution. Each time the solution scheme passes a time point where fishing/releasing took place we simply update the solution accordingly. If a large fraction of the total number of fish are caught each day we should include this information in the model, even though we re-release the fish we catch at the same position. From the solution of (I.7) the expected number of tagged fish in a trawl-width w around a position p at time t > 0 can be estimated. Let’s ew,p,t . denote this number N The negative log likelihood is derived as before. We get ew,p,t (α, σ ) − c p,t log(qN ew,p,t (α, σ )) . l(c; q, α, σ ) ∝ ∑ qN (p,t)
(I.6)
(I.8)
61 Numerical Solution of the PDE Model As the numerical solution scheme is needed as part of an optimization routine it is very important that it is efficient. It may be necessary to evaluate the solution hundreds of times. A relative fine spatial resolution of ∆x = 1 m was chosen because the trawl is only 4.5 m wide. The time resolution was chosen to be 1 minute. Finite-difference schemes are frequently used to solve differential equations like (I.7) (e.g., Sibert et al. 1999). Finite-difference schemes are efficient if the (approximate) solution is needed for every grid point. Several finite-difference schemes were implemented, but it became evident that it was too time consuming with the desired resolution. Fortunately, the solution in every grid point is not necessary. To optimize the likelihood it is sufficient to know the solution in catch points at catch times. This is utilized in the following solution scheme. The solution scheme needs to keep track of every single discrete impact on the system, which are denoted as impulses. An impulse is a set of three parameters I = {n, p,t}, where n is the number of fish we add to the system (catches are negative), p is the position and t is the time. The first impulse is the release of the N0,0 tagged fish at position p = 0 at time t = 0, or in other words I0 = {N0,0 , 0, 0}. If this impulse was the only one, we could calculate the solution in every future point from formula I.2. This is not the case in our experiment since the catches are additional impacts on the system. The first catch results in a number of impulses equal to the number of spatial grid points covered by the trawl. Each of these impulses is a fraction of the catch proportional to fraction of the trawl-width which cover the interval multiplied by the population density, which is known from previous impulses. This implies that the impulses can only be calculated forward in time. After the impulses from the first catch follow the impulse from the first re-release and so on. Each of these impulses contributes to the total solution, in any future points, in exactly the same way as the first impulse. The total solution is found as the sum of the contributions from each of the previous impulses. Given the set of impulses I0 , I1 , . . . , II we can calculate the numerical solution in any grid point (p,t) from
e∆x,p,t = N
∑
{i:t>ti }
ni Φ
(p + ∆x/2) − pi + α(t − ti ) p σ 2 (t − ti )
−Φ
!
(p − ∆x/2) − pi + α(t − ti ) p σ 2 (t − ti )
(I.9) !! .
This solution scheme dramatically reduces the calculation time in this example and the gain will be even greater if the spatial dimension was 2 or higher. The only drawback is that a bit more bookkeeping is required. (I.7)
62
Paper I. Diffusion of Fish from a Single Release Point Parameterization For simplicity, in the previous presentation of the models it has been assumed that the parameters q, α, σ are constant during the 9-day sampling period. This is not a realistic assumption, since the hydrographic parameters change during the experimental period and the released turbot adapt to their new environment. Therefore, in order to allow changes in these parameters, an NDA model where each of the three parameters is assumed constant during 24 h intervals (counting from release time) was fitted to the data. Hence qt , αt and σt are estimated where t = 1 . . . 9 equals days at liberty. The resulting model, called the full model, corresponds to estimating one normal distribution from each of the nine days’ catches, which produces a total estimation of 27 parameters. From this full model the goal was to find another model with fewer parameters needed to describe the density distributions. This reduced model was found by testing the hypotheses that advection was zero (αt = 0) and that the variance was the same all/some days (σ1 = · · · = σ9 ). The recaptured part (qt ) being connected to the trawl catchability (ρ) and the total mortality (Z) as: qt = ρe−Zt · N w,p,t .
(I.10)
was also analyzed, where it was assumed that mortality and catchability are constant. All hypotheses were tested by standard likelihood ratio tests.
Normal Distribution Approximation vs. Partial Differential Equation In order to evaluate if the two models gave identical results the parameter estimates were compared. This comparison was done only for the reduced model but for all parameters (αt , σt , ρt and Zt ). Since the NDA model is the solution to the PDE the differences between these two models do not lie in the basic assumption about diffusion. The main difference is that the PDE model incorporates our interference with the population density. To make the NDA model even simpler we assumed that the population density does not change between catches taken the same day. This assumption was not necessary to make the NDA model work, but it made the model very simple and intuitive. Unfortunately, this assumption makes it difficult to compare the estimates from the two models because the population density (and the resulting parameters) estimated from the NDA actually represents an average of several population densities. Therefore in order to facilitate a comparison we chose simply to compare the daily population estimates from the NDA to the population estimates from the PDE 18:00 each day. (I.8)
63
Wind Data Wind data, consisting of eight measurements per day, were purchased from the Danish Institute of Metrology, and from those, the water body displacement (wbdt ) in the east-west direction, that is parallel to this coast, was calculated. Since the thickness of the Ekman layer was much larger than the depth at the location, Coriolis forces could be excluded. Hence wbdt could be calculated as the steady state balance between wind stress and bottom friction with the surface and bottom drag coefficients set equal: r Z t ρa (I.11) wbdt = λ · v2 (s) · | sin(u(s))| · ds. ρw 0 Where λ = -1 or 1, in western and eastern wind respectively, ρa and ρw is the density of the air and water which equals 1.22 kg· m−3 and 1028 kg· m−3 respectively. Wind velocity is v and u is the wind direction. Each wind measurement is denoted by s. The integral was estimated by summing the average between two subsequent wind measurements. Two different estimates of water body displacements were calculated: one using the wind data from the entire day and one using only the wind in the period from 21:00 to 06:00, i.e., during the night. These data were compared to the advection found in the NDA model.
