Download A Hybrid Symbolic-Numerical Method for

Hybrid Symbolic-Numeric Method
for Detecting Parameter
Redundancy in Ecological Models’
Diana Cole, NCSE, University of Kent
Rémi Choquet, Centre d'Ecologie Fonctionnelle et Evolutive
Ben Hubbard, NCSE, University of Kent
Introduction – Example Capture-Recapture
• Cormack-Jolly-Seber (CJS) Capture-Recapture Model
• Parameters: 𝜙𝑗 survival of year j, 𝑝𝑗 recapture in year j
History
1111
1110
1101
⋮
Probability of History
𝜙1 𝑝2 𝜙2 𝑝3 𝜙3 𝑝4
𝜙1 𝑝2 𝜙2 𝑝3 (1 − 𝜙3 + 𝜙3 (1 − 𝑝4 )) = 𝜙1 𝑝2 𝜙2 𝑝3 (1 − 𝜙3 𝑝4 )
𝜙1 𝑝2 𝜙2 (1 − 𝑝3 )𝜙3 𝑝4
⋮
• The parameters 𝜙3 and 𝑝4 are confounded, can only estimate
the product never the parameters individually.
Introduction
• In some models it is not possible to estimate all the
parameters. This is termed parameter redundant / nonidentifiable.
• A model is parameter redundant if it can be reparameterised
in terms of a smaller number of parameters.
• Capture-recapture example:  = [1, 2, 3, p2, p3, p4 ]
R = [1, 2, p2, p3,  ]  = 3 p4
• Parameter redundancy can be due to the model (extrinsic) or
the data (intrinsic).
• Sometimes it is obvious that a model is parameter redundant
(e.g. capture-recapture example), but in more complex
models it is not necessarily obvious.
Symbolic Method
• Symbolic methods can be used to detect parameter redundancy
in less obvious cases (see for example Catchpole and Morgan,
1997, Cole et al, 2010).
• Firstly an exhaustive summary is required, 𝜿. An exhaustive
summary is a vector of parameter combinations that uniquely
define the model, e.g. probabilities of histories
• Let 𝜽 denote a vector of the p parameters.
𝜕𝜿
𝜕𝜽
• We then form a derivative matrix 𝑫 = and calculate its rank.
• When r = p, model is full rank; we can estimate all parameters.
• When r < p, model is parameter redundant with deficiency
d=p–r.
• In parameter redundant models we can also find a set of r
estimable parameter combinations by solving 𝜶𝑗 𝑫𝑻 = 𝟎 then
𝜕𝑓
𝒑
𝛼
𝒊=𝟏 𝑖𝑗 𝜕𝜃𝑖
2010).
= 0, 𝑗 = 1, … 𝑑 (Catchpole et al, 1998 or Cole et al,
Problems with the Symbolic Method
• In more complex models the derivative matrix is structurally
too complex. Computer runs out of memory calculating the
rank.
• Examples:
Wandering Albatross
Multi-state models for sea birds
Hunter and Caswell (2009)
Cole (2012)
Striped Sea Bass
Tag-return models for fish
Jiang et al (2007)
Cole and Morgan (2010)
• How do you proceed?
– Numerically – can give the wrong results.
– Symbolically – involves extending the theory and finding
simpler exhaustive summaries (Cole et al, 2010). However
this method is complex.
– Hybrid Symbolic-Numeric Method.
Hybrid-Symbolic Numeric Method
• Calculate the derivative matrix,
𝜕𝜿
𝑫=
,
𝜕𝜽
symbolically.
• Evaluate 𝑫 at a random point 𝜽𝑘 to give 𝑫𝑘 .
• Calculate 𝑟𝑘 the rank of 𝑫𝑘 .
• Repeat for 5 random points model, then 𝑟 = max 𝑟𝑘 .
• If the model is parameter redundant for any 𝑫𝑘 with 𝑟𝑘 = 𝑟
solve 𝜶𝑘 𝑫𝑇𝑘 = 0. The zeros in 𝜶𝑘 indicate positions of
parameters that can be estimated.
Example Capture-Recapture
•  = [1, 2, 3, p2, p3, p4 ]
Example – multi-site capture-recapture model
• The capture-recapture models can be extended to studies
with multiple sites (Brownie et al, 1993).
• Example Canada Geese in 3 different geographical regions T=6 years.
• Geese tend to return to the same site – memory model.
