1. Detecting Parameter Redundancy in Complex Ecological Models Diana Cole and Byron Morgan University of Kent

2. 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.

3. 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 p i – probability a bird is recaptured on occasion i  = [  1,  2,  3, p 2, p 3, p 4 ]

4. 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

5. 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   Q ij can be removed (Catchpole and Morgan, 1997), as long as there are no missing values.

6. Other tools to use with exhaustive summaries What can you estimate? (Generalisation from Catchpole et al, 1998.) Solve  T D = 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.

7. 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).

8. Reparameterisation Method 3.Calculate the derivative matrix D s 4.The no. of estimable parameters = min(q,rank(D s )) rank(D s ) = 5, no. est. pars = min(6,5) = 5 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 D s is full rank, so s is a reduced-form exhaustive summary

9. Reparameterisation Method 6.Use s re 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.

10. 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: survival breeding given survival successful breeding capture Wandering Albatross (Diomedea exulans) 1 3 2 4

11. Reparameterisation Method 1.Choose a reparameterisation, s, that simplifies the model structure 2.Rewrite the exhaustive summary,  (  ), in terms of the reparameterisation -  (s).

12. Reparameterisation Method 3.Calculate the derivative matrix D s 4.The no. of estimable parameters = min(p,rank(D s )) rank(D s ) = 12, no. est. pars = min(14,12) = 12 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

13. Reparameterisation 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 0 (11)0 (12) 2 (14)

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.

15. 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