Distributionfree Monte Carlo for population viability analysis

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Distribution-free Monte Carlo for population
viability analysis

• Janos G. Hajagos
• Stony Brook University, NY
• September 14, 2004

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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

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Interval Analysis
In interval analysis the atomic unit in the
calculation is the interval.
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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).
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Intervalized model of growth
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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.
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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.

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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
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Two models
Model I
Model II
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The algebraic trick
normali( ,s) s normali(0,1)
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A single realization
Model I.
Model II.
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Lots of realizations of the model condensed into
a probability distribution.
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Quasi-extinction decline curves
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Difference is similar to

• Let a2,3.
• Model I
• a-a0
• Model II
• a-a-1,1

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Which view of what an interval is correct?
Model I or Model II?
The correct answer depends on your operational
definition of an interval.
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Comparison to second-order
Model parameters N90,110 r-0.05,0.1 s0.05,
0.2 T20 years 1000 runs 100 inner samples
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Increasing number of realizations
Model parameters r0.05,0.01, s0.05,0.2,
N90,110, T20, outer runs1000
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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)

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Acknowledgements

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