Results No released turbot were caught in any of the hauls towed parallel to the coast at 6 m depth. During the sampling period a total of 332 catches was accomplished distributed on 105 hauls perpendicular to the coastline. Fewer than ten wild turbot of the same length class were caught, which, aside from the lack of a T-bar tag, were easily recognized by their darker pigmentation. Out of the total recaptures 136 were frozen for stomach content analysis in the laboratory. One haul on day 7 where two turbot were recaptured 1030 m west of the release position was excluded in the data analysis. The maximum catch was obtained on the first day where more than 40 individuals were caught in a single haul. During the following period the daily catches gradually decreased to a minimum on day 9. The first attempt to analyze the catch data was to fit a full NDA model, meaning that one normal distribution curve was fitted to the recaptures from each day and thus for each day one set of parameters describing a normal distribution curve was estimated (figure I.2). These parameters are (σt ), (αt ) and (qt ) (figure I.3), with 1 units m · day− 2 , m · day−1 , respectively for σt and αt , whereas qt is a fraction. The full model (figure I.4) was reduced to a model with fewer parameters where the fit was not significantly reduced (likelihood ratio test with p value = 47%). It was possible to describe the spreading of the released turbot using one single σ value each day from day 3 to 9 (table I.1). It was reasonable to describe qt with (I.9)
64
Paper I. Diffusion of Fish from a Single Release Point Day = 1
Day = 2
Day = 3
50
25
Catches in numbers
40 30
15
20
10
10
5
Day = 4
8
20
Day = 5
6 4 2 Day = 6
4 6
2
4
2
2
1
Day = 7
3
3
Day = 8
1
Day = 9 6 5 4 3 2 1
4 3 2 1
2
1
Shoreline
Figure I.2: Recaptured turbot per day (◦) with the density distribution estimated from the full NDA model (solid line). The solid vertical line is the release position. Distance between small dashed vertical lines represents 100 m whereas distance between large dashed lines represents 1 km. Day is days at liberty. one constant mortality (Z) and one constant catchability (ρ). The catchability was found to be 28%, indicating that 72% had escaped a direct contact with the trawl. The mortality was estimated to be 14%·day−1 , which means that at the end of the experimental period only 900 individuals out of the initial 3529 would still be alive. In table I.1 the parameter estimates from both the reduced NDA and PDE are shown. The estimates for mortality (Z) and catchability (ρ) derived from the two models were similar. Further, these two estimates are also determined with the same precision and the two models produced identical values for the standard deviation (σt ). For the advection estimated, minor differences were found between the two models, but it should be noted that the precision with which this parameter is estimated is quite low, and that it is mainly during the last day of sampling that differences are present. From the model we estimated the daily average movement for the released turbot (see Appendix B). During the first day the average distance moved was estimated to be 24 m·day−1 , 44 m·day−1 the second day and 151 m·day−1 the remaining days. These values corresponds to respectively 320, 587 and 2013 body lengths·day−1 . (I.10)
65
(a)
(b)
(c) 0.4
750 500
400
0.3
250 q1
alpha1
600
0
0.2
−250 200
0.1
−500 −750
0 800
0.0 0.4
750 500
400 200
0.3
250 q2
600 alpha2
sigma2
Reduced model
Parameter values sigma1
Full model
800
0 −250
0.1
−500 −750
0 1
3
5
7
9
0.2
0.0 1
3
5
7
9
1
3
5
7
9
Day
Figure I.3: Estimated values and 95% confidence limits for the parameters describing the normal distribution curves seen in figure I.2 and I.4. The parameters are: (a) the cumulated standard deviation (σt ), (b) the advection (αt ) and (c) the recaptured part (qt ). In the full model, parameters describing one normal distribution curve for each single day were estimated. In the reduced model σt and qt were restricted as described in the text. The proportion of fish that had started to feed was constant around 22% during the first two days at liberty (figure I.5). After day 2 the percentage that had started to feed increased abruptly to approximately 95% and remained constant during the following days, except during day 8 where only 75% of the recaptured fish had traces of food in their stomachs. It appears that displacement calculated using night wind only agrees better with the advection found in the reduced NDA model than when displacement is calculated using wind data from the entire day (figure I.6). But one should note that the difference between the two estimated wbdt is a result of an eastern wind during the two first days; after that the trend is similar, independent of whether wind data was taken all day or only during night for the calculation of wbdt .
