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Stochastic Process

The :py.stochastic_process module consists of classes and functions to generate samples of stochastic processes from prescribed properties of the process (e.g. power spectrum, bispectrum and/or autocorrelation function). The existing classes rely on stochastic expansions taking the following general form,

$$A(x) = \sum_{i=1}^N \theta(\omega) \phi(x),$$

such that the process can be expressed in terms of a set of uncorrelated random variables, θ(ω), and deterministic basis functions ϕ(x).

The :py.stochastic_process module supports simulation of uni-variate, multi-variate, multi-dimensional, Gaussian and non-Gaussian stochastic processes. Gaussian stochasitc processes can be simulated using the widely-used Spectral Representation Method (StochasticProcess1, StochasticProcess2, StochasticProcess3, StochasticProcess4) and the Karhunen-Loeve Expansion (StochasticProcess5, StochasticProcess6, StochasticProcess7). Non-Gaussian stochastic processes can be generated through higher-order spectral representations (StochasticProcess8, StochasticProcess9, StochasticProcess10) or through a nonlinear transformation from a Gaussian stochastic process to a prescribed marginal distribution using translation process theory StochasticProcess11. Modeling of arbitrarily distributed random processes with specified correlation and/or power spectrum can be performed using the Iterative Translation Approximation Method (ITAM) (StochasticProcess12, StochasticProcess13) for inverse translation process modeling.

This module contains functionality for all the stochastic process methods supported in UQpy.

The module currently contains the following classes:

  • .SpectralRepresentation: Class for simulation of Gaussian stochastic processes and random fields using the Spectral Representation Method.
  • .BispectralRepresentation: Class for simulation of third-order non-Gaussian stochastic processes and random fields using the Bispectral Representation Method.
  • .KarhunenLoeveExpansion: Class for simulation of stochastic processes using the Karhunen-Loeve Expansion.
  • .Translation: Class for transforming a Gaussian stochastic process to a non-Gaussian stochastic process with prescribed marginal probability distribution.
  • .InverseTranslation: Call for identifying an underlying Gaussian stochastic process for a non-Gaussian process with prescribed marginal probability distribution and autocorrelation function / power spectrum.

As with other modules of :py.UQpy, adding simulation methods requires the user to build a new class to support the desired functionality. It does not require modification of any existing classes or methods.

Spectral Representation Method <spectral_representation> Bispectral Representation Method <bispectral_representation> Karhunen Loeve Expansion <karhunen_loeve_1d> Karhunen Loeve Expansion 2D <karhunen_loeve_2d> Translation Processes <translation>