Detecting Parameter R edundancy in Integrated Population Models

1. Detecting Parameter Redundancy in Integrated Population 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 only ever be possible to estimate the product and never the two parameters separately. • This is an example of parameter redundancy.

3. Lapwing Example • Ring-recovery data on Lapwings 1963- • Probabilities of being ringed in year iand recovered in year j, Pij, • Ring-recovery model alone is not parameter redundant (Cole et al, 2012). • Integrated state-space model and ring-recovery model is not parameter redundant. • What happens if reporting probability is dependent on age-class? Ring-recovery model alone is parameter redundant. Is the integrated model parameter redundant?

4. Symbolic Method for Detecting Parameter Redundancy • In some models it is not possible to estimate all the parameters and the model can be written in terms of a smaller set of 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, , for each data set. An exhaustive summary is a vector of parameter combinations that uniquely define the model. • An exhaustive summary for ring-recovery data are the Pij. What exhaustive summary can be used for state-space models? • There are p parameters, . • We then form a derivative matrix, , where • 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).

5. 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.

6. Lapwing ExampleState-Space Model Rank(D2) = 2, therefore model is parameter redundant. (Solving a PDE shows estimable parameter combinations are and .)

7. Lapwing ExampleIntegrated Model • Probabilities of being ringed in year iand recovered in year j, Pij, form an exhaustive summary for the ring-recovery data (Cole et al, 2012). • Integrated model is not parameter redundant, so in theory it is possible to estimate all the parameters.

8. Lapwing ExampleIntegrated Model • Reporting probability is dependent on age class: • But there are 5 parameters so the integrated model is parameter redundant. • Estimable parameter combinations:

9. Symbolic method for more complicated models • For state-space models each terms is successively more complex than the previous. In more complex models (e.g. more than 4 states) it may not be possible to find Rank. • Suppose there are data sets in the integrated model with exhaustive summaries and, and parameters and of lengths and . • Rather than considering we can use the following result (which involves a combination of the extension theorem and reparameterisation theorem): • Let with rank . • If let If let be the estimable parameter combinations. • Then write as a function of s with extra parameters to given . • Then has rank . • The rank of the integrated model is . • If contains just one parameter no need to find rank of integrated model is .

10. Lapwing ExampleIntegrated Model again • Start with the ring-recovery model: • Then consider the state-space model: . • Rank of integrated model is .

11. Common Guillemots Example • Renolds et al (2009) examine four data sets on common guillemots. • Study extends for T years. • Data set 1: Productivity data, number of successful fledged chicks. • Parameters: ( productivity in year t). • Data set 2: Capture-recapture of adults. • Parameters: adult survival year t), ( recapture probability year t). • Data set 3: Mark-resight-recapture of chicks. • Parameters: , ,, ( reporting probability year t), (resighting probability aged jyear t), (tag loss), (prob bird does not emigrate). • Data set 4: Count of adults. • Parameters: , , ,

12. Deficiency = no. parameters – rank If deficiency = 0, model is not parameter redundant If deficiency > 0, model parameter redundant

13. Parameters that can be estimated:

14. Discussion • Parameter redundancy of state-space models can be investigated by expanding the expectation of the observation process. • It is possible to investigate parameter redundancy in integrated models by combining exhaustive summaries for each data set. • A result exists for more complex models, which allowed general parameter redundancy results for the common guillemots example to be found. • What if, for the common guillemots, there was no longer enough funding to collect all four data sets? In terms of what can be estimated would recommend keeping the chick MRR.

15. 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. • Reynolds, T. J., King, R., Harwood, J., Frederikesen, M., Harris, M. P. and Wanless, S. (2009) Integrated Data Analyses in the Presence of Emigration and Tag-loss. JABES, 14, 411-431.