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DIGITAL SIMULATION ALGORITHMS FOR SECOND-ORDER STOCHASTIC PROCESSES

DIGITAL SIMULATION ALGORITHMS FOR SECOND-ORDER STOCHASTIC PROCESSES. PHOON KK, QUEK ST & HUANG SP. WHY BET ON SIMULATION?. MOORE’S LAW - density of transistors doubles every 18 months Computing power will increase 1000-fold after 15 years

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DIGITAL SIMULATION ALGORITHMS FOR SECOND-ORDER STOCHASTIC PROCESSES

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  1. DIGITAL SIMULATION ALGORITHMS FOR SECOND-ORDER STOCHASTIC PROCESSES PHOON KK, QUEK ST & HUANG SP

  2. WHY BET ON SIMULATION? • MOORE’S LAW - density of transistors doubles every 18 months • Computing power will increase 1000-fold after 15 years • Common PC already comes with GHz processor, GB memory & hundreds of GB disk

  3. CHALLENGE • Develop efficient computer algorithms that can generate realistic sample functions on a modest computing platform • Should be capable of handling: • stationary or non-stationary covariance fns • Gaussian or non-Gaussian CDFs • short or long processes

  4. PROPOSAL Use a truncated Karhunen-Loeve (K-L) series for Gaussian process:

  5. K-L PROCESS uncorrelated zero-mean unit variance Gaussian random variables eigenvalues & eigenfunctions of target covariance function C(x1, x2)

  6. KEY PROBLEM are solutions of the homogenous Fredholm integral equation of the second kind Difficult to solve accurately & efficiently

  7. WAVELET-GALERKIN • Family of orthogonal Harr wavelets generated by shifting & scaling • Basis function over [0,1]

  8. 1 1= 0,1 j=0 0 -1 0 0.5 1 1 1 0 0 2= 1,0 3= 1,1 j=1 -1 -1 0 0.5 1 0 0.5 1 1 1 1 1 0 0 0 0 j=2 -1 -1 -1 -1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 6= 2,2 4= 2,0 5= 2,1 7= 2,3

  9. WAVELET-GALERKIN • Express eigenfunction as a truncated series of Harr wavelets • Apply Galerkin weighting

  10. NUMERICAL EXAMPLE (1) Stationary Gaussian process over [-5, 5] with target covariance:

  11. EIGENSOLUTIONS f(x) 

  12. COVARIANCE M = 10 M = 30

  13. NON-GAUSSIAN K-L For = zero-mean process with non-Gaussian marginal distribution = vector of zero-mean unit variance uncorrelated ?? random variables

  14. NON-GAUSSIAN K-L Can estimate using But integrand unknown – evaluate iteratively

  15. NUMERICAL EXAMPLE (2) Stationary non-Gaussian process over [-5, 5] with target covariance & marginal CDF: = 0.5816, = 0.4723, = -2

  16. MARGINAL CDF k = 1 k = 12

  17. CONCLUSIONS • K-L has potential for simulation • Eigensolutions can be obtained cheaply & accurately from DWT • Non-gaussian K-L can be determined by iterative mapping of CDF • Theoretically consistent way to generate stationary/non-stationary, Gaussian/non-Gaussian process over finite interval

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