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SIMULATION AND THE MONTE CARLO METHOD Episode 4

Tham khảo tài liệu 'simulation and the monte carlo method episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 70 RANDOM NUMBER RANDOM VARIABLE. AND STOCHASTIC PROCESS GENERATION 2.5.5 Generating Random Vectors Uniformly Distributed Over a Hyperellipsoid The equation for a hyperellipsoid centered at the origin can be written as xTEx r2 2.38 where E is a positive definite and symmetric n X n matrix x is interpreted as a column vector . The special case where E I identity matrix corresponds to a hypersphere of radius r. Since E is positive definite and symmetric there exists a unique lower triangular matrix B such that E BBT- see 1.25 . We may thus view the set 3C x xTEx r2 as a linear transformation y BTX of the n-dimensional ball y yTy c r2 . Since linear transformations preserve uniformity if the vector Y is uniformly distributed over the interior of an n-dimensional sphere of radius r then the vector X BT -1Y is uniformly distributed over the interior of a hyperellipsoid see 2.38 . The corresponding generation algorithm is given below. Algorithm 2.5.5 Generating Random Vectors Over the Interior of a Hyperellipsoid 1. Generate Y Y1 . Yn uniformly distributed over the n-sphere of radius r. 2. Calculate the matrix B satisfying E BBT. 3. Return X BT -1Y as the required uniform random vector. 2.6 GENERATING POISSON PROCESSES This section treats the generation of Poisson processes. Recall from Section 1.11 that there are two different but equivalent characterizations of a Poisson process Nt t 0 . In the first see Definition 1.11.1 the process is interpreted as a counting measure where Nt counts the number of arrivals in 0 t . The second characterization is that the interarrival times Ai of Nt t ỷ 0 form a renewal process that is a sequence of iid random variables. In this case the interarrival times have an Exp A distribution and we can write Aị lnL where the Ui are iid U 0 1 distributed. Using the second characterization we can generate the arrival times Tị Al Ai during the interval 0 T as follows. Algorithm 2.6.1 Generating a Homogeneous Poisson Process 1. Set To 0 and n 1. .