Parameter Redundancy and
Identifiability in Ecological Models
Diana Cole, University of Kent
Rémi Choquet, CEFE, CNRS, France.
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Introduction
Occupancy Model example
x
Species present
and detected
Prob = ππ
Prob seen = ππ
β’
β’
β’
β’
Species present
Species absent
but not detected
Prob = π 1 β π
Prob = 1 β π
Prob not seen = π 1 β π + 1 β π
= 1 β ππ
Parameters: π β site is occupied, π β species is detected.
Can only estimate ππ rather than π and π.
Model is parameter redundant or parameters are non-identifiable.
(Solution: MacKenzie et al, 2002, robust design; visit sites several
times in one season).
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Occupancy Model - Likelihood Surface
3/30
Occupancy Model - Likelihood Profile
Flat profile = Parameter Redundant
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Parameter Redundancy and Identifiability
β’ Suppose we have a model π(π) with parameters π. A model
is globally (locally) identifiable if π π1 = π(π2 ) implies that
π1 = π2 (for a neighbourhood of π).
β’ A model is parameter redundant if it can be reparameterised
in terms of a smaller set of parameters. A parameter
redundant model is non-identifiable.
β’ There are several different methods for detecting parameter
redundancy, including:
β numerical methods (e.g. Viallefont et al, 1998)
β symbolic methods (e.g. Cole et al, 2010)
β hybrid symbolic-numeric method (Choquet and Cole,
2012).
β’ Generally involves calculating the rank of a matrix, which gives
the number of parameters that can be estimated.
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Problems with Parameter Redundancy
β’ There will be a flat ridge in the likelihood of a parameter redundant
model (Catchpole and Morgan, 1997), resulting in more than one
set of maximum likelihood estimates.
β’ Numerical methods to find the MLE will not pick up the flat ridge,
although it could be picked up by trying multiple starting values and
looking at profile log-likelihoods. If a parameter redundant model is
fitted will get biased parameter estimates.
β’ The Fisher information matrix will be singular (Rothenberg, 1971)
and therefore the standard errors will be undefined.
β’ However the exact Fisher information matrix is rarely known.
Standard errors are typically approximated using a Hessian matrix
obtained numerically. Can get explicit (wrong) estimates of standard
errors in some cases.
β’ Model selection is based on identifiable models (e.g. AIC needs
number of parameters can estimate).
β’ Recommended you check the identifiability of a model before
fitting (or at least as part of model fitting process).
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Symbolic Method (Occupancy Example)
β’ Catchpole and Morgan (1997), Catchpole et al (1998), Cole et al (2010)
ππ
β’ Exhaustive summary: πΏ =
1 β ππ
β’ Parameters: π½ = [π π]
π βπ
ππΏ
β’ Derivative matrix: π« = = π βπ
ππ½
β’ Rank of D, r, is number of estimable parameters.
β’ If r is less than number of parameters, q, the model is parameter
redundant. Rank is 1, model parameter redundant.
β’ Can find if any of original parameters are individually identifiable by
solving πΆπ π« = π. Position of 0s indicate parameter identifiable.
πΆ = βπ/π 1 (no parameters identifiable)
β’ Can find estimable parameter combination by solving PDEs
π
π=1
ππ
πΌππ
= 0, π = 1, β¦ , π β π
πππ
solution shows can estimate ππ.
ππ π ππ
β
+
=0
ππ π ππ
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Symbolic Method (Occupancy Example)
β’ Also able to generalise results, e.g. in occupancy modelling
robust design involves visiting a site more than once in a short
space of time (within a season). Can estimate both
parameters if there are at least two samples per season. Can
extend results to multiple seasons.
β’ Requires symbolic algebra package, e.g. Maple, in most
models to find rank.
β’ Can run out of memory finding rank in complex models.
β’ Extended Symbolic method (Cole, et al. 2010) uses
reparameterisation to find simpler exhaustive summaries. For
example: multi-state models with unobservable states (Cole,
2012).
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Numerical Methods (Occupancy Example)
β’ Hessian Method (Viallefont et al, 1998):
β’ Hessian matrix should be singular and therefore have at least
one Eigenvalue that is 0. Found numerically the Hessian is not
necessarily singular.
180.25 179.99
β’ Hessian: π =
179.99 179.74
β’ Standardised Eigenvalues: 1, -0.000003
β’ Eigenvalue close to zero, model is parameter redundant.
β’ Inaccurate - can give wrong result (see Cole and Morgan,
2010)
β’ Not able to find identifiable parameter or estimable
parameter combinations.
β’ Not able to generalise results.
β’ But can be added to a software package (e.g. Mark).
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Other Numerical Methods
β’ Simulation (Gimenez et al., 2004)
β Simulate large data sets and compare bias and CV.