Discussion No released turbot were caught in any of the hauls towed parallel to the coast at 6 m depth. During the first period after release, liberated turbot are known to (I.11)
66
Paper I. Diffusion of Fish from a Single Release Point Day = 1
Day = 2
Day = 3
40
25
30
20
4
10
10
Catches in numbers
6
15
20
Day = 4
8
2
5 Day = 5
Day = 6 4
6
2
4
2
2
1
Day = 7
3
3
Day = 8
1
Day = 9 6 5 4 3 2 1
4 3 2 1
2
1
Shoreline
Figure I.4: Recaptured turbot per day (◦) with the density distribution estimated from the reduced NDA model (solid line). Solid vertical line is the release position. Distance between small dashed vertical lines represents 100 m whereas distance between large dashed lines represents 1 km. Day is days at liberty. remain in shallow waters (Støttrup et al. 2002) and natural turbot of this small size have only been observed at depths shallower than 2 m (Gibson 1973). Thus the experimental design, with fishing until the 6 m depth bathometer line is considered sufficient to cover the entire depth distribution of the released turbot, and provides the opportunity to analyze dispersal in one dimension only. One single haul was treated as an outlier and was excluded from the data analysis. This was a haul on day 7, where two tagged turbot were caught 1030 m to the west of the release position. A large eelgrass belt, located 1300 m from the release position limited the investigation area in that direction and the high catch may have been the result of an accumulation of turbot at the boundary of the sand area. During the first days after liberation it was clear that there was a good and highly significant agreement between the normal distribution curve estimated by the full NDA model and the observed data. As time at liberty increased, the visual presentation could give the impression of a decreasing fit of the model, but this was a result of the fact that only integer number of turbot could be caught. The model still described the low catches well. The reduced model was not significantly worsened compared to the full model, as (I.12)
67 Parameter name ρ Z σ1 σ2 σ3 = · · · = σ9 α1 α1 + α2 α1 + · · · + α3 α1 + · · · + α4 α1 + · · · + α5 α1 + · · · + α6 α1 + · · · + α7 α1 + · · · + α8 α1 + · · · + α9
Estimate 0.28 0.15 30.20 54.66 188.85 2.86 25.04 -88.67 -93.44 -286.51 -208.04 -476.91 -329.00 -118.04
NDA 95% C.I. (0.22 ; 0.34) (0.09 ; 0.21) (26.26 ; 34.15) (39.84 ; 69.47) (143.49 ; 234.21) (-2.27 ; 8.0) (9.39 ; 40.69) (-239.73 ; 62.38) (-245.44 ; 58.55) (-482.47 ; -90.55) (-424.78 ; 8.7) (-734.45 ; -219.37) (-641.86 ; -16.14) (-866.46 ; 630.38)
Estimate 0.26 0.15 37.04 59.22 192.03 5.95 30.96 -41.49 -127.61 -336.89 -171.57 -571.48 -193.37 -47.29
PDE 95% C.I. (0.2;0.31) (0.08;0.22) (32.37;41.71) (39.18;79.26) (107.15;276.91) (-1.8;13.7) (8.26;53.66) (-976.45;893.47) (-509.05;253.83) (-630.45;-43.33) (-465.85;122.71) (-981.7;-161.26) (-765.07;378.33) (-1331.89;1237.31)
Table I.1: Daily standard deviation (σt ), advection (αt ), mortality (Z) and catchability (ρ) estimates from both the Normal Distribution Approximation (NDA) and the Partial Differential Equation (PDE). The 95% confidence intervals (C.I.) are given in brackets. seen from the likelihood ratio test, but the number of parameters estimated was reduced from 27 to 14. This model revealed a catchability of turbot of 28% when using a young fish trawl. However, the last column on figure I.3 indicates that the reduced model overestimates the mortality during the last days and underestimates the mortality during the first days. In this experiment the daily mortality was estimated to be 14%·day−1 , which would result in no fish surviving beyond a year. That released turbot do survive beyond a year and can be recaptured in subsequent years has been demonstrated in a previous study (Støttrup et al. 2002). In that study the mortality estimated from the year following the release and over the subsequent years was 0.15%·day−1 . Although the study by Støttrup et al. (2002) was conducted on slightly larger fish, i.e., 15 cm as compared to 7–8 cm in this study, and size at release has been demonstrated to be important for post-release mortality (Yamashita et al. 1994), the size difference does not account for the large differences in the estimated mortality. These results suggest that there may be an initial ’adaptation’ period where the released fish are highly vulnerable and suffer high mortality rates. This is supported by the study of (Furuta et al. 1997). They estimated mortality rates of approximately 10%·day−1 for juvenile Japanese flounder (Paralichthys olivaceus) within the first week after release. Thus, measures to improve survival immediately following the release may help improve the outcome of fish releases. It appears from this study that the mortality provided by the model is overesti(I.13)
68
Percent of turbot feeding
Paper I. Diffusion of Fish from a Single Release Point
100 90 80 70 60 50 40 30 20 1
2
3
4
5
6
7
8
9
Day
Figure I.5: The percentage of turbot with food or traces of food items in their stomach. Day is days after release and all catches during the day are pooled. mated during the last six days at liberty and underestimated during the first three. Apart from this apparent repetitive change in mortality rate with days at liberty, there were other indications that pointed towards the possibility that what was actually observed was a mixture between two different activity levels and hence two different diffusion patterns. This was indicated by the fact that the model was not able to describe the diffusion using one single estimate on σ , but could only reduce the model down to having the same σ from day 3. One other indication of a switch in behavior and diffusion pattern was the sudden increase in the percentage of turbot that had started to feed between the second and the third day of liberty. At this time a change to a steady diffusion pattern is observed, which could be described by one σ . The event of feeding, the mortality, and the diffusion rate are not necessarily connected but they seem to occur simultaneously. Advection estimated from the reduced NDA was three times lower than the estimated displacement of the water body. Turbot, being a flatfish, spends a large proportion of its time inactive at the bottom and during that inactive period the current will not have any effect on the position of the fish. Hence if the advection of the released fish is a result of the current, it would mean that the fish is only present in the water column one third of the day. Whether the better fit using the night wind alone to describe the wbd is a result of the turbot being more active during night or a result of a lower activity level during the first two days (I.14)
69
1000
333
500
167
0
0
−500
−167
−1000
−333
−1500
−500 1
2
3
4
5
6
7
8
Advection (m)
WBD (m)
is difficult to tell. It is known that other flatfish species increase their activity level during dusk and dawn (Gibson et al. 1998), but the difference might well be a result of lower post release activity caused by stress. The constant speed at which the turbot dispersed after day three would, under the assumption that they were active one third of the day, be 0.07 body lengths·s−1 if they had moved in a straight line between release and recapture position. This is not a realistic swimming speed, since flatfish are negativly buoyant and the energy needed in order to lift the body from the bottom results in uneconomical swimming speeds below 0.6 body lengths·s−1 (Priede and Holliday 1980). Further, Priede and Holliday 1980 estimated the optimal swimming speed to be around 1.2 body lengths·s−1 and 1.6 body lengths·s−1 at 5 ◦ C and 15 ◦ C, respectively. Hence assuming a swimming speed on 1 body length·s−1 and that turbots move in a straight line, the resulting daily time spent in the water column from day three would be approximately 35 min. Juvenile Japanese flounder of the same size class as the turbot used in this experiment, fed on mysis which was also the main diet for turbot in our release. They were observed to increase their swimming activity as the level of hunger increased, by swimming for a longer period in the water column after an attack on a prey (Miyazaki et al. 2000).
