Detecting Parameter Redundancy in Complex

Ecological Models

Diana Cole and Byron MorganUniversity of Kent

Introduction

• If a model is parameter redundant or non-identifiable if you cannot estimate all the parameter in the model.

• Parameter redundancy can be detected by symbolic algebra.

• Ecological models are getting more complex – then computers cannot do the symbolic algebra and numerical methods are used instead.

• In this talk we show some of the tools that can be used to overcome this problem.

Example 1- Cormack Jolly Seber (CJS)

Model Herring Gulls (Larus argentatus) capture-recapture data for 1983 to 1987 (Lebreton, et al 1995)

i – probability a bird survives from occasion i to i+1

pi – probability a bird is recaptured on occasion i

= [1, 2, 3, p2, p3, p4 ]

etc1

00

0 22

43

433232

433221322121

pp

p

ppp

pppppp

Q

9100

31030

2467

N

r cn

ij ijicr c

ij

n

i

NRn

ij ij

n

i

n

ij

Nij QQL

11

1

111

123

78

R

Derivative Method (Catchpole and Morgan, 1997)

Calculate the derivative matrix D

rank(D) = 5 rank(D) = 5 Number estimable parameters = rank(D). Deficiency = q – rank(D) no. est. pars = 5, deficiency = 6 – 5 = 1

T

p

pp

p

ppp

pp

p

)ln(

)ln(

)ln(

)ln(

)ln(

)ln(

43

4332

32

433221

3221

21

κ

432321 ppp

j

i

D

T

ppppppR

pppR

ppR

pR

)1( 4332213221211

4332211

32211

211

μ

i

j

D

323211

4232112211

433211321111

432211

4323113211

432321322121

00

0

00

0

ppR

ppRpR

ppRpRR

pppR

pppRppR

pppRppRpR

14

14

14

13

13

13

13

12

12

12

13

13

13

12

12

12

12

11

11

11

000

00

000

000

00

000

ppp

pppp

ppp

Exhaustive Summaries• An exhaustive summary, , is a vector that uniquely defines

the model (Walter and Lecoutier, 1982).• The exhaustive summary is the starting point for finding the

derivative matrix.• More than one exhaustive summary exists for a model• Choosing a simpler exhaustive summary will simplify the

derivative matrix• Computer packages, such as Maple can find the symbolic rank

of the derivative matrix.• Exhaustive summaries can be simplified by any one-one

transformation such as multiplying by a constant, taking logs, and removing repeated terms.

• For multinomial models and product-multinomial models the more complicated 1 Qij can be removed (Catchpole and Morgan, 1997), as long as there are no missing values.

Other tools to use with exhaustive summaries

What can you estimate? (Generalisation from Catchpole et al, 1998.) Solve TD = 0. Zeros in indicate estimable pars. Solve PDE to find full set of estimable pars.

Extension theorem (Generalised from Catchpole and Morgan, 1997.) Usefully for generalising capture-recapture and ring-recovery models.

PLUR Decomposition. (Cole and Morgan, 2008) Useful for detecting points at which the model is parameter redundant or near parameter redundant, or sub models that are parameter redundant.

01

q

i iij

f

Reparameterisation Method(Cole and Morgan, 2008)

1. Choose a reparameterisation, s, that simplifies the model structure

2. Rewrite the exhaustive summary, (), in terms of the reparameterisation - (s).

43

32

32

21

21

5

4

3

2

1

p

p

p

p

p

s

s

s

s

s

s

)ln(

)ln(

)ln(

)ln(

)ln(

)ln(

)(

)ln(

)ln(

)ln(

)ln(

)ln(

)ln(

)(

5

54

3

542

32

1

43

4332

32

433221

3221

21

s

ss

s

sss

ss

s

p

pp

p

ppp

pp

p

sθ

Reparameterisation Method

3. Calculate the derivative matrix Ds

4. The no. of estimable parameters = min(q,rank(Ds))

rank(Ds) = 5, no. est. pars = min(6,5) = 5

5. If Ds is full rank s = sre is a reduced-form exhaustive summary. If Ds is not full rank solve set of PDE to find a reduced-form exhaustive summary, sre

Ds is full rank, so s is a reduced-form exhaustive summary

15

15

15

14

14

13

13

12

12

11

000

0000

0000

0000

00000

)(

sss

ss

ss

ss

s

s

s

i

js

D

Reparameterisation Method

6. Use sre as an exhaustive summary

A reduced-form exhaustive summary is

adding an extra year of capture and an extra year of recapture adds the extra exhaustive summary terms:

Then the extension theorem can be applied to show that the CJS is always parameter redundant with deficiency 1.

