Prof. Terrance J. Quinn II University of Alaska Fairbanks Juneau AK USA [email protected] [email protected] and Richard B. Deriso Inter-American.

Prof. Terrance J. Quinn II

University of Alaska FairbanksJuneau AK [email protected]

and

Richard B. Deriso

Inter-American Tropical Tuna CommissionLa Jolla CA USA

[email protected]

* 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?