Standardizing catch per unit effort data

2 Standardization of CPUE

Catch = catchability * Effort * Biomass CPUE = Catch/Effort = U = catchability * Biomass Ut = qBt

Ut: Catch per unit effort at time t q : catchability of the whole fleet

catchability: the proportion of the stock caught per one unit of effort

In most fisheries we normally have fleets with different catchabilities

Lets start by looking at a fishery on a stock where we have vessel types (i) that have different catchabilities:

Ut,i: Catch per unit effort of vessel type i at time t qi : catchability of vessel type i

tiit BqU ,

3 A very simplified artificial case: 2 Fleets

For illustration we create some CPUE data for 6 years for 2 fleets from known stock size, effort and catchability

Catchability is fleet specific, with fleet 2 having 3 times higher catchability than fleet 1

Effort in Fleet 1 declines while it increases in fleet 2. Total effort remains constant

i Fleet 1 Fleet 2 q1/q2q 0.00015 0.00045 3

Fleet 1 Fleet 2 Fleet 1 & 2 combined

YearStock size Effort Catch CPUE Effort Catch CPUE Effort Catch CPUE

2001 10000 400 600 1.50 100 450 4.50 500 1050 2.102002 12000 370 666 1.80 130 702 5.40 500 1368 2.742003 14000 340 714 2.10 160 1008 6.30 500 1722 3.442004 16000 310 744 2.40 190 1368 7.20 500 2112 4.222005 16000 280 672 2.40 220 1584 7.20 500 2256 4.512006 16000 250 600 2.40 250 1800 7.20 500 2400 4.80

ttiiti BEqC ,,

4 Stock size and overall unstandardize CPUE

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5 Relative values

lets standardize the overall CPUE series relative to that of the first year:

it is obvious that ignoring the different catchabilities of fleet 1 and 2 would lead wrong conclusion about biomass development

however, if we were to use either Fleet 1 OR 2 we would get accurate representation of the relative change in biomass

CPUE from some “selected” fleet, which is assumed to be homogenous over time is often used in practice

The problem is that the assumption of homogeneity is an assumption in real cases!

YearStock size

Relative stock

sizeOverall CPUE

Relative CPUE

2001 10000 1.00 2.10 1.002002 12000 1.20 2.74 1.302003 14000 1.40 3.44 1.642004 16000 1.60 4.22 2.012005 16000 1.60 4.51 2.152006 16000 1.60 4.80 2.29

6 Relative stock size and overall CPUE

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Relative stock size

it is obvious that ignoring the different catchabilities of fleet 1 and 2 would lead wrong conclusion about biomass development when using the

0.0

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2000 2002 2004 2006 2008

CPUE

7 Standardizing the catch rate of each fleet

lets standardized the CPUE series of each fleet relative to the first year:

if we were to use either Fleet 1 OR 2 we would get accurate representation of the relative change in biomass

CPUE from some “selected” fleet, which is assumed to be homogenous over time are often used in practice

CPUE Relative CPUE

YearStock size Fleet 1 Fleet 2 Total Fleet 1 Fleet 2 Total

2001 10000 1.50 4.50 2.10 1.00 1.00 1.002002 12000 1.80 5.40 2.74 1.20 1.20 1.302003 14000 2.10 6.30 3.44 1.40 1.40 1.642004 16000 2.40 7.20 4.22 1.60 1.60 2.012005 16000 2.40 7.20 4.51 1.60 1.60 2.152006 16000 2.40 7.20 4.80 1.60 1.60 2.29

8 Relative stock size and CPUE from Fleet 1 and 2

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9 General linear modeling of CPUE data – the math

Relative changes in biomass:

Lets first describe changes in biomass relative to the first year in the data series:

Bt – biomass at time t B1 – biomass in year 1 t – scaling factor where:

and hence 1 = 1.00

1BB tt

1BBtt

Time (year) Bt t1991 300 1.001992 290 0.971993 280 0.931994 270 0.901995 260 0.871996 250 0.831997 240 0.801998 260 0.871999 280 0.932000 300 1.002001 320 1.072002 340 1.132003 360 1.202004 380 1.272005 400 1.33

The aim is to use the t parameter in the relationship Ut = qBt

10 Biomass (Bt) and relative biomass (t)

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mass

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tive b

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ass

Btalpha t

11 General linear modeling of CPUE data – the math

We have

then

the 92, 93, 94, … are hence (again) a measure of biomass (B) relative to 91 = 1.00 but here we have related it to CPUE

more importantly we gotten rid of the actual biomass (B)!