43

32

32

21

21

s

p

p

p

p

p

re

54

43

p

p

Example 2 – Multi-state mark-recapture models for Seabirds

• Hunter and Caswell (2008) examine parameter redundancy of multi-state mark-recapture models, but cannot evaluate the symbolic rank of the derivative matrix (developed numerical method)

• 4 state breeding success model:

1)...()(

1

loglog

1211

1),(

1

1 1

4

1

4

1

),(),(,

rc

rc

mL

Trrcccc

Trrcr

N

r

N

rc i j

crij

crji

II

)1(0)1(0

0)1(0)1(

)1()1()1()1(

4422

3311

444333222111

444333222111

t

0000

0000

000

000

2

1

p

p

t

survival breeding given survival successful breeding capture

Wandering Albatross(Diomedea exulans)

1 3

2 4

Reparameterisation Method

1. Choose a reparameterisation, s, that simplifies the model structure

2. Rewrite the exhaustive summary, (), in terms of the reparameterisation - (s).

2

1

333

222

111

14

13

3

2

1

p

p

s

s

s

s

s

s

)1(

)(

)1(

)1(

)1(

)(

131321

146

132

145

131

121

21

211

2222

2221

1112

1111

sss

ss

ss

ss

ss

pp

p

p

p

p

sθ

Reparameterisation Method

3. Calculate the derivative matrix Ds

4. The no. of estimable parameters = min(p,rank(Ds))

rank(Ds) = 12, no. est. pars = min(14,12) = 12

5. If Ds is full rank s = sre is a reduced-form exhaustive summary. If Ds is not full rank solve set of PDE to find a reduced-form exhaustive summary, sre

Tre sssssssssssssssss 104934837141312116521 //

139

13145513

13131113

0000

)(000

)22(000

)(

ss

sssss

sssss

sD

i

js

s

Reparameterisation Method

6. Use sre as an exhaustive summary

etc..)1( where 11

22444113334

4

3

3214433222111222111

T

re pps

Breeding Constraint

Survival Constraint

1= 2=

3= 4

1= 3,

2= 4

1= 2,

3= 4

1, 2,

3,4

1= 2=3= 4 0 (8) 0 (9) 1 (9) 1 (11)

1= 3,2= 4 0 (9) 0 (10) 0 (10) 2 (12)

1= 2,3= 4 0 (9) 0 (10) 1 (10) 1 (12)

1,2,3, 4 0 (11) 0 (12) 0 (12) 2 (14)

Conclusion

• Exhaustive summaries can be used to detect parameter redundancy.

• The key to more complex problems is to find the exhaustive summary with the simplest structure.

• The most powerful method of finding an exhaustive summary is the reparameterisation method – which examines the basic building blocks of the model.

• These methods can be applied to any parametric model.

References• Catchpole, E. A. and Morgan, B. J. T. (1997) Detecting parameter

redundancy. Biometrika, 84, 187-196• Catchpole, E. A., Morgan, B. J. T. and Freeman, S. N. (1998) Estimation

in parameter redundant models. Biometrika, 85, 462-468• Hunter, C.M. and Caswell, H. (2008). Parameter redundancy in

multistate mark-recapture models with unobservable states. Ecological and Environmental Statistics - in press

• Cole, D. J. and Morgan, B. J. T (2008) Parameter Redundancy and Identifiability. University of Kent Technical Report UKC/IMS/08/022

• Lebreton, J. Morgan, B. J. T., Pradel R. and Freeman, S. N. (1995) A simultaneous survival rate analysis of dead recovery and live recapture data. Biometrics, 51, 1418-1428.

• Walter, E. and Lecoutier, Y (1982) Global approaches to identifiability testing for linear and nonlinear state space models. Mathematics and Computers in Simulations, 24, 472-482