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Hybrid Symbolic-Numeric Method for Detecting Parameter Redundancy in Ecological Models’ Diana Cole, NCSE, University of Kent Rémi Choquet, Centre d'Ecologie Fonctionnelle et Evolutive Ben Hubbard, NCSE, University of Kent Introduction – Example Capture-Recapture • Cormack-Jolly-Seber (CJS) Capture-Recapture Model • Parameters: 𝜙𝑗 survival of year j, 𝑝𝑗 recapture in year j History 1111 1110 1101 ⋮ Probability of History 𝜙1 𝑝2 𝜙2 𝑝3 𝜙3 𝑝4 𝜙1 𝑝2 𝜙2 𝑝3 (1 − 𝜙3 + 𝜙3 (1 − 𝑝4 )) = 𝜙1 𝑝2 𝜙2 𝑝3 (1 − 𝜙3 𝑝4 ) 𝜙1 𝑝2 𝜙2 (1 − 𝑝3 )𝜙3 𝑝4 ⋮ • The parameters 𝜙3 and 𝑝4 are confounded, can only estimate the product never the parameters individually. Introduction • In some models it is not possible to estimate all the parameters. This is termed parameter redundant / nonidentifiable. • 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. 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, 𝜿. An exhaustive summary is a vector of parameter combinations that uniquely define the model, e.g. probabilities of histories • Let 𝜽 denote a vector of the p parameters. 𝜕𝜿 𝜕𝜽 • We then form a derivative matrix 𝑫 = and calculate its rank. • 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 𝜶𝑗 𝑫𝑻 = 𝟎 then 𝜕𝑓 𝒑 𝛼 𝒊=𝟏 𝑖𝑗 𝜕𝜃𝑖 2010). = 0, 𝑗 = 1, … 𝑑 (Catchpole et al, 1998 or Cole et al, 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: 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) • 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. Hybrid-Symbolic Numeric Method • Calculate the derivative matrix, 𝜕𝜿 𝑫= , 𝜕𝜽 symbolically. • Evaluate 𝑫 at a random point 𝜽𝑘 to give 𝑫𝑘 . • Calculate 𝑟𝑘 the rank of 𝑫𝑘 . • Repeat for 5 random points model, then 𝑟 = max 𝑟𝑘 . • If the model is parameter redundant for any 𝑫𝑘 with 𝑟𝑘 = 𝑟 solve 𝜶𝑘 𝑫𝑇𝑘 = 0. The zeros in 𝜶𝑘 indicate positions of parameters that can be estimated. Example Capture-Recapture • = [1, 2, 3, p2, p3, p4 ] 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. (𝑡) • Initial state probabilities:𝜋𝑗 𝑡 𝑡 𝑡 𝑡 for 𝑗 = 1,2 & 𝑡 = 1, … 6 (𝜋3 = 1 − 𝜋1 − 𝜋2 ) 𝑡 • Transition probabilities: 𝜙∗𝑖𝑗 for 𝑖, 𝑗 = 1,2,3 & 𝑡 = 1, … , 5 and 𝜙𝑖𝑗𝑘 for 𝑖, 𝑗, 𝑘 = 1,2,3 & 𝑡 = 2, … , 5. 𝑡 • Capture probabilities: 𝑝𝑗 for 𝑖 = 1,2,3 , 𝑡 = 2, … , 6. (p = 180 Parameters) Example – Occupancy Models • Occupancy models considers whether or not a species is present at a particular site. • Parameters: 𝜓 − site is occupied, 𝑝 – species is detected. • Species detected at a site with probability 𝜓𝑝. • Species not detected at a site with probability 𝜓 1 − 𝑝 + 1 − 𝜓 = 1 − 𝜓𝑝 • Basic model is parameter redundant, so a robust design was developed, so that several surveys are conducted each season at each site, and assumed 𝜓 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). 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. • 𝜓 occupancy probabilities, 𝑝 detection probabilities. • (s) dependence on site, (t) dependence of time, ∙ dependent on neither site nor time. Model 𝜓 ∙ 𝑝 ∙ 𝜓 𝑠 𝑝 ∙ 𝜓 ∙ 𝑝 𝑠 𝜓 𝑡 𝑝 𝑡 𝜓 𝑡, 𝑠 𝑝 ∙ 𝜓 𝑡, 𝑠 𝑝 𝑡 𝜓 𝑡, 𝑠 𝑝 𝑡, 𝑠 Rank 20 65 35 59 161 176 236 Deficiency No. pars 0 20 0 65 0 35 0 59 17 178 17 193 67 303 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. 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. 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.