Detecting Parameter Redundancy in Complex Ecological Models Diana Cole and Byron Morgan University of Kent.
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Detecting Parameter Redundancy in Complex
Ecological Models
Diana Cole and Byron MorganUniversity of Kent
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.
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 i+1
pi – probability a bird is recaptured on occasion i
= [1, 2, 3, p2, p3, p4 ]
etc1
00
0 22
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p
ppp
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ij
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i
NRn
ij ij
n
i
n
ij
Nij QQL
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1
111
123
78
R
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
T
p
pp
p
ppp
pp
p
)ln(
)ln(
)ln(
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000
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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.
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.
01
q
i iij
f
Reparameterisation Method(Cole and Morgan, 2008)
1. Choose a reparameterisation, s, that simplifies the model structure
2. Rewrite the exhaustive summary, (), in terms of the reparameterisation - (s).
43
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5
4
3
2
1
p
p
p
p
p
s
s
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s
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s
)ln(
)ln(
)ln(
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)(
)ln(
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)(
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s
ss
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sss
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sθ
Reparameterisation Method
3. Calculate the derivative matrix Ds
4. The no. of estimable parameters = min(q,rank(Ds))
rank(Ds) = 5, no. est. pars = min(6,5) = 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
15
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Reparameterisation Method
6. 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.
43
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s
p
p
p
p
p
re
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p
p
Example 2 – Multi-state mark-recapture models for Seabirds
• 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)...()(
1
loglog
1211
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1
1 1
4
1
4
1
),(),(,
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survival breeding given survival successful breeding capture
Wandering Albatross(Diomedea exulans)
1 3
2 4
Reparameterisation Method
1. Choose a reparameterisation, s, that simplifies the model structure
2. Rewrite the exhaustive summary, (), in terms of the reparameterisation - (s).
2
1
333
222
111
14
13
3
2
1
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s
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s
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)1(
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sss
ss
ss
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pp
p
p
p
p
sθ
Reparameterisation Method
3. Calculate the derivative matrix Ds
4. The no. of estimable parameters = min(p,rank(Ds))
rank(Ds) = 12, no. est. pars = min(14,12) = 12
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
Tre sssssssssssssssss 104934837141312116521 //
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Reparameterisation Method
6. Use sre as an exhaustive summary
etc..)1( where 11
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re pps
Breeding Constraint
Survival 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)
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.
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
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