β Can be inaccurate.
β’ Likelihood Profile (Gimenez et al., 2004)
β Flat profile indicates parameter not identifiable.
β Difficult to distinguish between parameter redundancy and nearly
redundancy.
β’ Data Cloning (Lele et al., 2007, 2010)
β Bayesian MCMC with uninformative priors and K clones of data set.
β Variance β 0 as πΎ β β if parameter identifiable.
β Can be inaccurate, especially when calculating whether a particular
parameter is identifiable.
β’ Methods can be used to find individually identifiable parameters.
β’ None of these methods can find general results and estimable
parameter combinations.
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Hybrid Symbolic-Numeric Method
β’ Choquet and Cole (2012)
β’ Find derivative matrix, D, symbolically (or using automatic
differentiation).
β’ Find the rank of D at 5 random points, and the model rank is
the maximum of all 5 ranks.
β’ Occupancy model: rank 1 in all 5 cases, therefore parameter
redundant.
β’ Can also show if any of the original parameters are
identifiable, but not the estimable parameter combinations.
β’ Can be added to a software package (e.g. E-Surge and MSurge).
β’ Can this method be extended to find estimable parameter
combinations and general results?
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Comparison of Methods
Method00
Hessian
Simulation
Lik. Profile
Data Cloning
Symbolic
Ext. Symbolic
Hybrid
Ext. Hybrid
Identifiable
Accurate
Parameters
×
×
×
×
ο
ο
ο
ο
×
Estimable
General Complex
Parameter
Automatic
Results Models
Combinations
ο
ο
ο
×
×
×
×
×
×
×
×
ο
ο
ο
ο
ο
ο
×
?
ο
ο
×
?
ο
Slow
ο
Slow
×
ο
ο
ο
ο
×
×
×
×
×
ο
?
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Occupancy Model - Likelihood Profile
Flat profile = Parameter Redundant
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Occupancy Model - Likelihood Profile
Line is π =
π½
π
=
0.5
π
β 0.5 = ππ
Profile gives the estimable parameter combinations
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Extended Hybrid Method β Subset profiling
β’ Eisenberg and Hayashi (2014) use subset profiling to identify
estimable parameter combinations manually in compartment
modelling.
β’ Combine hybrid method (Choquet and Cole, 2012) with subset
profiling (Eisenberg and Hayashi, 2014) and extend to automatically
detect relationships between confounded parameters.
Step 1: Check whether model is parameter redundant using hybrid
method and remove any parameters that are individually identifiable.
Step 2: Find subsets of parameters that are nearly full rank. (Have π =
π β π = 1, and all subsets have π = 0).
Step 3: For each subset, fix all the parameters not in the subset. Then
for each pair of parameters, in the subset, produce a likelihood profile.
Fix one parameter, π1,π , and store the MLE of the other parameter(s) in
the subset, π2,π , π3,π , β¦ (π = 1, β¦ , π).
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Extended Hybrid Method β Subset profiling
Step 4: Identify the relationships. Find π½1 , π½2 , π½3 , π½4 that minimise
π
π=1
π½1 + π½2 π1,π
π2,π β
π½3 + π½4 π1,π
2
to identify most common relationships (should equal 0).
Occupancy Example: π1 = π, π2 = π, π½1 = 0.5, π½2 = 0, π½3 = 0, π½4 = 1
0.5 + 0 × π
0.5
πβ
=πβ
= 0 β ππ = 0.5
0+1×π
π
Step 5: Turn relationships between pairs of parameters to form
estimable parameter combinations.
Steps 1-4 can be programed to be automatic, step 5 not yet.
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Mark-Recovery Example
β’ Animals marked and recovered dead. Mallards 1963-65:
962
82 35 18
No. marked: 702 No. Recovered:
103 21
1132
82
β’ Probability animals marked in year π and recovered in year π:
1 β π1,1 π1 π1,1 1 β π2 π2 π1,1 π2 1 β π3 π3
π·=
β’
β’
β’
β’
β’
1 β π1,2 π1
π1,2 1 β π2 π2
1 β π1,3 π1
π1,π‘ probability animal survives 1st year at time t.
ππ probability an animal age π survives their π th year.
ππ probability animal aged π is recovered dead.
Model is parameter redundant with estimable parameter
combinations π1,1 , π1,2 , π1,3 , π1 , 1 β π2 π2 , π2 1 β π3 π3
(result found symbolically in Cole et al, 2012).
π1,1 , π1,2 , π1,3 , π1 β individually identifiable (can still be estimated).
