Statistical catch at age models

UALG

Statistical catch at age models

Einar Hjörleifsson

2

The data: Catches at age (here million of fish)

Age -->Year 0 1 2 3 4 5 6 7 8 9 10 Total1982 0 0 0 3 32 55 88 60 41 77 17 3801983 0 0 0 4 17 55 57 71 46 14 22 2981984 0 0 0 9 54 50 55 35 43 20 5 2821985 0 0 0 9 48 91 67 43 27 19 11 3231986 0 0 0 30 40 76 111 53 27 13 7 3651987 0 0 0 14 122 73 59 74 26 11 5 3901988 0 0 0 10 71 144 66 32 35 10 4 3781989 0 0 0 3 51 130 123 31 13 7 2 3631990 0 0 0 7 21 65 135 79 17 5 3 3351991 0 0 0 11 48 38 68 95 36 7 2 3081992 0 0 0 16 38 65 38 44 46 14 2 2651993 0 0 0 29 62 42 50 18 17 20 10 2511994 0 0 0 9 50 50 25 22 8 5 5 1781995 0 0 0 15 18 49 47 21 10 2 2 1691996 0 0 0 8 29 23 51 46 13 6 2 1811997 0 0 0 3 31 50 27 40 38 8 4 2031998 0 0 0 4 13 62 72 31 30 23 5 2441999 0 0 0 3 34 37 85 61 18 13 7 2602000 0 0 0 14 12 68 37 53 34 9 5 2352001 0 0 0 17 51 25 66 27 26 15 4 2342002 0 0 0 8 36 65 25 42 14 11 6 2082003 0 0 0 5 29 53 64 23 22 7 3 2082004 0 0 0 2 34 58 58 42 15 12 4 2272005 0 0 0 6 9 64 58 37 24 8 5 2132006 0 0 0 3 21 18 67 45 23 12 3 19620072008

Year effect

Age effect Cohort effect

Recall the lecture of structure fisheries data

3 The model in math

2 2 21 1 1 2 2 2 2

min 1ˆ ˆ ˆln ln ln ln ln lnS S S S S S S

C a ay ay S a ay ay a ay ayy a y a y a

SSE C C U U U U

ay a yF s F

ayMF

ayya

yaay Ne

MF

FC ayya )(

,

, ,1ˆ

1

2

1 1 1 1

2 2 2 2

ˆ where

ˆ where

Say ay

Say ay

p F MS S S Say a ay ay ay

p F MS S S Say a ay ay ay

U q N N N e

U q N N N e

1, 1 1, 1

1, 1 1, 1 , 1 , 1

,

( )

, 1, 1

( ) ( )

1, 1 , 1

a y a y

a y a y a y a y

a y

F M

a y a y

F M F M

a y a y

R

N N e

N e N e

1 or 1

1

11

plus

plus

a y

a a

a a

fullR

aa

full

aa

a

aae

aaes

full

L

full

for

for

2

2

4 The model in words

Make a separable model having: A fixed (constant through time) selection pattern (sa) for each age,

assume selection pattern follows double-half Gaussian Fixed selectivity with time is commonly referred to as a separable model.

Fishing mortality (for some reference age) for each year (Fy) Numbers of fish that enter the stock each year (year class size,

recruitment, N1,y) and in the first year (Na,1) A plus group: Catches of the oldest age groups are summed - Needs to

be taken into account in the model Calculate:

The number of fish caught each year and age by the fishermen (Cay-hat). This is the modeled Cay number.

The number of fish caught each year and age by the scientist (Uay-hat). This is the modeled Uay number.

Assume the relationship between stock size and survey indices as: Uay = qN

Set up an objective function (minimizing SS): Constrain the model such that we minimize the squared difference

between observed values (Cay and Uay) and predicted values (Cay-hat and Uay-hat)

5 Can you disentangle this?

6 The model as a mapYear Age -----> Year Age -----> Year Age ----->| | || | |V V V

Year Age -----> Year Age -----> Year Age ----->| | || | |V V V




Color codePopulation modelObservation modelYear Age -----> Year Age -----> Year Age ----->Measurements | | |Objectives | | |Parameters V V V

2. Selection pattern

9. S

olve

r 1

. Par

amet

ers

3. Fishing mortality Natural mortality

4a. Population numbers 4b. Nay at survey 1 time 4c. Nay at survey 2 time

5. Predicted catch 6a. Predicted survey 1 indices

6b. Predicted survey 2 indices

Observed catch at age Observed survey 1 indices Observed survey 2 indices

7. Catch residuals 8a. Survey 1 residuals 8b. Survey 2 residuals

7. Catch residuals squared 8a. Survey 1 residuals squared

8b. Survey 2 residuals squared

Popula

tion a

nd o

bse

rvatio

n m

odel

Obje

ctive fu

nctio

ns

Measu

rem

ents

7 The separable part

Selectivity describes the relative fishing mortality within each age group.