(𝑡)
• Initial state probabilities:𝜋𝑗
𝑡
𝑡
𝑡
𝑡
for 𝑗 = 1,2 & 𝑡 = 1, … 6 (𝜋3 = 1 − 𝜋1 − 𝜋2 )
𝑡
• Transition probabilities: 𝜙∗𝑖𝑗 for 𝑖, 𝑗 = 1,2,3 & 𝑡 = 1, … , 5 and 𝜙𝑖𝑗𝑘 for 𝑖, 𝑗, 𝑘 =
1,2,3 & 𝑡 = 2, … , 5.
𝑡
• Capture probabilities: 𝑝𝑗 for 𝑖 = 1,2,3 , 𝑡 = 2, … , 6. (p = 180 Parameters)
Example – Occupancy Models
• Occupancy models considers whether or not a species is present at
a particular site.
• Parameters: 𝜓 − site is occupied, 𝑝 – species is detected.
• Species detected at a site with probability 𝜓𝑝.
• Species not detected at a site with probability
𝜓 1 − 𝑝 + 1 − 𝜓 = 1 − 𝜓𝑝
• Basic model is parameter redundant, so a robust design was
developed, so that several surveys are conducted each season at
each site, and assumed 𝜓 is the same for each survey.
• More complex models consider multiple sites and interactions
between species.
• These models are not parameter redundant, but this assumes that
every possible combination of occupied and unoccupied is
observed. However parameter redundancy can be caused by the
data (intrinsic parameter redundancy).
Example – Occupancy models
• Monitoring of amphibians in the Yellowstone and Grand Teton
National Parks, USA (Gould et al, 2012).
• Two species: Columbian Spotted Frogs and Boreal Chorus Frogs.
• 𝜓 occupancy probabilities, 𝑝 detection probabilities.
• (s) dependence on site, (t) dependence of time, ∙ dependent
on neither site nor time.
Model
𝜓 ∙ 𝑝 ∙
𝜓 𝑠 𝑝 ∙
𝜓 ∙ 𝑝 𝑠
𝜓 𝑡 𝑝 𝑡
𝜓 𝑡, 𝑠 𝑝 ∙
𝜓 𝑡, 𝑠 𝑝 𝑡
𝜓 𝑡, 𝑠 𝑝 𝑡, 𝑠
Rank
20
65
35
59
161
176
236
Deficiency No. pars
0
20
0
65
0
35
0
59
17
178
17
193
67
303
Conclusion and future work
• The hybrid method can be used to find how many parameters can
be estimated in a model.
• Hybrid method is much simpler to use than extended symbolic
method.
• Can be added to standard software packages. For ecological
models it is available in M-surge and E-surge.
• It can quickly give results about whether a particular data set is
parameter redundant, even for several hundred parameters.
• However it currently is only applicable to a given number of years
of data. In the symbolic method there is an extension theorem that
allows general results to be developed. Expanding the hybrid
method to include the extension theorem is future work.
• In the parameter redundant model the hybrid method can currently
only determine which of the original parameters are identifiable.
Constraints needed to give an identifiable model can only be
obtained by trial and error. The symbolic method can also give
estimable parameter combinations.
References
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Hybrid Numeric-Symbolic Method:
Choquet, R. and Cole, D.J. (2012) A Hybrid Symbolic-Numerical Method for
Determining Model Structure. Mathematical Biosciences, 236, p117.
Symbolic Method:
Cole, D.J., Morgan, B.J.T., Titterington, D.M. (2010) Mathematical Biosciences, 228,
p16.
Cole, D.J., Morgan, B.J.T. (2010), JABES, 15, p431.
Catchpole, E. A., Morgan, B. J. T (1997) Biometrika, 84, p187.
Catchpole, E. A., Morgan, B.J.T., Freeman, S. N. (1998) Biometrika, 85, p42.
Cole, D.J. (2012) Journal of Ornithology , 152, p305.
Other:
Brownie, C. Hines, J., Nichols, J. et al (1993) Capture–recapture, Biometrics, 49, p1173.
Gould, W. R., Patla, D. A., Daley, R., et al. (2012). Wetlands, 32, p379.
Hunter, C., Caswell, H. (2009) Environmental and Ecological Statistics vol 3, p. 797.
Jiang, H.H., Pollock, K.H., Brownie, C. et al, (2007), JABES, 12, p 177
Lebreton, J. Morgan, B. J. T., Pradel R. and Freeman, S. N. (1995) Biometrics, 51, p1418.