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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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Rare event simulation splitting for variance reduction l.jpg

Rare-Event SimulationSplitting for Variance Reduction

IE 680, Spring 2007

Bryan Pearce

Splitting the beginning l.jpg
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 l.jpg

h(x) = l0

= l1

= l2

= l3

= lm = l

How to choose h l.jpg
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.

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First sub-interval

MC Sim N0 independent chains. R0 reach l1.





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Second sub-interval: Splitting

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

R1 reach l2.



…and so on for each sub-interval




Splitting policy fixed splitting l.jpg
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

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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

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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

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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.

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Truncation - Motivation

Simulation time spent reaching l1







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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.


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β = 2


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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)

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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:

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Dies with prob. (1 – 1/r3,2)

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

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Russian Roulette, cont.

  • There are various methods by which to use the chain weights can compensate for this truncation bias.

    • Probabilistic

    • Tag-based

    • Periodic

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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.

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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 l.jpg

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.