In this simplest model setup we assume that the selectivity is the same in all years. Fishing mortality by age and year can thus be described by:

Fay: Fishing mortality of age a year y sa: Selectivity of age a Fy: Fishing mortality (of some reference age) in

year y Note: The separability assumption reduces the number fishing

mortality parameters from:n = (#age groups x #years) ton = (#age groups + #years)

yaay FsF

8 Modelling selectivity

The selection pattern is a function of size/age sa = f(age)

Logistic Gaussian …

So instead of having independent values of s1, s2, .. sa we could use a function to describe the selection pattern as a function of age/size

we will use the normal distribution here for illustrative purpose, but that is not quite often applicable in practice

9 Assume double half-Gaussian

Lets make a further assumption here by letting selectivity follow:

afull: age at full selectivity R: Shape factor (standard deviation) for right hand curve L: Shape factor (standard deviation) for left side of curve

Note: by using this selection function we reduce the number of parameters from whatever number of age groups we have, to only 3 parameters.

But could just as well just estimate each Sa without resorting to a particular function.

The R, L and Afull are parameters that we estimate

fullR

aa

full

aa

a

aae

aaes

full

L

full

for

for

2

2

10 Selectivity - double half-Gaussian

0.00.10.20.30.40.50.60.70.80.91.0

0 5 10

Age

Sele

ctiv

ity

L=5, 10, 15

R=10000, 100, 15

afull = 8

Note asymmetry

11








9. S

olve

r 1

. Par

amet

ers









The map

Store Afull, Land R here

Calculate sa here

12 The map: selection pattern (say)

Param. y/a a a+1 a+2 a+3 a+4 a+5L y Sa Sa+1 Sa+2 Sa+3 Sa+4 Sa+5

R y+1 Sa Sa+1 Sa+2 Sa+3 Sa+4 Sa+5

afull y+2 Sa Sa+1 Sa+2 Sa+3 Sa+4 Sa+5

Fy y+3 Sa Sa+1 Sa+2 Sa+3 Sa+4 Sa+5

.. y+4 Sa Sa+1 Sa+2 Sa+3 Sa+4 Sa+5

.. y+5 Sa Sa+1 Sa+2 Sa+3 Sa+4 Sa+5

Fy+8 y+6 Sa Sa+1 Sa+2 Sa+3 Sa+4 Sa+5

Nay y+7 Sa Sa+1 Sa+2 Sa+3 Sa+4 Sa+5

Nay+1 y+8 Sa Sa+1 Sa+2 Sa+3 Sa+4 Sa+5

Nay+2

Nay+3

Nay+4

...

Parameters

fullR

aa

full

aa

a

aae

aaes

full

L

full

for

for

2

2

Excel speak: =exp(-((a-afull)^2/if(a=<afull;sL;sR)))

13 A word on nomenclature

Often make the following distinction: Selectivity: The probability of catching an individual

of a given age scaled to the maximum probability over all ages, given that all animals are available to be caught by a certain gear in a certain plaice.

This is what gear technologist study at lengths when they are studying the properties of various gears.

Availability: The relative probability, as a function of age, of being in the area in which catching occurs.

Vulnerability: The combination of selectivity and availability.

Thus should really refer to vulnerability but lets stick with the more ambiguous word selectivity, the reason being its wide usage.

14 Setting up Fy and calculating Fay

The fishing mortality each year (Fy) are parameters of the model that we want to estimate.