9
Day
Figure I.6: Water body displacement (wbd) calculated from wind data. The solid line indicates the displacement when only night wind was included in the calculation and dotted line indicates displacement calculated from the wind for the entire day. Dots are the advection (α) estimated from the normal approximation model. (I.15)
70
Paper I. Diffusion of Fish from a Single Release Point There were only minor differences between the ρ and Z values estimated from the NDA model and the PDE. So even though approximately 10% of the released turbot were caught within a 9-day period, this relatively high recapture rate did not result in any detectable effect on the distribution of the released turbot. This study has shown that applying diffusion theory is a very successful method for modeling movements of living organisms even when applied to small scale migration experiments. Further, it has been shown that even though the sampling intensity in this experiment was fairly high, it was sufficient to fit an NDA model instead of using the more complex PDE. It seems as if the dispersal took place in two steps and that the change from one dispersal rate to another happened simultaneous to a change in mortality and to the event of feeding. The current is one factor which may have determined the direction of turbot displacement. These results provide more information needed to ensure a successful stocking of marine fish species. In addition, this type of experiment provides knowledge on post release mortality and the catchability of the implemented gear. In the near future, improvements in the capability of Data Storage Tags may provide us with valuable additional knowledge that could be prosperous when combined with data obtained from studies similar to this.
Acknowledgments A special thanks to Dr. Andy Visser, Dr. Peter Lewy, Dr. Uffe Høgsbro Thygesen and Professor Ib Michael Skovgaard for valuable discussions. Further, the authors would like to thank Claus Petersen, Jesper Knudsen and Birtha M. Nielsen for technical assistance.
References Andow, D. A., Kareiva P. M., Levin S. A. and Okubo A. 1990. Spread of invading organisms. Landscape Ecol. 4: 177–188. Furuta, S., Watanabe T., Yamada H., Nishida T. and Miyanaga T. 1997. Changes in distribution, growth and abundance of hatchery-reared Japanese flounder Paralichtys olivaceus released in the costal area of Tottori prefecture. (In Japanese). Nippon Suisan Gakk. 63: 877–885. Gibson, R. N. 1973. The intertidal movements of young fish on a sandy beach with special reference to the plaice (Pleuronectes platessa L.). J. Exp. Mar. Biol. Ecol. 12: 79–102. Gibson, R. N., Pihl L., Burrows M. T., Modin J., Wennhage H. and Nickell L. A. 1998. Diel movements of juvenile plaice Pleuronectes platessa in relation to (I.16)
71 predators, competitors, food availability and abiotic factors on a microtidal nursery ground. Mar. Ecol. Prog. Ser. 165: 145–159. Kareiva, P. M. 1983. Local movements in herbivorous insects: applying a passive diffusion model to mark-recapture field experiments. Oceologia, 57: 322–327. Kareiva, P. M. and Shigesada N. 1983. Analyzing insect movements as correlated random walk. Oceologia, 56: 234–238. Levin, S. A., Cohen D. and Hastings A. 1984. Dispersal strategies in patchy environments. Theor. Popul. Biol. 26: 165–191 Levin, S. A. 1992. The problem of pattern and scale in ecology. Ecology, 73: 1943–1967. Lubina, J. A. and Levin S. A. 1988. The spreading of a reinvading species: range expansion in the california sea otter. Am. Nat. 131: 526–543. Metcalfe, J.D. and Arnold G. P. 1997. Tracking fish with electronic tags. Nature, 387: 665–666. Miyazaki, T., Masuda R., Furuta S. and Tsukamoto T. 2000. Feeding behaviour of hatchery-reared juveniles of the Japanese flounder following a period of starvation. Aquaculture 190: 129–138. Neuenfeldt, S. 2002. The influence of oxygen saturation on the distributional overlap of predator (cod, Gadus morhua) and prey (herring, Clupea harengus) in the Bornholm Basin of the Baltic Sea. Fish. Oceanogr. 11: 11–17. Okubo, A. 1980. Diffusion and ecological problems: mathematical models. Springer-Verlag, Berlin, Germany. Priede, I. G. and Holliday F. G. T. 1980. The use of a new tilting tunnel respirometer to investigate some aspects of the metabolism and swimming activity of the plaice (Pleuronectes Platessa L.). J. Exp. Biol. 85: 295–309. Rose, G. A. and W. C. Leggett. 1990. The importance of scale to predator-prey spatial correlations: an example of Atlantic fishes. Ecology, 71: 33–43. Sibert, J. R., Hampton J., Fournier D. A. and Bills P. J. 1999 An advectiondiffusion-reaction model for the estimation of fish movement parameters from tagging data, with application to skipjack tuna (Katsuwonus pelamis). Can. J. Fish. Aquat. Sci. 56: 925–938 Skellam, J. G. 1951. Random dispersal in theoretical populations. Biometrika, 38: 196–218. Støttrup, J. G., Sparrevohn C. R., Modin J. and Lehmann K. 2002. The use of reared fish to enhance natural populations. A case study on turbot Psetta maxima(Linné, 1758). Fish. Res. 1361: 1–20. (I.17)
72
Paper I. Diffusion of Fish from a Single Release Point Yamashita, Y., Nagahora S., Yamada H. and Kitagawa D. 1994. Effects of release size on the survival and growth of Japanese flounder Paralichthys olivaceus in coastal waters off Iwate Prefecture, northeastern Japan. Mar. Ecol. Prog. Ser. 105: 269–276. Viswanathan, G. M., Afanasyev V., Buldyrev S. V., Murphy E. J., Prince P. A. and Stanley H. E. 1996. Lévy flight search patterns of wandering albatrosses. Nature, 381: 413–415.
Appendix A Optimizing the Fishing Effort The obvious question is where should we trawl in order to get most information about the model parameters? In other words we wish to estimate (q, σ ) and need to decide on a number of positions p1 , . . . , pm .
