Parameter Redundancy and Identifiability

Parameter Redundancy

This work on parameter redundancy started as part of a National Centre for Statistical Ecology research associate project entitled Parameter redundancy and state-space modelling using the Kalman Filters, supervised by Byron Morgan and we have continued this research since then.

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, D, which has elements that are the partial derivative of the elements of the mean vector with respect to the derivatives of the parameters. If the rank of D is equal to the number of parameters, p, then the model is full rank. If the rank of D is less than p the model is parameter redundant.

Finding the rank of D can be executed using a symbolic computer package, such as Maple. Catchpole et al (2002) explain how Maple can be used to find the rank of D. Maple code for more recent versions of Maple is avaliable at Maple Code.

The rank of D is equal to how many parameters are estimable. It is possible to tell which of the original parameters are estimable by solving αT D = 0. The zero enteries in α correspond to a parameter which is estimable and α can used to for a set system of linear first-order particle differential equations, which can be solved to find which parameters can be estimated (see Catchpole et al, 1998 for more details).

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(pc,q), where pc is the numbr of parameters in the covariate model and q is the number of estimable parameters in the model without covariates (Cole and Morgan, 2010a). As adding covariates to a model can make derivative matrix calculations much more complex, this method simplifies the calculation considerably (Further details, examples and Maple code).

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:
  • Finding general exhaustive summaries for families of ecological models. Cole (2012) provides such an exhaustive summary for multistate mark-recapture models , which includes the more complicate problem of parameter redundancy when there are hidden states. Cole et al (2014) provides simpler exhaustive summaries for multi-site capture-recapture models, and memory models.
  • Creating taxonomy of parameter redundancy results. Ring recovery models are examined in Cole et al (2012). Parameter redundancy results for age-dependent models for ring-recovery data on animals marked at unknown age are given in McCrea et al (2013). Capture-recapture and capture-recapture-recovery models are examined Hubbard et al (2014) and Hubbard (2014).
  • Parameter redundancy in occupancy models, is considered in Hubbard (2014). In particular examining the effect the data has on parameter redundancy.
  • Chen Yu, a PhD student at Kent supervised by Byron Morgan and Diana Cole, has been examining parameter redundancy in capture-recapture, capture-recovery and capture-recapture-recovery mixture models.
  • Methods for detecting parameter redundancy in discrete state-space models are explored in Cole and McCrea (2012) and Newman et al (2014). Cole and McCrea (2012) also examined parameter reduancy in integrated population models.
  • The Hybrid Symbolic-Numerical combines the symbolic and numeric methods for detecting parameter redundancy. The symbolic method is used to calculate the derivative matrix and the rank is evaluated at about 5 random points. This work is described in Choquet and Cole (2012).

  • References

    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.

    Catchpole, E. A. and Morgan, B. J. T (2001) Deficiency of parameter redundant models. Biometrika, 88, 593-598.

    Catchpole, E. A., Morgan, B. J. T and Viallefont, A. (2002) Solving problems in parameter redundancy using computer algebra. Journal of Applied Statistics, 29, 625-636.

    Choquet, R. and Cole, D.J. (2012) A Hybrid Symbolic-Numerical Method for Determining Model Structure. Mathematical Biosciences, 236, 117-125. Download pre-print version of paper Download Maplecode

    Cole, D. J., Morgan, B. J. T and Titterington, D. M. (2010) Determining the Parametric Structure of Non-Linear Models. Mathematical Biosciences, 228, 1630. Download preprint version of paper Maple Code for examples

    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. Download pre-print version of paper Download pre-print version of Supplementary Material

    Cole, D. J. and Morgan, B. J. T (2010) Parameter Redundancy with Covariates. Biometrika, 97, 1002-1005 Pre-print version of paper Maple Code for examples

    Cole, D.J. (2012) Determining Parameter Redundancy of Multi-state Mark-recapture Models for Sea Birds. Journal of Ornithology , 152 (Suppl 2) 305315 Download a pre-print version of paper Download suplementary material Maple Code for examples

    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. Ecology and Evolution, 4, 2124-2133, Open access pdf version of paper.

    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. Statistical Methodology , 17, 17-29. Download preprint version of paper Download Supplementary Material Maple code

    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, Journal of the Royal Statistical Society: Series C (Applied Statistics), 62, 101-113. Download preprint version of paper Maple Code

    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)