Since we already calculated sa we can calculate fishing mortality by age and year from: Fay = saFy

15








9. S

olve

r 1

. Par

amet

ers









The map

F1F2....Fy

Calculate Fay here

16

Setting up Ninit and calculating Nay

The number of fish entering the system in first year and in the first age (Ninit) are parameters of the model that we want to estimate. Need: The number of fish in each age group in the

first year (Na,1) The number of recruits entering each year

(N1,y) Given the above we can then fill in the

abundance matrix by the conventional stock equation

( )

1, 1ay ayF M

a y ayN N e

17








9. S

olve

r 1

. Par

amet

ers









The map

Store Ninit

here

Calculate Nay here

18

The map: Population numbers details

Param. y/a a a+1 a+2 a+3 a+4 a+5L y Na,y Na+1,y Na+2,y Na+3,y Na+4,y Na+5,y

R y+1 Na,y+1 Na+1,y+1 Na+2,y+1 Na+3,y+1 Na+4,y+1 Na+5,y+1

afull y+2 Na,y+2 Na+1,y+2 Na+2,y+2 Na+3,y+2 Na+4,y+2 Na+5,y+2

Fy y+3 Na,y+3 Na+1,y+3 Na+2,y+3 Na+3,y+3 Na+4,y+3 Na+5,y+3

.. y+4 Na,y+4 Na+1,y+4 Na+2,y+4 Na+3,y+4 Na+4,y+4 Na+5,y+4

.. y+5 Na,y+5 Na+1,y+5 Na+2,y+5 Na+3,y+5 Na+4,y+5 Na+5,y+5

Fy+8 y+6 Na,y+6 Na+1,y+6 Na+2,y+6 Na+3,y+6 Na+4,y+6 Na+5,y+6

Nay y+7 Na,y+7 Na+1,y+7 Na+2,y+7 Na+3,y+7 Na+4,y+7 Na+5,y+7

Nay+1 y+8 Na,y+8 Na+1,y+8 Na+2,y+8 Na+3,y+8 Na+4,y+8 Na+5,y+8

Nay+2

Nay+3

Nay+4

...

Pastel green area: Estimated parameters

( )

1, 1ay ayF M

a y ayN N e

19 Predicting catch: Cay-hat

Once the population matrix is calculated it is simple to calculate the predicted catch (Cay-hat) according to the catch equation:

The C-hats are values that we will later “confront” with the measurements that we have.

ayMF

ayya

yaay Ne

MF

FC ayya )(

,

, ,1ˆ

20








9. S

olve

r 1

. Par

amet

ers









The map

Predicted Cay-hat

21 Confronting the model with data

Until now we have only set up equations that follow the progression of each year class and calculated catch.

This is more or less a population simulator. If we let recruitment be a function of SSB and we add some

noise to recruitment we have a closed system and thus almost a medium/long term simulator

This is also more or less the same thing as we do when we do a short term projection.

Or for that matter in a yield per recruit analysis, except that there we focus only on one cohort (here one diagonal line).

At present we are only interested in fitting the model to observations (measurements). Need thus some kind of objective function (minimizing sums of squares).

22 The objective function in words

Find the value of the parameters: fishing pattern by age, sa (controlled by L, R and Afull) yearly fishing mortality (Fy) population number in the first year (Na,1) recruitment (N1,y) in each year

that minimize the squared deviation of estimated catch (Cay-hat) and measured catch (Cay)

Note that here we assume a log-normal error distribution. Could easily be replace with other type of error structure.

2minˆlnln

y aayayC CCSSE

23








9. S

olve

r 1

. Par

amet

ers









The map

(obs-pre)

(obs-pre)^2

The sumto

minimize

24 Different weights by age

The catch of different age groups are often measured with different accuracy. Thus often set different weights to the residuals, so that the information from age groups that are measured with the most accuracy weigh more in the objective function:

2minˆlnln

y aayayaC CCSSE

is inversely related to the variance

25 If we only have Cay

If there are no other available data for a stock than catch at age one could attempt to fit the model to catches alone.