Frequentist Approach We start by assuming that we know the true parameter values (q0 , σ0 ). This may seem unreasonable, but as the experiment progresses we will improve our knowledge about the true parameter values. To make calculations simple we use the following approximation to the log likelihood for some fixed time point t0 app. logL(c; q, σ ) ∝ ∑ c p,t0 log(qϕ0,σ 2t0 (p)) − qN0,0 wϕ0,σ 2t0 (p) . p
We calculate the Fisher information I by I = −Eq0 ,σ0 D2(q,σ ) logL(c; θ , p) c p,t0 = − ∑ Eq0 ,σ0 p
q2
?
0 2 c p,t0 (ϕ0,σ 2 t (p)) 0
(ϕ0,σ 2 t (p))2 0
0 N0,0 wϕ0,σ 2 t (p) 0
00 + qN0,0 wϕ0,σ 2 t (p) − 0
00 c p,t0 ϕ0,σ 2 t (p) 0
ϕ0,σ 2 t (p) 0
In this case it is simple to evaluate the mean because I is a linear function of c. We get N wϕ (p) 0,0 0,σ 2 t0 0 N0,0 wϕ0,σ 2t (p) q 0 0 2 I = −∑ (ϕ0,σ 2 t (p)) p ? qN0,0 w ϕ 0 (p) 0,σ 2 t0
(I.18)
73 The covariance V of the parameter estimate is estimated by the inverse of the Fisher information V = I −1 . Hence we should choose the positions p1 , . . . , pm which minimize (some scalar extracts of) V . For instance if we want to get the most precise estimate of σ we should choose the points which minimize V2,2 . For overall optimization it is often chosen to minimize |V |.
Bayesian Approach As seen above the covariance V is a function of the positions and the parameter values. In the frequentist approach we simply assumed to know the true parameter values (q0 , σ0 ) and then minimized V with respect to the positions. In the Bayesian approach we try to express our knowledge/uncertainty about the parameters in terms of a prior distribution and then minimize the expected variance Eprior V p ; (q, σ ) with respect to the positions. As we do not really know the true values of the parameters this approach may seem more reasonable, but we must keep in mind that we do not know the true distribution of our knowledge either. A more pragmatic view is that the frequentist approach is the special case of the Bayesian approach where the variance of the prior distribution is (close to) zero.
Parameterization If we try to minimize |V | without any restrictions on p we will not be able to find a unique minimum, as any permutation of the elements in p will result in the same value of |V |. We have to find a different parameterization of the problem. To further simplify the problem we will only consider symmetric designs containing the point zero. We choose the following parameterization bm/2c
pdm/2e = 0, pdm/2e+1 = pe1 , pdm/2e+2 = pe1 + pe2 , . . . , pm =
∑
i=1
pei
where pei > 0 and pdm/2e+i = −pdm/2e−i .
Appendix B Calculation of Average Daily Movement The daily average movement is calculated from the daily standard deviation σt under the assumption that the daily advection αt is zero. Biologically this assumption (I.19)
74
Paper I. Diffusion of Fish from a Single Release Point means that any advection, if present, is caused by passive movement, hence the average movement we calculate equals the movement caused by active swimming. The daily average movement is calculated as the mean of absolute distance from the center of the normal distribution with standard deviation equal to the daily standard deviation. In other words the daily average movement (in meters) is calculated as r Z ∞ 2 σt ≈ 0.8σt . |x|ϕ0,σt2 ·1day (x)dx = π −∞
(I.20)
II Locating Point of Premature Detachment of Archival Tags or a Sudden Change in Movement Pattern
By Anders Nielsen
Abstract An approach to estimate the premature detachment location of an archival tag is presented. This approach extends a widely used state-space model to reconstruct the track of an archival-tagged individual from raw light based geo-locations. The method uses the parametric representation from the state-space model to detect changes in apparent movement behavior via the likelihood ratio test statistic and simulated confidence bands. The results are in agreement with the visual impression from the track, but the confidence is strengthened, as they are obtained from an objective method, and because the significance of an apparent change in movement behavior can now be quantified. Publication details The work described is part of a manuscript in preparation (Sibert et al. in prep.). The contribution made by this Ph.D. study is extracted for the purpose of the thesis.
76
Paper II. Locating Point of Premature Detachment of Archival Tags
Introduction Archival tagging devices can offer biologists a wealth of information about movement patterns and the spatial distribution of fishes (e.g. Arnold and Dewar 2001; Gunn and Block 2001). However, this potential is often not fulfilled because of two limitations. Huge geo-location errors from light based techniques (Musyl et al. 2001) and variable tag retention rates. The issue of premature tag detachment is a widespread, but seldom discussed problem in the realm of archival tagging. Some classes of externally attached tags have the capability to detect and report premature detachments, but this capability is not uniform, particular on older tag models. If this capability is not present, the most applied method has been visually inspecting the track of raw geo-locations and making a educated guess. This paper offers an objective approach based on the likelihood ratio test. Raw geo-locations derived from light based algorithms often have huge geo-location errors, and it is quite common that marine animals are mistakenly located hundreds of kilometers ashore (Musyl et al. 2001). One widely used method to obtain a more likely track from these raw geo-locations is based on Kalman filter estimation in a state-space model (Sibert et al. 2003). The model used here is extended in several ways from the previously published version. The equinox singularity correction is modified to better capture equinox errors in longer tracks and to displace these errors towards the winter months (Hill and Braun 2001). The linear approximation of the coordinate transformation is improved, and the ability to use known fixed pop-up position is included in the model. This parametric representation of the track offers the possibility to estimate the most likely time during the track, where an apparent change in movement behavior occurred. Furthermore, it offers a way to test if a change in movement behavior is significant, or just what should be expected from random fluctuations.
