Stochastic Modeling

There has been considerable discussion about how one might go about implementing stochastic models without using GridLAB-D in a Monte Carlo -like analysis, which could be computationality cost/time prohibitive.

= Askey Scheme =

A method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos. In their paper 1, Xiu and Karniadakis present a representation of stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of error. They treat several continuous and discrete processes, and provide numerical examples that show substantial speed-up compared to Monte Carlo simulations.

After lengthy discussion, the consensus at PNNL is that it may be desirable to create a new type of data that represents a stochastic variable. Such a property would capture the distribution shape and parameters for any given variable. Such a variable could be sampled using the random number generators when necessary, but the variable could be used to help construct and compute solutions to stochastic equations in modules where this was needed. The general scheme envisioned is as follows:


 * 1) Data is collected from observations are used to set stochastic variables to which the data is attributed using maximum likelihood fits.
 * 2) The stochastic variables are used to drive Kalman filters to adjust the parameters of the corresponding polynomial series.
 * 3) The polynomials are used to update dependent stochastic properties of objects.

In parallel, the polynomials can be converted to and from corresponding differential equations in order to create deterministic models that are typically available in GridLAB-D.

= References =

1 [http://portal.acm.org/citation.cfm?id=587325 Xiu and Karniadakis (2002) "The Wiener-Askey polynomial chaos for stochastic differential equations", SIAM J. Sci. Comput vol 24 no. 2 pp 619-644.]