Download Online Resource 1: Function Approach

Online Resource 1: Function Approach
Title: A New Method For Estimating Animal
Abundance with Two Sources of Data in CaptureRecapture Studies.
Journal: Methods in Ecology and Evolution
Author’s names: Bénédicte Madon, Olivier Gimenez, Brian McArdle, C. Scott Baker, Claire
Garrigue
Corresponding author:
Bénédicte Madon, School of Mathematics and Statistics, the University of New South Wales,
UNSW NSW 2052, Australia.
[email protected]
Description of the function approach used to estimate the probability of single identity Iid:
Following the work on the iteration, an algebraic function of Iid was written (Ross
Ihaka, pers. Comm., March 2007):
 1

1
f ( Iˆid )  1   2 a  3 ab 
ˆ
Iˆid 
 I id
where a  Pˆ1 Pˆ2 and b  P̂3
The idea here is to find the root(s) of f ( Iˆid ) . The root(s) I of a function of Iˆid can be
found at f ( Iˆid )  I , which is equivalent to f ( Iˆid )  I  0 for a continuous monotonic
function.
The parameter Iid belongs to [0,1] and f ( Iˆid ) is not necessarily monotonic on [0,1]: the
behavior of f ( Iˆid ) on [0,1] depends on the values of the probabilities a and b . It appears
that there are 4 possible scenarios denoted as A, B, C, D for the behavior of f ( Iˆid ) on [0,1]
that result in f ( Iˆid ) having either a single root (scenarios B, C, D and some cases of A) or
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three roots (in some cases of scenario A). The different scenarios A, B, C and D are illustrated
in Figure A1.
Fig.A1 - Root
of the function f ( Iˆid ) in scenario A, B, C and D.
For scenarios B, C, D and some cases of scenario A, the corrected value of Iˆid
corresponds to the root of f ( Iˆid ) in [0, 1]. For scenario B and C, the first and second
derivatives are used to obtain the root of f ( Iˆid ) . Here the first derivative f ( Iˆid ) is:
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2a 3ab
f ' ( Iˆid )  3  4
Iˆid Iˆid
and the second derivative of f ( Iˆid ) corresponds to:
 6a 12ab
f ' ( Iˆid )  4  5
Iˆ
Iˆ
id
id
If f ( Iˆid ) is monotonic on [0, 1] (scenario D), then the corrected value for Iˆid corresponds
to the root of f ( Iˆid ) in [0, 1].
If f ( Iˆid ) has an inflection point in [0, 1] and does not display any optimum in [0, root of
the second derivative] (scenario C), then the corrected value for Iˆid corresponds to the root of
f ' ( Iˆid ) in [0, 1].


If f ( Iˆid ) has an inflection point in [0, 1], has a local minimum in [0, 1] and f ( Iˆid )  I is
strictly positive (scenario B and in some cases of scenario A), then the corrected value for Iˆid
corresponds to the root of f ' ( Iˆid ) in [0, 1].
If f ( Iˆid ) has an inflection point in [0, 1], has a local maximum and a local minimum in


[0,1] and f ( Iˆid )  I is strictly negative (scenario A), then there are three possible roots. The
largest root was chosen in the event of scenario A, therefore the smallest correction in the
TSJS estimator. This choice was a conservative one as the trend in the TSJS estimator in
scenario A was unclear whatever root used. However, as shown in Figure A2, scenario A
would only occur in a real dataset for probabilities of belonging to list 3 smaller than 0.3 or
very high probabilities P̂1 and P̂2 . And simulations show anyway that the TSJS estimator is
most appropriate for probabilities of simultaneous double-tagging higher than 0.2.
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Fig.A2- Isopleths of the value of the parameter Iid and corresponding scenario
as a function of the probabilities a and b.
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