parameter redundancy and identifiability in ecological models

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Parameter Redundancy and Identifiability in Ecological Models. Diana Cole, University of Kent. Introduction. Occupancy Model example Parameters: – species is detected . C an only estimate rather than and . Model is parameter redundant or parameters are non-identifiable. - PowerPoint PPT Presentation

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The Problem with Parameter Redundancy

Parameter Redundancy and Identifiability in Ecological ModelsDiana Cole, University of Kent

Introduction

Species presentand detectedSpecies presentbut not detectedSpecies absent#/27Parameter Redundancy#/27Problems with Parameter RedundancyThere 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.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.The Fisher information matrix will be singular (Rothenberg, 1971) and therefore the standard errors will be undefined.However the exact Fisher information matrix is rarely known. Standard errors are typically approximated using a Hessian matrix obtained numerically. Can parameter redundancy be detected from the standard errors?#/27Is example 1 parameter redundant?

#/27Is example 2 parameter redundant?

#/27Is example 3 parameter redundant?

#/27Simulation Study for Examples 1 and 257% have defined standard errors

SVD threshold%age SVD test correct0.01100%0.00175%0.000115%0.000017%#/27Mark-Recovery Models

636465Ringing yr 63 64 65 Recapture yr63 64 64 #/27Mark-Recovery Models

#/27Symbolic Method(Cole et al, 2010 and Cole et al, 2012) Exhaustive summary unique representation of the model

Parameters

#/27Symbolic Method

#/27Estimable Parameter Combinations

#/27Other uses of symbolic methodUses of symbolic method:Catchpole and Morgan (1997) exponential family models, mostly used in ecological statistics,Rothenberg (1971) original general use, econometric examples,Goodman (1974) latent class models,Sharpio (1986) non-linear regression models,Pohjanpalo (1982) first use for compartment models,Cole et al (2010) General exhaustive summary framework,Cole et al (2012) Mark-recovery models.Finding estimable parameters:Catchpole et al (1998) exponential family models,Chappell and Gunn (1998) and Evans and Chappell (2000) compartment models,Cole et al (2010) General exhaustive summary framework.

Problem with Symbolic MethodThe key to the symbolic method for detecting parameter redundancy is to find a derivative matrix and its rank.Models are getting more complex.The derivative matrix is therefore structurally more complex.Maple runs out of memory calculating the rank.

How do you proceed?Numerically but only valid for specific value of parameters. But cant find combinations of parameters you can estimate. Not possible to generalise results.Symbolically involves extending the theory, again it involves a derivative matrix and its rank, but the derivative matrix is structurally simpler.Hybrid-Symbolic Numeric Method.

Wandering AlbatrossMulti-state models for sea birdsHunter and Caswell (2009)Cole (2012)Striped Sea BassTag-return models for fishJiang et al (2007)Cole and Morgan (2010)

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Multi-state capture-recapture example

Hunter and Caswell (2009) 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:

survivalbreeding given survivalsuccessful breedingrecapture

Wandering Albatross13241 success2 = failure3 post-success4 = post-failure#/27Extended Symbolic MethodCole et al (2010)Choose a reparameterisation, s, that simplifies the model structure.

Rewrite the exhaustive summary, (), in terms of the reparameterisation - (s).

#/27Calculate the derivative matrix Ds.

The no. of estimable parameters =rank(Ds)

rank(Ds) = 12, no. est. pars = 12, deficiency = 14 12 = 2

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.

Extended Symbolic Method

Use sre as an exhaustive summary.

Breeding ConstraintSurvival Constraint1= 2= 3= 41= 3, 2= 41= 2, 3= 41, 2, 3,41= 2= 3= 40 (8)0 (9)1 (9)1 (11)1= 3 ,2= 40 (9)0 (10)0 (10)2 (12)1= 2, 3= 40 (9)0 (10)1 (10)1 (12)1,2,3,40 (11)0 (12)0 (12)2 (14)

Extended Symbolic Method

Multi-state markrecapture models

State 1: Breeding site 1State 2: Breeding site 2State 3: Non-breeding, Unobservable in state 3 - survival - breeding - breeding site 1 1 - breeding site 2

#/27Multi-state markrecapture models General Model

General Multistate-model has S states, with the last U states unobservable with N years of data.Survival probabilities released in year r captured in year c:

t is an SS matrix of transition probabilities at time t with transition probabilities i,j(t) = ai,j(t).Pt is an SS diagonal matrix of probabilities of capture pt.pt = 0 for an unobservable state,

#/27r = 10N 17 d = N + 3

General simpler exhaustive summaryCole (2012)

#/27Hybrid Symbolic-Numeric MethodChoquet and Cole (2012) #/27Example multi-site capture-recapture model

#/27Example Occupancy models(Hubbard et al, in prep)

#/27ConclusionNumericSymbolicHybrid-SymbolicAccurate / correct answerNot alwaysYesYesGeneral Results (e.g. any no. of years)NoYesWork in progressEasy to use (e.g. for an ecologist)YesNo, but can develop simpler ex. sumYesPossible to add to existing computer packagesYesNo (needs symbolic algebra)Yes (E-surge and M-surge)Individually Identifiable ParametersNoYesYesEstimable parameter combinationsNoYesIn the future?Best for intrinsic PR and general resultsBest for extrinsic PR and a quick result#/27Referenceshttp://www.kent.ac.uk/smsas/personal/djc24/parameterredundancy.htmBrownie, C. Hines, J., Nichols, J. et al (1993) Biometrics, 49, p1173.Catchpole, E. A. and Morgan, B. J. T. (1997) Biometrika, 84, 187-196Catchpole, E. A., Morgan, B. J. T. and Freeman, S. N. (1998) Biometrika, 85, 462-468Chappell, M. J. and Gunn, R. N. (1998) Mathematical Biosciences, 148, 21-41. Choquet, R. and Cole, D.J. (2012) Mathematical Biosciences, 236, p117.Cole, D. J. and Morgan, B. J. T. (2010) JABES, 15, 431-434.Cole, D. J., Morgan, B. J. T and Titterington, D. M. (2010) Mathematical Biosciences, 228, 1630. Cole, D. J. (2012) Journal of Ornithology, 152, S305-S315.Cole, D. J., Morgan, B. J. T., Catchpole, E. A. and Hubbard, B.A. (2012) Biometrical Journal, 54, 507-523. Cole, D. J., Morgan, B.J.T., McCrea, R.S, Pradel, R., Gimenez, O. and Choquet, R. (2014) Ecology and Evolution, 4, 2124-2133,Evans, N. D. and Chappell, M. J. (2000) Mathematical Biosciences, 168, 137-159.Gould, W. R., Patla, D. A., Daley, R., et al (2012). Wetlands, 32, p379.Goodman, L. A. (1974) Biometrika, 61, 215-231.Hunter, C.M. and Caswell, H. (2009). Ecological and Environmental Statistics, 3, 797-825Jiang, H. Pollock, K. H., Brownie, C., et al (2007) JABES, 12, 177-194Pohjanpalo, H. (1982) Technical Research Centre of Finland Research Report No. 56.Rothenberg, T. J. (1971) Econometrica, 39, 577-591.Shapiro, A. (1986) Journal of the American Statistical Association, 81, 142-149.Viallefont, A., Lebreton, J.D., Reboulet, A.M. and Gory, G. (1998) Biometrical Journal, 40, 313-325.

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