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Mark-Recovery Example
Model has 8 parameters π½ = π1,1 , π1,2 , π1,3 , π2 , π3 , π1 , π2 , π3
Step 1: Hybrid method can be use to show model is parameter
redundant with rank 6. π1,1 , π1,2 , π1,3 , π1 are individually
identifiable. Non-identifiable parameters are π2 , π3 , π2 , π3 .
Step 2: Hybrid method used to find nearly full rank subsets:
{π3 , π3 },{π2 , π3 , π2 }, {π2 , π2 , π3 }
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Mark-Recovery Example
Step 3: x profiles for subset {π3 , π3 }:
Step 4:
π3 β
fitted relationship:
0.0736β0.0000π3
1.2267β1.2268π3
= 0 β 1 β π3 π3 = 0.06
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Mark-Recovery Example
Step 3: x profiles for subset {π2 , π3 , π2 }
Step 4:
fitted relationships:
π3 β
β0.1800+0.9999π2
0.0000+0.9999π2
π2 β
0.1200+0.0000π2
0.9999β0.9999π2
= 0 β 1 β π2 π2 = 0.12
π2 β
0.0013β0.0013π3
0.0088β0.0106π3
=0β
= 0 β π2 1 β π3 = 0.18
π2 0.83βπ3
1βπ3
= 0.12 (complex)
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Mark-Recovery Example
Step 3: x profiles for subset {π2 , π2 , π3 }
Step 4:
fitted relationships:
π2 β
π3 β
π3 β
0.1200+0.0000π2
=0β
0.9999β0.9999π2
0.1472β0.0000π2
=0β
0.0000+1.2267π2
β0.0000+0.0110π2
β0.0110+0.0913π2
=0β
π3
π2
1 β π2 π2 = 0.12
π2 π3 = 0.12
0.12 β π2 = β0.12 (complex)
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Mark-Recovery Example
Step 5:
β’ There could be more than one way of parameterising the
estimable parameter combinations. Looking for the simplest.
β’ From step 1 also can deduce we need to find 2 combinations of
parameters (rank 6, 4 identifiable parameters).
β’ π2 1 β π3 = 0.18, π2 π3 = 0.12, 1 β π3 π3 = 0.06
are consistent with: π2 1 β π3 π3
β’ 1 β π2 π2 = 0.12 is consistent with: 1 β π2 π2
β’ Estimable parameter combinations: π2 1 β π3 π3 & 1 β π2 π2
β’ The other (complex) relationships can be shown (with a bit of
algebra) to be consistent with estimable parameter combinations.
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Multi-State Mark Recovery Example
β’ Hunter and Caswell (2009) examine parameter redundancy of multistate mark-recapture models, but cannot evaluate the symbolic rank
of the derivative matrix, so use a version of the hybrid method.
β’ Cole et al (2010) and Cole (2012) developed an extend symbolic
method.
β’ 4 state breeding success model:
1 success
3 post-success
1
3
2
4
2 = failure
4 = post-failure
Wandering Albatross
successful breeding
recapture
breeding given survival
survival
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23/30
Multi-State Mark Recovery Example
β’ Model has 14 parameters
β’ π½ = π1 , π2 , π3 , π4 , π½1 , π½2 , π½3 , π½4 , πΎ1 , πΎ2 , πΎ3 , πΎ4 , π1 , π2
β’ The hybrid method can be used to show the rank is 12.
Therefore the model is parameter redundant.
β’ The hybrid method can also be used to show that the
parameters πΎ1 , πΎ2 , πΎ3 , πΎ4 , π1 , π2 are identifiable.
β’ Nearly full rank subsets are:
π1 , π3 , π½1 , π½3 , {π2 , π4 , π½2 , π½4 }
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Multi-State Mark Recovery Example
Subset {π1 , π3 , π½1 , π½3 }
β0.3 β 4.2π1
1 β 10π1
0.1
π½1 =
π1
π3 =
2.4
π½3 =
1 + 14π1
π½1 =
β0.42 + π3
0.3 + π3
β0.42 + π3
π½3 =
π3
π½3 =
1.7160π½1
1 + 0.7160π½1
25/30
Multi-State Mark Recovery Example
Subset {π2 , π4 , π½2 , π½4 }
0.18 β 1.05π2
1 β 4.1725π2
0.24
π½2 =
π2
π4 =
0.2963
π½4 =
β0.73 + 4.27π2
π½2 =
β0.18 + π4
β0.10 + π4
β0.18 + π4
π½4 =
π4
π½4 =
0.4367π½2
1 β 0.5633π½2
26/30
Multi-State Mark Recovery Example
β’ π½1 =
0.1
π1
β’ π½3 =
β0.42+π3
π3
β’ π½3 =
1.7160π½1
1.7160π½1
β π½3 =
1+0.7160π½1
1+1.7160π½1 βπ½1
0.24
β π2 π½2 = 0.24
π2
β0.18+π4
β π4 1 β π½4 = 0.18
π4
0.4367π½2
0.4367π½2
β π½4 =
1β0.5633π½2
1+0.4367π½2 βπ½2
β’ π½2 =
β’ π½4 =
β’ π½4 =
β π1 π½1 = 0.1
β π3 1 β π½3 = 0.42
β’ Estimable parameter combinations:
π½3 1βπ½1
π1 π½1 , π3 1 β π½3 ,
π½1 1βπ½3
β
π½3 1βπ½1
π½1 1βπ½3
= 1.7160
β
π½4 1βπ½2
π½2 1βπ½4
= 0.4367
π½4 1βπ½2
, π2 π½2 , π4 1 β π½4 ,
π½2 1βπ½4
β’ (Results are a reparameterisation of those found using extended
symbolic method.)
