Workshop on Parameter Redundancy Part II Diana Cole.

Workshop on Parameter Redundancy Part II

Diana Cole

Introduction

• The key to the symbolic method for detecting parameter redundancy is to find a derivative matrix and its rank.

• Models are getting more complex• The derivative matrix is therefore structurally more complex• Maple runs out of memory calculating the rank• Examples: Hunter and Caswell (2008), Jiang et al (2007)

• How do you proceed?– Numerically – but only valid for specific value of parameters– Symbolically – involves extending the theory, again it involves a

derivative matrix and it’s rank, but the derivative matrix is structurally simpler.

Wandering AlbatrossMulti-state models for sea birds

Striped Sea BassAge-dependent tag-return

models for fish

Exhaustive SummariesCJS Model (revisited)

Herring Gulls (Larus argentatus)

capture-recapture data for 1983

to 1986 (Lebreton, et al 1995)

Catchpole and Morgan (1997) theory

etc1

00

0 22

43

433232

433221322121

pp

p

ppp

pppppp

P

T

ppppppR

pppR

ppR

pR

)1( 4332213221211

4332211

32211

211

1

μκ

9100

31030

2467

N

111

123

78

R

i

kjμ

D

Exhaustive Summaries

13121111

3222

4332213221211

43322113

322112

2111

4

)ln(

)1ln(

)ln(

)ln(

)ln(

NNNRF

pN

ppppppF

pppN

ppN

pN

l

T

ij

κ

i

kjP

D

T

ij

p

pp

p

ppp

pp

p

P

43

4332

32

433221

3221

21

2

κ

T

ij

p

pp

p

ppp

pp

p

P

)ln(

)ln(

)ln(

)ln(

)ln(

)ln(

)ln(

43

4332

32

433221

3221

21

3

κ

r

cc

r c n

i

n

ij ijn

ij iji

n

i

n

ijijij PNRPNLl

11

1ln)ln(const)log(

Catchpole and Morgan (1997) - sufficient to examine or

(only if no missing data)

i

kjP

)ln(

D

Exhaustive Summaries

k Rank(Dk)

5

5

5

5

μκ 1

ijP2κ

)ln(3 ijPκ

ijl4κ

i

kjk

D

432321 ppp

432211432211

4323132114323113211

4323232221432321322121

00

)(0

)(

pppRpppR

pppppRpppRppR

ppppppRpppRppRpR

43221

43231321

432323222

00

0

ppp

ppppp

pppppp

13

12

12

11

11

11

00

0

432321322121

432211

3

3,1

432321322121

432313211

2

3,1

2

2,1

432321322121

4323232221

1

3,1

1

2,1

1

1,1

100

10

1

pppppp

pppFNpppppp

pppppFNNpppppp

ppppppFNNN

Exhaustive Summary• An exhaustive summary, , is a vector that uniquely defines the model (Cole

and Morgan, 2009a, generalises definition in Walter and Lecoutier, 1982).• Derivative matrix

• r = Rank(D) is the number of estimable parameters in a model• p parameters; d = p – r is the deficiency of the model (how many parameters

you cannot estimate). If d = 0 model is full rank (not parameter redundant) . If d > 0 model is parameter redundant.

• More than one exhaustive summary exists for a model• Choosing a structurally simpler exhaustive summary will simplify 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 Pij can be removed (Catchpole and Morgan, 1997), as long as there are no missing values. (If there are missing values the appropriate exhaustive summary is the complete set of log-likelihood terms)

i

j

D

Methods For Use With Exhaustive SummariesWhat can you estimate?

(Catchpole and Morgan, 1998 extended to exhaustive summaries in Cole and Morgan, 2009)

• A model: p parameters, rank r, deficiency d = p – r• There will be d nonzero solutions to TD = 0. • Zeros in s indicate estimable parameters. • Example: CJS, regardless of which exhaustive summary is used

• Solve PDEs to find full set of estimable pars.

• Example: CJS, PDE:

Can estimate: 1, 2, p2, p3 and 3p4

djfp

i iij ,...,10

1

10000

4

3

pT α

0434

3

p

ff

p

Methods For Use With Exhaustive SummariesExtension Theorem

• Suppose a model has exhaustive summary 1 and parameters 1.

• Now extend that model by adding extra exhaustive summary terms 2, and extra parameters 2. (Eg add more years of ringing/recovery) New model’s exhaustive summary is = [1 2]T and parameters are = [1 2]T.

• If D1 is full rank and D2 is full rank, the extended model will be full rank. The result can be further generalised by induction.

• Result is trivially always true, if you add zero or one extra parameter• Method can also be used for parameter redundant models by first rewriting the

model in terms of its estimable set of parameters.

i

j

,1

,11

D

2

,1

,21

,2

,2

,1

,2

,1

,1

00 D

DDD

i

j

i

j

i

j

i

j

Methods For Use With Exhaustive SummariesThe PLUR decomposition

• Ring-recovery Model

is not parameter redundant. (Call this top model)• However the nested model with 1,1 = 1,2 is parameter redundant,

deficiency = 1.• This information is in the top model’s derivative matrix

• Write derivative matrix which is full rank r as D = PLUR• P is a square permutation matrix • L is a lower diagonal square matrix, with 1’s on the diagonal• U is an upper triangular square matrix (any entry on the diagonal)• R is a matrix (size of D) in reduced echelon form• R will always have rank r, regardless of constraints.• By design P and L will always have rank r.• The rank of U can vary.

