doi:10.3850/978-981-08-7619-7_P054

ABSTRACT

The theory of non-Gaussian translation processes developed by Grigoriu imposes certain conditions between the non-Gaussian Power Spectral Density Function (PSDF) and the non-Gaussian marginal Probability Density Function (PDF) of the non-Gaussian process. When these two quantities are compatible, it is straightforward to estimate the PSDF of the underlying Gaussian process using translation process theory (the PSDF of the underlying Gaussian process is needed for simulation purposes). However, the most challenging case for applications arising from reallife problems is when the arbitrarily prescribed non-Gaussian PSDF and PDF are incompatible. In this case, the objective is to approximate the incompatible non- Gaussian PSDF with a compatible non-Gaussian PSDF that is as close as possible to the incompatible PSDF. To accomplish this, a number of methodologies have been proposed that involve either some iteration scheme on simulated sample functions or some general optimization approach. Although some of these techniques produce satisfactory results, they can be time consuming because of their nature (especially when the prescribed non-Gaussian PDF deviates significantly from the Gaussian PDF). In this work, a new iterative methodology is proposed that establishes the underlying Gaussian PSDF with high accuracy and truly superior computational efficiency. The basic idea is to iteratively update the Gaussian PSDF using the directly computed (through translation process theory) non-Gaussian PSDF at each iteration, rather than through expensive ensemble averaging of PSDFs computed from a large number of generated non-Gaussian sample functions. The proposed iterative scheme has the advantage of being very simple and converges extremely fast. Once the underlying Gaussian PSDF is established, simulation of non-Gaussian sample functions is straightforward without any need for iterations. Two numerical examples are provided demonstrating the capabilities of the methodology.

Keywords: Stochastic process, Translation process, Non-Gaussian process, Simulation, Spectral Representation Method.

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