rare event simulation splitting for variance reduction
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Rare-Event Simulation Splitting for Variance Reduction. IE 680, Spring 2007 Bryan Pearce. What is a Rare Event?. B. A. Ω. Formal Problem Definition. Splitting: the beginning. Importance function h Measures “how close” a state is to the rare event

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splitting the beginning
Splitting: the beginning
  • Importance function h
    • Measures “how close” a state is to the rare event
  • Divide the intermediary state space into m ‘levels’ according to the thresholds l0, l1, …, lm
slide5

h(x) = l0

= l1

= l2

= l3

= lm = l

how to choose h
How to choose h?
  • Defining the importance function can be difficult.
  • Ideally our h should reflect:
    • The most likely path to the rare event
    • pk(x) = pk (indep. of state)
    • pk = p (indep. of level)
  • Presumes apriori knowledge of the system.
first sub interval
First sub-interval

MC Sim N0 independent chains. R0 reach l1.

h

l1

0

time

second sub interval splitting
Second sub-interval: Splitting

MC Sim N1 chains, splitting from the previously achieved threshold states.

R1 reach l2.

h

l2

…and so on for each sub-interval

l1

0

time

splitting policy fixed splitting
Splitting policy – fixed splitting
  • Each chain that reaches level k is cloned ck times.
  • Nk will be random for each level k > 0
  • Stratified sampling from the entrance distribution of level k
splitting policy fixed effort
Splitting policy – fixed effort
  • Fix Nk in advance. Choose the states represented in the entrance distribution by:

Random assignment

    • Choose these Nk states randomly from the entrance distribution

Fixed assignment

    • Choose an equal quantity of each state
    • Better stratification
pros cons of splitting method
Pros & cons of splitting method
  • Fixed splitting –
    • Asymptotically more efficient under optimal conditions
    • Efficiency very sensitive to splitting factor ck
  • Fixed effort
    • Higher memory requirement
    • More robust
efficiency
Efficiency

Our hope is that splitting will allow our variance to shrink faster than our computational time grows. This has indeed been shown to be true in many cases.

truncation motivation
Truncation - Motivation

Simulation time spent reaching l1

h

l4

l3

l2

l1

0

simple biased truncation
Simple (biased) Truncation

Choose β:

  • If a chain falls below the level lk-β then terminate.
  • Estimator becomes biased, moreso with small β.
  • Large β does not reduce workload very much.
  • RESTART
slide17

h

l4

l3

l2

l1

0

}

β = 2

Terminate

unbiased truncation
Unbiased Truncation

Use the ‘Russian Roulette’ principle:

The first time a chain ‘down-crosses’ a level threshold it dies with probability (1 – 1/rk,j). If it survives then its weight is increased by a factor of rk,j.

(these rk,j are user-defined and determine the ‘strength’ of the truncation)

how to choose the r k j s
How to choose the rk,js
  • The selection of the rk,js at each level of the process will control the aggressiveness of the truncation policy.
  • A tried-and-true value:
slide20

h

l4

l3

l2

l1

0

Dies with prob. (1 – 1/r3,2)

Weight increases by a factor of r3,2 if the chain survives.

russian roulette cont
Russian Roulette, cont.
  • There are various methods by which to use the chain weights can compensate for this truncation bias.
    • Probabilistic
    • Tag-based
    • Periodic
truncation w o weights
Truncation w/o weights
  • Chain weighting truncation methods can inflate the variance of our gamma estimator.
  • We can avoid this problem by allowing our chains to probabilistically re-split upon re-achieving previously achieved goals.
conclusions and notes
Conclusions and notes
  • Potential performance
    • With γ = 10-20,

Var[MC] = 10-23 while Var[split] = 10-41

  • Poorly-behaved systems
    • Inefficient to apply
references
References

L’Ecuyer, P., V. Demers, B. Tuffin. 2006. Splitting for rare-event simulation.

Glasserman, P., P. Heidelberger,and T. Zajic. 1998. A large deviations perspective on the efficiency of multilevel splitting.

L’Ecuyer, P., V. Demers, B. Tuffin. 2006. Rare-events, splitting, and quasi-Monte Carlo.

Garvels, M. J. J. 2000. The splitting method in rare event simulation.

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