Materials and Methods Materials The three tracks presented here are part of a larger study of large-scale movement, distribution, and changes in behavior of Atlantic bluefin tuna. These fish were all tagged near Cape Cod within a month of the autumnal equinox. Two in 1999 and one in 2000. The tags were programmed to release themselves from the fish at preselected times at liberty. The tags recorded light intensities and time of day, and transmitted the data to the tag manufacturer via satellite at the preselected dates. The data were used to estimate raw light based geo-locations via a proprietary, and to this author unknown, algorithm. (II.2)
77
Basic Model The model is a state-space model, where the transition equation is describing the movements along the sphere. A random walk model is assumed: αi = αi−1 + ci + ηi ,
i = 1, . . . , T
(II.1)
Here αi is a two dimensional vector containing the coordinates (αi,1 , αi,2 ) in nautical miles along the sphere from longitude=0 and latitude=0 at time ti , ci is the drift vector describing the deterministic part of the movement, and ηi is the noise vector describing the random part of the movement. The deterministic part of the movement is assumed to be proportional to time ci = (u∆ti , v∆ti )0 . The random part is assumed to be serially uncorrelated and follow a two dimensional Gaussian distribution with mean vector 0 and covariance matrix Qi = 2D∆ti I2×2 . The measurement equation of the state-space model is a non-linear function describing the expected observation at a given state (α position). Each observation yi consists of two coordinates a longitude and a latitude. The two coordinates are derived from a light based geo-location algorithm. The measurement equation describing yi is: yi = z(αi ) + di + εi ,
i = 1, . . . , T
(II.2)
The observational bias di = (blon , blat )0 , if present, describe systematic measurement errors, for instance if the internal clock in the tag is not set absolutely correct. The coordinate change function z is given by: z(αi ) =
αi,1 60 cos(αi,2 π/(180·60)) αi,2 60
(II.3)
The measurement error εi is assumed to follow a Gaussian distribution with mean vector 0 and covariance matrix Hi = diag(σ 2 , σ 2 ). The variance of the latilon lati tude measurements is closely related to the equinox. Measurements close to the equinox have large latitude errors. The following variance structure is assumed: . 2 2 σlat = σlat cos2 2π(Ji + (−1)si b0 )/365.25 + a0 (II.4) i 0 where Ji is the number of days since last solstice prior to all observations, si is the season number since the beginning of the track (one for the first 182.625 days, then two for the next 182.625, then three and so on). a0 , b0 and σ 2 are model lat0 parameters.
Detachment Model If at any point during the observed track the tag came off the fish, the movement pattern would be expected to change instantaneously. The movement pattern (II.3)
78
Paper II. Locating Point of Premature Detachment of Archival Tags is described by the transition equation (II.1). The measurement equation (II.2) would remain unchanged, as it only describes how the true position of the tag is translated into the observed measurements. Consider a given detachment time τ. If the tag came off the fish after τ days at liberty, the parameters describing the movement of the tag (u, v and D) would be expected to be different while the tag was on the fish, than while the tag was simply floating by itself. Denote the parameters before detachment as u1 , v1 and D1 , and after detachment as u2 , v2 and D2 . Observations are only available at discrete time points, so the movement between two observations may be partly before the detachment and partly after. As a consequence the terms ci and ηi should be calculated as weighted averages. For instance, the first coordinate of the drift term ci should be uei ∆ti , where , if τ ≥ ti u1 def. τ−ti−1 ti −τ u1 + ∆ti u2 , if ti−1 < τ < ti (II.5) uei = wave(u1 , u2 ,ti−1 ,ti , τ) = ∆ti u2 , if τ ≤ ti−1 Following this example the second coordinate of ci should be vei ∆ti , and the diagoei ∆ti , where nal elements of the covariance matrix Qi of ηi should be 2D vei = wave(v1 , v2 ,ti−1 ,ti , τ) and
ei = wave(D1 , D2 ,ti−1 ,ti , τ). D
(II.6)
In the actual implementation of the detachment model a smooth (differentiable) version swave of the wave function was used. This modification allows for automatic differentiation of the model and the approximation error is negligible. This function is derived in appendix A.
The Likelihood Function Both the basic model and the detachment model are state-space models and the only non-linearity in the models is introduced by the z function (II.3). The extended Kalman filter (Harvey 1990) is used to calculate the likelihood function from both models. Let `1 denote the negative log-likelihood function for the basic one segment model, and `2 denote the negative log-likelihood function for the two segment detachment model. Let θ1 and θ2 denote the vectors of model parameters for the two models respectively. The basic model is clearly a sub-model of the detachment model, as it correspond to the case where u1 = u2 , v1 = v2 and D1 = D2 . Compared to the basic model, the detachment model has four additional parameters (τ and the three extra movement parameters).
Locating the Detachment Position τ The most probable detachment position is the maximum likelihood estimate of the model parameter τ in the detachment model. τb denotes the maximum likelihood (II.4)
79 estimate of τ. τb describes the optimal position to divide the movement model of the track into two, or in other words the detachment position resulting in the highest likelihood. It proved difficult to do a standard minimization of the negative log-likelihood `2 corresponding to the detachment model, as random fluctuations of the track can result in many local minima. To ensure that a global minimum was found the following “brute force” algorithm was used. Initial scan: For a number Nτ (20 or so, but should depend on the length of the track) of different fixed values τ1 < · · · < τNτ covering the entire track, `2 is minimized by estimating the remaining parameters u1 , u2 , v1 , v2 , D1 , D2 , blon , blat , σlon , σlat , a0 and b0 . 0
Final minimization: Starting from the detachment position τ where the initial scan returned the smallest `2 value, a final minimization of `2 is done. In this minimization all model parameters u1 , u2 , v1 , v2 , D1 , D2 , blon , blat , σlon , σlat , a0 , b0 and τ are included. 0
If enough fixed values of τ are included in the initial search, this algorithm will return the global minimum.
