A Hybrid Symbolic-Numerical Method for Determining Model Structure Diana Cole, NCSE, University of Kent Rémi Choquet, Centre d'Ecologie Fonctionnelle et Evolutive Ben Hubbard, NCSE, University of Kent

Introduction – Example Capture-Recapture  83  84  85 Yr released 83  84  85  Recapture yr

Introduction – Example Capture-Recapture

Introduction In some models it is not possible to estimate all the parameters. This is termed parameter redundant / non- identifiable. A model is parameter redundant if it can be reparameterised in terms of a smaller number of parameters. Capture-recapture example:  = [  1,  2,  3, p 2, p 3, p 4 ]  R = [  1,  2, p 2, p 3,  ]  =  3 p 4 Parameter redundancy can be due to the model (extrinsic) or the data (intrinsic). Sometimes it is obvious that a model is parameter redundant (e.g. capture-recapture example), but in more complex models it is not necessarily obvious.

Symbolic Method

Problems with the Symbolic Method In more complex models the derivative matrix is structurally too complex. Computer runs out of memory calculating the rank. Examples: How do you proceed? – Numerically – can give the wrong results. – Symbolically – involves extending the theory and finding simpler exhaustive summaries (Cole et al, 2010). However this method is complex. – 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) Bio-kinetic compartment model of sludge respiration Douchain et al (2007) Cole et al (2010)

Hybrid-Symbolic Numeric Method

Example Capture-Recapture  = [  1,  2,  3, p 2, p 3, p 4 ]

Example – multi-site capture-recapture model

Example – Occupancy Models

Example – Occupancy models ModelRankDeficiencyNo. pars

Example - Bio-kinetic compartment model of sludge respiration

Conclusion and future work The hybrid method can be used to find how many parameters can be estimated in a model. Hybrid method is much simpler to use than extended symbolic method. Can be added to standard software packages. For ecological models it is available in M-surge and E-surge. It can quickly give results about whether a particular data set is parameter redundant, even for several hundred parameters. However it currently is only applicable to a given number of years of data (ecological models) or substrates (sludge model). In the symbolic method there is an extension theorem that allows general results to be developed. Expanding the hybrid method to include the extension theorem is future work. In the parameter redundant model the hybrid method can currently only determine which of the original parameters are identifiable. Constraints needed to give an identifiable model can only be obtained by trial and error. The symbolic method can also give estimable parameter combinations.

References Hybrid Numeric-Symbolic Method: Choquet, R. and Cole, D.J. (2012) A Hybrid Symbolic-Numerical Method for Determining Model Structure. Mathematical Biosciences, 236, p117. Symbolic Method: Cole, D.J., Morgan, B.J.T., Titterington, D.M. (2010) Mathematical Biosciences, 228, p16. Cole, D.J., Morgan, B.J.T. (2010), JABES, 15, p431. Catchpole, E. A., Morgan, B. J. T (1997) Biometrika, 84, p187. Catchpole, E. A., Morgan, B.J.T., Freeman, S. N. (1998) Biometrika, 85, p42. Cole, D.J. (2012) Journal of Ornithology, 152, p305. Other: Brownie, C. Hines, J., Nichols, J. et al (1993) Capture–recapture, Biometrics, 49, p1173. Dochain, D, Vanrolleghem, P.A., Van Dale, M. (1995) Water Research, 29, p2571. Gould, W. R., Patla, D. A., Daley, R., et al. (2012). Wetlands, 32, p379. Hunter, C., Caswell, H. (2009) Environmental and Ecological Statistics vol 3, p Jiang, H.H., Pollock, K.H., Brownie, C. et al, (2007), JABES, 12, p 177 Lebreton, J. Morgan, B. J. T., Pradel R. and Freeman, S. N. (1995) Biometrics, 51, p1418.