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T Sw = Hw Hw , T Sb = Hb Hb ,

. . . . . . . . . . . .

. . . . . . . . . . . .

(1.7)

Distribution type, mean, variance, & skew Statistical regression parameters Statistical regression parameters Statistical regression parameters ditto

. . . . . . . . . . . .

x c(j) 2 ,

. . . . . . . . . . . .

. . . . . . . . . . . .

(1.8)

. . . . . . . . . . . .

Spatial variability of soil parameters Ave number per reach and size Spatial variability of permeability Spatial variability of soil parameters

1

. . . . . . . . . . . .

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. . . . . . . . . . . .

trace(Sb ) =

. . . . . . . . . . . .

. . . . . . . . . . . .

(1.9)

Chance exploration detected Soil parameters and spatial model parameters Statistics of soil properties, model uncertainty Chance exploration detected Soil parameters and spatial model parameters Statistics of soil properties, model uncertainty

. . . . . . . . . . . .

Thus trace(Sw ) measures the distance between the data points and their corresponding class centroid, and trace(Sb ) captures the distance between the class centroids and the global centroid. In the lower-dimensional space resulting from the linear transformation G, the scatter matrices become

. . . . . . . . . . . .

. . . . . . . . . . . .

L Sw = GT Sw G, L Sb = GT Sb G,

. . . . . . . . . . . .

The reason for this result is that the existence of the soft lenses makes both events simultaneously more likely, while the non-existence makes both simultaneously less likely. Since the two events are no longer independent, their marginal probabilities can no longer be simply summed to give the probability that either one or the other occurs. The correlation due to a common dependence reduces the risk to the structure in this case, but the outcome could be the reverse in other circumstances. Branch probabilities associated with separate event nodes can be correlated through of any of the following: 1. Causal dependence, meaning that one event physically causes another. For example, liquefaction-induced settlement may directly lead to overtopping of an embankment, thus the liquefaction event and overtopping event would not be independent of one another. If the liquefaction settlement occurs, the probability of overtopping might be greatly enhanced. 2. Probabilistic correlation, meaning that two uncertainties may share a common dependence on a third uncertainty, as in the case of the low-density soil lenses in the example above. Whether the low-density soil lenses exist or not simultaneously changes the probability of liquefaction cracking and of overtopping.

(1.10)

. . . . . . . . . . . .

. . . . . . . . . . . .

L L An optimal transformation G would maximize trace(Sb ) and minimize trace(Sw ) siL ) and minimizing trace(S L ) multaneously, which is equivalent to maximizing trace(Sb t L L simultaneously, since StL = Sw + Sb . The optimal transformation, GLDA , of LDA is computed by solving the following optimization problem [8, 16]: L GLDA = arg max trace Sb StL G 1

. . . . . . . . . . . .

. . . . . . . . . . . .

(1.11)

1.0 E-4 -

. . . . . . . . . . . .

It is known that the optimal solution to the optimization problem in Eq. (1.11) can be obtained by solving the following generalized eigenvalue problem [8]: Sb x = St x. (1.12)

. . . . . . . . . . . .

. . . . . . . . . . . .

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