Extended and generalized fragility functions

Fragility functions indicate the probability of a system exceeding certain damage states given some appropriate measures that characterize recorded or simulated data series. Presented in two main parts, this paper develops fragility functions in their utmost generality, accounting for both (1) multivariate intensity measures with multiple damage states and (2) longitudinal damage state dependencies in time. Without adopting the limiting assumption of common variance to avoid improper function crossings, the first part presents what is here compactly termed as extended fragility functions. As shown, these are best supported by the softmax function for any arbitrary distribution of the exponential family to which the intensity measures of different states may belong, including the typically used normal distribution in the logarithmic scale of intensity measures. In the second part, generalized fragility functions are introduced for cases where multiple system state transitions need to be captured. To that end, dependent Markov and hidden Markov models are employed because they are able to portray longitudinal data dependencies and reveal intrinsic deterioration trends for multiple sequential events. Numerical results are presented, together with underlying implementation details, statistical properties, and practical suggestions.