Detecting Parameter Redundancy in Ecological

State-Space Models

Diana Cole and Rachel McCrea National Centre for Statistical Ecology, University of Kent

Lapwing Example

• Lapwing (Vanellus vanellus) 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). Besbeas et 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.

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).

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 mm 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.

Lapwing ExampleState-Space Model

Rank D1 = 2, therefore model is parameter redundant.

(Solving a PDE shows estimable parameter combinations are and .)

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 Besbeas et 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,

Lapwing ExampleState-Space Model Combined with Ring-Recovery Data

• Probabilities of being ringed in year i and 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.

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, .

Multi-Site Model• McCrea et al (2010) consider a multi-site state-space model for great cormorants (Phalacrocorax carbo).

• Census data consisted of annual nest counts at 2 different sites (simplification of 3 sites).

• State-space model,

where 1,k survival prob. of immature animals, 2,k survival prob. of breeders, ρk productivity, k prob. becoming a breeder and kj transition prob. (site k).

new born immature breeders 1 2 1 2 1 2

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.

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.

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.