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Distributionfree Monte Carlo for population viability analysis

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Title: Distributionfree Monte Carlo for population viability analysis


1
Distribution-free Monte Carlo for population
viability analysis
  • Janos G. Hajagos
  • Stony Brook University, NY
  • September 14, 2004

2
Two types of uncertainty
  • Epistemic is uncertainty in knowledge.
  • It can be reduced
  • Modeled with intervals or subjective probability
  • Stochastic is inherent uncertainty in a process.
  • It cannot be reduced
  • Modeled with random variables

3
Interval Analysis
In interval analysis the atomic unit in the
calculation is the interval.
4
Interval Arithmetic
Addition a,bc,d ac,bd. Multiplication
a,bc,d min(ac,ad,bc,bd),
max(ac,ad,bc,bd). The function exp exp(a,b)
exp(a),exp(b).
5
Intervalized model of growth
6
Stochastic population growth
where normal() generates a normal deviate with an
average growth rate (rln(R)) and sr is the
standard deviation of the growth rate.
7
Second-order Monte Carlo
  • What if we dont know the exact value of or
    s?
  • We can guess the value
  • Or we can sample from a second statistical
    distribution
  • If we only know the range of the value then we
    can sample the value from a uniform distribution.

8
Sampling from a uniform distribution
Twenty random samples from uniform(2,3) 1
2.041 2.078 2.193 2.201 2.295 2.311 2.545 2.548
9 2.571 2.590 2.594 2.596 2.650 2.767 2.775
2.840 17 2.874 2.893 2.915 2.915
9
Two models
Model I
Model II
10
The algebraic trick
normali( ,s) s normali(0,1)
11
A single realization
Model I.
Model II.
12
Lots of realizations of the model condensed into
a probability distribution.
13
Quasi-extinction decline curves
14
Difference is similar to
  • Let a2,3.
  • Model I
  • a-a0
  • Model II
  • a-a-1,1

15
Which view of what an interval is correct?
Model I or Model II?
The correct answer depends on your operational
definition of an interval.
16
Comparison to second-order
Model parameters N90,110 r-0.05,0.1 s0.05,
0.2 T20 years 1000 runs 100 inner samples
17
Increasing number of realizations
Model parameters r0.05,0.01, s0.05,0.2,
N90,110, T20, outer runs1000
18
Implications for Risk Assessment
  • Need to account for uncertainty in estimation of
    risk parameters.
  • Second-order Monte Carlo can underestimate bounds
    on extinction risk.
  • Interval Monte Carlo offers an alternative with
    no additional distributional assumptions.
  • Interval Monte Carlo is conservative no need for
    biased sampling (e.g. Latin hypercube sampling)

19
Acknowledgements
  • Lev Ginzburg of Stony Brook University
  • Scott Ferson of Applied Biomathematics
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