Variable Support

Different nondeterministic methods have differing support for uncertain variable distributions. Table 13, Table 14, and Table 15 summarize the uncertain variables that are available for use by the different methods, where a “-” indicates that the distribution is not supported by the method, a “U” means the uncertain input variables of this type must be uncorrelated, a “C” denotes that correlations are supported involving uncertain input variables of this type, and an “A” means the appropriate variables must be specified as active in the variables specification block. For example, if one wants to support sampling or a stochastic expansion method over both continuous uncertain and continuous state variables, the specification active all must be listed in the variables specification block. Additional notes include:

• we have four variants for stochastic expansions (SE), listed as Wiener, Askey, Extended, and Piecewise which draw from different sets of basis polynomials. The term stochastic expansion indicates polynomial chaos and stochastic collocation collectively, although the Piecewise option is only currently supported for stochastic collocation. Refer to polynomial_chaos and stoch_collocation for additional information on these three options.

• methods supporting the epistemic interval distributions have differing approaches:

  • sampling and global_interval_est lhs model the interval basic probability assignments (BPAs) as continuous histogram bin distributions for purposes of generating samples;

  • local_interval_est and global_interval_est ego ignore the BPA details and models these variables as simple bounded regions defined by the cell extremes;

  • and local_evidence and global_evidence model the interval specifications as true BPAs.

Table 13 Part I: Continuous Aleatory Types

Distribution Type	Sampling	Local Reliability	Global Reliability	Wiener SE	Askey SE	Extended SE	Piecewise SE	Local Interval	Global Interval	Local Evidence	Global Evidence
Normal	C	C	C	C	C	C	-	-	-	-	-
Bounded Normal	C	U	U	U	U	U	U	-	-	-	-
Lognormal	C	C	C	C	C	U	-	-	-	-	-
Bounded Lognormal	C	U	U	U	U	U	U	-	-	-	-
Uniform	C	C	C	C	U	U	U	-	-	-	-
Loguniform	C	U	U	U	U	U	U	-	-	-	-
Triangular	C	U	U	U	U	U	U	-	-	-	-
Exponential	C	C	C	C	U	U	-	-	-	-	-
Beta	C	U	U	U	U	U	U	-	-	-	-
Gamma	C	C	C	C	U	U	-	-	-	-	-
Gumbel	C	C	C	C	C	U	-	-	-	-	-
Frechet	C	C	C	C	C	U	-	-	-	-	-
Weibull	C	C	C	C	C	U	-	-	-	-	-
Continuous Histogram Bin	C	U	U	U	U	U	U	-	-	-	-

Table 14 Part II: Discrete Aleatory Types

Distribution Type	Sampling	Local Reliability	Global Reliability	Wiener SE	Askey SE	Extended SE	Piecewise SE	Local Interval	Global Interval	Local Evidence	Global Evidence
Poisson	C	-	-	-	-	-	-	-	-	-	-
Binomial	C	-	-	-	-	-	-	-	-	-	-
Negative Binomial	C	-	-	-	-	-	-	-	-	-	-
Geometric	C	-	-	-	-	-	-	-	-	-	-
Hypergeometric	C	-	-	-	-	-	-	-	-	-	-
Discrete Histogram Point	C	-	-	-	-	-	-	-	-	-	-

Table 15 Part III: Epistemic, Design, and State Types

Distribution Type	Sampling	Local Reliability	Global Reliability	Wiener SE	Askey SE	Extended SE	Piecewise SE	Local Interval	Global Interval	Local Evidence	Global Evidence
Interval	U	-	U,A	U,A	U,A	U,A	U,A	U	U	U	U
Continuous Design	U,A	-	U,A	U,A	U,A	U,A	U,A	-	-	-	-
Discrete Design Range, Int Set, Real Set	U,A	-	-	-	-	-	-	-	-	-	-
Continuous State	U,A	-	U,A	U,A	U,A	U,A	U,A	-	-	-	-
Discrete State Range, Int Set, Real Set	U,A	-	-	-	-	-	-	-	-	-	-