A parameter redundant model is a model for which it is not possible to estimate all the parameters in the model. The resulting model is not identifiable. A model that is not parameter redundant is termed full rank. Catchpole and Morgan (1997) and Catchpole et al (1998) developed a symbolic method detecting parameter redundancy within exponential family models. This method involves calculating the derivative matrix,

Finding the rank of

The rank of

A limitation of Maple is that matrix dimensions cannot be given symbolically: it is only possible to calculate the rank of a matrix of a given size, and eventually for very large derivative matrices the computer will not have enough memory to be able to calculate the rank. This problem is solved for product-multinomial models using the extension theorem (Catchpole and Morgan, 1997 and Catchpole and Morgan, 2001).

The parameter redundancy status of a model with covariates can be determined from the same model without covariates. The number of estimable parameters in the model with covariates is equal to min(

However models are getting more complex. These more complex models model can result in derivative matrices that are structurally too complex; Maple is unable to calculate the rank of a derivative matrix that is structurally too complex. Instead of having to resort to numerical methods we have developed methods to give structrally simpler derivative matrices using an exhaustive summary framework (Cole et al, 2010).

Exhaustive summaries are vectors that uniquely define a model. Any exhaustive summary can be differtiated with respect to parameters to form a derivative matrix. The rank of the derivative matrix is still equal to the number of estimable parameters, and similar results to those above still hold, regardless of the exhaustive summary (Cole et al, 2010). The key to being able to evaluate the symbolic rank is to choose an exhaustive summary that results in a structurally simple derivative matrix. Structrally simpler exhaustive summaries can be found by any one-to-one transformation, such as removing terms or taking logs. If this is not sufficent reparameterisation can be used to find a simpler exhaustive summary (Cole et al, 2010). Several examples are given in Cole et al (2010), a further more complex example can be found in (Cole and Morgan, 2010b).

Other research includes:

Catchpole, E. A., Morgan, B. J. T and Freeman, S. N. (1998) Estimation in parameter redundant models

Catchpole, E. A. and Morgan, B. J. T (2001) Deficiency of parameter redundant models.

Catchpole, E. A., Morgan, B. J. T and Viallefont, A. (2002) Solving problems in parameter redundancy using computer algebra.

Choquet, R. and Cole, D.J. (2012) A Hybrid Symbolic-Numerical Method for Determining Model Structure.

Cole, D. J., Morgan, B. J. T and Titterington, D. M. (2010) Determining the Parametric Structure of Non-Linear Models.

Cole, D.J., Morgan, B.J.T., Catchpole, E.A. and Hubbard, B. A. (2012) Parameter Redundancy in Mark-Recovery Models.

Cole, D. J. and Morgan, B. J. T (2010) Parameter Redundancy with Covariates.

Cole, D.J. (2012) Determining Parameter Redundancy of Multi-state Mark-recapture Models for Sea Birds.

Cole, D. J., Morgan, B.J.T., McCrea, R.S, Pradel, R., Gimenez, O. and Choquet, R. (2014) Does your species have memory? Capture-Recapture Data with Memory Models.

Hubbard, B. A. (2014) Parameter Redundancy with Applications in Statistical Ecology University of Kent Thesis. Hubbard, B. A., Cole, D.J. and Morgan, B.J.T. (2014) Parameter Redundancy in Capture-Recapture-Recover Models.

McCrea, R.S., Morgan, B.J.T. and Cole, D.J. (2013) Age-dependent mixture models for recovery data on animals marked at unknown age,

Newman, K.B., Buckland, S.T., Morgan, B.J.T., King, R., Borchers, D.L. and Cole, D.J. and Besbeas, P. and Gimenez, O. and Thomas, L. (2014) Modelling Population Dynamics: model formulation, fitting and assessment using state-space methods. Springer. Springer Book Webpage Computer Code for Book (Maple Code for Parameter Redundancy Section)