Notation and probability spaces#
Symbol |
Meaning in these docs |
|---|---|
\(\mathbf{x}\), \(\mathbf{X}\) |
Physical input values and random variables, with their engineering units. |
\(\mathbf{u}\), \(\mathbf{U}\) |
Independent standard normal coordinates in the usual transformation. |
\(\mathbf{z}\), \(\mathbf{Z}\) |
Independent normal coordinates in older derivations that use z-space for u-space. In the sensitivity derivation, \(\mathbf{z}\) instead denotes correlated normal coordinates, related by \(\mathbf{z}=\mathbf{L}_0\mathbf{u}\). Check each local definition. |
\(g(\mathbf{x})\), \(p_f\) |
Limit-state function and probability of the specified failure event. |
\(\beta=-\Phi^{-1}(p_f)\) |
Normal-equivalent reliability index. |
Subscripts and symbols in reproduced papers retain the source’s definitions. In the continuous examples, failure is described by \(g\leq0\); strict and nonstrict inequalities have the same probability when the boundary has zero probability. Check the event convention when introducing point masses.
A normal-equivalent index can summarize any probability. It equals the signed FORM design-point distance when the approximation uses independent normal coordinates. For explicit generalized Nataf with a Student-t copula, the standard coordinates are spherical Student-t: their components share a mixing variable and are dependent (with zero cross-correlation when second moments exist). The half-space tail is Student-t, so geometric distance and normal-equivalent beta generally differ. Ordinary Rosenblatt with that same copula instead produces independent normal coordinates.
The probability integral transform also uses uniforms on the unit hypercube as intermediate coordinates. A change of coordinates preserves the physical event probability when the complete joint law is held fixed; a local FORM approximation need not be invariant under a nonlinear change of coordinates. See Rosenblatt ordering in system reliability for the practical consequence.
Continue: Probability transformations and copulas · Copulas and isoprobabilistic transformations · Interpreting results and convergence