Parameter Redundancy and Identifiability in Ecological Models

Diana Cole, University of Kent

2/27

Occupancy Model example

• Parameters: – species is detected.• Can only estimate rather than and .• Model is parameter redundant or parameters are non-identifiable.

Introduction

Species presentand detected

Species presentbut not detected

Species absent

Prob Prob Prob

Prob detected Prob not detected

3/27

Parameter Redundancy• Suppose we have a model with parameters . A model is

globally (locally) identifiable if implies that (for a neighbourhood of ).

• A model is parameter redundant model if it can be written in terms of a smaller set of parameters. A parameter redundant model is non-identifiable.

• There are several different methods for detecting parameter redundancy, including: – numerical methods (e.g. Viallefont et al, 1998),– symbolic methods (e.g. Cole et al, 2010),– hybrid symbolic-numeric method (Choquet and Cole,

2012).• Generally involves calculating the rank of a matrix, which

gives the number of parameters that can be estimated.

4/27

Problems with Parameter Redundancy

• There will be a flat ridge in the likelihood of a parameter redundant model (Catchpole and Morgan, 1997), resulting in more than one set of maximum likelihood estimates.

• Numerical methods to find the MLE will not pick up the flat ridge, although it could be picked up by trying multiple starting values and looking at profile log-likelihoods.

• The Fisher information matrix will be singular (Rothenberg, 1971) and therefore the standard errors will be undefined.

• However the exact Fisher information matrix is rarely known. Standard errors are typically approximated using a Hessian matrix obtained numerically. Can parameter redundancy be detected from the standard errors?

5/27

Is example 1 parameter redundant?Parameter Estimate Standard Error

0.39 imaginary0.64 0.0610.09 imaginary0.18 imaginary

• Hessian () computed numerically has rank 4 (exact Hessian would have rank < 4 if parameter redundant)

• Single Value Decomposition• Write , Matrix is diagonal matrix (Eigen values), the number of

non-zero values is the rank of the matrix.

• Standardised • Hybrid Symbolic-Numeric method: rank 3, only is estimable.• Symbolic Method: rank 3, estimable parameter combinations

.

6/27

Is example 2 parameter redundant?Parameter Estimate Standard Error

0.41 0.700.83 0.070.10 0.110.19 0.33

• Hessian (H) computed numerically has rank 4 (exact would have rank < 4 if parameter redundant).

• Standardised Single Value Decomposition

• Hybrid-Symbolic Numeric method: rank 3, only is estimable.• Symbolic Method: rank 3, estimable parameter combinations

.

7/27

Is example 3 parameter redundant?

• Standardised Single Value Decomposition [1.00 0.65 0.11 0.096 0.074 0.039 0.034 0.0011]• Hybrid-Symbolic Numeric method: rank 8 so is not parameter redundant.• Symbolic method: rank 8 so is not parameter redundant, but a further

test reveals that model could be near redundant, as when model is the same as example 1.

Parameter Estimate Standard Error0.37 0.190.48 0.190.39 0.200.34 0.170.40 0.200.65 0.060.10 0.030.18 0.09

8/27

Simulation Study for Examples 1 and 2

57% have defined standard errors

Parameter True Value Average MLE St. Dev. MLE0.4 0.49 0.32

0.7 0.70 0.06

0.1 0.28 0.320.2 0.33 0.32

SVD threshold %age SVD test correct0.01 100%

0.001 75%0.0001 15%

0.00001 7%

9/27

Mark-Recovery ModelsAnimals are marked and then when the animal dies its mark is recovered. E.g. Lapwings

¬63¬64¬65

Ringing yr 63 64 65

Recapture yr63 64 64

10/27

Mark-Recovery Models

• 1st year survival probability, adult year survival• 1st year recovery probability, adult year recovery

•

• 𝛽1

𝛽2

11/27

Symbolic Method(Cole et al, 2010 and Cole et al, 2012)

• Exhaustive summary – unique representation of the model

• Parameters

12/27

Symbolic Method• Form a derivative matrix

• Calculate rank. Number estimable parameters = rank(D). Deficiency = p – rank(D). Deficiency > 0 model is parameter redundant.

• Rank Rank Rank but there are 4 parameters, so model is parameter redundant.

13/27

Estimable Parameter Combinations• For a parameter redundant model with deficiency d, solve .

There will be d solutions, . If for all j, then is estimable.

• Estimable parameter combinations can be found by solving a set of PDEs:

• Estimable parameter combinations: .

Other uses of symbolic method

• Uses of symbolic method:– Catchpole and Morgan (1997) exponential family models,

mostly used in ecological statistics,– Rothenberg (1971) original general use, econometric

examples,– Goodman (1974) latent class models,– Sharpio (1986) non-linear regression models,– Pohjanpalo (1982) first use for compartment models,– Cole et al (2010) General exhaustive summary framework,– Cole et al (2012) Mark-recovery models.

• Finding estimable parameters:– Catchpole et al (1998) exponential family models,– Chappell and Gunn (1998) and Evans and Chappell (2000)

compartment models,– Cole et al (2010) General exhaustive summary framework.

15/27

Problem with Symbolic Method• 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.

• How do you proceed?– Numerically – but only valid for specific value of

parameters. But can’t find combinations of parameters you can estimate. Not possible to generalise results.

– Symbolically – involves extending the theory, again it involves a derivative matrix and its rank, but the derivative matrix is structurally simpler.

– Hybrid-Symbolic Numeric Method.