Determining If the Proposed Detachment Point is Significant As the basic model is a sub-model of the detachment model, the natural test statistic is the likelihood ratio test statistic −2 log Q = 2(`1 (θb1 ) − `2 (θb2 ))
(II.7)
This is a so called change–point problem (Carlstein et al. 1994), where the usual asymptotics does not apply because an abrupt change is part of what is estimated. For a fixed detachment point τ selected before data was known −2 log Q follows a chi-square distribution, but this is not very helpful as the point where −2 log Q is maximal is deliberately selected. In this setting only few asymptotic results exist (James et al. 1987; Siegmund 1988), and in simpler models than the present. Furthermore, the finite-sample properties of these asymptotic results are not well investigated. Instead of developing extensions of these asymptotic results the following Monte Carlo approach is applied. First, the real observed track is used to estimate the model parameters of the (o) (o) basic one segment model θb1 and of the two segment detachment model θb2 , and to calculate the likelihood ratio test statistic −2 log Q(o) . Next, the hypothesis of no detachment is assumed and a number Nmc of tracks are simulated forward from the basic one segment state-space model (equations II.1 and II.2). Each track is simulated from the same initial date and the same initial position as the real observed track, and if observations are missing at some dates for the real (II.5)
Paper II. Locating Point of Premature Detachment of Archival Tags observed track the same dates are excluded from the simulated tracks. Finally, the likelihood ratio test statistic −2 log Q( j) , j = 1, . . . , Nmc is calculated from each of these simulated tracks, following the exact same procedure as for the real observed track. This simulated, or parametric bootstrapped, distribution of the likelihood ratio statistic is used to evaluate if the observed likelihood ratio statistic −2 log Q(o) is larger than what should be expected if the hypothesis was true, which would indicate a significant detachment point.
(o )
1 ●
− log L
80
●
●
● ● ● ● ●
●
●
● ● ● ● ● ●
●
●
● ●
(o )
2
5% 1%
τ
Figure II.1: The negative log-likelihood for the two segment detachment model `2 is plotted against potential detachment times τ. `2 is evaluated in all points indicated by circles. The horizontal line above indicate the value of the negative log-likelihood `1 for the basic one segment model (independent of τ). The horizontal lines below indicates the simulated confidence limits for the difference between the negative log-likelihoods.
To visualize this search for premature detachment the following plot can be useful. For each point in the initial scan the negative log-likelihood for the two segment (o) detachment model `2 is plotted against potential detachment times τ. Horizontal (o) lines are added corresponding to the negative log-likelihood `1 for the basic one segment model, and the simulated confidence limits for the difference between (o) the negative log-likelihoods (annotated illustration see figure II.1). If `2 at any point is below the dashed line a detachment point is diagnosed at the five percent significance level (similar for the solid one percent line below). This strict criteria can naturally also be evaluated directly from the simulated and observed likelihood ratio test statistics, but the plot can be useful for judging if a significant change really is an abrupt detachment. (II.6)
81
Results The first track (number 4934 see figure II.2) shows clear evidence of an abrupt change in movement behavior after slightly less than 50 days at liberty. The plot of the two segment negative log-likelihood `2 indicate this by dropping below both the simulated 5% and 1% quantiles. The time of the change is well determined as `2 only drops in one place, which could imply that the track is well described by one set of movement parameters on each side of the change–point. Visually inspecting the longitude coordinate show the same pattern. Up till around 50 days at liberty the fish is moving east much faster than after. The latitude coordinate shows no clear pattern except that the geo-location error is much higher around the equinox (20 March). The 20 equidistant points selected to evaluate `2 in the initial scan is, judged by the plot, sufficient to ensure that the global minimum is found. The two segment negative log-likelihood `2 of the second track (number 4364 see figure II.3) is slightly below or only slightly above the simulated 5% quantile in the entire interval between 40 and 120 days at liberty. This indicates that the track is not well described by one constant set of movement parameters for the entire track. The method estimate (somewhat arbitrarily) the potential change–point to around 110 days at liberty, but the negative log-likelihood value at the minimum point is not really different from the value at 50 days at liberty. This indicates that the change is not sudden, but gradual and that a change–point anywhere between 40 and 120 will give a significantly or almost significantly better fit. Visually inspecting the longitude coordinate shows that the fish is moving east, but the velocity is gradually decreasing. The four rightmost evaluated points on the `2 curve are plotted identical to the basic one segment `1 curve. This is because the two segment detachment model did not converge with those fixed detachment points, which happens eventually. Lack of convergence is diagnosed by extreme estimated movement or variance parameters, or too high gradient components at the suggested minimum. Placing these points on the `1 curve ensures that a false detachment is not diagnosed at such points. The third track (number 3816 see figure II.4) shows no evidence of a change in movement pattern. The two segment negative log-likelihood `2 is above the simulated 5% quantile for the entire track, which indicates that no two segment model is significantly better than the basic one segment model. The latitude coordinate shows huge geo-location errors around the two equinoxes (22 September and 20 March).
