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Detecting Parameter R edundancy in Ecological State-Space M odels

Detecting Parameter R edundancy in Ecological State-Space M odels. Diana Cole and Rachel McCrea National Centre for Statistical Ecology, University of Kent. Lapwing Example.

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Detecting Parameter R edundancy in Ecological State-Space M odels

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  1. Detecting Parameter Redundancy in Ecological State-Space Models Diana Cole and Rachel McCrea National Centre for Statistical Ecology, University of Kent

  2. Lapwing Example • Lapwing (Vanellusvanellus) census data consists of a yearly index of abundance derived from counts of adult Lapwings. • Let denote the number of 1 year old birds (unobserved) and number of adults (observed). Besbeaset al (2002) considered the following state-space model juvenile survival probability; adult survival; productivity; and error processes. • The two parameters and only ever appear as a product. It will never be possible to estimate the product and never the two parameters separately. • This is an example of parameter redundancy.

  3. Symbolic Method for Detecting Parameter Redundancy • In some models it is not possible to estimate all the parameters. This is termed parameter redundancy. • 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. For example the probabilities of different life histories form an exhaustive summary. • There are p parameter, . • We then form a derivative matrix, • 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, and we can also find a set of r estimable parameter combinations (see Catchpole et al, 1998 or Cole et al, 2010 for details).

  4. Parameter Redundancy in State-Space Models • Linear state-space model format: observation process, state equation, measurement matrix, transition matrix, and are error processes. • An exhaustive summary can be obtained by expanding the observation process, • If is an mm matrix then need to expand up to . Otherwise an extension theorem (Catchpole and Morgan, 1997, Cole et al, 2010) can be used. • If the error processes involve parameters, then we can also expand the variance to extend the exhaustive summary. • Method also extends to non-linear models.

  5. Lapwing ExampleState-Space Model Rank D1 = 2, therefore model is parameter redundant. (Solving a PDE shows estimable parameter combinations are and .)

  6. Integrated Models • State-space models may be parameter redundant because not all states are observed or due to the underlying state equation. However, there may still be interest in estimating all the parameters. A possible solution is then to combine two or more different types of data, and describe them with an integrated population model (see for example Besbeaset al, 2002). • An exhaustive summary is required for each data set, .   • The joint exhaustive summary, , is differentiated with respect to the p parameters, , to form the derivative matrix,

  7. Lapwing ExampleState-Space Model Combined with Ring-Recovery Data • Probabilities of being ringed in year iand recovered in year j, Pij, form an exhaustive summary for the ring-recovery data, with additional parameter , the reporting probability (Cole et al, 2012). • Combined model is not parameter redundant, so in theory it is possible to estimate all the parameters.

  8. Extended Symbolic Method • The key to the symbolic method for detecting parameter redundancy is to find a derivative matrix and its rank. • Note that in the exhaustive summary for the state-space model, each successive term is more complex than the last. For state-space models with more than a few states the resulting derivative matrix is structurally too complex and Maple runs out of memory calculating the rank. • In such cases we can use the extended symbolic method (Cole et al, 2010). • This involves choosing a reparameterisation, s, that simplifies the model structure. • We then rewrite the exhaustive summary, (), in terms of the reparameterisation, (s). • Calculate the derivative matrix, .

  9. Multi-Site Model • McCrea et al (2010) consider a multi-site state-space model for great cormorants (Phalacrocoraxcarbo). • Census data consisted of annual nest counts at 2 different sites (simplification of 3 sites). • State-space model, where 1,ksurvival prob. of immature animals, 2,ksurvival prob. of breeders, ρkproductivity, k prob. becoming a breeder and kjtransition prob. (site k). new born immature breeders 1 2 1 2 1 2

  10. Multi-Site Model • Reparameterisation: • is 10 by the reparameterisation theorem, 10. There are 10 parameters in the original parameterisation so this model is not parameter redundant.

  11. Discussion • Parameter redundancy of state-space models can be investigate by expanding the expectation of the observation process. • It is not always necessary to combine state-space models with other types of data, even when not all states are observed. It is often possible to estimate more than expected. • It is possible to investigate parameter redundancy in integrated models by combining exhaustive summaries for each data set. • The reparameterisation theorem and extension theorem have been combined to create a simpler method to investigate parameter redundancy in integrated models.

  12. References • Besbeas, P., Freeman, S. N., Morgan, B. J. T. and Catchpole, E. A. (2002) Integrating Mark-Recapture-Recovery and Census Data to Estimate Animal Abundance and Demographic Parameters. Biometrics, 58, 540-547. • 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. • Cole, D. J., Morgan, B. J. T. and Titterington, D. M. (2010) Determining the parametric structure of models. Mathematical Biosciences, 228, 16-30. • Cole, D. J. and McCrea, R. S. (2012) Parameter Redundancy in Discrete State-Space and Integrated Models. • Cole, D.J., Morgan, B.J.T., Catchpole, E.A. and Hubbard, B. A. (2012) Parameter Redundancy in Mark-Recovery Models. Biometrical Journal, 54, 507-523. • McCrea, R. S., Morgan, B. J. T., Gimenez, O, Besbeas, P., Lebreton, J. D., Bregnballe, T. (2010) Multi-Site Integrated Population Modelling. JABES, 15, 539-561.

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