Wandering AlbatrossMulti-state models for sea birds

Hunter and Caswell (2009)Cole (2012)

Striped Sea BassTag-return models for fish

Jiang et al (2007)Cole and Morgan (2010)

16/27

0000

0000

000

000

2

1

p

p

Multi-state capture-recapture example

• Hunter and Caswell (2009) 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

survival breeding given survival successful breeding recapture

Wandering Albatross

1 3

2 4

1 success

2 = failure

3 post-success

4 = post-failure

17/27

Extended Symbolic MethodCole et al (2010)

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

3. 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

Extended Symbolic Method

6. Use sre as an exhaustive summary.T

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)

Extended Symbolic Method

20/27

Multi-state mark–recapture models

State 1: Breeding site 1State 2: Breeding site 2State 3: Non-breeding, Unobservable in state 3 - survival - breeding - breeding site 1 1 – - breeding site 2

21/27

Multi-state mark–recapture models – General Model

• General Multistate-model has S states, with the last U states unobservable with N years of data.

• Survival probabilities released in year r captured in year c:

• t is an SS matrix of transition probabilities at time t with transition probabilities i,j(t) = ai,j(t).

• Pt is an SS diagonal matrix of probabilities of capture pt.

• pt = 0 for an unobservable state,

22/27

r = 10N – 17 d = N + 3

General simpler exhaustive summaryCole (2012)

23/27

Hybrid Symbolic-Numeric MethodChoquet and Cole (2012)

• Calculate the derivative matrix,

symbolically.• Evaluate at a random point to give .• Calculate the rank of.• Repeat for 5 random points model, then .• If the model is parameter redundant for any with solve . The

zeros in indicate positions of parameters that can be estimated.

24/27

Example – multi-site capture-recapture model• The capture-recapture models can be extended to studies with multiple sites (Brownie et al, 1993).• Example Canada Geese in 3 different geographical regions T=6 years.• Geese tend to return to the same site – memory model.• Initial state probabilities: for ) • Transition probabilities: for and for .• Capture probabilities: for (p = 180 Parameters)• (General simpler exhaustive summary, Cole et al, 2014)

25/27

Example – Occupancy models(Hubbard et al, in prep)

• Robust design used to remove PR in occupancy models• Monitoring of amphibians in the Yellowstone and Grand Teton

National Parks, USA (Gould et al, 2012).• Two species: Columbian Spotted Frogs and Boreal Chorus Frogs. • occupancy probabilities, detection probabilities.• (s) dependence on site, (t) dependence of time, dependent on

neither site nor time.Model Rank Deficiency No. pars

20 0 2065 0 6535 0 3559 0 59

161 17 178176 17 193236 67 303

26/27

ConclusionNumeric Symbolic Hybrid-Symbolic

Accurate / correct answer

Not always Yes Yes

General Results (e.g. any no. of years)

No Yes Work in progress

Easy to use (e.g. for an ecologist)

Yes No, but can develop simpler ex. sum

Yes

Possible to add to existing computer

packages

Yes No (needs symbolic algebra)

Yes (E-surge and M-surge)

Individually Identifiable Parameters

No Yes Yes

Estimable parameter combinations No Yes In the future?

Best for intrinsic PR and general results

Best for extrinsic PR and a quick result

27/27

Referenceshttp://www.kent.ac.uk/smsas/personal/djc24/parameterredundancy.htm– Brownie, C. Hines, J., Nichols, J. et al (1993) Biometrics, 49, p1173.– Catchpole, E. A. and Morgan, B. J. T. (1997) Biometrika, 84, 187-196– Catchpole, E. A., Morgan, B. J. T. and Freeman, S. N. (1998) Biometrika, 85, 462-468– Chappell, M. J. and Gunn, R. N. (1998) Mathematical Biosciences, 148, 21-41. – Choquet, R. and Cole, D.J. (2012) Mathematical Biosciences, 236, p117.– Cole, D. J. and Morgan, B. J. T. (2010) JABES, 15, 431-434.– Cole, D. J., Morgan, B. J. T and Titterington, D. M. (2010) Mathematical Biosciences, 228, 16–30. – Cole, D. J. (2012) Journal of Ornithology, 152, S305-S315.– Cole, D. J., Morgan, B. J. T., Catchpole, E. A. and Hubbard, B.A. (2012) Biometrical Journal, 54, 507-

523. – Cole, D. J., Morgan, B.J.T., McCrea, R.S, Pradel, R., Gimenez, O. and Choquet, R. (2014) Ecology

and Evolution, 4, 2124-2133,– Evans, N. D. and Chappell, M. J. (2000) Mathematical Biosciences, 168, 137-159.– Gould, W. R., Patla, D. A., Daley, R., et al (2012). Wetlands, 32, p379.– Goodman, L. A. (1974) Biometrika, 61, 215-231.– Hunter, C.M. and Caswell, H. (2009). Ecological and Environmental Statistics, 3, 797-825– Jiang, H. Pollock, K. H., Brownie, C., et al (2007) JABES, 12, 177-194– Pohjanpalo, H. (1982) Technical Research Centre of Finland Research Report No. 56.– Rothenberg, T. J. (1971) Econometrica, 39, 577-591.– Shapiro, A. (1986) Journal of the American Statistical Association, 81, 142-149.– Viallefont, A., Lebreton, J.D., Reboulet, A.M. and Gory, G. (1998) Biometrical Journal, 40, 313-325.