Standardizing catch per unit effort data
Standardization of CPUE
2



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  qi Bt


Ut,i: Catch per unit effort of vessel type i at time t
qi : catchability of vessel type i
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
q 0.00015 0.00045
Fleet 1
Year
2001
2002
2003
2004
2005
2006
Ci,t  qi Ei,t Bt
Stock
size
10000
12000
14000
16000
16000
16000
Effort Catch CPUE
400
600
1.50
370
666
1.80
340
714
2.10
310
744
2.40
280
672
2.40
250
600
2.40
Fleet 2
Effort
100
130
160
190
220
250
q1/q2
3
Fleet 1 & 2 combined
Catch CPUE Effort Catch CPUE
450
4.50
500
1050
2.10
702
5.40
500
1368
2.74
1008
6.30
500
1722
3.44
1368
7.20
500
2112
4.22
1584
7.20
500
2256
4.51
1800
7.20
500
2400
4.80
4
Stock size and overall unstandardize CPUE
CPUE
Stock size
6.00
18000
16000
5.00
14000
4.00
12000
10000
3.00
8000
6000
2.00
4000
1.00
2000
0
2000
2002
2004
2006
2008
0.00
2000
2002
2004
2006
2008
5
Relative values

lets standardize the overall CPUE series relative to that of the first year:
Year
2001
2002
2003
2004
2005
2006


Stock
size
10000
12000
14000
16000
16000
16000
Relative
stock
size
1.00
1.20
1.40
1.60
1.60
1.60
Overall Relative
CPUE CPUE
2.10
1.00
2.74
1.30
3.44
1.64
4.22
2.01
4.51
2.15
4.80
2.29
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!
Relative stock size and overall CPUE
6
Relative stock size
CPUE
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
2000
2002
2004
2006
0.0
2008 2000
2002
2004
2006
2008
it is obvious that ignoring the different catchabilities of fleet 1 and 2 would
lead wrong conclusion about biomass development when using the
7
Standardizing the catch rate of each fleet

lets standardized the CPUE series of each fleet relative to the
first year:
CPUE
Year
2001
2002
2003
2004
2005
2006

Stock
size
10000
12000
14000
16000
16000
16000
Fleet 1
1.50
1.80
2.10
2.40
2.40
2.40
Fleet 2
4.50
5.40
6.30
7.20
7.20
7.20
Relative CPUE
Total
2.10
2.74
3.44
4.22
4.51
4.80
Fleet 1
1.00
1.20
1.40
1.60
1.60
1.60
Fleet 2 Total
1.00
1.00
1.20
1.30
1.40
1.64
1.60
2.01
1.60
2.15
1.60
2.29
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
Relative stock size and CPUE from Fleet 1 and 2
8
Relative stock size
2.5
2.50
2.0
2.00
1.5
1.50
1.0
1.00
0.5
0.50
0.0
2000
0.00
2000
2002
2004
2006
2008
Fleet 1
Fleet 2
Total
2002
2004
2006
2008
General linear modeling of CPUE data – the math
9

Relative changes in biomass:

Lets first describe changes in
biomass relative to the first
year in the data series:
Bt  at B1




Bt – biomass at time t
B1 – biomass in year 1
at – scaling factor
where:
a t  Bt B1

and hence
Time
(year)
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
Bt
300
290
280
270
260
250
240
260
280
300
320
340
360
380
400
a1 = 1.00
The aim is to use the at parameter in the relationship Ut = qBt
at
1.00
0.97
0.93
0.90
0.87
0.83
0.80
0.87
0.93
1.00
1.07
1.13
1.20
1.27
1.33
Biomass (Bt) and relative biomass (at)
450
1.4
400
1.2
1.0
300
0.8
250
Bt
alpha t
200
150
0.6
0.4
100
0.2
50
Year
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
0.0
1991
0
Relative biomass
350
Biomass
10
General linear modeling of CPUE data – the math
11

We have
Bt  at B1 , Ut  qBt and U1  qB1

then
U t  qBt
 qa t B1
 a t qB1
 a tU1


Time
(year)
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
Ut
3.0
2.9
2.8
2.7
2.6
2.5
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
at
1.00
0.97
0.93
0.90
0.87
0.83
0.80
0.87
0.93
1.00
1.07
1.13
1.20
1.27
1.33
the a92, a93, a94, … are hence (again) a measure of biomass (B)
relative to a91 = 1.00 but here we have related it to CPUE
more importantly we gotten rid of the actual biomass (B)!
General linear modeling of CPUE data – the math
12

The relationship:


Ut  atU1
only applies to homogenous fleet
Lets revisit the imaginary 2 vessel class fisheries (where we “know” that q
within a fleet has remained constant):
U t ,1  q1 Bt  Bt  U t ,1 q1
U t , 2  q2
Bt
 q2 U t ,1 q1
 q2 q1 U t ,1
 q2 q1 a tU1,1
 b 2|1 a tU1,1


Here the b2|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 (at)
since first year
General linear modeling of CPUE data (4)
13


In general for multivessel fisheries we can write
where






Ut ,i  at biU1,1
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
bi: The efficiency of vessel class i relative to vessel class 1
at: Relative abundance
Food for thought:


What is the value of at when t = 1?
What is the value of bi when i = 1?
General linear modeling of CPUE data (5)
14

To take into account measurement errors the statistical model becomes:

Ut ,i  at biU1,1e

The error can be normalized by transformation:
lnUt ,i   lnat   lnbi   lnU1,1    t ,i


