Although the soil moisture conditions are not explicitly considered as stochastic variables
, the wetting and drying sequences are described by a continuous rainfall-runoff model based on historic records.
In a Monte Carlo simulation approach stochastic variables
are those variables that the decision maker cannot foresee with certainty.
The next step is to assign appropriate probability distributions to each of these stochastic variables
. A multivariate normal probability distribution following the approach of Ince (1990) was assumed.
And in Section 3, the SFD is constructed by a local linear model with stochastic variables
. Section 4 details the variational Bayesian learning method for the SFD's parameter distribution inference.
There are N individual stochastic variables
[[xi].sub.1],[[xi].sub.2], ..., [[xi].sub.N] generated from probability distribution [PHI]([xi]).
In a similar situation, we consider stochastic variables
[alpha](k) and [beta](k) to describe successive packet dropouts in a random way:
In this paper, the stochastic variable
[[alpha].sub.k] is assumed to be a Bernoulli distributed sequence, which represents whether the communication environment changes or not at each nonnegative integer time k.
Since SNM calculates the stochastic variables
at the equilibrium, the stability as well as the uniqueness of the solution is guaranteed in SNM.
where w = [([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T] is the weight vector of the stochastic variable
[X.sub.j] (j = 1,2,n) attribute, [w.sub.j] [member of] [0,1], [SIGMA] [w.sub.j] = 1.
It is assumed that the data packet dropout can be described by a stochastic variable
; that is,
The study of Kenyon and Morton , from their experiments of small stochastic vehicle routing problems (SVRP) with only nine nodes, indicates that solutions to the stochastic model, in which the travel time [t.sup.m.sub.ij], for each arc (i, j) and slice m, is considered as a stochastic variable
, can be significantly better than solutions obtained by solving the associated mean-value model (i.e., the deterministic vehicle routing problem in which all random parameters are replaced with their population means).
Typically, a white noise stochastic variable
represented by the differential form of Brownian motion or Weiner process is contained in a stochastic differential equation.