Importance switching based hybrid IPS algorithm in .NET

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Although in theory the IPS approach is applicable virtually to any strong Markov process, in practice the straightforward application of this approach to stochastic hybrid processes may fail to produce reasonable estimates within a reasonable amount of simulation time First, there may be few or no particles in modes with small probabilities (ie, light modes) This happens because each resampling step tends to sample more heavy particles from modes with higher probabilities, thus, light particles in the light modes tend to be discarded Second, if the switching rate is small then it is highly unlikely to observe even one switch during a simulation run In such cases, the possible switching between modes is not properly taken into account Together with the rst problem, this badly affects IPS estimation performance By increasing the number of particles the IPS estimates should improve but only at the cost of substantially increased simulation time which makes the performance of IPS approach similar to that of the standard Monte Carlo The HIPS algorithm of [33] incorporates sampling per mode (strati ed sampling with modes de ning the strata) to cope with large differences in mode weights, and importance switching (a form of importance sampling for the discrete-valued component { t }) to cope with rare mode switching In what follows, we outline the HIPS algorithm If the initial probabilities of some particular modes are very small then it is highly unlikely that particles will be drawn in these modes To avoid this, at the initial sampling step we start with a xed number of particles in each mode however small the initial probability is and adjust the weights appropriately Let
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Np denote the initial number of particles in each mode M In total the system of particles will consist of N = Np |M| particles Let J denote the ordered set of indices of particles which are in mode (J J = for = ) The whole set of indices is de ned by J
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J = {1, 2, , N },
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|J | = N
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At the initial sampling step we will have |J | = Np particles in each mode M As particles evolve and switch from one mode to another, the numbers of particles in different modes will change, as will the index sets J But at each resampling step we will again sample Np particles for each mode M from a conditional empirical distribution Let k and k denote numerical approximations of k and p k | k ( |1), respec tively We choose these numerical approximations in the form of the weighted i i empirical distributions associated with the particle system { k , k }N , where i=1 k (x k T , k T ) and [0, 1] When simulating from k 1 T to k T , only a fraction k of the Monte Carlo simulated trajectories will reach Qk The HIPS algorithm estimates these fractions and their product in a recursive way using the following steps: Step 0 generates, for each M value, Np initial particles at k = 0 and then starts the cycling through steps 1 through 3 for k := 1, 2, , m Step 1 extrapolates each particle from k 1 T to k T in time steps of length h, using importance switching for the new value of the { t } component Step 2 evaluates the particles that have arrived at Qk For this, use is made of equations 94 95 Step 3 resamples with replacement, for each M, Np particles that have arrived at Qk ; the weights must be adjusted accordingly Each of these steps is speci ed in detail below Hybrid interacting particle system (HIPS) HIPS Step 0: Initial sampling for k = 0 For each M, sample Np independent initial Rn values outside D1 : x0 px0 | 0 ( | ) Set 0 = , then 0 = (x0 , 0 ), j J Assign initial weights: 0 = P 0 ( ) Np ,
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