May need a extra “stabilizer”: The brave one may assume that fishing mortality does not change much between consecutive years:

yyyC

FC

FFSSE

SSESSESSE2

1

min

lnln

26 Tuning with survey indices

If additional information are available it is relatively easy to add them to the model. If age-based survey indices are available one may use:

where qa is a parameter (catchability). The minimization is by (again assuming log-normal errors):

ayaay NqU ˆ

2minˆlnln

y aayayaU UUSSE

27 iCod: Age based survey indicesAge -->

Year 0 1 2 3 4 5 6 7 8 9 10198219831984 01985 0 17 111 35 48 64 23 15 5 3 21986 0 15 61 96 22 21 26 7 2 1 11987 0 4 29 103 82 21 12 12 3 1 01988 0 3 7 72 102 67 8 6 6 1 01989 0 4 16 22 78 68 34 4 1 1 01990 0 6 12 26 14 27 32 14 2 1 01991 0 4 16 18 30 15 18 21 4 1 01992 0 1 17 33 19 16 7 6 5 1 01993 0 4 5 31 36 13 10 2 2 1 01994 0 14 15 9 27 22 6 4 1 0 01995 0 1 29 25 9 24 18 4 2 0 01996 0 4 5 43 29 13 15 14 4 1 01997 0 1 22 14 56 29 9 9 7 1 01998 0 8 6 30 16 62 28 7 5 3 11999 0 7 33 7 42 13 24 11 2 1 12000 0 19 28 55 7 30 8 8 4 1 02001 0 12 22 37 38 5 15 3 2 1 02002 0 1 38 41 40 36 7 8 1 1 02003 0 11 4 46 39 32 19 4 5 1 02004 0 7 25 8 62 35 25 14 3 3 02005 0 3 15 39 10 43 23 11 6 1 12006 0 9 7 23 38 11 28 10 4 1 02007 0 6 18 9 21 28 9 10 5 2 12008

28

Population numbers at survey time

If survey time is not in the beginning of the year we need to take that into account by:

Where N’ is the population size at survey time p is the fraction of the year when the survey

takes place.

' ( )

'ˆ

p Fay Mayay ay

ay a ay

N N e

U q N

29








9. S

olve

r 1

. Par

amet

ers









The map

obs-pre

(obs-pre)^2

The sum

Predict Uay-hat

Store qa

here

Calculate N’ay

30 Objective function I

Simple to combine the two objective functions:

2

2

min

ˆlnln

ˆlnln

y aayaya

y aayaya

UC

UU

CC

SSESSESSE

Lets not worry about pa for now

31 Getting it all together

The heart of thesetup lies in the left side of the spread-sheet. There we have thethe objective functions(minimize SS) in one place.The only thing leftis to setup the solversuch that it minimizesthe total SSE bychanging the para-meters. And this youwere introduced already at day 2 of thecourse.

THE MINIMIZATION STUFF

Sum of squares LambdaC@A 48.419 1

U@A - Survey 1 53.900 1

U@A - Survey 2 49.084 1

SSE total 151.4023

PARAMETERS

Name Ln(parameter) Switches ParameterLn Afull 2.3900 10.91Ln L 1.4481 4.26Ln R 5.0000 148.41

33 The output: Historical development of the stock

1905190619071908190919101911191219131914191519161917191819191920192119221923192419001901

0

50

100

150

200

250

300

350

400

1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924

Recruitment

0

100

200

300

400

500

600

700

800

900

1000

1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924

SSB

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924

F

0

50

100

150

200

250

1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924

Yield

0

50

100

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200

250

300

1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924

Survey 1 numbers

0

10

20

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50

60

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100

1900 1902 1904 1906 1908 1910 1912 1914 1916 1918 1920 1922 1924

Survey 2 numbers

0

2

4

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12

14

16

1895 1900 1905 1910 1915 1920 1925 1930

Catch residuals

0

2

4

6

8

10

12

1895 1900 1905 1910 1915 1920 1925 1930

Survey 1 residuals

0

2

4

6

8

10

12

1895 1900 1905 1910 1915 1920 1925 1930

Survey 2 residuals

34 Age/size structured models

Advantages Populations do have age/size structure Basic biological processes are age/size specific

Growth Mortality Fecundity

The process of fishing is age/size specific Relatively simple to construct mathematically Model assumption not as “strict” as in e.g. logistic models

Disadvantages Sample intensive Data often not available Mostly limited to areas where species diversity is low Have to have knowledge of natural mortality For long term management strategies have to make model

assumptions about the relationship between stock and recruitment

Often not needed to address the question at hand

UALG

Some addition points on Y/R and then on reference points in light of

model uncertainty

36

37 Atlantic mackerel

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40 NEA cod

41 NEA haddockFind the Fmsy and the Bmsy!

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Statistical catch at age models