Discussion The results from the presented methods are consistent with the appearance of the analyzed tracks, but the method offers substantial improvements to visually in(II.7)
82
Paper II. Locating Point of Premature Detachment of Archival Tags
45
Longitude
290 300 310 320
Track number 4934
0
50
100
150
Latitude
40 35 30
Latitude
45
40
Days at liberty
50
100
150 35
0
430
●
●
● ●
●
●
●
● ●
●
425
● ●
● ●
420
●
●
● ●
30
−log likelihood
Days at liberty
●
6Mar2000
16Jan2000
27Nov99
8Oct99
● ●
290
Date
300
310
320
Longitude
Figure II.2: Estimated track of tag number 4934 (right). Raw geo-locations are marked by crosses, deployment point is marked by “5”, and known recapture/pop-up is marked by “4”. The big cross indicate the estimated most likely potential detachment point. Left panel shows the longitude coordinate (upper), the latitude coordinate (middle), and the search for premature detachment (lower), as illustrated in figure II.1. specting the tracks. First, it gives an objective criteria for locating the most likely potential detachment point. In a setting with varying variance structure and periods where geo-locations are missing, it can be very difficult to “visually estimate” the most likely detachment point reliably. Second, it quantifies the significance of the potential detachment point. Visual inspection offers no alternative for this. The method is designed to detect and verify a sudden change in the movement pattern of the fish, which could indicate a premature tag detachment, but it has no strict way to distinguish between a premature detachment and the fish actually changing its behavior. The plot of the search for detachment (explained in figure II.1) does offer a little advice. If the negative log-likelihood `2 only takes a single drop below the simulated confidence limit(s) and has a consistent behavior before and a different consistent behavior after (like in figure II.2) it could indicate (II.8)
83
60
330 310
55
290
Longitude
Track number 4364
0
50
100
150
200
150
200
45
Latitude
50 40 30
Latitude
60
50
Days at liberty
50
100
40
0
Days at liberty
35
265
●
● ●
● ● ●
Date
30
●
12Apr2000
14Nov99
25Sep99
●
22Feb2000
●
● ●
●
● ●
3Jan2000
260
● ●
●
255
−log likelihood
● ● ● ●
290
300
310
320
330
340
Longitude
Figure II.3: Estimated track of tag number 4364 (right). Raw geo-locations are marked by crosses, deployment point is marked by “5”, and known recapture/pop-up is marked by “4”. The big cross indicates the estimated most likely potential detachment point. The left panel shows the longitude coordinate (upper), the latitude coordinate (middle), and the search for premature detachment (lower), as illustrated in figure II.1. a premature detachment, but if `2 drops below several times during the track (like in figure II.3) it is most likely not a premature detachment, as it can happen only once for each track. If `2 is always above the simulated confidence limits there is no evidence of premature detachment. The method is highly computational demanding because of the simulated confidence limits. Finding theoretical approximative confidence limits would certainly make this approach much faster. In real applications this is a minor concern, as any scientist capable of waiting many months for the return of a tag also should be capable of waiting a few extra hours for the computer to finish. For archival tags that are surgically implanted, or have the capability to detect and report a premature detachment, the method is still highly relevant. If the method detects a significant change–point in such a track, it means that the fish changed its (II.9)
84
Paper II. Locating Point of Premature Detachment of Archival Tags
310
40
300 290
Longitude
60
Track number 3816
0
50
100
150
200
250
300
350
250
300
350
0 Latitude
20 −20
−20
−60
Latitude
60
20
Days at liberty
0
50
100
150
200
●
● ●
●
●
●
●
●
●
● ●
470
●
●
●
● ● ●
●
−60
● ●
●
11Jul2001
30Aug2001
Date
22May2001
2Apr2001
11Feb2001
23Dec2000
3Nov2000
14Sep2000
460
−log likelihood
480
−40
Days at liberty
290
295
300
305
310
315
Longitude
Figure II.4: Estimated track of tag number 3816 (right). Raw geo-locations are marked by crosses, deployment point is marked by “5”, and known recapture/pop-up is marked by “4”. The big cross indicates the estimated most likely potential detachment point. The left panel shows the longitude coordinate (upper), the latitude coordinate (middle), and the search for premature detachment (lower), as illustrated in figure II.1. behavior, or put differently, that the assumption of constant movement parameters for the entire track is not valid for the track. This should lead to a change in the movement model. Detecting changes in behavior from the observed tracks could potentially reveal population level features of the movement pattern, for instance delimit spawning locations or migration corridors, but it would require long tracks from numerous individuals.
Acknowledgements The research was supported by the SLIP research school under the Danish Network for Fisheries and Aquaculture Research (www.fishnet.dk) financed by the (II.10)
85 Danish Ministry for Food, Agriculture and Fisheries and the Danish Agricultural and Veterinary Research Council. The author thanks Molly Lutcavage for kindly providing the data sets used as examples.
References Arnold, G., and Dewar, H. 2001. Electronic tags in marine fisheries research: a 30-year perspective. In: Electronic Tagging and Tracking in Marine Fisheries Reviews: Methods and Technologies in Fish Biology and Fisheries. J.R. Sibert & J.L. Nielsen (eds) Dordrecht: Kluwer Academic Press, pp. 7–64. Carlstein, E., Muller H. G., and Siegmund, D. 1994. Change-Point Problems. Institute of Mathematical Statistics Lecture Notes Series, Vol. 23, Institute of Mathematical Studies. Gunn, J., and Block, B. 2001. Advances in acoustic, archival, and satellite tagging of tunas. In: Tuna Physiology, Ecology, and Evolution. B.A. Block & E.D. Stevens (eds) NY: Academic Press, pp. 167–224. Harvey, A. C. 1990. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press, Cambridge. Hill, R. D., and Braun, M. J. 2001. Geolocation by Light Level - The Next Step: Latitude. In J.R. Sibert and J. Nielsen (Eds.), Electronic Tagging and Tracking in Marine Fisheries (pp. 315-330). Kluwer Academic Publishers, The Netherlands. James, B., James, K. L., and Siegmund, D. 1987. Tests for change–point. Biometrika 74:71–83. Musyl, M. K., Brill, R. W., Curran, D. S., Gunn, J. S., Hartog, J. R., Hill, R. D., Welch, D. W., Eveson, J. P., Boggs, C. H., and Brainard, R.E. 2001. Ability of archival tags to provide estimates of geographical position based on light intensity. In: Electronic Tagging and Tracking in Marine Fisheries Reviews: Methods and Technologies in Fish Biology and Fisheries. J.R. Sibert & J.L. Nielsen (eds) Dordrecht: Kluwer Academic Press, pp. 343–368. Sibert, J., Musyl, M. K. and Brill, R. W. 2003. Horizontal movements of bigeye tuna (Thunnus obesus) near Hawaii determined by Kalman filter analysis of archival tagging data. Fish. Oceanogr. 12(3):141–151. Sibert, J., Nielsen, A., Lutcavage, M. and Bril, R. W. in prep. Large-scale movement, distribution and changes in behavior of Atlantic bluefin tuna determined from analysis of pop-up archival tag data. Siegmund, D. O. 1988. Confidence sets in in change–point problems Int. Statist. Rev. 56:31–48. (II.11)
86
Paper II. Locating Point of Premature Detachment of Archival Tags
Appendix A Definition of the Smooth Weighted Average Function A smooth (differentiable) approximation of the wave function defined in (II.5) is found in the following way. The wave function can also be defined as: uei = wave(u1 , u2 ,ti−1 ,ti , τ) =
1 ∆ti
Z ti ti−1
1{t
