Prof. Terrance J. Quinn II
University of Alaska FairbanksJuneau AK [email protected]
and
Richard B. Deriso
Inter-American Tropical Tuna CommissionLa Jolla CA USA
* Combining the Cohen-Fishman growth increment model with a Box-Cox transformation: flexibility and uncertainty
*Modeling Growth*L = length, a = age, Y, x = size (any function of L)
* Naive: L = growth model + (normal) error
* Less naive: L = growth model * (lognormal) error
* Standard: L a through growth transition matrix
* Stochastic:
*L(this time) = L(last time) + growth increment
*PART 1. Cohen-Fishman Model (1)
),0(~, 2111 Nxx aaaa
Initial condition: ),(~ 2111 Nx
Iterative solution for constant time increments (e.g., age)
and resultant expected value
*PART 1. Cohen-Fishman Model (2)
11
1
1
01
1
01
1
11
series) geometric(
11
11
aa
a
aa
a
a
i
iaa
i
ia
aa
x
xx
*PART 1. Cohen-Fishman Model (3)Variance
aa
aa
2212
22
2221
1
1
Can also be written as a function of mean size
and release size. Chris, the refutation of the comment in your 1988 paper is indeed on page 192!
Generalizable to variance as a function of age,
viz. 2a , maybe length, viz. )(2 L .
*PART 1. Cohen-Fishman Model (4)Example EBS Pacific cod
L 105.4 0.78923 0.2367 22.2154 t0 1.057
2 9
21.1 1
2 1
*PART 1. Cohen-Fishman Model (5)Example EBS Pacific cod
*PART 1. Cohen-Fishman Model (6)
Likelihood Estimation Size/Age Data {Li, ai, i=1,…, n}
F({L}) =
n
i
N1
(VBGM(ai), 2a )
*PART 1. Cohen-Fishman Model (7)Mark-recapture data : Mark at time t1,
recapture at time t2, iii ttt 12 {L2i | L1i, it , i=1,…, n}
Replace x with L; 1 with L1i.
),0(~, 212 NLL iiii
2
22
1222
1122
1
1),|var(
11
),|(E
i
ii
t
iiii
it
t
iiii
tLL
LtLL
F({L2}) = ),(1
222
n
iiiN
*PART 2. Box-Cox Transformation (1)Let
0 ,ln
0 ,/1
Y
Yx .
Assume x ~ N(x, 2x ).
Then 2
22
1
12/122)(x
x
x
x eYYf
.
Expected values, medians, modes x: ).(mode)(med)(E xxx x
*PART 2. Box-Cox Transformation (2)Y:
on.distributi normal a offunction generating
moment thefrom found is for which
1 So
integer. /1for solution form closed has )(E
Y
I
Y
Ix
Y
IY
Med(Y), mode(Y) are the inverse Box-Cox
transformations of med(x), mode(x).
*Combining Cohen-Fishman and Box-Cox
*PART 3. Combining Cohen-Fishman and Box-Cox
*PART 3. Combining Cohen-Fishman and Box-Cox
In terms of the transform variable x ~~
1 aa xx , So the Box-Cox transformation preserves the Cohen-Fishman formulation.
*PART 3. Combining Cohen-Fishman and Box-Cox
Final steps
1. Estimation of through the likelihood outside the data, or
2. Incorporate into the age and length model of Deriso and Parma (1987), using {xa}.
3. Plots of the transformed data and density functions for {x} and {Y}.
Illustration
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
pd
f f(
L)
Length (cm)
EBS Pacific cod
2
3
4
5
6
7
L
*Questions or comments?