We hence have a general linear model which can be used to estimate the
parameters. For stock assessment purposed the parameters at is of most
interest. However, one could consider that the bi 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(at) when t = 1?
What is the value of ln(bi) when i = 1?
General linear modeling of CPUE data (6)
15

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.:
U t ,i  a t b iU1,1e
U t ,i
U1,1
 a t bie
 t ,i
 t ,i
lnU t ,i U1,1   lna t   ln(b t )   t ,i
16
Our example

Lets first add some measurement noise (stochasticity) to our
artificial deterministic CPUE data:
U STO  U DET e
Fleet 1
Year Stock
(t) size (Bt)
2001
10000
2002
12000
2003
14000
2004
16000
2005
16000
2006
16000
Year
2001
2002
2003
2004
2005
2006
Effort
400
370
340
310
280
250
Catch det CPUE sto CPUE
600
1.50
1.57
666
1.80
1.71
714
2.10
2.09
744
2.40
2.57
672
2.40
2.49
600
2.40
2.22
Fleet 2
Effort
100
130
160
190
220
250
Catch det CPUE sto CPUE
450
4.50
4.21
702
5.40
5.16
1008
6.30
6.30
1368
7.20
7.47
1584
7.20
7.05
1800
7.20
7.51
standardized CPUE relative to ln standardized CPUE relative
U1,2001
to U1,2001
Fleet 1
Fleet 2
Fleet 1
Fleet 2
1.00
2.68
0.00
0.98
1.08
3.28
0.08
1.19
1.33
4.00
0.29
1.39
1.64
4.75
0.49
1.56
1.58
4.48
0.46
1.50
1.41
4.77
0.35
1.56
17
Spreadsheet schematics of the model for the simplified example
lnUt ,i U1,1   lnat   ln(bt )   t ,i
Minimization
SSE
0.0586
Parameters
Name
a2001
a2002
a2003
a2004
a2005
a2006
b1
b2
Just moving the
parameter values here
for clarity/convenience
GLM model
numeric
2001
2002
2003
2004
2005
2006
1
2
ln value
0.00
0.18
0.34
0.47
0.47
0.47
0.00
1.10



value
1.00
1.20
1.40
1.60
1.60
1.60
1.00
3.00
time (t) vessel (i)
2001
1
2002
1
2003
1
2004
1
2005
1
2006
1
2001
2
2002
2
2003
2
2004
2
2005
2
2006
2
observed
ln cpue
0.00
0.08
0.29
0.49
0.46
0.35
0.98
1.19
1.39
1.56
1.50
1.56
ln(at)
0.00
0.18
0.34
0.47
0.47
0.47
0.00
0.18
0.34
0.47
0.47
0.47
ln(bi)
0.00
0.00
0.00
0.00
0.00
0.00
1.10
1.10
1.10
1.10
1.10
1.10
predicted
ln cpue
0.00
0.18
0.34
0.47
0.47
0.47
1.10
1.28
1.44
1.57
1.57
1.57
2
obs-pre (obs-pre)
0.00
0.000
-0.10
0.010
-0.05
0.003
0.02
0.001
-0.01
0.000
-0.12
0.015
-0.12
0.013
-0.09
0.009
-0.05
0.002
-0.01
0.000
-0.07
0.005
-0.01
0.000
The ln-value for t=1 and i=1 is by definition zero
The b value for the reference fleet i=1 is always zero, irrespective of the year
(t)
The a value for the reference fleet i=2 is the same as for i=1 within each
year
Best fit
20000
Biomass
18000
Minimized the squared residuals to
obtain the best parameter estimates
Minimization
SSE
0.0197
1.6
14000
1.4
12000
1.2
10000
1.0
8000
0.8
6000
0.6
4000
0.4
2000
0.2
0
Parameters
numeric
2001
2002
2003
2004
2005
2006
1
2
1.8
16000
2000
Name
a2001
a2002
a2003
a2004
a2005
a2006
b1
b2
2.0
Bt
at
Relative CPUE
18
0.0
2001
2002
2003
2004
2005
2006
2007
GLM model
ln value
0.00
0.10
0.30
0.49
0.44
0.42
0.00
1.07
value
1.00
1.10
1.35
1.63
1.56
1.52
1.00
2.92
time (t) vessel (i)
2001
1
2002
1
2003
1
2004
1
2005
1
2006
1
2001
2
2002
2
2003
2
2004
2
2005
2
2006
2
observed
ln cpue
0.00
0.08
0.29
0.49
0.46
0.35
0.98
1.19
1.39
1.56
1.50
1.56
ln(at)
0.00
0.10
0.30
0.49
0.44
0.42
0.00
0.10
0.30
0.49
0.44
0.42
ln(bi)
0.00
0.00
0.00
0.00
0.00
0.00
1.07
1.07
1.07
1.07
1.07
1.07
predicted
ln cpue
0.00
0.10
0.30
0.49
0.44
0.42
1.07
1.17
1.37
1.56
1.51
1.49
2
obs-pre (obs-pre)
0.00
0.000
-0.02
0.000
-0.01
0.000
0.00
0.000
0.01
0.000
-0.07
0.005
-0.09
0.008
0.02
0.000
0.01
0.000
-0.00
0.000
-0.01
0.000
0.07
0.005
Expanding the GLM model
19

The expansion of the GLM model to take into account:

Area
Season/month
…

is mathematically straightforward:


U t ,i , j ,k ,..  a t b i j k ...U1,1,1,1e
 t ,i , j ,k ,...
ln(U t ,i , j ,k ,..)  ln(a t )  ln(b i )  ln( j )  ln( k )  ...  ln(U1,1,1,1 )   i ,i , j ,k ,...

the model fitting process is the same?
Where it goes wrong …
20

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