11t1 and qB , qBUUBB ttt

1

1

1

U

qB

Bq

qBU

t

t

t

tt

Time (year) Ut t1991 3.0 1.001992 2.9 0.971993 2.8 0.931994 2.7 0.901995 2.6 0.871996 2.5 0.831997 2.4 0.801998 2.6 0.871999 2.8 0.932000 3.0 1.002001 3.2 1.072002 3.4 1.132003 3.6 1.202004 3.8 1.272005 4.0 1.33

12 General linear modeling of CPUE data – the math

The relationship: only applies to homogenous fleet

Lets revisit the imaginary 2 vessel class fisheries (where we “know” that q within a fleet has remained constant):

Here the 2|1 is the efficiency of vessel class 2 relative to vessel class 1.

The mathematical formula is effectively saying: the CPUE of vessel class 2 at any one time t is just a multiplier of CPUE of

vessel class 1 at time t taking into account changes relative changes in biomass (t) since first year

1UU tt

1,11|2

1,112

1,12

11,2

22,

11,11,

U

Uqq

Uqq

qUq

BqU

qUBBqU

t

t

t

t

tt

tttt

13 General linear modeling of CPUE data (4)

In general for multivessel fisheries we can write

where i: vessel size class i Ut,i : CPUE of the for time t and vessel class i U1,1: CPUE of the 1st vessel class in the 1st time period i: The efficiency of vessel class i relative to vessel class

1 t: Relative abundance

Food for thought: What is the value of t when t = 1? What is the value of i when i = 1?

1,1, UU itit

14 General linear modeling of CPUE data (5)

To take into account measurement errors the statistical model becomes:

The error can be normalized by transformation:

We hence have a general linear model which can be used to estimate the parameters. For stock assessment purposed the parameters t is of most interest. However, one could consider that the i parameters may be of interest in terms of understanding the fishery and for management

What we have here is nothing more than:Yi = Ŷi + i

… and we know how we estimates the parameters of such a simple model

Food for thought: What is the value of ln(t) when t = 1? What is the value of ln(i) when i = 1?

eUU itit 1,1,

ititit UU ,1,1, lnlnlnln

15 General linear modeling of CPUE data (6)

The GLIM model fit is often done by rescaling all the cpue observation to that of U1,1 (as we have already done) I.e.:

itttit

itit

itit

UU

eU

U

eUU

it

it

,1,1,

1,1

,

1,1,

)ln(lnln

,

,

16 Our example

Lets first add some measurement noise (stochasticity) to our artificial deterministic CPUE data:

eUU DETSTO

Fleet 1 Fleet 2Year

(t)Stock

size (Bt) Effort Catch det CPUE sto CPUE Effort Catch det CPUE sto CPUE2001 10000 400 600 1.50 1.57 100 450 4.50 4.212002 12000 370 666 1.80 1.71 130 702 5.40 5.162003 14000 340 714 2.10 2.09 160 1008 6.30 6.302004 16000 310 744 2.40 2.57 190 1368 7.20 7.472005 16000 280 672 2.40 2.49 220 1584 7.20 7.052006 16000 250 600 2.40 2.22 250 1800 7.20 7.51

