Parameter Redundancy and Identifiability in Ecological Models

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

2. Introduction Occupancy Model example • Parameters: – species is detected. • Can only estimate rather than and . • Model is parameter redundant or parameters are non-identifiable. Species present but not detected Species present and detected Species absent Prob Prob Prob Prob not detected Prob detected

3. 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. 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. Is example 1 parameter redundant? • 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. Is example 2 parameter redundant? • 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. 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.

8. Simulation Study for Examples 1 and 2 57% have defined standard errors

9. Mark-Recovery Models Animals are marked and then when the animal dies its mark is recovered. E.g. Lapwings Recapture yr 63 64 64 Ringing yr 63 64 65 • 63 • 64 • 65

10. Mark-Recovery Models • 1st year survival probability, adult year survival • 1st year recovery probability, adult year recovery

11. Symbolic Method(Cole et al, 2010 and Cole et al, 2012) • Exhaustive summary – unique representation of the model • Parameters

12. Symbolic Method • Form a derivative matrix • Calculate rank. Number estimable parameters = rank(D). Deficiency = p – rank(D). Deficiency > 0 model is parameter redundant. • RankRankRank but there are 4 parameters, so model is parameter redundant.

13. 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: .

14. 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. 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. Multi-state capture-recapture example Wandering Albatross • 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: 3 post-success 1 success 3 1 4 2 4 = post-failure 2 = failure recapture successful breeding breeding given survival survival

17. Extended Symbolic MethodCole et al (2010) • Choose a reparameterisation, s, that simplifies the model structure. • Rewrite the exhaustive summary, (), in terms of the reparameterisation - (s).

18. Extended Symbolic Method • Calculate the derivative matrix Ds. • The no. of estimable parameters =rank(Ds) rank(Ds) = 12, no. est. pars = 12, deficiency = 14 – 12 = 2 • 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.

19. Extended Symbolic Method • Use sre as an exhaustive summary.

20. 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. 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) = ai,j(t). • Pt is an SS diagonal matrix of probabilities of capture pt. • pt = 0 for an unobservable state,

22. General simpler exhaustive summaryCole (2012) r = 10N – 17 d = N + 3

23. 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. 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. 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.

26. Conclusion

27. References http://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.