Standardizing catch per unit effort data. 2 Standardization of CPUE Catch = catchability * Effort *...
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Transcript of Standardizing catch per unit effort data. 2 Standardization of CPUE Catch = catchability * Effort *...
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
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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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Fleet 1Fleet 2Total
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
0.0
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Rela
tive b
iom
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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CP
UE
Bt
at
MinimizationSSE 0.0197
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