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General Results
β’ In symbolic method general results found using the extension
theorem (Catchpole and Morgan, 1997, Cole et al, 2010).
β’ This can be extended to the Hybrid method.
Mark Recovery Example:
Reparameterise in terms of estimable parameters,
π1,1 , π1,2 , π1,3 , π1 , π1 = 1 β π2 π2 and π2 = π2 1 β π3 π3 .
Hybrid Method shows we can estimate π1,1 , π1,2 , π1,3 , π1 , π½1 , π½2 .
Add an extra year of ringing and recovery adds extra parameters
π1,4 and π3 = π2 π3 1 β π4 π4 .
Hybrid method shows can also estimate π1,4 , π3 .
By the extension theorem, for n years of ringing and recovery,
model is always parameter redundant, estimable parameter
combinations will be
πβ1
π1,1 , π1,2 , β¦ , π1,π , π1 , 1 β π2 π2 , π2 1 β π3 π3 , β¦ ,
ππ
1 β ππ ππ
π=2
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Conclusion
Method
Hessian
Simulation
Lik. Profile
Identifiable
Accurate
Parameters
×
×
Data Cloning
×
×
Symbolic
ο
Ext. Symbolic
ο
ο
ο
Hybrid
Ext. Hybrid
×
ο
ο
ο
ο
ο
ο
ο
Estimable
General Complex
Parameter
Automatic
Results Models
Combinations
ο
×
×
×
×
×
×
×
×
Slow
ο
ο
×
ο
×
ο
×
ο
ο
ο
ο
ο
Slow
ο
ο
×
×
×
×
×
ο
Semi
29/30
References
β’
β’
β’
β’
β’
β’
β’
β’
β’
β’
β’
β’
β’
Catchpole, E. A. and Morgan, B. J. T. (1997) Detecting parameter redundancy. Biometrika, 84, 187-196.
Catchpole, E. A. et al. (1998) Estimation in parameter redundant models. Biometrika, 85, 462-468.
Choquet, R. and Cole, D.J. (2012) A Hybrid Symbolic-Numerical Method for Determining Model
Structure. Mathematical Biosciences, 236, p117.
Cole, D. J. and Morgan, B. J. T. (2010) A note on determining parameter redundancy in age-dependent
tag return models for estimating fishing mortality, natural mortality and selectivity. JABES, 15, 431-434.
Cole, D. J. (2012) Determining parameter redundancy of multi-state mark-recapture models for sea
birds. Journal of Ornithology, 152, 305β315.
Cole, D. J., et al. (2012) Parameter redundancy in mark-recovery models. Biometrical Journal, 54, 507523.
Eisenberg, M. C. and Hayashi, M. A. L. (2014) Determining identifiable parameter combinations using
subset profiling. Mathematical Biosciences, 256, 116β126.
Gimenez, O., et al. (2004), Methods for investigating parameter redundancy. Animal Biodiversity and
Conservation, 27, 1β12.
Hunter, C. M. and Caswell, H. (2009) Rank and redundancy of multi-state mark-recapture models for
seabird populations with unobservable states. In Environmental and Ecological Statistics Series: Volume
3. Eds., D.L. Thomson, E.G. Cooch and M.J. Conroy, 797-826.
Lele, S. R., et al. (2007) Data Cloning: Easy Maximum Likelihood Estimation for Complex Ecological
Models Using Bayesian Markov Chain Monte Carlo Methods. Ecology Letters, 105, 551β563.
Lele, S. R., et al. (2010) Estimability and Likelihood Inference for Generalized Linear Mixed Models Using
Data Cloning. JASA, 10, 1617β1625.
MacKenzie, D. I., et al (2002) Estimating site occupancy rates when detection probabilities are less than
one. Ecology, 83, 2248-2255.
Viallefont, A., et al. (1998) Parameter identifiability and model selection in capture-recapture models: A
numerical approach. Biometrical Journal, 40, 313-325.
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