This presentation is the property of its rightful owner.
1 / 24

# Rare-Event Simulation Splitting for Variance Reduction PowerPoint PPT Presentation

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

Rare-Event Simulation Splitting for Variance Reduction

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## Rare-Event SimulationSplitting for Variance Reduction

IE 680, Spring 2007

Bryan Pearce

B

A

Ω

### 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

h(x) = l0

= l1

= l2

= l3

= lm = l

### 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

MC Sim N0 independent chains. R0 reach l1.

h

l1

0

time

### 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

• 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

• 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

• Fixed splitting –

• Asymptotically more efficient under optimal conditions

• Efficiency very sensitive to splitting factor ck

• Fixed effort

• Higher memory requirement

• More robust

### 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

Simulation time spent reaching l1

h

l4

l3

l2

l1

0

### 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

h

l4

l3

l2

l1

0

}

β = 2

Terminate

### 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 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:

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.

• 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

• 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

• Potential performance

• With γ = 10-20,

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

• Poorly-behaved systems

• Inefficient to apply

### 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.