Parameter Redundancy and Identifiability in Ecological Models Diana Cole, University of Kent

2/27 Introduction Species present and detected Species present but not detected Species absent

3/27 Parameter Redundancy

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? ParameterEstimateStandard Error 0.39imaginary imaginary 0.18imaginary

6/27 Is example 2 parameter redundant? ParameterEstimateStandard Error

7/27 Is example 3 parameter redundant? ParameterEstimateStandard Error

8/27 Simulation Study for Examples 1 and 2 57% have defined standard errors ParameterTrue ValueAverage MLESt. Dev. MLE SVD threshold%age SVD test correct % % % %

9/27 Mark-Recovery Models  63  64  65 Ringing yr 63  64  65  Recapture yr

10/27 Mark-Recovery Models

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

13/27 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 Albatross Multi-state models for sea birds Hunter and Caswell (2009) Cole (2012) Striped Sea Bass Tag-return models for fish Jiang et al (2007) Cole and Morgan (2010)

16/27 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: survival breeding given survival successful breeding recapture Wandering Albatross success 2 = failure 3 post-success 4 = post-failure

17/27 Extended Symbolic Method Cole et al (2010) 1.Choose a reparameterisation, s, that simplifies the model structure. 2.Rewrite the exhaustive summary,  (  ), in terms of the reparameterisation -  (s).

3.Calculate the derivative matrix D s. 4.The no. of estimable parameters =rank(D s ) rank(D s ) = 12, no. est. pars = 12, deficiency = 14 – 12 = 2 5.If D s is full rank s = s re is a reduced-form exhaustive summary. If D s is not full rank solve set of PDE to find a reduced-form exhaustive summary, s re. Extended Symbolic Method

6.Use s re as an exhaustive summary. 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) 2 (12)  1 =  2,  3 =  4 0 (9)0 (10)1 (10)1 (12) 1,2,3,41,2,3,4 0 (11)0 (12) 2 (14) Extended Symbolic Method

20/27 Multi-state mark–recapture models State 1: Breeding site 1 State 2: Breeding site 2 State 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 S  S matrix of transition probabilities at time t with transition probabilities  i,j (t) = a i,j (t). P t is an S  S diagonal matrix of probabilities of capture p t. p t = 0 for an unobservable state,

22/27 r = 10N – 17 d = N + 3 General simpler exhaustive summary Cole (2012)

23/27 Hybrid Symbolic-Numeric Method Choquet and Cole (2012)

24/27 Example – multi-site capture-recapture model

25/27 Example – Occupancy models (Hubbard et al, in prep) ModelRankDeficiencyNo. pars

26/27 Conclusion NumericSymbolicHybrid-Symbolic Accurate / correct answer Not alwaysYes General Results (e.g. any no. of years) NoYesWork in progress Easy to use (e.g. for an ecologist) YesNo, but can develop simpler ex. sum Yes Possible to add to existing computer packages YesNo (needs symbolic algebra) Yes (E-surge and M- surge) Individually Identifiable Parameters NoYes Estimable parameter combinationsNoYesIn the future? Best for intrinsic PR and general results Best for extrinsic PR and a quick result

27/27 References – Brownie, C. Hines, J., Nichols, J. et al (1993) Biometrics, 49, p1173. – Catchpole, E. A. and Morgan, B. J. T. (1997) Biometrika, 84, – Catchpole, E. A., Morgan, B. J. T. and Freeman, S. N. (1998) Biometrika, 85, – Chappell, M. J. and Gunn, R. N. (1998) Mathematical Biosciences, 148, – Choquet, R. and Cole, D.J. (2012) Mathematical Biosciences, 236, p117. – Cole, D. J. and Morgan, B. J. T. (2010) JABES, 15, – 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, – Cole, D. J., Morgan, B.J.T., McCrea, R.S, Pradel, R., Gimenez, O. and Choquet, R. (2014) Ecology and Evolution, 4, , – Evans, N. D. and Chappell, M. J. (2000) Mathematical Biosciences, 168, – Gould, W. R., Patla, D. A., Daley, R., et al (2012). Wetlands, 32, p379. – Goodman, L. A. (1974) Biometrika, 61, – Hunter, C.M. and Caswell, H. (2009). Ecological and Environmental Statistics, 3, – Jiang, H. Pollock, K. H., Brownie, C., et al (2007) JABES, 12, – Pohjanpalo, H. (1982) Technical Research Centre of Finland Research Report No. 56. – Rothenberg, T. J. (1971) Econometrica, 39, – Shapiro, A. (1986) Journal of the American Statistical Association, 81, – Viallefont, A., Lebreton, J.D., Reboulet, A.M. and Gory, G. (1998) Biometrical Journal, 40,