Year Fleet 1 Fleet 2 Fleet 1 Fleet 22001 1.00 2.68 0.00 0.982002 1.08 3.28 0.08 1.192003 1.33 4.00 0.29 1.392004 1.64 4.75 0.49 1.562005 1.58 4.48 0.46 1.502006 1.41 4.77 0.35 1.56

standardized CPUE relative to U1,2001

ln standardized CPUE relative to U1,2001

17

Spreadsheet schematics of the model for the simplified example

The ln-value for t=1 and i=1 is by definition zero The value for the reference fleet i=1 is always zero, irrespective

of the year (t) The value for the reference fleet i=2 is the same as for i=1 within

each year

itttit UU ,1,1, )ln(lnln MinimizationSSE 0.0586

Parameters GLM model

Name numeric ln value value time (t) vessel (i)observed

ln cpue ln(t) ln(i)predicted

ln cpue obs-pre (obs-pre)2

2001 2001 0.00 1.00 2001 1 0.00 0.00 0.00 0.00 0.00 0.0002002 2002 0.18 1.20 2002 1 0.08 0.18 0.00 0.18 -0.10 0.0102003 2003 0.34 1.40 2003 1 0.29 0.34 0.00 0.34 -0.05 0.0032004 2004 0.47 1.60 2004 1 0.49 0.47 0.00 0.47 0.02 0.0012005 2005 0.47 1.60 2005 1 0.46 0.47 0.00 0.47 -0.01 0.0002006 2006 0.47 1.60 2006 1 0.35 0.47 0.00 0.47 -0.12 0.0151 1 0.00 1.00 2001 2 0.98 0.00 1.10 1.10 -0.12 0.0132 2 1.10 3.00 2002 2 1.19 0.18 1.10 1.28 -0.09 0.009

2003 2 1.39 0.34 1.10 1.44 -0.05 0.0022004 2 1.56 0.47 1.10 1.57 -0.01 0.0002005 2 1.50 0.47 1.10 1.57 -0.07 0.0052006 2 1.56 0.47 1.10 1.57 -0.01 0.000

Just moving the parameter values here for clarity/convenience

18 Best fit

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ss

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UE

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at

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Parameters GLM model

Name numeric ln value value time (t) vessel (i)observed

ln cpue ln(t) ln(i)predicted

ln cpue obs-pre (obs-pre)2

2001 2001 0.00 1.00 2001 1 0.00 0.00 0.00 0.00 0.00 0.0002002 2002 0.10 1.10 2002 1 0.08 0.10 0.00 0.10 -0.02 0.0002003 2003 0.30 1.35 2003 1 0.29 0.30 0.00 0.30 -0.01 0.0002004 2004 0.49 1.63 2004 1 0.49 0.49 0.00 0.49 0.00 0.0002005 2005 0.44 1.56 2005 1 0.46 0.44 0.00 0.44 0.01 0.0002006 2006 0.42 1.52 2006 1 0.35 0.42 0.00 0.42 -0.07 0.0051 1 0.00 1.00 2001 2 0.98 0.00 1.07 1.07 -0.09 0.0082 2 1.07 2.92 2002 2 1.19 0.10 1.07 1.17 0.02 0.000

2003 2 1.39 0.30 1.07 1.37 0.01 0.0002004 2 1.56 0.49 1.07 1.56 -0.00 0.0002005 2 1.50 0.44 1.07 1.51 -0.01 0.0002006 2 1.56 0.42 1.07 1.49 0.07 0.005

Minimized the squared residuals to obtain the best parameter estimates

19 Expanding the GLM model

The expansion of the GLM model to take into account:

Area Season/month …

is mathematically straightforward:

the model fitting process is the same?

,...,,,1,1,1,1,..,,,

1,1,1,1,..,,,

)ln( ... )ln()ln()ln()ln()ln(

... ,...,,,

kjiikjitkjit

kjitkjit

UU

eUU kjit

20 Where it goes wrong …

Catch rate may not be proportional to abundance Hence the abundance trend from GLM will not be

proportional to abundance Any changes unrelated to quantifiable effects will

not be captured in the GLM analysis. In such cases the change will wrongly be ascribed to

changes in abundance e.g. increase in vessel efficiency within a fleet class due to

increased skill

ERGO: GLM analysis is the best tool available to calculate

standardized catch rate Weather the actual abundance trend form GLM

represents true changes in stock abundance will always be a subjective call