aa

aaaaaP

2,112,1

1,11,111,1

0

aa 12,11,1

00000

10000

1000

100

1

g

fe

dcba

R

Methods For Use With Exhaustive Summaries The PLUR decomposition

• A square matrix which is not full rank will have determinant 0.• If Det(U) = 0 at any point, model is parameter redundant at that point (as

long as R is defined). The deficiency of U evaluated at that point is the deficiency of that nested model (Cole and Morgan, 2009a)

• Ring-recovery example top model:

5)(Rank

00

000

00

000

00

111

11

11

1111

12,1

12,1

11,1

11,1

11,1

DD

aaa

aaaa

10000

01000

00100

00010

00001

R)1)(1(

)(Det2,11,112,11,1

2,11,1

aa

U

0)(DetIf 2,11,1 U

Therefore nested model is parameter redundant with deficiency 1

4)(Rank2,11,1U

Finding simpler exhaustive summaries Reparameterisation

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

CJS Model (revisited):

2. Reparameterise the exhaustive summary. 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(

)(

43

4332

32

433221

3221

21

p

pp

p

ppp

pp

p

θ

)ln(

)ln(

)ln(

)ln(

)ln(

)ln(

)(

5

54

3

542

32

1

s

ss

s

sss

ss

s

s

43

433232

433221322121

00

0

p

ppp

pppppp

P

Reparameterisation3. Calculate the derivative matrix Ds

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

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

5. If Ds is full rank ( Rank(Ds) = Dim(s) ) 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

There are 5 si and the Rank(Ds) = 5, so Ds is full rank. 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

)(DimRankif si

js

Reparameterisation

6. Use sre as an exhaustive summary

A reduced-form exhaustive summary is

Rank(D2) = 5; 5 estimable parameters.Solve PDEs: estimable parameters are 1, 2, p2, p3 and 3p4

43

32

32

21

21

s

p

p

p

p

p

re

3

22

11

4

33

22

2

0000

000

000

0000

000

000

p

pp

pp

s

i

rejD

The PLUR decomposition• CJS with survival covariates.

• Let i = {1 + exp(a +bxi)}-1 where xi is a covariate

• Parameters: = [a b p2 p3 p4]

• Use exhaustive summary:

• Rank(D) = 5 not parameter redundant

• PLUR decomposition Det(U) =

• Parameter redundant if x1 = x2

41

3

31

2

31

2

21

1

21

1

43

32

32

21

21

)exp(1

)exp(1

)exp(1

)exp(1

)exp(1

s

pbxa

pbxa

pbxa

pbxa

pbxa

p

p

p

p

p

re

i

jres

D

)exp(1)exp(1)exp(1

))(exp()exp(

33

23

1

1221

bxabxabxa

xxbxabxa

Reparameterisation

• 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

0000

0000

000

000

2

1

p

psurvival breeding given survival successful breeding capture

Wandering Albatross(Diomedea exulans)

1 3

2 4

1 success

2 = failure

3 post-success

4 = post-failure

Reparameterisation

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(

)(

121

21

211

2222

2221

1112

1111

pp

p

p

p

p

θ

)1(

)(

131321

146

132

145

131

sss

ss

ss

ss

ss

s

Reparameterisation3. Calculate the derivative matrix Ds

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

rank(Ds) = 12, no. est. pars = 12, deficiency = 14 – 12 = 2

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

)(DimRankif si

js

Reparameterisation Method

6. Use sre as an exhaustive summaryT

re pp

2244411333

4

4

3

3214433222111222111

s

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)

Reparameterisation Method

• Jiang et al (2007) age-dependent fisheries model is more complex, but essentially uses reparameterisation method (Cole and Morgan, 2009b)

• Rachel’s talk looked at multi-state analysis of Great Crested Newts. The parameter redundancy of the more complex models can be examined using the reparameterisation method to find a simpler exhaustive summary

Conclusion• Exhaustive summaries offer a more general framework

for symbolic detection of parameter redundancy• Parameter redundancy can be investigated symbolically

by examining a derivative matrix and its rank.• Parameter redundant nested models can be found using

a PLUR decomposition of any full rank derivative matrix.• The use of reparameterisation allows us to produce

structurally much simpler exhaustive summaries, allowing us to examine parameter redundancy of much more complex models symbolically.

• Methods are general and can in theory be applied to any parametric model

References• Original Symbolic Approach:

– 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– Catchpole, E. A. and Morgan, B. J. T. (2001) Deficency of parameter redundant models. Biometrika, 88,

593-598• Recent Advances in Symbolic Approach:

– Cole, D. J. and Morgan, B. J. T (2009a) Determining the Parametric Structure of Non-Linear Models IMSAS, University of Kent Technical report UKC/IMS/09/005

– Cole, D. J. and Morgan, B. J. T. (2009b) A note on determining parameter redundancy in age-dependent tag return models for estimating fishing mortality, natural mortality and selectivity. IMSAS, University of Kent Technical report UKC/IMS/09/003 (To appear in JABES)

– Cole, D.J. and Morgan, B.J.M (2007) Detecting Parameter Redundancy in Covariate Models. IMSAS, University of Kent Technical report UKC/IMS/07/007,

– See http://www.kent.ac.uk/ims/personal/djc24/publications.htm for papers• Other references:

– Hunter, C.M. and Caswell, H. (2008). Parameter redundancy in multistate mark-recapture models with unobservable states. In Ecological and Environmental Statistics Volume 3. Eds., D. L. Thomson, E. G. Cooch and M. J. Conroy, 797-826

– Jiang, H. Pollock, K. H., Brownie, C., Hightower, J. E., Hoenig, J. M. and Hearn, W. S. (2007) Age-dependent tag return models for estimating fishing mortality, natural mortality and selectivity. JABES, 12, 177-194

– 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 identifiably testing for linear and nonlinear state space models. Mathematics and Computers in Simulations, 24, 472-482