Detecting Parameter Redundancy in Complex Ecological Models

1
Detecting Parameter Redundancy in Complex
Ecological Models

• Diana Cole and Byron Morgan
• University of Kent

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Introduction

• If a model is parameter redundant or
  non-identifiable if you cannot estimate all the
  parameter in the model.
• Parameter redundancy can be detected by symbolic
  algebra.
• Ecological models are getting more complex then
  computers cannot do the symbolic algebra and
  numerical methods are used instead.
• In this talk we show some of the tools that can
  be used to overcome this problem.

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Example 1- Cormack Jolly Seber (CJS) Model

• Herring Gulls (Larus argentatus)
  capture-recapture data for 1983 to 1987
  (Lebreton, et al 1995)
• ?i probability a bird survives from occasion i
  to i1
• pi probability a bird is recaptured on
  occasion i
• ? ?1, ?2, ?3, p2, p3, p4

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Derivative Method (Catchpole and Morgan, 1997)

• Calculate the derivative matrix D
• rank(D) 5 rank(D) 5
• Number estimable parameters rank(D).
  Deficiency q rank(D)
• no. est. pars 5, deficiency 6 5 1

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Exhaustive Summaries

• An exhaustive summary, ?, is a vector that
  uniquely defines the model (Walter and Lecoutier,
  1982).
• The exhaustive summary is the starting point for
  finding the derivative matrix.
• More than one exhaustive summary exists for a
  model
• Choosing a simpler exhaustive summary will
  simplify the derivative matrix
• Computer packages, such as Maple can find the
  symbolic rank of the derivative matrix.
• Exhaustive summaries can be simplified by any
  one-one transformation such as multiplying by a
  constant, taking logs, and removing repeated
  terms.
• For multinomial models and product-multinomial
  models the more complicated 1 ? ?Qij can be
  removed (Catchpole and Morgan, 1997), as long as
  there are no missing values.

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Other tools to use with exhaustive summaries

• What can you estimate? (Generalisation from
  Catchpole et al, 1998.) Solve ?TD 0. Zeros in ?
  indicate estimable pars.
• Solve PDE to find full set of estimable pars.
• Extension theorem (Generalised from Catchpole
  and Morgan, 1997.) Usefully for generalising
  capture-recapture and ring-recovery models.
• PLUR Decomposition. (Cole and Morgan, 2008)
  Useful for detecting points at which the model is
  parameter redundant or near parameter redundant,
  or sub models that are parameter redundant.

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Reparameterisation Method(Cole and Morgan, 2008)

• Choose a reparameterisation, s, that simplifies
  the model structure
• Rewrite the exhaustive summary, ?(?), in terms of
  the reparameterisation - ?(s).

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Reparameterisation Method

• Calculate the derivative matrix Ds
• The no. of estimable parameters min(q,rank(Ds))
• rank(Ds) 5, no. est. pars min(6,5) 5
• If Ds is full rank s sre is a reduced-form
  exhaustive summary. If Ds is not full rank solve
  set of PDE to find a reduced-form exhaustive
  summary, sre
• Ds is full rank, so s is a reduced-form
  exhaustive summary

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Reparameterisation Method

• Use sre as an exhaustive summary
• A reduced-form exhaustive summary is
• adding an extra year of capture and an extra
  year of recapture adds the extra exhaustive
  summary terms
• Then the extension theorem can be applied to
  show that the CJS is always parameter redundant
  with deficiency 1.

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Example 2 Multi-state mark-recapture models for
Seabirds
Wandering Albatross (Diomedea exulans)

• Hunter and Caswell (2008) examine parameter
  redundancy of multi-state mark-recapture models,
  but cannot evaluate the symbolic rank of the
  derivative matrix (developed numerical method)
• 4 state breeding success model

1
3
4
2
breeding given survival
successful breeding
survival
capture
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Reparameterisation Method

• Choose a reparameterisation, s, that simplifies
  the model structure
• Rewrite the exhaustive summary, ?(?), in terms of
  the reparameterisation - ?(s).

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Reparameterisation Method

• Calculate the derivative matrix Ds
• The no. of estimable parameters min(p,rank(Ds))
• rank(Ds) 12, no. est. pars min(14,12) 12
• If Ds is full rank s sre is a reduced-form
  exhaustive summary. If Ds is not full rank solve
  set of PDE to find a reduced-form exhaustive
  summary, sre

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Reparameterisation Method

• Use sre as an exhaustive summary

Breeding Constraint Survival Constraint Survival Constraint Survival Constraint Survival Constraint
Breeding Constraint ?1 ?2 ?3 ?4 ?1 ?3, ?2 ?4 ?1 ?2, ?3 ?4 ?1, ?2, ?3,?4
?1 ?2?3 ?4 0 (8) 0 (9) 1 (9) 1 (11)
?1 ?3,?2 ?4 0 (9) 0 (10) 0 (10) 2 (12)
?1 ?2,?3 ?4 0 (9) 0 (10) 1 (10) 1 (12)
?1,?2,?3, ?4 0 (11) 0 (12) 0 (12) 2 (14)
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Conclusion

• Exhaustive summaries can be used to detect
  parameter redundancy.
• The key to more complex problems is to find the
  exhaustive summary with the simplest structure.
• The most powerful method of finding an exhaustive
  summary is the reparameterisation method which
  examines the basic building blocks of the model.
• These methods can be applied to any parametric
  model.

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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
• Hunter, C.M. and Caswell, H. (2008). Parameter
  redundancy in multistate mark-recapture models
  with unobservable states. Ecological and
  Environmental Statistics - in press
• Cole, D. J. and Morgan, B. J. T (2008) Parameter
  Redundancy and Identifiability. University of
  Kent Technical Report UKC/IMS/08/022
• Lebreton, J. Morgan, B. J. T., Pradel R. and
  Freeman, S. N. (1995) A simultaneous survival
  rate analysis of dead recovery and live recapture
  data. Biometrics, 51, 1418-1428.
• Walter, E. and Lecoutier, Y (1982) Global
  approaches to identifiability testing for linear
  and nonlinear state space models. Mathematics and
  Computers in Simulations, 24, 472-482