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chosen at random (without replacement), so these chosen trajectories would have Nk /Rk 1 + 1 offspring assigned to them [25, 26] In a xed success implementation [24], a different perspective is considered The idea is to create and simulate suf ciently many offspring, from time Tk 1 onward, so that a xed and predetermined number Hk of trajectories actually manage to reach the intermediate region Bk before time T The issue here is to control the computational effort, because the number Nk of replicas needed to achieve exactly Hk successes is random On the positive side, this implementation sorts out the extinction problem automatically, by construction, that is, the simulation will always run until it reaches the rare event a suf cient number of times On the other hand, the computing effort may have a large variance In a xed probability of success implementation proposed in [8], the level sets are constructed recursively such that the probability of reaching one level from the previous level is approximately q, where q a xed constant such that 0 < q < 1 In this variant, assuming E[T ] < , each of the N = N0 chains is simulated until it reaches the recurrent set Let us denote by Xi ( ) the trajectory of the ith chain, T i its stopping time, and SN,i = sup0 t T i h(Xi (t)) the maximum value of the importance function over its entire trajectory Sort in increasing order the values (SN,1 , , SN,N ), to obtain SN,(1) SN,(N) The K = N q chains yielding the largest values SN,(N K+1) , , SN,(N) are kept, and in order to maintain a population of N chains, N K new trajectories are simulated with initial state the state at which the value SN,(N K) was recorded, and until they reach Combining the maximum value of the importance function of these N K new trajectories with the K values recorded previously, we obtain a new sample of N values that we sort again in increasing order We repeat the procedure while SN,(N K) L, that is, while at least N K chains have not reached B The number n of iterations of the algorithm, and the number R of chains reaching B when the algorithm stops, are random variables, and the estimator of the probability of the rare event is (K/N )n (R/N ) This estimator is biased but consistent It also achieves the same asymptotic variance as N as the xed-effort algorithm, with a probability q of going from any given level to the next Another important issue, from a practical viewpoint, is the computational effort required at each level If T is the return time to a given set A of initial states, for example, then the average time before either reaching the next level or going back to A is likely to increase signi cantly when k increases Several techniques can be designed to alleviate this problem A simple heuristic is to pick a positive integer and just kill (truncate) the chains that go down by levels or more below the current level Lk 1 , based on the idea that they are very unlikely to come up again and reach level k This reduces the computation time, but on the other hand introduces a bias One way to deal with this bias. java ean 13 reader Java Barcode Reader Tutorial to scan, read linear, 2d barcodes in ...
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