A Hybrid Symbolic-Numerical Method for Determining Model Structure

1. 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

2. Introduction – Example Capture-Recapture Herring Gulls (Larusargentatus) capture-recapture data for 1983 to 1986 (Lebreton, et al 1995) Numbers Released: Numbers Recaptured: • 83 • 84 • 85 Recapture yr 84 85 86 Yr released 83 84 85

3. Introduction – Example Capture-Recapture i – probability a bird survives from occasion i to i+1 pi – probability a bird is recaptured on occasion i  = [1, 2, 3,p2, p3, p4 ] recapture probabilities Can only ever estimate 3p4 - model is parameter redundant or non-identifiable.

4. 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,p2, p3, p4 ] R = [1, 2, p2, p3,  ]  = 3 p4 • 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.

5. Symbolic Method • Symbolic methods can be used to detect parameter redundancy in less obvious cases (see for example Catchpole and Morgan, 1997, Cole et al, 2010). • Firstly an exhaustive summary is required, . An exhaustive summary is a vector of parameter combinations that uniquely define the model, e.g. recapture probabilities, Q. • Let denote a vector of the p parameters. • We then form a derivative matrix,

6. Symbolic Method • Then calculate the rank, r, of . • When r = p, model is full rank; we can estimate all parameters. • When r < p, model is parameter redundant with deficiency d = p – r . • In parameter redundant models we can also find a set of r estimable parameter combinations by solving then (Catchpole et al, 1998 or Cole et al, 2010).

7. 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. Bio-kinetic compartment model of sludge respiration Douchainet al (2007) Cole et al (2010) 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)

8. Hybrid-Symbolic Numeric Method • 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.

9. Example Capture-Recapture •  = [1, 2, 3,p2, p3, p4 ]

10. Example – multi-site capture-recapture model • The capture-recapture models can be extended to studies with multiple site (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)

11. Example – Occupancy Models • Rather than marking animals, occupancy models looks at whether or not a species is present at a particular site. • Parameters: – species is detected. • Species detected at a site with probability . • Species not detected at a site with probability • Basic model is parameter redundant, so a robust design was developed, so that several surveys are conducted each season at each site, and assumed is the same for each survey. • More complex models consider multiple sites and interactions between species. • These models are not parameter redundant, but this assumes that every possible combination of occupied and unoccupied is observed. However parameter redundancy can be caused by the data (intrinsic parameter redundancy).

12. Example – 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.

13. Example - Bio-kinetic compartment model of sludge respiration • Non-linear compartment models can be used to describe the activated sludge-process (Dochainet al, 1995). • The exogenous oxygen uptake, U, depends on the bio-degredation of two substrates, S. • Parameters: where Yi is the fraction of the pollutant Si which is not oxidised but converted into a new biocatalyst, X. The parameters max1 and max2 are rate constants, Km1and Km2 are affinity constants. • From extended symbolic method estimable parameter combinations:

14. 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.

15. 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. 797. • 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.