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 Albatross Multi-state models for sea birds Striped Sea Bass Age-dependent tag-return models for fish

Exhaustive Summaries CJS Model (revisited) Herring Gulls (Larus argentatus) capture-recapture data for 1983 to 1986 (Lebreton, et al 1995) Catchpole and Morgan (1997) theory

Exhaustive Summaries Catchpole and Morgan (1997) - sufficient to examine or (only if no missing data)

Exhaustive Summaries kk Rank(D k )

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   P ij 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)

Methods For Use With Exhaustive Summaries What 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  T D = 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, p 2, p 3 and  3 p 4

Methods For Use With Exhaustive Summaries Extension 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 D 1 is full rank and D 2 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.

Methods For Use With Exhaustive Summaries The 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.

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: Therefore nested model is parameter redundant with deficiency 1

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).

Reparameterisation 3.Calculate the derivative matrix D s 4.The no. of estimable parameters = rank(D s ) rank(D s ) = 5, no. est. pars = 5 5.If D s is full rank ( Rank(D s ) = Dim(s) ) 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 There are 5 s i and the Rank(D s ) = 5, so D s is full rank. s is a reduced- form exhaustive summary

Reparameterisation 6.Use s re as an exhaustive summary A reduced-form exhaustive summary is Rank(D 2 ) = 5; 5 estimable parameters. Solve PDEs: estimable parameters are  1,  2, p 2, p 3 and  3 p 4

The PLUR decomposition CJS with survival covariates. Let  i = {1 + exp(a +bx i )} -1 where x i is a covariate Parameters:  = [a b p 2 p 3 p 4 ] Use exhaustive summary: Rank(D) = 5  not parameter redundant PLUR decomposition Det(U) = Parameter redundant if x 1 = x 2

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: survival breeding given survival successful breedingcapture Wandering Albatross (Diomedea exulans) 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).

Reparameterisation 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

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1,2,3,4 0 (11)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, –Catchpole, E. A., Morgan, B. J. T. and Freeman, S. N. (1998) Estimation in parameter redundant models. Biometrika, 85, –Catchpole, E. A. and Morgan, B. J. T. (2001) Deficency of parameter redundant models. Biometrika, 88, 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 for papershttp:// 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, –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, –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, –Walter, E. and Lecoutier, Y (1982) Global approaches to identifiably testing for linear and nonlinear state space models. Mathematics and Computers in Simulations, 24,